tag:blogger.com,1999:blog-267065642016-05-01T06:48:18.791-04:00Thoughts On EconomicsRobert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.comBlogger989125tag:blogger.com,1999:blog-26706564.post-1151835560707333482016-12-31T03:00:00.000-05:002015-01-06T06:09:59.760-05:00WelcomeI study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.<br /><br />The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.<br /><br />In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.<br /><br />I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.<br /><br /><B>Comments Policy:</B> I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.Robert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.com64tag:blogger.com,1999:blog-26706564.post-31083396772773635712016-04-30T11:20:00.000-04:002016-04-30T14:31:32.142-04:00A STEAM Experience For A Flash Mob<B>1.0 Introduction</B><P>STEAM stands for Science, Technology, Engineering, Arts, and Mathematics. This post describes a possible plan for a crowd of many people to participate in. Roles for players consist of: </P><UL><LI>A Recorder.</LI><LI>State Actors.</LI><LI>Holders of letters in a line.</LI></UL><P>I once read Terry Eagleton suggesting that part of the definition of art is that it be "<A HREF="http://www.moma.org/collection/artists/3528">somewhat pointless</A>." </P><B>2.0 Equipment</B><P>Equipment to be provided consists of: </P><UL><LI>A six-sided die.</LI><LI>Two balls. They could be soccer balls, beach balls, volley balls, or so on. One ball is called the Head, and the other ball is called the State Pointer.</LI><LI>Six sets of equipment, labelled 1 through 6. A set of equipment consists of:</LI><UL><LI>A set of cards, where each card is a "letter" from an alphabet. Letters can be, for example, "Blank", "(", ")", ";", "End", "0", and "1". Many letters must have many cards with that letter.</LI><LI>A set of state placards. Each state placard contains:</LI><UL><LI>An arbitrary label. These labels are arbitrary, but not repeated. They could be in high elvish, for all it matters, as long as participants can pronounce each label.</LI><LI>Either the word "Halt" or a set of rules. The placards for the halting states may also contain a short phrase. Each rule in a set of rules is designated by a letter from the alphabet.</LI></UL><LI>Guidelines for setting up. These guidelines include:</LI><UL><LI>Optional guidelines for the geographical distribution of states.</LI><LI>A specification of which State Actor initially holds the State Pointer.</LI><LI>Guidelines for forming an initial line of letters from the alphabet. These guidelines must include a specification of which holder of a letter initially also holds the Head.</LI></UL></UL></UL><B>3.0 Playing the Game</B><P></P><B>3.1 Preliminaries</B><P>The Recorder throws the die and chooses the corresponding set of equipment. One might create only one set of equipment, and this step would be omitted. </P><P>The Recorder distributes the state placards. A audience member comes up for each placard. He collects it, and becomes a State Actor. The State Actors all gather, with some distance between them, in a designated region. (One might break down the region into sub-regions, for subsets of the states, if one wants.) </P><P>The Recorder gives the State Pointer to the State Actor holding the placard for the initial state. </P><P>The Recorder reads out the guidelines for the initial line of letters. Audience members come up and form a line, accordingly. As an example, the guidelines might say: </P><BLOCKQUOTE>The first player sits in the line and holds the "End" letter. The second player stands behind the first player. He holds a "Blank" and the Head. A number of players sit in the line behind the second player. They should each hold "0" or "1", as they choose. A person should sit after these players, and she holds a ";". Another number of players sit in in a row behind her. They also should each hold a "0" or "1". </BLOCKQUOTE><P>The Recorder writes down the sequence of letters in the initial set up. This step is optional. </P><P>Play can now commence. Play consists of a sequence of clock cycles. </P><B>3.2 A Clock Cycle</B><P>The player holding the Head commences a clock cycle. This player calls out the letter he is holding. </P><P>The state actor holding the State Pointer now plays. He looks at his rules and finds the rule corresponding to the letter that has been called out. Each rule has two parts. The first part is either a letter from the alphabet or the word "Forward" or the word "Backward". The second part is the name of a state. That state could be the label on the state placard that this State Actor is holding. Or it could be another state. </P><P>If the State Actor calls out a letter, an audience member comes up. He selects that letter from leftover letters in the initial set of equipment. He replaces the player holding the Head in the line. And that player hands the new player the Head. </P><P>If the State Actor calls out Forward, the player holding the Head hands it to the player holding a letter in front of him and sits down. The player now holding the Head stands up. There would be no such player if the player holding the Head at the start of the cycle is standing at the front of the line. In this case, an audience member picks up a "Blank" from the leftover set of equipment. That player accepts the Head and stands at the front of the line. </P><P>If the State Actor calls out Backward, the player holding the Head hands it to the player holding a letter behind him and sits down. As you might expect, that player now holding the Head stands up. This step might also result in a new player coming up from the audience and joining the line. And this new player would join the line at the back. </P><P>The State Actor holding the State Pointer now calls out the state listed on the second part of the rule he is executing. If that state is not the state listed on his state placard, he hands the State Pointer to the appropriate State Actor. </P><P>The Recorder writes down the new state that the State Pointer has now transitioned to. (This step is optional.) </P><B>3.3 Ending the Game</B><P>The game ends either when the players become convinced it could go on forever, or it ends when a State Actor holding a placard for a halting state receives the State Pointer. If the game ends in a halting state, the State Actor reads the corresponding phrase from the state placard. That phrase might be something like: </P><BLOCKQUOTE>You have been a Turing machine computing the sum of two non-negative integers, written in binary. </BLOCKQUOTE><P>Or it could be: </P><BLOCKQUOTE>You had at least one unmatched parentheses in your initial line. </BLOCKQUOTE><P>If you want, the Recorder could have more audience members come up to recreate the initial line. You can then review, if you like, the computation. For example, you might check that the sum of the two numbers separated by a comma in the initial line up is equal to the number now represented by the final line up on the stage. </P><B>4.0 Much To Do</B><P>Obviously, much would need to be done to flesh this out. In particular, equipment sets need to be constructed. Some choices to think about: </P><UL><LI>Would one want to include an equipment set in which the simulated Turing machine does not terminate for some initial line of letters? Or would one want to, at least for the first performance, only have rules that are guaranteed termination for all (valid?) inputs?</LI><LI>Might one want to emulate automata for languages lower down on the Chomsky hierarchy? For example, one might create a stack to be pushed and popped before the start of the line. Here I envision that a subset of the states specify subroutines. And the State Actors defining these subroutines might be grouped separately from the other actors.</LI><LI>Would one want to share alphabets among more than one equipment set? Maybe all six sets should have the same alphabet.</LI><LI>How would one describe the initial line up for a Turing machine that is to decide or semi-decide whether a given string is in a given language? The specification of a grammar for generating a string can be quite confusing to beginners.</LI><LI>I am thinking that one would not want to create rules for a universal Turing machine. Even some of the suggestions above might be too long to play.</LI></UL><P>An interesting variation would be to simulate a non-deterministic Turing machine. For some clock cycles, the line would be duplicated. And one would introduce another Head and State Pointer. </P><B>5.0 Instruction and Theatrics</B><P>This activity could serve pedagogical purposes. Suppose the players are different cohorts of students. Could the older students be directed to write the rules for some other computation at the next meeting? Could a set of recursive functions be built up over many meetings? Maybe one would end up with a group engaging in real-time debugging in a joint activity. </P><P>One could set up an accompanying talk or lecture. Many topics could be broached: The Church-Turing thesis and universality, uncomputable functions and the halting problem, computational complexity and the question of whether P equals NP, linguistics and the Chomsky hierarchy, etc. Or one might talk about the British secret service and reading the Nazi's mail. I guess there is both a Broadway play and a movie to go along with this activity. </P><P>One could introduce some sophistication in showmanship, depending on where this concept is instantiated. I like the idea of the alphabet players wearing different colored shirts, with each color corresponding to a character. Zero could be red, and one could be green. Blanks would be a neutral color, such as white. The State Actors could be in a dim area, with a spotlight serving as the State Pointer. The State Actors or the letter holders could be members of an orchestra, with some tune being played for every state transition or invoked rule. At termination, the entire derivation written down by the Recorder could be run-through. I imagine it would be difficult to design a set of rules that results in an interesting tune. At any rate, I guess the interests of an observing mathematician, the participants, and a theatergoer would be in tension. </P><P>I hope if somebody was to try this project, they would give me appropriate acknowledgement. </P><B>Reference</B><UL><LI>Lou Fisher (1975). "Nobody Named Gallix", <I>The Magazine of Fantasy and Science Fiction</I> (Jan.): pp. 98-109.</LI><LI>Andrew Hodges (1983). <I>Alan Turing: The Enigma</I>, Princeton University Press.</LI><LI>HarryR. Lewis and Christos H. Papadimitriou (1998). <I>Elements of the Theory of Computation</I>, 2nd edition. Prentice Hall.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-47279410192700315972016-04-13T15:32:00.000-04:002016-04-18T06:58:21.378-04:00Math Is Power<B>1.0 Introduction</B><P>A common type of post in this blog is the presentation of concrete numerical examples in economics. Sometimes I present examples to illustrate some principle. But usually I try to find examples that are counter-intuitive or perverse, at least from the perspective of economics as mainstream economists often misteach it. </P><P>Voting games provide an arena where one can find surprising results in political science. I am thinking specifically of power indices. In this post, I try to explain two of the most widely used power indices by means of an example. </P><B>2.0 Me and My Aunt: A Voting Game</B><P>For purposes of exposition, I consider a specific game, called <I>Me and My Aunt</I>. There are four players in this version of the game, represented by elements of the set: </P><BLOCKQUOTE><I>P</I> = The set of players = {0, 1, 2, 3} </BLOCKQUOTE><P>Out of respect, the first player gets two votes, while all other players get a vote each (Table 1). A coalition, <I>S</I>, is a set of players. That is, a coalition is a subset of <I>P</I>. A coalition passes a resolution if it has a majority of votes. Since there are four players, one of who has two votes, the total number of votes is five. So a majority, in this game of weighted voting, is three votes. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 1: Players and Their Votes</b></CAPTION><TR align="CENTER"><TD><B>Players</B></TD><TD><B>Votes</B></TD></TR><TR align="CENTER"><TD>0 (Aunt)</TD><TD>2</TD></TR><TR align="CENTER"><TD>1 (Me)</TD><TD>1</TD></TR><TR align="CENTER"><TD>2</TD><TD>1</TD></TR><TR align="CENTER"><TD>3</TD><TD>1</TD></TR></tbody></table><P>One needs to specify the payoff to each coalition to complete the definition, in characteristic function form, of this game. The <A HREF="http://robertvienneau.blogspot.com/2008/12/dont-say-there-must-be-something-common.html">characteristic function</A>, <I>v</I>(<I>S</I>) maps the set of all subsets of <I>P</I> to the set {0, 1}. If the players in <I>S</I> have three or more votes,<I>v</I>(<I>S</I>) is 1. Otherwise, it is 0. That is, a winning coalition gains a payoff of one to share among its members. </P><B>3.0 The Penrose-Banzhaf Power Index</B><P>Power for a player, in this mathematical approach, is the ability to be the decisive member of a coalition. If, for a large number of coalitions, you being in or out of a coalition determines whether or not that coalition can pass a resolution, you have a lot of power. Correspondingly, if the members of most coalitions do not care whether you join, because your presence has no influence on whether or not they can put their agenda into effect, you have little power. </P><P>The Penrose-Banzhaf power index is one (of many) attempts to quantify this idea. Table 2 lists all 16 coalitions for the voting game under consideration. (The number of coalitions is the sum of a row in Pascal's triangle.) The second column in Table 2 specifies the value for the characteristic function for that coalition. Equivalently, the third column notes which eight coalitions are winning coalitions, and which eight are losing. The last two columns are useful for tallying up counts needed for the Penrose-Banzhaf index. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 2: Calculations for Penrose-Banzhaf Power Index</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Coalition</B></TD><TD ROWSPAN="2"><B>Characteristic<BR>Function</B></TD><TD ROWSPAN="2"><B>Winning<BR>or Losing</B></TD><TD COLSPAN="2"><B>Player</B></TD></TR><TR align="CENTER"><TD><B>Aunt (0)</B></TD><TD><B>Me (1)</B></TD></TR><TR align="CENTER"><TD>{}</TD><TD><I>v</I>( {} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0}</TD><TD><I>v</I>( {0} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1}</TD><TD><I>v</I>( {1} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{2}</TD><TD><I>v</I>( {2} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{3}</TD><TD><I>v</I>( {3} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1}</TD><TD><I>v</I>( {0, 1} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>1</TD></TR><TR align="CENTER"><TD>{0, 2}</TD><TD><I>v</I>( {0, 2} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 3}</TD><TD><I>v</I>( {0, 3} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 2}</TD><TD><I>v</I>( {1, 2} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 3}</TD><TD><I>v</I>( {1, 3} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{2, 3}</TD><TD><I>v</I>( {2, 3} ) = 0</TD><TD>Losing</TD><TD>0</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1, 2}</TD><TD><I>v</I>( {0, 1, 2} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 1, 3}</TD><TD><I>v</I>( {0, 1, 3} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{0, 2, 3}</TD><TD><I>v</I>( {0, 2, 3} ) = 1</TD><TD>Winning</TD><TD>1</TD><TD>0</TD></TR><TR align="CENTER"><TD>{1, 2, 3}</TD><TD><I>v</I>( {1, 2, 3} ) = 1</TD><TD>Winning</TD><TD>0</TD><TD>1</TD></TR><TR align="CENTER"><TD>{0, 1, 2, 3}</TD><TD><I>v</I>( {0, 1, 2, 3} ) = 1</TD><TD>Winning</TD><TD>0</TD><TD>0</TD></TR></tbody></table><P>The Penrose-Banzhaf index, ψ(<I>i</I>) is calculated for each player <I>i</I>. It is defined, for a given player, to be the ratio of the number of winning coalitions in which that player is decisive to the total number of coalitions, winning or losing. A player is decisive for a coalition if: </P><UL><LI>The coalition is a winning coalition.</LI><LI>The removal of the player from the coalition converts it to a losing coalition.</LI></UL><P>From the table above, one can see that player 0 is decisive for six coalitions, while player 1 is decisive for only two coalitions. Hence, the Penrose-Banzhaf index for "my aunt" is: </P><BLOCKQUOTE>ψ(0) = 6/16 = 3/8 </BLOCKQUOTE><P>By symmetry, the index values for players 2 and 3 are the same as the value for player 1: </P><BLOCKQUOTE>ψ(1) = ψ(2) = ψ(3) = 2/16 = 1/8 </BLOCKQUOTE><P>More than one player can be decisive for a winning coalition. No need exists for the Penrose-Banzhaf index to sum up to one. How much one's vote is weighted does not bear a simple relationship to how much power one has. Also note that the definition of this power index is not confined to simple majority games. Power indices can be calculated for voting games in which a super-majority is required to pass a measure. For example, in the United States Senate, 60 senators are needed to end a filibuster. </P><B>4.0 The Shapley-Shubik Power Index</B><P>The Shapley-Shubik power index is an application of the calculation of the Shapley value to voting games. The Shapley value applies to cooperative games, in general. For its use as a measure of power in voting games, it matters in which order players enter a coalition. Accordingly, Table 3 lists all 24 permutations of all four players in the voting game being analyzed. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 3: Calculations for the Shapley-Shubik Power Index</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Permutation</B></TD><TD COLSPAN="2"><B>Player</B></TD></TR><TR align="CENTER"><TD><B>Aunt (0)</B></TD><TD><B>Me (1)</B></TD></TR><TR align="CENTER"><TD>(0, 1, 2, 3)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1} ) - <I>v</I>( {0} ) = 1</TD></TR><TR align="CENTER"><TD>(0, 1, 3, 2)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1} ) - <I>v</I>( {0} ) = 1</TD></TR><TR align="CENTER"><TD>(0, 2, 1, 3)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 2} )<BR> - <I>v</I>( {0, 2} ) = 0</TD></TR><TR align="CENTER"><TD>(0, 2, 3, 1)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(0, 3, 1, 2)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {0, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(0, 3, 2, 1)</TD><TD><I>v</I>( {0} ) - <I>v</I>( {} ) = 0</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 0, 2, 3)</TD><TD><I>v</I>( {0, 1} )<BR>- <I>v</I>( {1} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 0, 3, 2)</TD><TD><I>v</I>( {0, 1} )<BR>- <I>v</I>( {1} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 2, 0, 3)</TD><TD><I>v</I>( {0, 1, 2} )<BR>- <I>v</I>( {1, 2} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 2, 3, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 3, 0, 2)</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {1, 3} ) = 1</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(1, 3, 2, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1} ) - <I>v</I>( {} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 0, 1, 3)</TD><TD><I>v</I>( {0, 2} )<BR>- <I>v</I>( {2} ) = 1</TD><TD><I>v</I>( {0, 1, 2} )<BR>- <I>v</I>( {0, 2} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 0, 3, 1)</TD><TD><I>v</I>( {0, 2} )<BR>- <I>v</I>( {2} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 1, 0, 3)</TD><TD><I>v</I>( {0, 1, 2} )<BR>- <I>v</I>( {1, 2} ) = 1</TD><TD><I>v</I>( {1, 2} ) - <I>v</I>( {2} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 1, 3, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 2} ) - <I>v</I>( {2} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 3, 0, 1)</TD><TD><I>v</I>( {0, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(2, 3, 1, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD></TR><TR align="CENTER"><TD>(3, 0, 1, 2)</TD><TD><I>v</I>( {0, 3} )<BR>- <I>v</I>( {3} ) = 1</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {0, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 0, 2, 1)</TD><TD><I>v</I>( {0, 3} )<BR>- <I>v</I>( {3} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 1, 0, 2)</TD><TD><I>v</I>( {0, 1, 3} )<BR>- <I>v</I>( {1, 3} ) = 1</TD><TD><I>v</I>( {1, 3} ) - <I>v</I>( {3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 1, 2, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 3} ) - <I>v</I>( {3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 2, 0, 1)</TD><TD><I>v</I>( {0, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {0, 2, 3} ) = 0</TD></TR><TR align="CENTER"><TD>(3, 2, 1, 0)</TD><TD><I>v</I>( {0, 1, 2, 3} )<BR>- <I>v</I>( {1, 2, 3} ) = 0</TD><TD><I>v</I>( {1, 2, 3} )<BR>- <I>v</I>( {2, 3} ) = 1</TD></TR></tbody></table><P>Table 3 shows some initially confusing calculations in the last two columns, where each of these columns is defined for a given player. Suppose a player and a permutation are defined. For that permutation, let the set <I>S</I><SUB>π, <I>i</I></SUB> contain those players in the permutation π to the left of the given player <I>i</I>. The difference, in the last two columns, is the following, for <I>i</I> equal to 0 and to 1, respectively: </P><BLOCKQUOTE><I>v</I>(<I>S</I><SUB>π, <I>i</I></SUB> ∪ {<I>i</I>}) - <I>v</I>(<I>S</I><SUB>π, <I>i</I></SUB>) </BLOCKQUOTE><P>The Shapley-Shubik power index, for a player, is the ratio of a sum to the number of permutations of players. And that sum is calculated for each player, as the sum over all permutations, of the above difference in the value of the value of the characteristic function. </P><P>If I understand correctly, given a permutation, the above difference can only take on values of 0 or 1. And it will only be 1 for one player, where that player determines whether the formation of the coalition in the order given will be a winning coalition. As a consequence, the Shapley-Shubik power index is guaranteed to sum over players to unity. In this case, power is a fixed amount, with each player being measured as having a defined proportion of that power. </P><B>5.0 Both Power Indices</B><P>The above has stepped through the calculation of two power indices, for all players, in a given game. Table 4 lists their values, as well as a normalization of the Penrose-Banzhauf power index such that the sum of the power, over all players, is unity. (I gather that the Penrose-Banzhauf index and the normalized index do not have the same properties.) As one might expect from the definition of the game, "my aunt" has more power than "me" in this game. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 4: The Penrose-Banzhaf and Shapley-Shubik Power Indices</b></CAPTION><TR align="CENTER"><TD ROWSPAN="2"><B>Player</B></TD><TD COLSPAN="2"><B>Penrose-Banzhaf Power Index</B></TD><TD ROWSPAN="2"><B>Shapley-Shubik<BR>Power Index</B></TD></TR><TR align="CENTER"><TD><B>Index</B></TD><TD><B>Normalized</B></TD></TR><TR align="CENTER"><TD>0</TD><TD>6/16 = 3/8</TD><TD>6/12 = 1/2</TD><TD>12/24 = 1/2</TD></TR><TR align="CENTER"><TD>1</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>2</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR><TR align="CENTER"><TD>3</TD><TD>2/16 = 1/8</TD><TD>2/12 = 1/6</TD><TD>4/24 = 1/6</TD></TR></tbody></table><P>In many voting games, the normalized Penrose-Banzhauf and Shapley-Shubik power indices are not identical for all players. In fact, suppose the rules for the above variation of <I>Me and my Aunt</I> voting game are varied. Suppose now that four votes - a supermajority - are needed to carry a motion. The normalized Penrose-Banzhaf index for player 0 becomes 1/3, while each of the other players have a normalized Penrose-Banzhaf index of 2/9. Interestingly enough, the Shapley-Shubik indices for the players do not change, if I have calculated correctly. But the values assigned to rows in Table 3 do sometimes vary. Anyways, that one tweak of the rules results in different power indices, depending on which method one adopts. A more interesting example would be one in which the rankings vary among power indices. </P><P>Other power indices, albeit less common, do exist. Which one is most widely applicable? I would think that mainstream economists, given game theory and marginalism, would tend to prefer the Shapley-Shubik power index. Felsenthal and Machover (2004) seem to be widely recognized experts on measures of voting power, and they have come to prefer the Penrose-Banzhaf index over the Shapley-Shubik index. </P><B>6.0 Where To Go From Here</B><P>I have described above a couple of power indices in voting games. As I understand it, many have tried to write down reasonable axioms that characterize power indices. One challenge is to specify a set of axioms such that your preferred power index is the only one that satisfies them. But, as I understand it, some sets of reasonable axioms are open insofar as more than one power index would satisfy them. I seem to recall a theorem that one could create a power index for a reasonable set of axioms such that whichever player you want in a voting game is the most powerful. Apparently, a connection can be drawn between a power index and a voting procedure. And Donald Saari <A HREF="http://robertvienneau.blogspot.com/2008/11/militant-voting.html">boasts</A> that he could create an apparently fair voting procedure that would result in whatever candidate you like being elected. </P><P>I gather that many examples of voting games have been presented in which apparently paradoxical or perverse results arise. And these do not seem to be merely theoretical results. Can I find some such examples? Perhaps, I should look here at some of <A HREF="http://robertvienneau.blogspot.com/2011/02/daron-acemoglu.html">Daron Acemoglu's</A> work. </P><P>I am aware of three types of examples to look for. One is that of a dummy. A dummy is a player that, under the weights and the rule for how many votes are needed for passage, can never be decisive in a coalition. Whether this player drops out or joins a coalition can never change whether or not a resolution is passed, even though the player has a positive weight. A second odd possibility arises as the consequence of adding a new member to the electorate: </P><BLOCKQUOTE>"...power of a weighted voting body may increase, rather than decrease, when new members are added to the original body." -- Steven J. Brams and Paul J. Affuso (1976). </BLOCKQUOTE><P>A third odd possibility apparently can arise on a council when one district annexes another. Suppose, the district annexing the other consequently increases the weight of its vote accordingly. One might think a greater weight leads to more power. But, in certain cases, the normalized Penrose-Banzhaf index can decrease. </P><P>The above calculations for the Penrose-Banzhaf and Shapley-Shubik power indices treat all coalitions or permutations, respectively, as equally likely to arise. Empirically, this does not seem to be true. And this has an impact on how one might measure power. For example, since voting is unweighted on the Supreme Court of the United States, all justices might be thought to be equally powerful. But, because of the formation of well-defined blocks, Anthony Kennedy was often described as being particularly powerful in deciding court decisions, at least when Antonin Scalia was still alive. So empirically, one might include some assessment of the affinities of the players for one another and, thus, some influence on the probabilities of each coalition forming. This will have consequences on the calculation of power indices. But why stop there? In the United States these days, <A HREF="http://robertvienneau.blogspot.com/2013/04/political-elites-bowing-down-before.html">politicians</A> only seem to <A HREF="http://robertvienneau.blogspot.com/2012/09/your-opinion-does-not-matter.html">represent</A> the most <A HREF="http://robertvienneau.blogspot.com/2013/05/our-rulers-do-not-know-why-they-dislike.html">wealthy</A>. </P><P><B>Update:</B> This <A HREF="http://homepages.warwick.ac.uk/~ecaae/#Progam_List">page</A>, from the University of Warwick, has links to utilities for calculating various power indices. </P><B>References</B><UL><LI>Steven J. Brams and Paul J. Affuso (1976). Power and Size: A New Paradox, <I>Theory and Decision</I>. V. 7, Iss. 1 (Feb.): pp. 29-56.</LI><LI>Dan S. Felsenthal and Moshé Machover (2004). Voting Power Measurement: A Story of Misreinvention, London Scool of Economics and Political Science</LI><LI>Andrew Gelman, Jonathan N. Katz, and Joseph Bafumi (2004). Standard Voting Power Indexes Do Not Work: An Empirical Analysis, <I>B. J. Pol. S.</I>. V. 34: pp. 657-674.</LI><LI>Guillermo Owen (1971) Political Games, <I>Naval Research Logistics Quarterly</I>. V. 18, Iss. 3 (Sep.): pp. 345-355.</LI><LI>Donald G. Saari and Katri K. Sieberg (1999). Some Surprising Properties of Power Indices.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-21179391124817521102016-04-11T13:00:00.000-04:002016-04-11T13:00:05.632-04:00Inane Responses To The Cambridge Capital Controversy<P>I consider the following views, if unqualified and without caveats, just silly: </P><UL><LI>The Cambridge Capital Controversy (CCC) was only attacking aggregate neoclassical theory.</LI><LI>The CCC is just a General Equilibrium argument, and it has been subsumed by General Equilibrium Theory. (Citing Mas Colell (1989) here does not help.)</LI><LI>The CCC does not have anything to say about partial, microeconomic models.</LI><LI>Perverse results, such as reswitching and capital-reversing, only arise in the special case of Leontief production functions. If you adopt widely used forms for production functions, the perverse results go away.</LI><LI>It is an empirical question whether non-perverse results follow from neoclassical assumptions. And nobody has ever found empirical examples of capital-reversing or reswitching.</LI><LI>Mainstream economists have moved on since the 1960s, and their models these days are not susceptible to the Cambridge critique.</LI></UL><P>I would think that one could not get such ideas published in any respectable journal. On the other hand, Paul Romer did get his <A HREF="https://criticalfinance.org/2016/03/27/economics-science-or-politics-a-reply-to-kay-and-romer/">ignorance</A> about Joan Robinson into the <I>American Economic Review</I></P><B>References</B><UL><LI>Andreu Mas-Colell (1989). Capital theory paradoxes: Anything goes. In <I>Joan Robinson and Modern Economic Theory</I> (ed. by George R. Feiwel), Macmillan.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-38947436794814467082016-03-19T16:54:00.000-04:002016-03-22T07:04:07.010-04:00Post Keynesianism From The Outside<P>Post Keynesian economics has been under development for about three quarters of a century now. Academics in countries around the world have made contributions to the theory and to its application. And they have participated in many common practices of academics, including economists<SUP>1</SUP>. </P><P>Post Keynesians have written and published papers in peer-reviewed journals. Over this time-span, such journals include widely referenced mainstream journals, such as the <I>American Economic Review</I>, the <I>Economic Journal</I>, the <I>Journal of Economic Literature</I>, the <I>Journal of Political Economy</I>, and the <I>Quarterly Journal of Economics</I><SUP>2</SUP>. Lately, certain specialized journals have proven more sympathetic to publishing Post Keynesians. Such journals include, for example, the <I>Cambridge Journal of Economics</I>, the <I>Journal of Post Keynesian Economics</I>, <I>Kyklos</I>, and the <I>Review of Political Economy</I>. The list suggests two other activities: the founding and editing of journals. As I understand it, Joan Robinson, among other economists now thought of as Post Keynesian, participated in the founding of the <I>Review of Economic Studies</I>, while the <I>Review of Keynesian Economics</I> is a more recent academic journal with an analogous start. The <I>Banca Nazionale del Lavoro Quarterly Review</I>, the <I>Canadian Journal of Economics</I>, and <I>Metroeconomica</I> are some journals, while not being specifically heterodox, I guess, had Post Keynesians as editors for some time<SUP>3, 4</SUP>. </P><P>Participation in professional societies, as officers, as organizers of conferences and conference sessions, and as presenters at conferences, provides another typical venue for academic activities. Naturally, Post Keynesians have performed such activities. For example, John Kenneth Galbraith was president of the American Economic Society, and the annual meeting of the Allied Social Sciences Association (ASSA), held in conjunction with the American Economics Association, regularly holds sessions dedicated to Post Keynesian topics. Recently, heterodox economics have become interested in pluralism and how their concerns overlap. These concerns have been reflected in much work in many professional societies relating to heterodox economics. </P><P>I began this article with journal publications because economics has become less focused on books and more focused on journal publications during the period in which Post Keynesianism grew. But during this period, Post Keynesians have also made original contributions in books published by prestige university and academic presses. I think, for example, of presses associated with Cambridge, Columbia, and Harvard, to pick some examples at random<SUP>5</SUP>. </P><P>After decades of work, a need will arise to introduce others to it. And Post Keynesians have addressed this need with anthologies of classic papers, introductory <A HREF="http://robertvienneau.blogspot.com/2014/05/paradigming-is-easy.html">works</A> for other economists, and <A HREF="http://robertvienneau.blogspot.com/2006/08/textbooks-for-teaching-non.html">textbooks</A>. One can also find Post Keynesians editing, or participating in the development, of standard reference works<SUP>6</SUP>. </P><P>On a more local level, Post Keynesian economists have participated in the governance of economic departments around the world<SUP>7</SUP>. And they have provided governments with advice many a time, from within and without<SUP>8</SUP>. </P><P>I have deliberately not written about substantial Post Keynesian ideas in this post. If one is aware of the history mentioned in this post, even if one had never been exposed to Post Keynesian ideas, one must conclude Post Keynesian theory is much like any other set of academic ideas. One would have difficulty in seeing how academics could justify dismissing these ideas without engaging with them Likewise, one might wonder how, perhaps, those aspiring to be professional economists might not even be exposed to Post Keynesianism in gaining a post graduate degree. Yet, apparently, such a happenstance seems to be not at all rare among mainstream economists. </P><B>Footnotes</B><OL><LI>I recognize my post is biased towards the English language. It is also quite impressionistic and selective. I am taking Sraffians as Post Keynesians for the purposes of this post.</LI><LI>Major contributions to the Cambridge Capital Controversy are to be found in these journals.</LI><LI>For example, Paolo Sylos Labini for the <I>Banca Nazionale del Lavoro Quarterly Review</I>, Athanasios Asimakopulos for the <I>Canadian Journal of Economics</I>, and Neri Salvadori for <I>Metroeconomica</I>.</LI><LI>For what it is worth, I am published in the <I>Manchester School</I>.</LI><LI>How would one characterize Edward Elgar and Routledge, for example?</LI><LI>The first edition of the <I>New Palgrave</I> is an obvious example.</LI><LI>Economics at Cambridge University is an obvious case. Albert Eichner chairing the Rutgers economics department is another case.</LI><LI> Examples include Nicholas Kaldor's work with the Radcliffe Committee, John Kenneth Galbraith giving advice to John F. Kennedy, the advocacy of Tax-based Income Policies (TIPs) in the 1970s to fight stagflation, and policy suggestions associated with Modern Monetary Theory (MMT).</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-90437476509717743982016-03-02T08:04:00.000-05:002016-03-02T08:04:04.692-05:00Romer And Romer Stumble<P>A <A HREF="http://billmoyers.com/story/the-battle-over-reagans-economic-plan/">debate</A> <A HREF="http://www.truth-out.org/news/item/34978-the-dominant-media-left-leaning-economists-and-the-illusion-of-consensus">has</A> <A HREF="http://big.assets.huffingtonpost.com/ResponsetoCEA.pdf">recently</A> <A HREF="http://jwmason.org/slackwire/plausibility/">arisen</A> <A HREF="http://econbrowser.com/archives/2016/02/what-is-the-assumed-output-gap-in-the-friedman-projections">about</A> Gerald Friedman's <A HREF="http://www.dollarsandsense.org/What-would-Sanders-do-013016.pdf">analysis</A> of Bernie Sanders' proposed economic program. In a welcome turn of events, two defenders of the establishment, Christina and David Romer, finally offer some substance, instance of just relying on their authority as Very Serious People. </P><P>In this post, I ignore most of the substance of the argument. I want to focus on three errors I find in this passage: </P><BLOCKQUOTE>"Potentially more worrisome are the extensive interventions in the labor market. The experiences of many European countries from the 1970s to today show that an overly regulated labor market can have severe consequences for normal unemployment. There are strong arguments for raising the minimum wage; and over the range observed historically in the United States, the short-run employment effects of moderate increases appear negligible. But doubling the minimum wage nationwide, adding new requirements for employer-funded paid vacations and sick leave, and increasing payroll taxes substantially would take us into uncharted waters. Obviously, these changes would not bring the United States all the way to levels of labor market regulation of many European countries in the 1970s. But they are large enough that one can reasonably fear that they could have a noticeable impact on capacity growth." -- Christina D. Romer and David H. Romer, <A HREF="http://ineteconomics.org/uploads/general/romer-and-romer-evaluation-of-friedman1.pdf">Senator Sander's Proposed Policies and Economic Growth</A> (5 February 2016) p. 10-11. </BLOCKQUOTE><P>First, the reference to "interventions in the labor market" and an "overly regulated labor market" imposes a false dichotomy. An unregulated labor market cannot exist. Certainly this is so in an advanced capitalist economy. Possible choices are among sets of regulations and norms, not among intervention or not. Calling one set of regulations an example of government non-intervention is to disguise taking a side under obfuscatory verbiage. </P><P>Second, Romer and Romer presuppose a consensus about the empirical effects of different regulations on the labor market in Europe and the United States that I do not think exists. If I wanted to find empirically based arguments countering Romer and Romer's claim, I would look through back issues of the <I>Cambridge Journal of Economics</I>. Perhaps at <A HREF="http://cje.oxfordjournals.org/content/39/2/467.abstract">least</A> <A HREF="https://cje.oxfordjournals.org/content/37/4/845">one</A> <A HREF="https://cje.oxfordjournals.org/content/35/2/437.abstract?sid=21976fe7-e9d2-4a61-9793-e674e80ab7f3">of</A> <A HREF="https://cje.oxfordjournals.org/content/33/1/51.abstract?sid=21976fe7-e9d2-4a61-9793-e674e80ab7f3">these</A> <A HREF="https://cje.oxfordjournals.org/content/27/1/123.abstract?sid=21976fe7-e9d2-4a61-9793-e674e80ab7f3">articles</A> might be helpful. </P><P>Third, Romer and Romer suggest that, given the set of regulations they like to think of as government non-intervention, markets for labor and goods would have a tendency to clear. Otherwise, economic growth would be jeopardized. No theoretical foundation exists for thinking so. </P><P>Even the best mainstream economists seem incapable of writing ten pages without spouting ideological claptrap and propagating silly errors exposed more than half a century ago. Something seems terribly wrong with economics profession. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-50803031409156361302016-02-29T08:03:00.000-05:002016-02-29T08:03:03.523-05:00Conservatism According to Corey Robin<P>I have been re-reading Corey Robin's <I>The Reactionary Mind</I>. According to this book, defending arbitrary hierarchies is the first priority among conservatives. They believe: </P><UL><LI>Workers should obey their masters.</LI><LI>Wives should obey their husbands.</LI><LI>Downtrodden ethnic groups should obey socially privileged ethnic groups.</LI><LI>The laity should obey priests.</LI><LI>The non-affluent should show proper deference towards those with great wealth (who could never be malefactors).</LI></UL><P>These hierarchies have implications for daily lives, not just political rule. For the right, liberty is liberty for the rulers to do as they will, not for those who suffer what they must. </P><P>I deliberately do not write about slavery. According to Robin, conservatism is literally reactionary. Conservatives defend hierarchies that are currently threatened or recently overthrown. They focus on restoring what was recently lost. Maybe this has something to do with widespread fear and resentment on the right. </P><P>Conservatives often do not have admiration for the rulers of the ancient regime. If those rulers were willing to do what needed to be done to preserve their power, the threats would never have gotten so far, and losses would not have been suffered. The conservative, unlike his popular and complimentary image, is willing to make radical changes so as to reconstruct society as it once was. This seems to go along with the awareness of some contemporary neoliberals that market societies are not natural formations, but must be constructed and maintained by state power. But is this aping of the left consistent with the conservative's encouragement of anti-intellectualism and stupidity? Perhaps the idea is that only an elite need understand the goal, while widespread ignorance among the masses can only help the cause. </P><P>The hierarchies that conservatives seek to defend or restore are not meritocracies, in the sense that those on top are expected to have superior intellect, wisdom, or morals. Rulers should demonstrate their fitness to rule by seizing what they can, in war or business. Maybe this has something to do with why many conservatives endorse the supposed "free market", without worrying about externalities, information asymmetries, transaction costs, or market power. One can also see here an echo of Friedrich Nietzsche's overman. </P><P>Much of the above comes from the introduction and first couple of chapters of Robin's book. Much of the rest consists of case studies of particular thinkers and polemicists. </P><B>Reference</B><UL><LI>Corey Robin (2013). <A HREF="http://www.amazon.com/The-Reactionary-Mind-Conservatism-Edmund/dp/0199959110"><I>The Reactionary Mind: Conservatism from Edmund Burke to Sarah Palin</I></A>, Oxford University Press.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-66542433942826114432016-02-17T08:08:00.000-05:002016-02-17T08:08:02.217-05:00Classification of Finite Simple Groups: A Proved Theorem?<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><tr><td align="center"><a href="https://2.bp.blogspot.com/-2u-dgwXt7Mo/Vr3RzZOW5gI/AAAAAAAAAoc/rB6zXwFt2mc/s1600/D4Lattice.JPG" imageanchor="1"><img border="0" src="https://2.bp.blogspot.com/-2u-dgwXt7Mo/Vr3RzZOW5gI/AAAAAAAAAoc/rB6zXwFt2mc/s320/D4Lattice.JPG" /></a></td></tr><tr><td align="center"><b>Figure 1: Lattice Diagram for Group of Symmetries of the Square</b></td></tr></tbody></table><Blockquote>"I shall now mention something I obviously do not understand." - Ian Hacking (2014, p. 18) </BLOCKQUOTE><b>1.0 Introduction</b><P>This has nothing to do with economics. It is my attempt to get my mind around a place where I can get a glimmer of some exciting mathematics being done in my lifetime. </P><P>Mathematicians have stated a theorem for classifying finite simple groups. Whether they have proven this theorem is an intriguing question in the philosophy of mathematics. </P><P>A finite simple group is a group with a finite number of elements and no proper normal subgroup. This definition contains several technical terms. In this post, I try to explain these terms and the setting of the theorem for classifying simple groups. This preamble raises several questions: <P><ul><li>What is a group? A proper subgroup? A normal subgroup?</li><li>How can a finite, non-simple group be factored into a composition of simple groups?</li></ul><P>I try to clarify the answers to these questions by means of a lengthy example. You can probably find this better expressed elsewhere. In working this out, I relied heavily on Fraleigh's textbook, which is the only book in the references that I have read, albeit mostly in the second edition. </P><b>2.0 The Group of Symmetries of the Square</b><P>A <i>group</i> is a generalization, in some sense, of a multiplication table. Formally, it is a set with a binary operation, in which the binary operation satisfies three axioms. A <i>finite group</i> is a group in which the set contains a finite number of elements. </P><P>To illustrate, I consider the set of symmetries of the square (Figure 2). These eight elements of the set are like the numbers along the top and left side of a multiplication table. Each element is an operation that can be performed on a square, leaving the square superimposed on itself. Each operation is described in the right column of Figure 2. The third column provides a picture of the operation. The four vertices of the square are numbered so that one can see the result of the operation. The second column specifies each operation as a permutation of the numbered vertices. The first row in each permutation lists the vertices, while the second row shows which of the original vertices ends up in the place of each vertex. The first column introduces a notation for naming each operation. The remainder of this post is expressed in this notation. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><tr><td align="center"><a href="https://4.bp.blogspot.com/-MDXjdMdZx64/Vr3RrfNqLtI/AAAAAAAAAoY/BlP91-FWgVI/s1600/D4Definition.JPG" imageanchor="1"><img border="0" src="https://4.bp.blogspot.com/-MDXjdMdZx64/Vr3RrfNqLtI/AAAAAAAAAoY/BlP91-FWgVI/s320/D4Definition.JPG" /></a></td></tr><tr><td align="center"><b>Figure 2: Elements of a Group</b></td></tr></tbody></table><P>The group operation, *, is function composition. Let <i>a</i> and <i>b</i> be elements of the set {ρ<sub>0</sub>, ρ<sub>1</sub>, ρ<sub>2</sub>, ρ<sub>0</sub>, μ<sub>0</sub>, μ<sub>1</sub>, σ<sub>0</sub>, σ<sub>1</sub>}. The product <i>a</i>*<i>b</i> is defined to be the single operation that is equivalent to first performing the operation <i>a</i> on the square and then performing the operation <i>b</i> on the result. (Many textbooks define functional composition from right-to-left, instead.) Table 1 is the multiplication table for this group, under these definitions. For example, rotating a square 90 degrees clockwise twice is equivalent to rotating the square clockwise through 180 degrees. Thus:</P><BLOCKQUOTE>ρ<SUB>1</SUB> * ρ<SUB>1</SUB> = ρ<SUB>2</SUB></BLOCKQUOTE><table align="CENTER" border=""><tbody><CAPTION><b>Table 1: The Group D<sub>4</sub></b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>ρ<sub>0</sub></b></td><td><b>ρ<sub>1</sub></b></td><td><b>ρ<sub>2</sub></b></td><td><b>ρ<sub>3</sub></b></td><td><b>μ<sub>0</sub></b></td><td><b>μ<sub>1</sub></b></td><td><b>σ<sub>0</sub></b></td><td><b>σ<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>ρ<sub>0</sub></b></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td><td>σ<sub>0</sub></td><td>σ<sub>1</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>1</sub></b></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>σ<sub>0</sub></td><td>σ<sub>1</sub></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>2</sub></b></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td><td>σ<sub>1</sub></td><td>σ<sub>0</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>3</sub></b></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>σ<sub>1</sub></td><td>σ<sub>0</sub></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td></tr><tr align="CENTER"><td><b>μ<sub>0</sub></b></td><td>μ<sub>0</sub></td><td>σ<sub>1</sub></td><td>μ<sub>1</sub></td><td>σ<sub>0</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>1</sub></td></tr><tr align="CENTER"><td><b>μ<sub>1</sub></b></td><td>μ<sub>1</sub></td><td>σ<sub>0</sub></td><td>μ<sub>0</sub></td><td>σ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>3</sub></td></tr><tr align="CENTER"><td><b>σ<sub>0</sub></b></td><td>σ<sub>0</sub></td><td>μ<sub>0</sub></td><td>σ<sub>1</sub></td><td>μ<sub>1</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td></tr><tr align="CENTER"><td><b>σ<sub>1</sub></b></td><td>σ<sub>1</sub></td><td>μ<sub>1</sub></td><td>σ<sub>0</sub></td><td>μ<sub>0</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td></tr></tbody></table><P>A group is defined by the following three axioms: </P><ul><li>The binary operation in the group is <i>associative</i>. That is, for all <i>a</i>, <i>b</i>, and <i>c</i> in the group:</li></ul><blockquote><blockquote>(<i>a</i> * <i>b</i>) * <i>c</i> = <i>a</i> * (<i>b</i> * <i>c</i>) </blockquote></blockquote><ul><li>The group contains an <i>identity element</i>. There exists an element <i>e</i> in the group such that for all <i>a</i> in the group:</li></ul><blockquote><blockquote><i>e</i> * <i>a</i> = <i>a</i> * <i>e</i> = <i>a</i></blockquote></blockquote><ul><li>Every element of the group has an <i>inverse</i>. For all <i>a</i> in the group, there exists an element <i>a</i><sup>-1</sup> in the group such that:</li></ul><blockquote><blockquote><i>a</i> * <i>a</i><sup>-1</sup> = <i>a</i><sup>-1</sup> * <i>a</i> = <i>e</i></blockquote></blockquote><P>Associativity is tedious to check for <b>D<sub>4</sub></b>. Associativity implies that one can drop parenthesis below. ρ<sub>0</sub> is the identity element. Every row and column in the multiplication table for <b>D<sub>4</sub></b> contains ρ<sub>0</sub>; thus, every element has an inverse. </P><P>An <i><a href="https://en.wikipedia.org/wiki/Niels_Henrik_Abel">Abelian</a> group</i> is one in which the binary operation is commutative. The group of symmetries of the square is not Abelian. For an Abelian group, the multiplication table is symmetric across the principal diagonal; it does not matter to the result in which order one performs the operation for two arguments. The following two equations illustrates that <b>D<sub>4</sub></b> is not Abelian: </P><blockquote>μ<sub>0</sub>*ρ<sub>1</sub> = σ<sub>1</sub></blockquote><blockquote>ρ<sub>1</sub>*μ<sub>0</sub> = σ<sub>0</sub></blockquote><P>In words, flipping a square around its horizontal axis of symmetry and then rotating it ninety degrees clockwise is not equivalent to rotating it ninety degrees clockwise and then then reflecting it across that axis. The result of the first composition of operations is equivalent to reflecting the square across the diagonal axis of symmetry running from the south west to the north east. The second composition of operations is equivalent to flipping the square across the other diagonal. </P><P>One can also set up equations in a group, for example: </P><blockquote>ρ<sub>1</sub>*ρ<sub>2</sub>*<i>x</i> = μ<sub>0</sub></blockquote><P>Then <i>x</i> must be σ<sub>0</sub>. Solving a Rubik's cube is analogous to solving such an equation. </P><b>3.0 Proper and Improper Subgroups</b><P>Some rows and columns in Table 1 can stand alone as a group. The entries in these restricted row and columns all appear as headings in the rows and columns. These entries form a <I>subgroup</I> of the original group. One-fourth of the table in the upper left of Table 1 provides an example. {ρ<SUB>0</SUB>, ρ<SUB>1</SUB>, ρ<SUB>2</SUB>, ρ<SUB>3</SUB>} is a subgroup of <B>D<SUB>4</SUB></B> (Table 2). </P><table align="CENTER" border=""><CAPTION><b>Table 2: A Subgroup of D<sub>4</sub> with Four Elements</b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>ρ<sub>0</sub></b></td><td><b>ρ<sub>1</sub></b></td><td><b>ρ<sub>2</sub></b></td><td><b>ρ<sub>3</sub></b></td></tr><tr align="CENTER"><td><b>ρ<sub>0</sub></b></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>1</sub></b></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>2</sub></b></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>3</sub></b></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td></tr></table><P>The group <B>D<SUB>4</SUB></B> has ten subgroups, as shown in the <I>Lattice Diagram</I> in Figure 1 above. Subgroups have been defined such that, for any group <B>G</B>, the group <B>G</B> is a subgroup of itself. Another trivial case, the one-element group consisting of the identity element, also provides a subgroup of <B>G</B>. These two subgroups are known as <I>improper subgroups</I>. All other subgroups are <I>proper subgroups</I>. </P><P>One can make a couple of observations about subgroups. The binary operation in the group is the same as the binary operation in the subgroup. The property of associativity carries over from the group to the subgroup. Since a subgroup is a group, it must contain an identity element. And that identity element must also be the identity element for the group containing the subgroup. Thus, every subgroup of <B>D<SUB>4</SUB></B> contains ρ<SUB>0</SUB>. Likewise, for every element of a subgroup, the subgroup must also contain its inverse. Finally, the number of elements in a subgroup must evenly divide the number of elements in the group. </P><P>I have shown above how the eight elements of <B>D<SUB>4</SUB></B> can be defined in terms of permutations. As a matter of fact, the set of permutations of (1, 2, ..., <I>n</I>) form a group under the operation of function composition. This <I>permutation group</I> is designated as <B>S<SUB><I>n</I></SUB></B>, and it contains <I>n</I>! elements. Thus, <B>S<SUB>4</SUB></B> contains 24 (= 4x3x2x1) elements. Not only can one find all the subgroups of <B>D<SUB>4</SUB></B>, one can extend the group such that <B>D<SUB>4</SUB></B> is a subgroup of that extended group. </P><b>4.0 Isomorphic Groups</b><P>In a group, the order of rows and columns in the multiplication table are of no matter. Likewise, the names of the elements are irrelevant to the structure of the group. Two groups are <I>isomorphic</I> if the multiplication table for one group can be mapped into the multiplication table for another group by reordering and renaming the elements of, say, the first group. As an example, consider the groups {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, μ<SUB>0</SUB>, μ<SUB>1</SUB>} and {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, σ<SUB>0</SUB>, σ<SUB>1</SUB>}. They each have the same number of elements, which is necessary for an isomorphism. Table 3 defines the group operation for the first group. Suppose that, in Table 3, μ<sub>0</sub> is renamed σ<sub>0</sub>, and μ<sub>1</sub> is renamed σ<sub>1</sub> throughout. The resulting table will match the operation for the second group. Thus, the two groups are isomorphic. </P><table align="CENTER" border=""><CAPTION><b>Table 3: The Group {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, μ<SUB>0</SUB>, μ<SUB>1</SUB>}</b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>ρ<sub>0</sub></b></td><td><b>ρ<sub>2</sub></b></td><td><b>μ<sub>0</sub></b></td><td><b>μ<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>ρ<sub>0</sub></b></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>2</sub></b></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td></tr><tr align="CENTER"><td><b>μ<sub>0</sub></b></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td></tr><tr align="CENTER"><td><b>μ<sub>1</sub></b></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td></tr></table><P>The groups in Tables 2 and 3 are NOT isomorphic. They each contain four elements. Each element, however, in the group in Table 3 is its own inverse. This is an algebraic property, preserved no matter how the elements of the group are renamed. And the group in Table 2 does not have this property. As a matter of fact, only two groups containing four elements exist, up to an isomorphism. In other words, any group with four elements is isomorphic to either the group in Table 2 or to the group in Table 3. </P><P>Furthermore, only one group, up to isomorphism, contains two elements. Its operation is defined by Table 4. All the subgroups of <B>D<SUB>4</SUB></B> containing two elements are isomorphic to this group and, ipso facto, to each other. The text colors of the subgroups in the lattice diagram (Figure 1) express these isomorphisms. </P><table align="CENTER" border=""><CAPTION><b>Table 4: The Unique Group (Up To Isomorphism) With Two Elements</b></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>0</b></td><td><b>1</b></td></tr><tr align="CENTER"><td><b>0</b></td><td>0</td><td>1</td></tr><tr align="CENTER"><td><b>1</b></td><td>1</td><td>0</td></tr></table><b>5.0 Normal Subgroups, Factor Groups, and Homomorphisms</b><P>Certain additional patterns are apparent in Table 1. I have already pointed out that the first four rows and columns constitute the subgroup with the operation shown in Table 2. Notice that none of the entries in the last four columns for the first four rows are in this subgroup. Likewise, none of the entries in the first four columns for the last four rows are in this subgroup. On the other hand, the entries in the remaining rows and columns in the lower right are all in this subgroup. Can you see that these observations reveal the pattern expressed in Table 4? Mathematicians express this by saying that the <I>factor group</I> <B>D<SUB>4</SUB></B>/{ρ<SUB>0</SUB>, ρ<SUB>1</SUB>, ρ<SUB>2</SUB>, ρ<SUB>3</SUB>} is isomorphic to the group with two elements. </P><P>A subgroup is <I>normal</I> if it can be used to divide up the rows and columns in the multiplication table for the group like this. For another example, consider the subgroup {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>}. Table 5 shows a reordering of the rows and columns in Table 1 to facilitate the calculation of the factor group for this subgroup. Consider dividing this grid up into 16 blocks of two rows and two columns each. Each block will contain two elements of the group <B>D<sub>4</sub></B>, and which element is paired with each element does not vary among these blocks. </P><table align="CENTER" border=""><CAPTION><b>Table 5: The Group D<sub>4</sub> Reordered</B></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>ρ<sub>0</sub></b></td><td><b>ρ<sub>2</sub></b></td><td><b>ρ<sub>1</sub></b></td><td><b>ρ<sub>3</sub></b></td><td><b>μ<sub>0</sub></b></td><td><b>μ<sub>1</sub></b></td><td><b>σ<sub>0</sub></b></td><td><b>σ<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>ρ<sub>0</sub></b></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>3</sub></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td><td>σ<sub>0</sub></td><td>σ<sub>1</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>2</sub></b></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>1</sub></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td><td>σ<sub>1</sub></td><td>σ<sub>0</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>1</sub></b></td><td>ρ<sub>1</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td><td>σ<sub>0</sub></td><td>σ<sub>1</sub></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td></tr><tr align="CENTER"><td><b>ρ<sub>3</sub></b></td><td>ρ<sub>3</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td><td>σ<sub>1</sub></td><td>σ<sub>0</sub></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td></tr><tr align="CENTER"><td><b>μ<sub>0</sub></b></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td><td>σ<sub>1</sub></td><td>σ<sub>0</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>1</sub></td></tr><tr align="CENTER"><td><b>μ<sub>1</sub></b></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td><td>σ<sub>0</sub></td><td>σ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>3</sub></td></tr><tr align="CENTER"><td><b>σ<sub>0</sub></b></td><td>σ<sub>0</sub></td><td>σ<sub>1</sub></td><td>μ<sub>0</sub></td><td>μ<sub>1</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>0</sub></td><td>ρ<sub>2</sub></td></tr><tr align="CENTER"><td><b>σ<sub>1</sub></b></td><td>σ<sub>1</sub></td><td>σ<sub>0</sub></td><td>μ<sub>1</sub></td><td>μ<sub>0</sub></td><td>ρ<sub>3</sub></td><td>ρ<sub>1</sub></td><td>ρ<sub>2</sub></td><td>ρ<sub>0</sub></td></tr></table><P>These observations can be formalized by the function defined in Table 6. For an element <I>a</I> of <B>D<SUB>4</SUB></B>, let <I>f</I>(<I>a</I>) denote the map defined in Table 6. To find the value of this function, locate <I>a</I> in the first column. Whether this value is 0, 1, 2, or 3 is determined by the corresponding entry in the second column. For all <I>a</I> and <I>b</I> in <B>D<SUB>4</SUB></B>: </P><BLOCKQUOTE><I>f</I>(<I>a</I> * <I>b</I>) = <I>f</I>(<I>a</I>) o <I>f</I>(<I>b</I>) </BLOCKQUOTE><P>A map from one group to another with this property is a <I>homomorphism</I>. An isomorphism is a homomorphism, but a homomorphism is a more general concept. Homomorphisms do not need to leave the number of elements in the group invariant. </P><TABLE ALIGN="CENTER" border=""><CAPTION><b>Table 6: A Homomorphism from D<SUB>4</SUB> to {0, 1, 2, 3}</b></CAPTION><TR ALIGN="CENTER"><TD><B>Elements of D<SUB>4</SUB></B></TD><TD><B>Image</B></TD></TR><TR ALIGN="CENTER"><TD>ρ<SUB>0</SUB>, ρ<SUB>2</SUB></TD><TD>0</TD></TR><TR ALIGN="CENTER"><TD>ρ<SUB>1</SUB>, ρ<SUB>3</SUB></TD><TD>1</TD></TR><TR ALIGN="CENTER"><TD>μ<SUB>0</SUB>, μ<SUB>1</SUB></TD><TD>2</TD></TR><TR ALIGN="CENTER"><TD>σ<SUB>0</SUB>, σ<SUB>1</SUB></TD><TD>3</TD></TR></TABLE><P>The factor group <B>D<SUB>4</SUB></B>/{ρ<SUB>0</SUB>, ρ<SUB>2</SUB>} is easily calculated. Replace each element of <B>D<SUB>4</SUB></B> in Table 5 by its image under the homomorphism in Table 6. Collapse each pair of rows and columns. One ends up with Table 7, where I have renamed the group operation, as above. The factor group <B>D<SUB>4</SUB></B>/{ρ<SUB>0</SUB>, ρ<SUB>2</SUB>} is isomorphic to the group with four elements with the operation shown in Table 3 above. The number of elements in a factor group is the quotient of the number of elements in the original group and the number of elements in the subgroup used to form the factor group. </P><table align="CENTER" border=""><CAPTION><b>Table 7: The Factor Group D<SUB>4</SUB>/{ρ<SUB>0</SUB>, ρ<SUB>2</SUB>}</b></CAPTION><tr align="CENTER"><td><b>o</b></td><td><b>0</b></td><td><b>1</b></td><td><b>2</b></td><td><b>3</b></td></tr><tr align="CENTER"><td><b>0</b></td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr align="CENTER"><td><b>1</b></td><td>1</td><td>0</td><td>3</td><td>2</td></tr><tr align="CENTER"><td><b>2</b></td><td>2</td><td>3</td><td>0</td><td>1</td></tr><tr align="CENTER"><td><b>3</b></td><td>3</td><td>2</td><td>1</td><td>0</td></tr></table><P>The two improper subgroups for any group are normal and yield trivial factor groups. The factor group <B>D<SUB>4</SUB></B>/<B>D<SUB>4</SUB></B> is isomorphic to the one-element group whose only member is the identity element. The factor group <B>D<SUB>4</SUB></B>/{ρ<SUB>0</SUB>} is isomorphic to <B>D<SUB>4</SUB></B>. The factor groups for improper subgroups provide no information about the structure of a group. </P><B>6.0 A Subgroup that is Not Normal</B><P>Not all subgroups are normal. The subgroup {ρ<SUB>0</SUB>, μ<SUB>0</SUB>}, for example, is not a normal subgroup of <B>D<SUB>4</SUB></B>. Table 8 proposes a map from the elements of the group to the first four natural numbers. And Table 9 illustrates another reordering of the rows and columns in Table 1, with the entries replaced by the natural numbers to which they map. If one confines oneself to the first two columns, each pair of rows could be collapsed into one, with the label from the row taken from the map. But this process breaks down for the next two and the last two columns. </P><TABLE ALIGN="CENTER" border=""><CAPTION><b>Table 8: A Map from D<SUB>4</SUB> to {0, 1, 2, 3} that is Not a Homomorphism</b></CAPTION><TR ALIGN="CENTER"><TD><B>Elements of D<SUB>4</SUB></B></TD><TD><B>Image</B></TD></TR><TR ALIGN="CENTER"><TD>ρ<SUB>0</SUB>, μ<SUB>0</SUB></TD><TD>0</TD></TR><TR ALIGN="CENTER"><TD>ρ<SUB>1</SUB>, σ<SUB>0</SUB></TD><TD>1</TD></TR><TR ALIGN="CENTER"><TD>ρ<SUB>2</SUB>, μ<SUB>1</SUB></TD><TD>2</TD></TR><TR ALIGN="CENTER"><TD>ρ<SUB>3</SUB>, σ<SUB>1</SUB></TD><TD>3</TD></TR></TABLE><table align="CENTER" border=""><CAPTION><b>Table 9: Another Reodering of The Group D<sub>4</sub></B></CAPTION><tr align="CENTER"><td><b>*</b></td><td><b>ρ<sub>0</sub></b></td><td><b>μ<sub>0</sub></b></td><td><b>ρ<sub>1</sub></b></td><td><b>σ<sub>0</sub></b></td><td><b>ρ<sub>2</sub></b></td><td><b>μ<sub>1</sub></b></td><td><b>ρ<sub>3</sub></b></td><td><b>σ<sub>1</sub></b></td></tr><tr align="CENTER"><td><b>ρ<sub>0</sub></b></td><td>0</td><td>0</td><td>1</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td></tr><tr align="CENTER"><td><b>μ<sub>0</sub></b></td><td>0</td><td>0</td><td>3</td><td>3</td><td>2</td><td>2</td><td>1</td><td>1</td></tr><tr align="CENTER"><td><b>ρ<sub>1</sub></b></td><td>1</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td><td>0</td><td>0</td></tr><tr align="CENTER"><td><b>σ<sub>0</sub></b></td><td>1</td><td>1</td><td>0</td><td>0</td><td>3</td><td>3</td><td>2</td><td>2</td></tr><tr align="CENTER"><td><b>ρ<sub>2</sub></b></td><td>2</td><td>2</td><td>3</td><td>3</td><td>0</td><td>0</td><td>1</td><td>1</td></tr><tr align="CENTER"><td><b>μ<sub>1</sub></b></td><td>2</td><td>2</td><td>1</td><td>1</td><td>0</td><td>0</td><td>3</td><td>3</td></tr><tr align="CENTER"><td><b>ρ<sub>3</sub></b></td><td>3</td><td>3</td><td>0</td><td>0</td><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr align="CENTER"><td><b>σ<sub>1</sub></b></td><td>3</td><td>3</td><td>2</td><td>2</td><td>1</td><td>1</td><td>0</td><td>0</td></tr></table><P>Suppose a subgroup contains <I>n</I> elements. To determine if the subgroup is normal, it is sufficient to examine the first <I>n</I> rows and the first <I>n</I> columns in the reordered table. This capability follows from a theorem about what are known as left and right cosets for a subgroup. </P><P>The permuation group <B>S<SUB>4</SUB></B> provides another example of a subgroup that is not normal. By my calculations, <B>D<SUB>4</SUB></B> is NOT a normal subgroup of <B>S<SUB>4</SUB></B>. </P><b>7.0 The Composition Series of a Group</b><P>At this point, I have completed my explanation of the lattice diagram at the top of this post, including circles, text colors, and boxes. I draw from these results to illustrate how a non-simple group, namely <B>D<SUB>4</SUB></B>, can be expressed as a composition of factor groups. </P><P>Table 10 lists twelve series of subgroups of the group of symmetries of the square. Each series has the following properties: </P><UL><LI>The leftmost group in the series is the one-element group containing the identity element.</LI><LI>The rightmost group is <B>D<SUB>4</SUB></B>.</LI><LI>Each group in the series (except <B>D<SUB>4</SUB></B>) is a proper normal subgroup of the group immediately to the right of it in the series.</LI></UL><P>A series with these properties is known as a <I>subnormal series of the group</I> <B>D<SUB>4</SUB></B>. If every group in the series is also a normal subgroup of <B>D<SUB>4</SUB></B>, the series is a <I>normal series of the group</I> <B>D<SUB>4</SUB></B>. By the last property in the bulleted list, one can calculate a factor group for each pair of immediately successive groups in the series. </P><TABLE align="CENTER" border=""><CAPTION><B>Table 10: Twelve Normal and Subnormal Series for <B>D<SUB>4</SUB></B></B></CAPTION><TR align="CENTER"><TD><B>Number<BR>Factor Groups</B></TD><TD><B>Series</B></TD><TD><B>Normal<BR>Series</B></TD></TR><TR align="CENTER"><TD>1</TD><TD>{ρ<SUB>0</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>2</TD><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>1</SUB>, ρ<SUB>2</SUB>, ρ<SUB>3</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD ROWSPAN="3">2</TD><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, μ<SUB>0</SUB>, μ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, σ<SUB>0</SUB>, σ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD ROWSPAN="7">3</TD><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>1</SUB>, ρ<SUB>2</SUB>, ρ<SUB>3</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, μ<SUB>0</SUB>, μ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, σ<SUB>0</SUB>, σ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>Yes</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, μ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, μ<SUB>0</SUB>, μ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, μ<SUB>1</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, μ<SUB>0</SUB>, μ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, σ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, σ<SUB>0</SUB>, σ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR><TR align="CENTER"><TD>{ρ<SUB>0</SUB>} < {ρ<SUB>0</SUB>, σ<SUB>1</SUB>} < {ρ<SUB>0</SUB>, ρ<SUB>2</SUB>, σ<SUB>0</SUB>, σ<SUB>1</SUB>} < <B>D<SUB>4</SUB></B></TD><TD>No</TD></TR></TABLE><P>The definition of an isomorphism for a subnormal series builds on the definition of isomorphism for groups. Consider the factor groups arising in each series from successive pairs of subgroups in each series. Two series are isomorphic if they contain the same of number of factor groups, in this sense, and these factor groups are isomorphic. The order in which the factor groups arise can vary among isomorphic subnormal series. </P><P>I have collected isomorphic series together, in Table 10, by means of horizontal lines in the first column. The series with one factor group is not isomorphic to any other series. The first series shown with two factor groups is not isomorphic to the other three series with two factor groups. And those three series are isomorphic to one another. All of the series with three factor groups are isomorphic to one another. </P><P>The series with three factor groups have another property. All factor groups in these series with three factor groups are simple groups. That is, they contain no proper normal subgroups. A subnormal series of a group in which all factor groups formed by the series are simple is known as a <I>composition series</I>. By the Jordan-Hölder Theorem, all compositions series for a group are isomorphic series. This theorem justifies one in speaking of THE composition series for a group. Finding the factor groups in a the composition series for a group is somewhat analogous to factoring a natural number. Note that <B>D<SUB>4</SUB></B> contains eight elements and each of the three factor groups in the composition series contain two elements. Furthermore, </P><BLOCKQUOTE>8 = 2<SUP>3</SUP></BLOCKQUOTE><P>For a natural number, the prime factors can be combined to yield the original number. Here the analogy apparently breaks down. The factor groups in a composition series for a group constrain the structure of the group, but two non-isomorphic groups can have the same composition series. But still, mathematicians have solved various problems in group theory for finite non-simple groups by use of the classification of finite simple groups. </P><P>Composition series apparently have an application in solving polynomial equations. The composition series for the permutation group <B>S<SUB>5</SUB></B> contains a factor group that is non-Abelian. This is connected with the insolvability of the quintic. There are formulas for zeros for cubic and fourth order polynomial, analogous to the quadratic formula. But there is no such formulas for poynomials of the fifth degree and higher. </P><b>8.0 Classification of Finite Simple Groups</b><P>At this point, I have explained how finite simple groups arise as factor groups for the composition series of any finite group. I hope that this gives some hint of why the following theorem is of interest. </P><P><B>Theorem:</B> Each finite simple group is one of the following, up to an isomorphism: </P><UL><LI>A group of prime order.</LI><LI>An alternating group.</LI><LI>A Lie group.</LI><LI>One of 26 sporadic groups not otherwise classified.</LI></UL><P>I am aware that this this theorem uses technical terms I still have not explained, including one that I simply do not understand myself. </P><P>The sporadic groups are finite simple groups that do not fall into the other categories, although, I gather, some sporadic groups are related to one another.The sporadic group with the largest number of elements is called the Monster group. It has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements. </P><B>9.0 History of the Theorem</B><P>In 1972, Daniel Gorenstein proposed that mathematicians could complete a classification of all simple groups. By the early 1980s, mathematicians had stated the theorem and those specialists who had pursued Gorenstein's program believed they had proven it. The proof, however, was scattered among (tens of?) thousands of pages in hundreds(?) of papers in many mathematics journals. No one person had probably ever understood the proof or read it in its entirety. </P><P>The proof, however, was discovered even then to be incomplete. Steve Smith and Michael Aschbacher worked on closing this gap, relating to <I>quasithin</I> groups. They succeeded by 2004. </P><P>Meanwhile, a number of mathematicians have been trying to simplify the proof and to restate it in one location. The ambition of these mathematicians is to produce a "second generation" proof of only a couple thousand pages. </P><P>Has a theorem been proven if only one or two mathematicians have read the proof in its entirety? How about if nobody has, which would have been the case in the 1980s if the proof had indeed been valid? Certainly, the proof of the classification theorem is not surveyable, in Wittgenstein's sense. Do mathematical results need to be established by a social process? If so, how can such social processes be characterized? </P><b>Appendix: Terms Defined or Illustrated Above</b><P>Abelian group, Associativity, Composition Series, Factor Group, Finite Group, Group, Homomorphism, Identity Element, Improper Subgroup, Inverse, Isomorphic Groups, Isomorphic Subnormal Series, Lattice Diagram, Normal Series, Normal Subgroup, Permutation Group, Proper Subgroup, Subgroup, Subnormal Series. </P><b>References</b><ul><LI>Michael Aschbacher (2004). The Status of the Classification of the Finite Simple Groups, <I>Notices of the AMS</I>, V. 51, No. 7 (Aug.): pp. 736-740.</LI><li>Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon (2011). <a href="http://www.amazon.com/Classification-Finite-Simple-Groups-Characteristic/dp/0821853368/"><i>The Classification of Finite Simple Groups: Groups of Characteristic 2 Type</i></a>, American Mathematical Society.</li><li>Nicolas Bourbaki (1943). <i>Elements of Mathematics: Algebra I: Chapters 1-3</i>.</li><LI>J. H. Conway and S. P. Norton (1979). <A HREF="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</A>, <I>Bulletin of the London Mathematical Society</I>, V. 11, no. 3: pp. 308-339.</LI><li>John B. Fraleigh (2002). <i>A First Course in Abstract Algebra</i>, 7th Edition, Pearson.</li><li>Daniel Gorenstein, Richard Lyons, and Ronald Solomon (1994). <a href="http://www.amazon.com/gp/product/0821809601/ref=pd_lpo_sbs_dp_ss_3?pf_rd_p=1944687742"><i>The Classification of the Finite Simple Groups</i></a>, American Mathematical Society.</li><LI>Ian Hacking (2014). <I>Why is there Philosophy of Mathematics at all?</I>, Cambridge University Press.</LI><LI>Daniel Kunkle and Gene Cooperman (2007). Twenty-Six Moves Suffice for Rubik's Cube, <I>ISSAC'07</I>, 29 Jul. - 1 Aug., Waterloo, Canada.</LI><LI>Tomas Rokicki (2008). Twenty Five Moves Suffice for Rubik's Cube.</LI></ul>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-64730449728154567612016-02-11T14:51:00.000-05:002016-02-11T14:51:04.826-05:00European Monetary Union Without Political Union<P>I recently read Richard Davenport-Hines' <A HREF="http://www.amazon.com/Universal-Man-Lives-Maynard-Keynes/dp/0465060676"><I>Universal Man: The Lives of John Maynard Keynes</I></A>. One thing I learned was of the existence of the <A HREF="https://en.wikipedia.org/wiki/Latin_Monetary_Union">Latin Monetary Union</A>. </P><P>Apparently, in the latter half of the nineteenth century, gold and silver coins circulated in a number of European countries in which they speak Romance languages. And the amount of gold or silver in these coins was specified. I guess this is part of being on the gold standard. I gather the countries in the Latin Monetary Union agreed on a fixed ratio of silver to gold. As part of this agreement, coins from all these countries circulated freely throughout these countries. You could spend a franc coin in Italy just as conveniently as a lira coin. </P><P>I am surprised that this union lasted past World War I. From Keynes' <A HREF="http://delong.typepad.com/keynes-1923-a-tract-on-monetary-reform.pdf"><I>Tract on Monetary Reform</I></A> (1924), I recall something about the European inflations and deflations that hit Europe after World War I. Yet from my limited reading, I do not recall much about the stresses that must have arisen in this monetary union. Larger issues seem to me to revolve around how the allies in the United States in the war could pay off their loans and how Germany could pay their reparations, agreed to at Versailles, while abiding by the limitations on their economy - such as the occupation of the Ruhr - imposed by the allies. My interest here might be biased by my interest in Keynes, since these issues were a major point of <I>Economic Consequences of the Peace</I>. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-45941549291139715282016-01-23T11:39:00.000-05:002016-01-25T07:13:52.285-05:00Two Views On Introductory Economics<P>Recently, two bloggers have commented on what is taught in college classes for introduction to economics<SUP>1</SUP>. Noah Smith <A HREF="http://noahpinionblog.blogspot.com/2016/01/101ism.html">accepts</A> simple partial equilibrium models of perfect competition as internally valid<SUP>2</SUP>. He argues, however, that "Economics 101" models should be complemented, especially in policy applications, with complications introduced in more advanced models. Robert Paul Wolff, on the other hand, <A HREF="http://robertpaulwolff.blogspot.com/2016/01/i-yield-to-thundering-demand.html">uses</A> <A HREF="http://robertpaulwolff.blogspot.com/2016/01/i-start-to-respond-to-comments.html">introductory</A> <A HREF="http://robertpaulwolff.blogspot.com/2016/01/more-responses-to-comments.html">economics</A> as an <A HREF="http://robertpaulwolff.blogspot.com/2016/01/a-lengthy-response-to-wallace-stevens.html">example</A> of ideological bullshit, to use Frankfort's technical term. </P><P>As far as I am concerned, simplistic supply-and-demand reasoning has been shown to be an incoherent mishmash decades ago. Like Prof. Wolff, I like to justify this view by referring to accepted findings of research literature. I particularly like to emphasize the supposed <A HREF="http://robertvienneau.blogspot.com/2006/12/wages-and-employment-not-determined-by.html">market</A> for <A HREF="http://robertvienneau.blogspot.com/search/label/Labor%20Markets">labor</A>. Why do economists not revise their teaching<SUP>3</SUP> so it is not susceptible to being criticized as ideology? I offer three suggestions to complement Wolff's treatment. </P><P>First, perhaps economists who teach outdated nonsense are just doing their job. Introductory courses are followed by later courses. And teachers of later courses expect students who have satisfied the prerequisites to have been exposed to graphs of supply and demand functions, the theory of utility maximization, marginal cost, marginal revenue, the First Order Conditions for maximization, consumer and producer surplus, etc. You might hope for teachers who introduce a bit of pluralism. But even economists who agree with me might find it challenging for the students to be both exposed to critiques and alternatives, and yet gain a command over the conventional material. </P><P>Second, perhaps the situation might be thought of as a type of coordination game, as in modeling a totalitarian society. Maybe the majority of economists privately think that they are being asked to teach balderdash. But, with the profession being the way it is, they see little benefit in saying so. Each sees others as publicly accepting what is being taught. So they put their doubts aside. If all were to be forthright at once, the situation would be different. But how could teaching transverse from the current equilibrium to that new one? </P><P>Third, maybe many economists come to accept what they are teaching as a way of managing cognitive dissonance. It must be an uncomfortable feeling to know one is spouting nonsense and, if one wants to advance in the profession, to be impotent to change it. Better come to accept the nonsense<SUP>4</SUP>. </P><B>Footnotes</B><OL><LI>Both bloggers seem to be concentrating on microeconomics.</LI><LI>Is Noah's conflation of <I>elasticity</I> with the slope of a function an acceptable simplification for a mass audience? Or just muddle?</LI><LI>I do not teach.</LI><LI>I guess this is related to the <A HREF="https://en.wikipedia.org/wiki/Just-world_hypothesis">just world fallacy</A>.</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-36788554163357181012016-01-18T11:24:00.000-05:002016-01-18T11:24:38.194-05:00Krugman On Robert Reich's New Book<B>1.0 Introduction</B><P>Robert B. Reich has a new book,<I>Saving Capitalism: For the Many, Not the Few</I> out last year. Paul Krugman <A HREF="http://www.nybooks.com/articles/2015/12/17/robert-reich-challenging-oligarchy/">reviewed</A> it, on 17 December 2015, in <I>The New York Review of Books</I>. In this post, I record a negative reaction I have to this review. I do not think I am formulating a strong argument, rather merely making a claim that needs more justification. </P><B>2.0 Review of Reich's Book</B><P>Reich notes that many people portray the major political economic choice in the United States of America as between free markets and government intervention. Reich rightfully rejects this false dichotomy and argues that government creates the markets. Consider such matters as the definition of property rights; what practices are permitted in the market by, say, antitrust law; what contracts will be enforced in courts of law; what legal options, such as bankruptcy, agents can resort to when unforeseen circumstances arise; and the distribution of the allotment of resources to enforcement of various laws. Decisions along these lines create markets, and government can choose various sides. These choices are not necessarily interventions, but constitutive of the definition of markets. </P><P>Many examples can be cited. Think of intellectual property, such as copyrights and patents. Consider how markets arise, from cap and trade polices, for pollution permits. Or think of the labor market. Some states will not permit corporations and unions to agree to contracts in which every worker at some specified rank must be a union member; rather, corporations are permitted to hire workers that get the benefit of union wages without making contributions. One could simplify voting for unions by instituting card check. And, if workers choose to join a union, why shouldn't that union be able to freely choose the portion of their budget to spend on political lobbying? </P><P>Various myths follow from an acceptance of the false dichotomy. For example, the theory of marginal productivity has been read by many since its creation to say workers are paid in the market what they are worth. Reich also looks at the reality of how corporate executives have increased their pay.</P><P>Market processes and their outcomes refract social and legal norms, not <A HREF="http://robertvienneau.blogspot.com/2013/07/against-biotechnological-determinism.html">natural laws</A>. These norms and their <A HREF="http://robertvienneau.blogspot.com/2006/12/income-inequality-in-usa.html">outcomes</A> <A HREF="http://robertvienneau.blogspot.com/2006/07/reversal-of-great-compression-in.html">differ</A> a lot between the post-(World) war (II) golden age and the neoliberal world established after the end of Bretton Woods. Capitalism is a dynamic system, and the current rules are always changing. I do not see why, with lots of struggle, vicious circles currently enriching the few cannot be overthrown and <A HREF="http://robertvienneau.blogspot.com/2012/03/thomas-palleys-book-on-little.html">shared prosperity</A> be re-established to some extent. </P><P>I have some suggestions for how Reich could strengthen his arguments. I think Reich slips into polemics sometimes when I would prefer more analysis<SUP>1</SUP>. I wish Reich would reference more scholars and traditions developing similar points<SUP>2</SUP>. I think John Kenneth Galbraith shows an awareness of traditions I like, and Reich does have Galbraith's notion of countervailing power as a major theme in his book. Maybe explorations of these traditions would lead Reich to more radical conclusions<SUP>3</SUP>. I think Reich still has a hankering for the theory of perfect competition. Even if markets were perfect and corporate boards did not consist of overlapping sets of cronies, neither wages nor executive pay would be determined by marginal productivity. </P><B>4.0 Krugman's Review</B><P>Paul Krugman's review is generally positive<SUP>4</SUP>. This contrasts with how Krugman used to write about Reich back in the 1980s and 1990s. For Krugman then, Reich was a policy entrepreneur who did not measure up to the supposedly rigorous standards of mainstream economists. </P><P>A major theme of Reich's book is power. Krugman, by casting this theme in terms of market power, asserts (mainstream) economists have long addressed this issue. I agree that mainstream economists have models addressing this idea: </P><BLOCKQUOTE>"Market power has a precise definition: it’s what happens whenever individual economic actors are able to affect the prices they receive or pay, as opposed to facing prices determined anonymously by the invisible hand." -- Paul Krugman </BLOCKQUOTE><P>Given this orientation, Krugman can argue against Stigler's claim that Chicago school models of perfectly competitive markets are empirically adequate. Krugman also takes the opportunity of Reich's book to argue that the theory of Skill-Biased Technical Change (SBTC) is mistaken. I think Krugman is reading Reich's book in a more mainstream economist's world of discourse<SUP>5</SUP> than, in fact, is and should be the case. </P><B>Footnotes</B><OL><LI>Maybe this is a matter of contrasting tastes. I'm less likely to draw policy conclusions. Reich certainly knows more about Washington than I do.</LI><LI>For example, institutional economics; Karl Polanyi's <I>The Great Transformation</I>; Hacker and Pierson's <I>Winner Take All Politics</I>; theories of adminstrative, full-cost, or markup pricing.</LI><LI>What substantive disagreement is involved in saying your goal is saving capitalism, as opposed to instituting social democracy?</LI><LI>The back cover of Reich's book features blurbs from Laura D'Andrea Tyson, Joseph Stiglitz, and Lawrence Summers, economists all.</LI><LI>Reich does, in fact, address the (incoherent and incorrect) theory of SBTC.</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-40990606930591084372016-01-09T13:09:00.000-05:002016-03-30T09:29:06.078-04:00Marxist-Feminist-Anti-racist-Ecological Economics<P>I have recently read Julie Nelson's 1995 essay in the <I>Journal of Economic Perspectives</I>. She thinks - and this is a well-established idea among academics - that gender and sex are not the same. One is socially constructed, and the other relates more to a physical substratum. This concept goes back as far as Simone de Beauvoir's <I>The Second Sex</I>. She argues that woman is defined as the negative of man: </P><BLOCKQUOTE>"Humanity is male, and man defines woman, not in herself, but in relation to himself; she is not considered an autonomous being... she is nothing other than what man decides; she is thus called 'the sex,' meaning that the male sees her essentially as a sexed being; for him she is sex, so she is it in the absolute. She is determined and differentiated in relation to man, while he is not in relation to her; she is the inessential in front of the essential. He is the Subject; he is the Absolute. She is the Other." -- Simone de Beauvoir </BLOCKQUOTE><TABLE BORDER ALIGN="CENTER"><CAPTION><B>Table 1: Gender-Coded Dualisms</B></CAPTION><TR ALIGN="CENTER"><TD><B>Male</B></TD><TD><B>Female</B></TD></TR><TR ALIGN="CENTER"><TD>Objectivity</TD><TD>Subjectivity</TD></TR><TR ALIGN="CENTER"><TD>Strength</TD><TD>Weakness</TD></TR><TR ALIGN="CENTER"><TD>Self-Interested</TD><TD>Caring</TD></TR><TR ALIGN="CENTER"><TD>Thinking</TD><TD>Feeling</TD></TR></TABLE><P>In this way of analyzing social customs, one might see homo economicus as gendered male. One might wonder if the traditional neoclassical analysis of the optimizing, but constrained, agent is only a partial viewpoint. Do the objective functions in typical neoclassical models miss goals that are often coded as feminine, for example, altruism? (Might your answer <A HREF="http://scholar.harvard.edu/rabin/home">have</A> <A HREF="http://robertvienneau.blogspot.com/2013/11/mainstream-and-non-mainstream-economics.html">varied</A> <A HREF="http://robertvienneau.blogspot.com/2013/11/thoughts-on-davis-individuals-and.html">since</A> the publication of Nelson's essay?) </P><P>Thinking about how certain dualisms are gender-coded might lead one to thinking about other groups that are taken by hegemonic groups as Other. Socially constructed race is an obvious category in the United States in my lifetime. Looking about, I might think that intellect versus physicality is an analogous dualism for race, with intellect being assigned to whites and physicality assigned to blacks. But reading Eldridge Cleaver's <I>Soul on Ice</I> long ago taught me that such assignments vary with time and space. Cleaver thought that both superior intellect and superior physical fitness were assigned to whites. I suppose you can see such tropes in old books, say, Edgar Rice Burroughs' <I>Tarzan</I> series. </P><P>Feminist economics also points to the need for economists to analyze the household. This idea of looking outside a narrow definition of economic activity for a full understanding of markets reminds me of another argument, namely Schumacher's in <I>Small is Beautiful</I>. Economists need to also look outside markets to natural ecologies to fully understand markets. </P><P>Suppose one is interested in how an advanced capitalist economy, such as in the United States, can sustain itself. How is <A HREF="http://robertvienneau.blogspot.com/2009/02/simple-and-expanded-reproduction.html">reproduction</A>, either on the same or an expanded scale, possible? This question was explored by Marx. Furthermore, to fully address this question, one must look <A HREF="http://robertvienneau.blogspot.com/2012/09/reproducing-civil-society.html">beyond</A> the economy of the advanced country, narrowly defined. For an economy to be reproduced, preconditions must be met in: </P><UL><LI>The households, in which workers are <A HREF="http://robertvienneau.blogspot.com/2007/07/ten-principles-of-feminist-economics.html">reproduced and cared</A> for. Households are outside markets, but provide a necessary foundation on which markets rest.</LI><LI>Other economies, particularly in the third world, where many resources are extracted and production for the market is off-shored these days. That is, the activities in other countries, outside the United States, provide a foundation on which American capitalism rests.</LI><LI>Nature, which also lies outside markets and provides a necessary foundation on which markets rest.</LI></UL></P>Thus, there is a need for a <A HREF="http://www.wellesley.edu/economics/faculty/matthaeij">Marxist-Feminist-Anti-racist-Ecological economics</A>. </P><B>References</B><UL><LI>Simone de Beauvoir (1949, 2009). <I>The Second Sex</I>, Trans. by Constance Borde and Sheila Malovany-Chevallier.</LI><LI>Eldridge Cleaver (). <I>Soul on Ice</I>.</LI><LI>Robin Hahnel (2016). <A HREF="https://www.aeaweb.org/aea/2016conference/program/preliminary.php?search_string=Sraffian&search_type=session&association=&jel_class=&search=Search#search_box">Environmental Sustainability in a Sraffa Framework</A>, <I>Proceedings of the American Economic Association</I>.</LI><LI>Julie A. Nelson (1995). Feminism and Economics, <I>Journal of Economic Perspectives</I>, V. 9, No. 2 (Spring): pp. 131-148</LI><LI>E. F. Schumacher (1973). <I>Small is Beautiful: Economics as if People Mattered.</I></LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-37605729189214547122015-12-30T09:39:00.000-05:002015-12-30T09:39:36.444-05:00Frugal Science<P>Carolyn Kormann has an article, <A HREF="http://www.newyorker.com/magazine/2015/12/21/through-the-looking-glass-annals-of-science-carolyn-kormann">Through the Looking Glass</A>, in this week's <I>New Yorker</I>. This article profiles Manu Prakash, a biophysicist at Stanford and his invention of the Foldscope. The Foldscope is a small, foldable microscope, with the case made of paper. It is an example of <A HREF="http://frugal-science.com/">frugal</A> <A HREF="http://blog.path.org/2014/11/what-is-frugal-science/">science</A>. Prakash hopes to make these microscopes widely available to people in third world countries. One impact might be that residents in, say, African countries will be more conscious of disease-causing micro-organisms, since they can now see such. But, it is not clear to me, what the overall impact of this project might be. </P><P>Frugal science reminds me somewhat of E.F. Schumacher's "appropriate technology". It seems to me that in the last few years I've read articles about people developing new <A HREF="http://www.cnn.com/2015/01/30/africa/eco-stove-kampala-sustainable-cooking/">stoves</A> and <A HREF="http://www.scidev.net/global/innovation/news/cheap-waterless-toilet-african-trial.html">toilets</A> without water targeted to have very low cost and for distribution among the global poor. (THose links are the result of googling now - not where I first read about them.) It seems to me solar power now gives isolated communities a capability to have power without being hooked up to an extensive infrastructure. I like to look for hopeful stories. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-16478312310214719192015-12-19T12:56:00.000-05:002016-02-17T06:21:46.057-05:00Obscure Postmodern Language<P>I try here to outline certain postmodern<SUP>1</SUP> doctrines that, in a full development, might result in one using obscure terminology. None of this is to say that every postmodern writer using polysyllabic terminology is expressing complicated ideas in the most effective way. Nor do I want to argue that it is impossible to ever write clearly<SUP>2</SUP> about (some subset) of these ideas. </P><P>People have a tendency towards reification<SUP>3</SUP>, towards talking as if certain abstract ideas are concrete realities. For example, they might tend to confuse relationships between people with relationships between things<SUP>4</SUP>. And people tend to think dualistically, or at least to categorize things into pre-existing categories. And with dividing things into two categories, one may tend to elevate one over the other, or to define the inferior in terms of the negation of the properties of the superior<SUP>5</SUP>. One might think that these confusions become embedded in our language<SUP>6</SUP>. It is not as if we have access to a language appropriate for a "view from nowhere", where nature is carved at its joints<SUP>7</SUP>. </P><P>Furthermore, current classifications and fundamental ideas embodied in current language have a history; our current language does not reflect how people always thought. In looking at past patterns of language and governance, one should try not to read our current way of thinking into the past<SUP>8</SUP>. </P><P>One might also think current classifications have a functional relationship to class structure, hegemonic<SUP>9</SUP> ethnicities, patriarchal relationships, or whatever<SUP>10</SUP>. </P><P>I have deliberately been abstract here. But, I suppose, I might mention some examples. In economics, I think one is confused if one looks at capitalism as catallaxy, that is, purely in terms of market relationships, in which all parties are free. Furthermore, many things have been said to be socially constructed. I think here of money<SUP>11</SUP>, race<SUP>12</SUP>, gender<SUP>13</SUP>, and sex<SUP>14</SUP>. </P><P>In fully trying to explicate these ideas, one can be expected to struggle with bewitchments brought about by language. One might look for multivocalities in past texts. How have current suppositions been read into them? How might they be read from a subaltern position? How might language be expanded so as not to deny normalcy to currently marginalized groups? So reasons exist why academics thinking along postmodern trends might express themselves obscurely. </P><P>The above is not to say that these ideas cannot be criticized<SUP>15</SUP>. </P><B>Update (21 December 2015):</B><UL><LI>Am I agreeing or disaggreeing with what Robert Paul Wolff says <A HREF="http://robertpaulwolff.blogspot.com/2015/12/some-reflections-resurrected-on.html">here</A>?</LI><LI>Noah Smith has a knee-jerk <A HREF="http://noahpinionblog.blogspot.com/2015/12/academic-bs-as-artificial-barriers-to.html">reaction</A> to postmodernism.</LI><LI>The blogger with the pseudonym "Lord Keynes" has <A HREF="http://socialdemocracy21stcentury.blogspot.com/2015/12/gad-saad-on-postmodernism.html">often</A> <A HREF="http://socialdemocracy21stcentury.blogspot.com/2015/12/chomsky-defends-enlightenment-from.html">complained</A> about left-leaning postmoderns.</LI></UL><B>Footnotes</B><OL><LI>For purposes of this post, I do not distinguish between deconstruction, post structuralism, various trends in the social studies of science, etc.</LI><LI>Richard Rorty is an example of a postmodern philosopher known for clear - but not necessarily easy - writing.</LI><LI>The popularity of the term "reification", in postmodern discourse, comes from Georg Lukás.</LI><LI>This is how Marx defined commodity fetishism.</LI><LI>I am thinking of how Simone de Beauvoir, early in <I>The Second Sex</I>, describes women being defined as the Other.</LI><LI>Here I point to Ludwig Wittgenstein's later work, unpublished in his lifetime.</LI><LI>I guess this relates to Jacques Derrida's claim, "There is no outside the text."</LI><LI>Michel Foucault, in particular, offers provocative studies of changing European thought in the classical age, between the Renaissance and the nineteenth century.</LI><LI>The popularity of the term "hegemony", in postmodern discourse, comes from Antonio Gramsci.</LI><LI>As Marx said, "The ruling ideas are the ideas of the ruling classes."</LI><LI>This is an example of how something can both be socially constructed and real. Obviously, money has quite real effects in modern societies.</LI><LI>Think of the use of the words "Black" and "Colored" in South Africa and in the USA. In the former, they are not synonyms, while among older Americans of a certain sort, they are.</LI><LI>I gather Judith Butler originated the concept of gender as performative.</LI><LI>Judith Butler also questions whether sex is necessarily a biological division. People might be classified based on chromosomes, hormones, genitalia, and secondary sex characteristics. More than two categories exist in many of these classifications, and they do not always line up. Philip Mirowski observes somewhere that, for the International Olympic Committee (and the International Association of Athletics Federations), these classifications are a quite <A HREF="http://www.newyorker.com/magazine/2009/11/30/eitheror">practical</A> issue. After all, they are structured to find exceptional humans.</LI><LI>For explicit references below, I only give critiques. I am sympathetic to the idea that the popularity of postmodernism among academics was connected to an inability to successfully improve material conditions for many.</LI></OL><B>References</B><UL><LI>Samir Amin (1998). <I>Spectres of Capitalism: A Critique of Current Intellectual Fashions</I>, Monthly Review Press.</LI><LI>Terry Eagleton (1996). <I>The Illusions of Postmodernism</I>, Blackwell.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-756039994814997082015-12-03T08:05:00.000-05:002015-12-03T08:05:00.699-05:00Keynes On Rational Expectations And Policy Ineffectiveness<P>John Maynard Keynes' famous saying, "<I>In the long run</I> we are all dead", is from Chapter III of <I>A Tract on Monetary Reform</I>. He describes, in Chapter II of this 1924 book, how governments can obtain resources from their citizens through a deliberate policy of inflation. In this sense, inflation is like taxation. He also discusses how people might react to such a policy, making it difficult for the government to "tax" at the same rate without constantly raising the rate of inflation. </P><P>In Chapter III, Keynes states a general principle: </P><BLOCKQUOTE>"...a large change in [the quantity of cash], which rubs away the initial friction, and especially a change in [the quantity of cash] due to causes which set up a general expectation of a further change in the same direction, may produce a <I>more</I> than proportionate effect on the [price level]. After the general analysis of Chapter I. and the narratives of catastrophic inflations given in Chapter II., it is scarcely necessary to illustrate this further, - it is a matter more readily understood than it was ten years ago. A large change in [the price level] greatly affects individual fortunes. Hence a change after it has occurred, or sooner in so far as it is anticipated, may greatly affect the monetary habits of the public in their attempt to protect themselves from a similar loss in future, or to make gains and avoid loss during the passage from the equilibrium corresponding to the old value of [the quantity of cash] to the equilibrium corresponding to its new value. Thus after, during, and (so far as the change is anticipated) before a change in the value of [the quantity of money], there will be some reactions on the values of [the parameters of the quantity equation in Keynes' Cambridge formulation], with the result that changes in the value of [the price level], at least temporarily and perhaps permanently (since habits and practices, once changed with not revert to exactly their old shape), will not be precisely in proportion to the change in [the quantity of cash]." -- J. M. Keynes, pp. 81-82. </BLOCKQUOTE><P>It seems to me that the above is the Lucas critique, but with a more realistic understanding of human behaviour. What exactly did Lucas contribute again? </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com5tag:blogger.com,1999:blog-26706564.post-1300472478232404432015-11-24T18:18:00.001-05:002015-11-30T13:13:34.099-05:00Herbert Scarf (1930-2015)<P>Herbert Scarf died this 15th of November. I think of Scarf as the economist who first <A HREF="http://robertvienneau.blogspot.com/2008/06/moving-finger-writes.html">demonstrated</A> that general equilibria need not be stable. Something more, some special case assumption or another approach entirely, is needed. </P><P>From his Wikipedia <A HREF="https://en.wikipedia.org/wiki/Herbert_Scarf">page</A>, I learned that have been exposed to more of Scarf's work than I knew. Long ago I took a course in Operations Research, in which we were taught queuing theory and how to find policies for optimal inventory management. Apparently, that approach to the study of inventory policies comes from Scarf. </P><P>I did not find the <I>New York Times</I> <A HREF="http://www.nytimes.com/2015/11/23/business/herbert-scarf-an-economists-mathematician-dies-at-85.html">obituary</A> enlightening. I wish they had mentioned that his algorithm was for finding so-called Computable General Equilibrium (CGE). I have never quite got CGE models. The ones I have seen do not have the dated commodities of the Arrow-Debreu model of intertemporal equilibrium. I have never been sure that they really belong with that tradition, or, like Leontief's model, really fit with a revival of classical economics. Perhaps they are an example of temporary equilibria, as put forth by J. R. Hicks in <A HREF="http://www.amazon.com/Value-Capital-Fundamental-Principles-Economic/dp/0198282699"><I>Value and Capital</I></A>. </P><P>Quite some time ago, Rajiv Sethi <A HREF="http://rajivsethi.blogspot.com/2010/11/herbert-scarfs-1964-lectures-eyewitness.html">discussed</A> Duncan Foley's appreciation of Scarf as a teacher. </P><P><B>Update:</B> Barkley Rosser provides some <A HREF="http://econospeak.blogspot.com/2015/11/what-has-not-been-said-about-later.html">comments</A> on Scarf (hat tip to Blissex). <A HREF="https://theoryclass.wordpress.com/2015/11/16/herbert-scarf-1930-2015/">Here</A> is an obituary from the blog, Leisure of the Theory Class. </P><P>(Unrelated to the above, Cameron Murray recently <A HREF="http://www.fresheconomicthinking.com/2015/11/economic-capital-is-like-pornography.html">comments</A> on economists confusion about what is meant by "capital".) </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com7tag:blogger.com,1999:blog-26706564.post-29989655931230770562015-11-18T06:51:00.000-05:002015-11-18T06:51:34.721-05:00"Those to whom evil is done/Do evil in return"<BLOCKQUOTE><P>"...I spent the evening walking round the streets, especially in the neighbourhood of Trafalgar Square, noticing cheering crowds, and making myself sensitive to the emotions of passers-by. During this and the following days I discovered to my amazement that average men and women were delighted at the prospect of war. I had fondly imagined what most pacifists contended, that wars were forced upon a reluctant population by despotic and Machiavellian governments. I had noticed during previous years how carefully Sir Edward Grey lied in order to prevent the public from knowing the methods by which he was committing us to support France in the event of war. I naïvely imagined that when the public discovered how he had lied to them, they would be annoyed; instead of which, they were grateful to him for having spared them the moral responsibility..."</P><P>Meanwhile, I was living at the highest emotional tension. Although I did not foresee anything like the full disaster of the war, I foresaw a great deal more than most people did. The prospect filled me with horror, but what filled me with even more horror was the fact that the anticipation of carnage was delightful to something like ninety percent of the population. I had to review my views on human nature. At that time I was wholly ignorant of psychoanalysis, but I arrived for myself at a view of human passions not unlike that of the psychoanalysts. I arrived at this view in an endeavour to understand popular feeling about the War. I had supposed until that time that it was quite common for parents to love their children, but the War persuaded me that it is a rare exception. I had supposed that most people liked money better than almost anything else, but I discovered that they liked destruction even better. I had supposed that intellectuals loved truth, but I found here again that not ten per cent of them prefer truth to popularity. Gilbert Murray, who had been a close friend of mine since 1902, was a pro-Boer when I was not. I therefore naturally expected that he would again be on the side of peace; yet he went out of his way to write about the wickedness of the Germans, and the superhuman virtue of Sir Edward Grey. I became filled with despairing tenderness towards the young men who were to be slaughtered, and with rage against all the statesmen of Europe. For several weeks I felt that if I happen to meet Asquith or Grey I should be unable to refrain from murder. Gradually, however, these personal feelings disappeared. They were swallowed up by the magnitude of the tragedy, and by the realization of the popular forces which the statesmen merely let loose.</P><P>-- Bertrand Russell (1951). <I>The Autobiography of Bertrand Russell: The Middle Years: 1914-1944</I></P></BLOCKQUOTE>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-65147021179849763722015-11-03T08:15:00.000-05:002015-11-03T08:15:00.738-05:00Update to my Paper on Pension Capitalism<P>I have updated my <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2663984">paper</A>, "A Neoclassical Model of Pension Capitalism in Which <I>r</I> > <I>g</I>". Changes include: </P><UL><LI>Deletion of the claim that, in general, inequality increases in a steady state when the real rate of return on finance exceeds the rate of growth.</LI><LI>Deletion of states of portfolio indifference, in which the real rates of return on money and on bonds are equal, from the model.</LI><LI>Addition of illustrations of the solution to the (nonlinear) model with some graphs of some state variables along dynamic equilibrium paths.</LI><LI>Inclusion of a description of one method for finding such solutions numerically.</LI><LI>Many minor corrections and rewording.</LI></UL><P>In general, I try to write papers so anybody, including me several months hence, can follow all the details all they want. I realize in submissions to publication, my appendices would have to be drastically shortened or deleted altogether. My typesetting of the mathematics in this paper needs modification, but it is kind to those with old eyes. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-16684910242990690402015-10-21T08:32:00.000-04:002015-10-21T08:32:00.084-04:00Feels Like Science<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://4.bp.blogspot.com/-D0xYt47kBAI/ViYaiEgp_aI/AAAAAAAAAoE/CCJVMzxJ4f8/s1600/StableLCS.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-D0xYt47kBAI/ViYaiEgp_aI/AAAAAAAAAoE/CCJVMzxJ4f8/s320/StableLCS.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Evolution of Two State Variables along Two Dynamic Equilibrium Paths</B></TD></TR></TABLE><P>I <A HREF="http://robertvienneau.blogspot.com/2015/09/failure-to-replicate-hahn-and-solow.html">continue</A> <A HREF="http://robertvienneau.blogspot.com/2015/09/for-technical-discussions-of-cavalry.html">to</A> <A HREF="http://robertvienneau.blogspot.com/2015/10/a-bifurcation-diagram-for-hahn-and-solow.html">explore</A> a micro-founded macroeconomic model from Frank Hahn and Robert Solow, generalized to allow a positive rate of growth of households. Hahn and Solow put forth this model as a strawman, to show that even with perfectly flexible prices and wages, markets clearing always, and rational expectations, room for government macroeconomic management can arise. In their book, they then move on to consider imperfectly competitive markets, norms for wages, and so on. </P><P>A dynamic equilibrium path, in the model, defines the values of three state variables at the end of each time period in the model. One of these state variables, the real quantity of money in circulation is easily calculated from the other two. The other two, taken here as the real rate of return on corporate bonds and on money, must be found, in general, by solving a recursive system of two equations at each point in time. I found the code I wrote for this <A HREF="http://robertvienneau.blogspot.com/2015/03/newton-method-re-iterated.html">post</A> helpful here. </P><P>Figure 1 illustrates the evolution of two state variables for two dynamic equilibrium paths. (The model parameters are β = 2/5, ξ = 2.11, and <I>G</I> = 2. The household utility function is of the form specified by Example 1 in Hahn and Solow, with ε = -1/2.) The stationary, dashed-line, path is for a steady state, which is asymptotically approached by the other dynamic equilibrium path. The oscillations seen in this approach are not in the linear approximation about the steady state. One might view these oscillations as a decaying business cycle. One should be clear, however, that even though economic output varies along such a path, neither unemployment nor disappointed plans arise in this model. Households foresee all future variations in prices and quantities along a dynamic equilibrium path. </P><P>One could add various complications to make the model more realistic. Households could live for multiple periods more than two, thereby perhaps modifying the time period for the business cycle. One could add leisure into the utility function and model the supply of labor as the result of <A HREF="http://robertvienneau.blogspot.com/2006/07/can-one-respect-henry-hazlitt.html">trading off</A> the earning of wages for consumption and leisure. Employment would then vary along a business cycle; in this theory, recessions are long vacations. One could add noise terms, from known probability distributions, for various terms. So agents would be continually adjusting their plans to accommodate realizations of stochastic processes. One could add imperfect competition, as modeled by Avinash Dixit and Joseph Stiglitz. I suppose one could describe the parameters of utility functions as lying along a continuum, therefore adding a sort of diversity in the model of households. And so on. </P><P>I suppose one would find it difficult to add all of these refinements at once. So one could empirically compare a basic model with each refinement. And a model with one refinement might fit better here and with another there. Room for technical innovation for modelling then arises. Can you add two or more refinements, perhaps simplified, where others could could only add one before? Can you take a model that previously was only described for a linear approximation and analyze at least some non-linearities (as I do above)? </P><P>I gather I have just briefly outlined the direction of research in <A HREF="http://robertvienneau.blogspot.com/2010/11/slight-illness-doctor-jests-king-today.html">mainstream macroeconomics</A> over the last third of a century, albeit the freshwater school did not start, I take it, with overlapping generations models and a Clower constraint. </P><P>None of these refinements would even hint at an approach to addressing the <A HREF="http://uneasymoney.com/2015/08/13/romer-v-lucas/">question</A> of how economies get into equilibrium. At the end of each year, the economy is automatically in equilibrium in the model, and this instantaneous magic has been foreseen for all time. Head of households and managers of firms have no need to learn a model of the economy. Agents never have disagreements among themselves about what is the true model. And they never change their minds about the structure of the model. J. R. Hicks, the inventor of the model of temporary equilibrium, came to see that it was set in logical time, not historical time. In other words, John Hicks chose to ally himself with Joan Robinson on this theoretical point. </P><P>Without an acceptable understanding of disequilibria, mainstream economists should be tolerant of polyvocality in methodology. Why should some economists not be exploring models that are not microfounded? Why not consider the impact and evolution of social norms, without first insisting that they they be justified by methodological individualism? I consider some work in complexity and <A HREF="http://www2.econ.iastate.edu/tesfatsi/amulmark.htm">agent based modeling</A> to be of interest along these lines and not even all that non-mainstream. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-68196617256944578832015-10-05T15:51:00.000-04:002015-10-07T09:38:41.593-04:00A Bifurcation Diagram for Hahn and Solow<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-A945S_IBs3c/VhJMxPN-NpI/AAAAAAAAAns/44LXS11zckE/s1600/BifurcationDiagramWithStability.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-A945S_IBs3c/VhJMxPN-NpI/AAAAAAAAAns/44LXS11zckE/s320/BifurcationDiagramWithStability.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Bifurcation Diagram for Hahn and Solow, Example 1, Generalized</B></TD></TR></TABLE><P>I have <A HREF="http://robertvienneau.blogspot.com/2015/09/failure-to-replicate-hahn-and-solow.html">been</A> <A HREF="http://robertvienneau.blogspot.com/2015/09/for-technical-discussions-of-cavalry.html">writing</A> a draft <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2663984">paper</A>, "A Neoclassical Model of Pension Capitalism in which <I>r</I> > <I>g</I>". In my latest iteration, I have developed the bifurcation diagram shown above. This is a generalization for the overlapping generations model, in which the number of households can grow, but specialized to Hahn and Solow's Example 1. Example 1 specifies the form of the utility function. </P><P>One can define dynamic equilibrium paths for the model. And given the values of certain parameters, one can locate a steady state in a certain range of parameters. Always being happy to examine a model, whether it can or cannot ever be instantiated in an actually existing economy, I have identified types of steady states and their stability in certain parameter ranges. I was able to establish analytically the boundary between steady Portfolio Indifferent and Liquidity Constrained States. I located the curved dashed and solid lines towards the south east of the diagram through a mixture of analysis and numeric experimentation. This is also true for my identification of types of stability (saddle-point, locally stable, locally unstable). </P><P>I do not fully understand the topological variation in flows for the bifurcations that I have identified. I think I understand the bifurcation, shown by the dashed line, in which a steady Liquidity Constrained State loses stability. This bifurcation most likely results from the steady state ejecting a stable or absorbing an unstable two-period business cycle. The former case is analogous to the <A HREF="http://robertvienneau.blogspot.com/2013/12/period-doubling-as-route-to-chaos.html">logistic equation</A> for a parameter <I>a</I> of 3. I can understand the bifurcation in which the steady state disappears in terms of the diagram in this <A HREF="http://robertvienneau.blogspot.com/2015/09/failure-to-replicate-hahn-and-solow.html">post</A>. But I find it difficult to understand how dynamic equilibrium paths differ across this bifurcation. And I have not previously gone into the details of the analysis of how two dynamic systems - in this case, for Portfolio Indifferent and Liquidity Constrained States are patched together across a bifurcation. But the linked paper illustrates what I have so far. </P><P>More complete details are provided in the linked <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2663984">paper</A>. I provide more details than anybody can want in appendices so as to be able to step through the model myself, if I look at this stuff later. </P><B>Reference</B><UL><LI>Hahn, Frank and Robert Solow (1995). <I>A Critical Essay on Modern Economic Theory</I>, MIT Press</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-51785647970387676512015-09-23T18:15:00.000-04:002015-09-23T18:15:02.310-04:00For Technical Discussions Of Cavalry Tactics At The Battle Of Austerlitz?<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://1.bp.blogspot.com/-Rzhk1e92NXI/Vfw6jYie_kI/AAAAAAAAAm8/HudK3KKpULk/s1600/SteadyStateExistence.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-Rzhk1e92NXI/Vfw6jYie_kI/AAAAAAAAAm8/HudK3KKpULk/s320/SteadyStateExistence.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Steady States As Function Of Effective Return On Savings</B></TD></TR></TABLE><BR><B>1.0 Introduction</B><P>I have previously <A HREF="http://robertvienneau.blogspot.com/2013/02/against-science-reality-and-free-will.html">said</A> I am not thrilled about arguments about whether or not assumptions are realistic. In this post, I describe some <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2663984">analysis</A>I have done with a model of a world that does not exist and analysis I may do in the future with some variation on such a world. The title of this post refers to this quote from Bob Solow, talking about how to respond to Robert Lucas and the new "classical" school: </P><BLOCKQUOTE>"Suppose someone sits down where you are sitting right now and announces to me that he is Napoleon Bonaparte. The last thing I want to do with him is to get involved in a technical discussion of cavalry tactics at the battle of Austerlitz." -- Robert Solow </BLOCKQUOTE><B>2.0 Generalization of Hahn and Solow's Model of Overlapping Generations</B><P>I have previously <A HREF="http://robertvienneau.blogspot.com/2015/09/failure-to-replicate-hahn-and-solow.html">outlined</A>a micro-founded macroeconomic model of overlapping generations, presented in Hahn and Solow (1995). They use this model to show that claims, from new classical economists and their followers, of the desirability of perfectly flexible prices and wages are unjustified, even on their own theory. They do not think of this model as a good empirical description of any actually existing economy. Hahn and Solow present another model as a prototype of the direction in which they thought macroeconomics should have developed. </P><P>Hahn and Solow consider case where one household is born at the start of each year. Under their assumptions, a stationary state is characterized by an equality between a certain function of the effective rate of return on savings and certain model parameters: </P><BLOCKQUOTE><I>g</I>(<I>Q</I>) = [ξ/(ξ - 1)] [β/(1 - β)] </BLOCKQUOTE><P>The parameter ξ relates to the Clower cash-in-advance contraint. The parameter β is for the aggregate Cobb-Douglas production function. Parameters and the form of the utility function are embodied in the function <I>g</I>. </P><P>I consider a slight modification to this model. Suppose the number of households born each year is no longer constant. Specifically, let the number of households born at the start of year <I>t</I>, <I>h</I><SUB><I>t</I></SUB>, grow at the rate <I>G</I>: </P><BLOCKQUOTE><I>h</I><SUB><I>t</I></SUB> = <I>G</I><SUP><I>t</I></SUP>, </BLOCKQUOTE><P>where: </P><BLOCKQUOTE><I>G</I> ≥ 1. </BLOCKQUOTE><P>I have worked through this model somewhat. A steady state exists if only if the following equality holds for the effective rate of return on savings: </P><BLOCKQUOTE><I>g</I>(<I>Q</I>) = <I>G</I> [ξ/(ξ - 1)] [β/(1 - β)] </BLOCKQUOTE><P>Along a steady state growth path, the nominal price of corn declines so as to maintain a constant real money supply. Hahn and Solow also have that the supply of money is a fixed quantity. They need this assumption, I guess, for their abstract discussion of policy responses to a shock to make sense. </P><B>3.0 Other Generalizations</B><P>Here are some other possible generalizations and explorations one might make to the model: </P><UL><LI>Household lives more than two years.</LI><LI>Endogenous supply of labor, with leisure entering the utility function.</LI><LI>Introduction of a bequest motive.</LI><LI>Heterogeneous households.</LI><LI>Non-homothetic preferences.</LI><LI>Various specific forms of utility functions.</LI><LI>Multiple sectors in production, instead of the production of a single good.</LI><LI>Introduction of fixed capital (with radioactive depreciation), instead of only circulating capital.</LI><LI>Various specific forms of production functions.</LI><LI>Introduction of stochastic noise.</LI><LI>Analysis of reactions to different kind of shocks.</LI><LI>Introduction of government, foreign trade.</LI><LI>More detailed analysis of money, finance, and banks.</LI></UL><P>The above outlines a research program, not necessarily original. Econometricians can go through models in this family in the literature, trying to find the best fit for some time period and country. From what little I know, one can find models with one generalization and not another, or vice versa. A theoretician might want to try to develop a model that combines some generalizations, thereby advancing the field. </P><B>4.0 Empirical Applicability of Generalized Model?</B><P>This program entails lots of work, some of it empirical. How could an outsider have standing to criticize this approach? </P><P>Truthfully, the mathematics is mostly tedious algebra, only not at a high school level because of the length of the derivations. I suppose the concepts I am applying <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2663984">here</A>are deeper than that. Sometimes one gets to the level of high school calculus, what with LaGrangians and all. (If I can develop a fairly comprehensive and interesting <A HREF="http://robertvienneau.blogspot.com/2013/02/empirical-bifurcation-diagram-for.html">bifurcation</A> diagram for some models, I will consider myself to be approaching <A HREF="http://robertvienneau.blogspot.com/2013/12/steve-keen-economists-are.html">advanced mathematics</A>.) Some conventional concepts from economics (marginal conditions, excess demand functions, Walras' law, steady states) help organize the approach. </P><P>One who has learned the details of such a program might react negatively to criticism. The supposedly unrealistic assumptions you object to are maintained for analytical tractability. Past developments have supposedly shown us how to relax assumptions. One can be confident that future developments will continue to show us how to generalize the models and how to remove more scaffolding, leaving the building untouched. And, if analytical developments, such as tractable models of imperfect competition, lead to widescale changes, we will adopt them if empirical data shows such changes to be warranted. </P><P>But are there some assumptions that are untouched by such a program, that are always maintained, and that render all models (admittedly, internally consistent) developed along these lines forever empirically inapplicable? </P><B>4.1 How Are Dynamic Equilibrium Paths Found?</B><P>Under the assumption of perfect competition, prices and wages are assumed to be flexible. This is assumed to imply that markets in each period instantaneously clear. I do not understand why anybody up-to-date on economic theory should believe this? </P><B>4.2 No Keynesian Uncertainty</B><P>Households and firms are assumed to know what the usual range of interest rates, for example, will be in 60 years, in only probabilistically. This does not seem to be plausible to me. </P><B>5.0 Conclusions</B><P>I intend to pursue some generalizations suggested above. (I could be distracted by trying to develop a <A HREF="http://robertvienneau.blogspot.com/2013/02/empirical-bifurcation-diagram-for.html">bifurcation diagram</A>by a Hahn and Solow model in a later chapter.) The point of the mathematics is to tell a story of some fantasy or science fiction world. This sort of project, to me, does not to make empirical claims. Rather I am interested in whether qualitatively similar stories can be told with some complications. Which, if any, generalizations undermine such stories? </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-18798244301383736672015-09-14T15:33:00.000-04:002015-09-14T15:33:00.173-04:00Paul Krugman Stumbles<P>In his editorial in the <I>New York Times</I> this morning (14 September 2015), Paul Krugman writes about Jeremy Corbyn and the British Labour Party. The establishment politicians in Labour are none too happy about Corbyn's victory. Krugman criticizes these establishment politicians for accepting Tory canards on recent economic history in the United Kingdom, with the former Labour government supposedly being at fault. Krugman's concluding paragraph is: </P><BLOCKQUOTE>"Beyond that, however, Labour's political establishment seems to lack all conviction, for reasons I don't fully understand. And this means that the Corbyn upset isn't about a sudden left turn on the part of Labour supporters. It's mainly about the strange, sad moral and intellectual collapse of Labour moderates." -- Paul Krugman </BLOCKQUOTE><P>I have no comment on the substance of Krugman's editorial. However, when I read "lack all conviction", I hear an echo of W. B. Yeat's poem, "<A HREF="http://www.poetryfoundation.org/poem/172062">The Second Coming</A>". I have in mind the following lines: </P><BLOCKQUOTE>"The best lack all conviction, while the worst<BR>Are full of passionate intensity." -- W. B. Yeats </BLOCKQUOTE><P>This allusion, if intended, is backwards from the article. That is, it would suggest that Labour establishment is composed of the best, contradicting the rest of the article. </P><P>I do like Krugman's previous allusions to Talking Heads lyrics. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-88010160566842797142015-09-03T16:07:00.000-04:002015-09-10T09:35:33.138-04:00Failure To Replicate Hahn And Solow (1995), Figure 2.1<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-GQXa_N73AzA/VehfGm1vhPI/AAAAAAAAAmg/8X4XSNDA-KI/s1600/HahnSolow1995.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-GQXa_N73AzA/VehfGm1vhPI/AAAAAAAAAmg/8X4XSNDA-KI/s320/HahnSolow1995.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Stationary States As Function Of Effective Return On Savings</B></TD></TR></TABLE><BR><B>1.0 Introduction</B><P>In Chapter 2 of their <I>Critical Essay</I>, Frank Hahn and Robert Solow present an overlapping generations model<SUP>1</SUP>. This model exhibits rational expectations and perfectly flexible wages and prices. Thus, all markets, including the labor market clear. Hahn and Solow argue that even in such a model, unacceptable fluctuations in national income can arise. Room arises, even under these severe assumptions, for a national government to pursue macroeconomic policy. </P><P>I am interested in how mainstream models can exhibit counter-intuitive behavior, including bifurcations of steady states and interesting non-steady state dynamics. The endogenous generation of cyclical or aperiodic orbits is among the dynamics in which I am interested. Hahn and Solow suggest that this model can have different numbers of stationary states and can have orbits that fail to converge to stationary states. </P><P>I have looked at other <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1307930">models</A> of <A HREF="http://robertvienneau.blogspot.com/2007/02/bifurcations-and-perverse-switch-points.html">overlapping</A> <A HREF="http://robertvienneau.blogspot.com/2010/07/manifestations-of-sraffa-effects-in.html">generations</A> before. So I thought I would look into Hahn and Solow's model. They provide two examples of specific forms of utility functions for their model. This post documents my reasons for thinking their first example cannot replicate certain qualitative properties of their model that they claim can arise in general. </P><B>2.0 Overlapping Generations Model</B><P>The model consists of four markets, for a consumer good, for corporate bonds ("real capital"), for money, and for labor. The supply and demands in these markets are generated by two institutions, households and firms. In this section, I basically echo Hahn and Solow's description of their model. I am particularly interested in three parameters, one for the utility function, one for the production function, and the last for characterizing a liquidity constraint. </P><B>2.1 Households</B><P>Every year, one household is born. Households live two years. During the first year, they supply one person-year of labor, and they are paid their wages at the end of the year. At the end of the first year, they consume some of their wages and save the rest. They are retired and do not labor<SUP>2</SUP> during their second year. At the end of the second year, they consume all of their savings, and then die. <P><P>Households can save their income in the form of two assets: </P><UL><LI>Money, which earns a real return only if prices decline while a household holds it<SUP>3</SUP>.</LI><LI>Corporate bonds, which at the end of each year are paid off with the full (accounting) profits earned by firms.</LI></UL><P>Households would prefer to hold their savings only in the form of the asset with the larger real return. However, a transactions demand for money is introduced in the form of a Clower cash-in-advance constraint<SUP>4</SUP>. </P><P>Formally, the household born at the start of year <I>t</I> must choose decision variables to solve the following non-linear program: </P><BLOCKQUOTE>Maximize <I>u</I>(<I>c</I><SUB><I>t</I>,<I>t</I></SUB>, <I>c</I><SUB><I>t</I>,<I>t</I> + 1</SUB>) </BLOCKQUOTE><P>such that: </P><BLOCKQUOTE><I>c</I><SUB><I>t</I>,<I>t</I></SUB> + <I>s</I><SUB><I>t</I></SUB> ≤ <I>w</I><SUB><I>t</I></SUB></BLOCKQUOTE><BLOCKQUOTE><I>c</I><SUB><I>t</I>,<I>t</I> + 1</SUB> ≤ <I>Q</I><SUB>ξ</SUB>(<I>R</I><SUB><I>t</I></SUB>) <I>s</I><SUB><I>t</I></SUB></BLOCKQUOTE><BLOCKQUOTE><I>c</I><SUB><I>t</I>,<I>t</I> + 1</SUB> ≤ ξ <I>m</I><SUB><I>t</I></SUB> <I>p</I><SUB><I>t</I></SUB>/<I>p</I><SUB><I>t</I> + 1</SUB></BLOCKQUOTE><P>The first constraint specifies that the sum of the consumption and savings at the end of the household's first year cannot exceed the wages received by the household at that point in time. The second constraint states that the consumption at the end of the second year cannot exceed savings, accumulated during that year at the effective rate of return on savings, <I>Q</I><SUB>ξ</SUB>(<I>R</I><SUB><I>t</I></SUB>). The notation for the effective rate of return reflects the dependence of that rate on the real rate of return, <I>R</I>, on corporate bonds and a parameter, ξ, arising in the third constraint. The third constraint is the Clower cash-in-advance condition. The household must hold at least some given fraction (namely, 1/ξ) of the consumption planned at the end of the last period in the form of money during this period<SUP>5</SUP>, where </P><BLOCKQUOTE>ξ > 1 </BLOCKQUOTE><P>In a state of Portfolio Indifference (PI), the real rate of return for money and for corporate bonds are equal. On the other hand, if households are Liquidity Constrained (LC), they would prefer to hold savings at the higher rate of return provided by corporate bonds, but cannot because of the Clower constraint. The effective rate of return on savings is therefore less than the rate of return on real capital. </P><B>2.1.1 Hahn and Solow's First Example</B><P>To be a bit more concrete, Hahn and Solow gives two examples of possible forms of the utility function. The first is: </P><BLOCKQUOTE><I>u</I>(<I>c</I><SUB><I>t</I>,<I>t</I></SUB>, <I>c</I><SUB><I>t</I>,<I>t</I> + 1</SUB>) = (1/α)(<I>c</I><SUB><I>t</I>,<I>t</I></SUB>)<SUP>α</SUP> + (1/α)(<I>c</I><SUB><I>t</I>,<I>t</I> + 1</SUB>)<SUP>α</SUP></BLOCKQUOTE><P>where, </P><BLOCKQUOTE>α < 1 </BLOCKQUOTE><P>Sometimes it is more convenient to express the solution of the household's program in terms of the parameter ε: </P><BLOCKQUOTE>ε = α/(α - 1) </BLOCKQUOTE><B>2.2 An Aggregate Cobb-Douglas Production Function</B><P>The firms are characterized by an aggregate production function<SUP>6</SUP>. To be concrete, they specify a Cobb-Douglas form: </P><BLOCKQUOTE><I>y</I><SUB><I>t</I></SUB> = (<I>k</I><SUB><I>t</I> - 1</SUB>)<SUP>β</SUP> (<I>l</I><SUB><I>t</I></SUB>)<SUP>β + 1</SUP> </BLOCKQUOTE><P>where: </P><BLOCKQUOTE>0 < β < 1 </BLOCKQUOTE><P>The wage, the real rate of return on corporate bonds, the demand for labor, and the supply of corporate bonds (also known as the demand for capital) come out of the usual profit-maximizing analysis. The demand for labor is constrained to match the households' supply of one person-year per year. That is, with flexible wages and prices, the labor market is assumed to clear. </P><B>3.0 Stationary States</B><P>By solving the above model, one can find excess demands, at the end of each year, for the produced commodity, corporate bonds, and money. Along a dynamic equilibrium path, excess demands in all three markets are zero. As I understand it, solving for one state variable, the rate of return on corporate bonds, in each year is sufficient to trace out such paths. Stationary states, if any exist, are found by dropping time indices. </P><P>Stationary states are conveniently expressed in terms of the following function. </P><BLOCKQUOTE><I>g</I>(<I>Q</I>) = <I>Q</I> <I>s</I>(<I>Q</I>) </BLOCKQUOTE><P>where <I>s</I>(<I>Q</I>) is the stationary state savings found by solving the household's constrained maximization problem and substituting in a wage of unity in the solution<SUP>7</SUP>. </P><P>Exactly one real rate of return, <I>R</I>, corresponds to each each stationary state value of <I>Q</I>, and vice versa. The parameters α and ξ enter into this invertible function. The following equation is a necessary and sufficient condition for a stationary state: </P><BLOCKQUOTE><I>g</I>(<I>Q</I>) = [ξ/(ξ - 1)] [β/(1 - β)] </BLOCKQUOTE><P>Figure 1 graphs <I>g</I>(<I>Q</I>) and the Right Hand Side of the above equation for given parameters in Example 1. The horizontal line can be lowered or raised, within a certain range, by varying, β the parameter in the production function, while leaving other curves unchanged. It is a bit more complicated to analyze the effects of varying ξ. α enters into the shapes of the upward-sloping curves. For this example, they all take on a value of 1/2 at <I>Q</I> = 1. </P><P>Anyways, Hahn and Solow present a figure showing possible shapes and locations of <I>g</I>(<I>Q</I>). And they comment on the number and types of possible stationary state equilibria. Table 2 summarizes and compares and contrasts their and my results. I have been unable to find an example with two LCS in their example. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Number of Stationary States</B></CAPTION><TR><TD ALIGN="center"><B>Hahn and Solow<BR>Possibilities</B></TD><TD ALIGN="center"><B>Example 1<BR>Possibilities</B></TD></TR><TR><TD ALIGN="left"><UL><LI>None.</LI><LI>No PIS, Exactly one LCS.</LI><LI>Exactly one PIS, No LCS.</LI><LI>Exactly one PIS, two LCS.</LI></UL></TD><TD ALIGN="left"><UL><LI>None.</LI><LI>No PIS, Exactly one LCS.</LI><LI>Exactly one PIS, No LCS.</LI></UL></TD></TR></TABLE><P></P><B>4.0 Conclusion</B><P>I was hoping to find a model with multiple equilbria for some subset of the parameter space. Perhaps I have made some simple error in algebra, but I was disappointed to not find such. This post does not say that Hahn and Solow are in error. They do not claim multiple equilibrium can arise for every conventional form of the utility function in their problem. I guess I'll have to focus on their second example<SUP>8</SUP>. </P><P><B>Update (10 September 2015):</B> I've convinced myself that neither Hahn and Solow's Example 1 or Example 2 can exhibit one PIS and two LCS. The derivative of <I>g</I>(1) is upward-sloping in both cases, unlike in Hahn and Solow's diagram for the case of three equilibria. (I do not see off-hand why Hahn and Solow rule out a case of in which no PIS exists, but two LCS do.) </P><B>Footnotes</B><OL><LI>This model is in the style of the macroeconomics that they are criticizing from the inside. Chapter 6 presents a prototype model more in the spirit of how Hahn and Solow think macroeconomics should be pursued. This model is without an exact reduction to microeconomics, with a labor market which is justified by an earlier game-theoretic analysis of social norms, and with imperfect competition in product markets.</LI><LI>In other models of overlapping generations, how much labor a household supplies each year is a decision variable.</LI><LI>In a stationary state, prices are stationary and money earns a real return of unity.</LI><LI>I had not recognized a Clower constraint before. Presumably, it is not original with this book; Robert Clower's work in macroeconomics goes back to at least the 1960s.</LI><LI>Hahn and Solow suggest this unrealistic approach to the transactions demand for money can be justified by a deeper analysis.</LI><LI>Sometimes economists justify ignoring the Cambridge Capital Controversy on the grounds that there are so many other problems with mainstream economics that one need not focus on capital theory. This model illustrates this claim.</LI><LI>This definition only works for homothetic utility functions, another unrealistic assumption justified here by the critical intent of the model.</LI><LI>I like that their second household has a parameter for time-discounting for households, anyways.</LI></OL><B>Reference</B><UL><LI>Hahn, Frank and Robert Solow (1995). <I>A Critical Essay on Modern Economic Theory</I>, MIT Press</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-87966770132148239442015-08-21T12:15:00.000-04:002015-09-01T14:23:50.910-04:00Paul Romer Gyring In A Cul-De-Sac<P>Paul Romer continues to display his confusion. In reverse chronological order, you can look <A HREF="http://paulromer.net/reactions-to-solows-choice/">here</A>, <A HREF="http://paulromer.net/solows-choice/">here</A>, <A HREF="http://paulromer.net/what-went-wrong-in-macro-historical-details/">here</A>, <A HREF="http://paulromer.net/what-went-wrong-in-macro-overview/">here</A>, and so on. Also see <A HREF="http://noahpinionblog.blogspot.com/2015/08/science-vs-politics.html">Noah</A> <A HREF="http://noahpinionblog.blogspot.com/2015/08/translating-paul-romer.html">Smith</A>. </P><P>Romer continues to put forward ever more false dichotomies and other simple-minded logical fallacies. For example, he seems to say economics has a choice between talky, non-scientific political advocacy or rigorous mathematical economics. And he gets his history wrong: </P><BLOCKQUOTE>"Over the five decades from 1890 to 1940 (a time when physicists developed mathematical theories of statistical mechanics, quantum mechanics and both special and general relativity) economists avoided the use even of calculus and spent 50 years mired in the confusion spawned by the talky, market-by-market, supply-and-demand-ish approach to economic analysis codified in 1890 in Alfred Marshall's <I>Principles of Economics</I>." -- <A HREF="http://paulromer.net/what-went-wrong-in-macro-historical-details/">Paul Romer</A></BLOCKQUOTE><P>I suppose one can be generous and take Romer to be confining himself to Anglo-American economics. Obviously, economists such as Leon Walras, Gustav Cassel, and Frederick Zeuthen were analyzing mathematical models. (As I understand it, Zeuthen was the first to formulate the Walras-Cassel model with inequalities.) And, I guess in this tradition, Abraham Wald, in 1935, provided the first rigorous proof of the existence of a general equilibrium. </P><P>But even when restricted to Anglo-American economics, Romer is not quite correct. J. R. Hicks, with his 1939 edition of <I>Value and Capital</I> and earlier papers with R. G. D. Allen, reintroduced General Equilibrium theory into Anglo-American economics, with as many derivatives, matrices, etc. as you please. </P><P>Romer's comments about "talkiness" are silly. I would be embarrassed to dismiss a scholar like Fernand Braudel on the grounds that he did not put forth mathematical models, as in physics. </P><P>Romer is just as silly on the other side of his false dichotomy. He's seems to think that as long as a model is put forth in terms of valid mathematics, it is rigorous. Here's what he writes about Solow's growth model: </P><BLOCKQUOTE>"Robert Solow (a close colleague of Samuelson's at MIT) ... showed how to describe the behavior of an economy in which things did change. By restricting attention to a single type of output, Solow developed a workable framework for talking about changes in wages, the return to capital, and total output." -- <A HREF="http://paulromer.net/what-went-wrong-in-macro-historical-details/">Paul Romer</A></BLOCKQUOTE><P>When I read that in context, I thought Romer was just expressing himself badly. This is in the midst of a short overview about Paul Samuelson's contributions to economics, a <A HREF="http://robertvienneau.blogspot.com/2009/12/paul-samuelson-1915-2009.html">task</A> I would find Herculean. Maybe Romer knows that Solow's model is, at best, a non-rigorous, rough-and-ready framework for empirical work. But he really does think otherwise, that Solow's model is rigorous: </P><BLOCKQUOTE>"Solow's explicit dynamic model of growth based on an aggregate production function was a solid piece of SAGE [Simple, Applied General Equilibrium] theory. After all, if new Chicago and the rest of the profession agree on one part of good theoretical practice, this has to signal something." -- <A HREF="http://paulromer.net/reactions-to-solows-choice/">Paul Romer</A></BLOCKQUOTE><P>The above is just false. The rest of the profession do not agree. </P><P>What would have to be the case for Solow's model to apply in a world in which more than one commodity is produced? One set of assumptions is that, in some sense, effectively one commodity is produced. At any given time, the capital stock could be disassembled and costlessly transmuted into either any consumption good or any other collection of capital goods, and vice versa. Then, the historical cost of capital goods, the current prices of capital goods, and their present value would not diverge. On the other hand, these costs do diverge in actual economies set in historical time. The above is a summary of a substantive argument from Joan Robinson, who jokingly claimed that neoclassical economists thought of capital goods as meccano sets or ectoplasm. </P><P>Romer resolutely refuses to address the substance of either side of the Cambridge Capital Controversy. (And there are other points than the above. Is Romer even aware of the existence of Piero Sraffa or <A HREF="http://robertvienneau.blogspot.com/2011/10/pierangelo-garegnani-1930-14-october.html">Pierangelo Garegnani</A>?) Instead, he whines about Robinson's tone: </P><BLOCKQUOTE>"...the sarcasm and put-downs that were a part of British intellectual life that <A HREF="http://robertvienneau.blogspot.com/2006/07/response-to-comments-on-steve-keens.html">Solow</A> had to confront in his exchanges with Joan Robinson." -- <A HREF="http://paulromer.net/what-went-wrong-in-macro-historical-details/">Paul Romer</A></BLOCKQUOTE><P>And he attacks Joan Robinson's motives: </P><BLOCKQUOTE>"In so doing, he used the same techniques that economists from Cambridge England used to attack his model of output as a function of a stock of capital. Joan Robinson probably had the same concern. What will young Samuelson and Solow do with all their maths? Because an aggregate production function might lend support for a marginal productivity theory of the distribution of income, perhaps we should strangle it in the crib." -- <A HREF="http://paulromer.net/solows-choice/">Paul Romer</A></BLOCKQUOTE><P>The above is simply ad hominem. Apparently, some have sent email to Romer with similar points. He then <A HREF="http://paulromer.net/reactions-to-solows-choice/">cites</A> Roger Backhouse as an authority, while doubling down on the ad hominem. </P><P>I suppose I cannot complain about Romer's treatment of Robinson. Romer's knowledge of General Equilbrium theory seems to be lacking, and he treats Frank Hahn and Robert Solow's objections to macroeconomics after Lucas no more seriously. He complains about their tone, but pretends they had no substance to their complaints. Is Romer even aware of Hahn's attempts to integrate money into the Arrow-Debreu model and his outline of the difficulties? Is Romer even aware of the existence of Hahn and Solow's 1995 monograph? To be generous to Romer, I suppose one could say the latter is only of retrospective importance when considering the controversies in macroeconomics in the 1970s. </P><P>I might as well conclude with another example of silliness from Romer. Here Romer tries to explain one of Lucas's contributions: </P><BLOCKQUOTE>"Then Robert Lucas showed how to add uncertainty to a version of the Samuelson and Diamond models. This let him pin down loose conjectures from Keynes about the role of expectations." -- <A HREF="http://paulromer.net/what-went-wrong-in-macro-historical-details/">Paul Romer</A></BLOCKQUOTE><P>Now, Chapter 12 in the <I>General Theory</I> is often turned to when one wants to read Keynes on expectations. And in that chapter, one finds: </P><BLOCKQUOTE>"By 'very uncertain' I do not mean the same thing as 'very improbable'. <I>Cf</I>. my <I>Treatise on Probability</I>..." -- John Maynard Keynes (1936, p. 148). </BLOCKQUOTE><P>Romer is equivocating. As far as I know, Lucas did not introduce uncertainty in any mathematical models in economics. (Can anybody find Lucas explicitly discussing the inconsistency between rational expectations and non-ergodic time series?) So Romer should either not reference Keynes at all (with silliness about "loose conjectures") or talk about Lucas modeling probability (also known as risk) or expand on his text to show how Lucas was actually modeling Keynes's uncertainty. That is, Romer should if he has any interest in the truth value of his statements. </P><P>I think the above is not one of my better posts. Too uniformly negative even for me and too wandering. But I think Romer should try not to commit simple logical fallacies in his complaints about lack of scholarship and rigor among economists. </P><B>References</B><UL><LI>Braudel, Fernand (). <I>Civilization and Capitalism, 15th - 18th Century, Volume 1: The Structure of Everyday Life.</I></LI><LI>Hahn, Frank and Robert Solow (1995). <I>A Critical Essay on Modern Macroeconomic Theory</I>, MIT Press.</LI><LI>Hicks, J. R. (1939). <I>Value and Capital</I> (1st edition).</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com12tag:blogger.com,1999:blog-26706564.post-8414889381054562092015-07-27T08:05:00.000-04:002015-07-27T08:05:00.104-04:00Labor Reversing Without Capital: An Example<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-jG1P4nhW2jQ/VbYDboaVs8I/AAAAAAAAAl4/wKWCb1zCpLk/s1600/SkilledLaborDemand.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-jG1P4nhW2jQ/VbYDboaVs8I/AAAAAAAAAl4/wKWCb1zCpLk/s320/SkilledLaborDemand.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Skilled Labor Hired by Firms per Unit Output</B></TD></TR></TABLE><BR><B>1.0 Introduction</B><P>This example is from Opocher and Steedman (2015). They present many examples in which the reader is expected to work them out, as illustrated in this post. </P><P>This is an example in which cost-minimizing firms desire to hire more labor (of a specific type) for an increased wage, around a specific wage. This example is of a firm producing a single commodity from inputs of specific types of land and specific types of labor. No produced capital goods exist in this example, and the interest rate is assumed to be zero. Yet perverse behavior arises on the demand side of markets for factors of production anyway - where results are called <I>perverse</I> merely if they violate neoclassical intuitions shown to be mistaken half a century ago. The most complicated aspect of this example is that some techniques of production are specific to specific types of land. </P><B>2.0 Indirect Average Cost Functions</B><P>Consider a firm that produces widgets from inputs of skilled labor, unskilled labor, and land of one of two types. Suppose the price of widgets is unity. Define: </P><UL><LI><I>p</I><SUB>α</SUB> is the rent for alpha-type land.</LI><LI><I>p</I><SUB>β</SUB> is the rent for beta-type land.</LI><LI><I>w</I><SUB>1</SUB> is the wage for unskilled labor.</LI><LI><I>w</I><SUB>2</SUB> is the wage for skilled labor.</LI></UL><P>The indirect average cost function for widgets produced on land of type alpha is: </P><BLOCKQUOTE><I>c</I><SUB>α</SUB>(<I>p</I><SUB>α</SUB>, <I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (1/2)[(<I>w</I><SUB>1</SUB> <I>p</I><SUB>α</SUB>)<SUP>1/2</SUP> + (<I>w</I><SUB>1</SUB> <I>w</I><SUB>2</SUB>)<SUP>1/2</SUP><BLOCKQUOTE>+ (<I>w</I><SUB>2</SUB> <I>p</I><SUB>α</SUB>)<SUP>1/2</SUP>] </BLOCKQUOTE></BLOCKQUOTE><P>The indirect average cost function for widgets produced on land of type beta is: </P><BLOCKQUOTE><I>c</I><SUB>β</SUB>(<I>p</I><SUB>β</SUB>, <I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (3/5)(<I>w</I><SUB>1</SUB> <I>p</I><SUB>α</SUB>)<SUP>1/2</SUP> + (3/10)(<I>w</I><SUB>1</SUB> <I>w</I><SUB>2</SUB>)<SUP>1/2</SUP><BLOCKQUOTE>+ (11/20)(<I>w</I><SUB>2</SUB> <I>p</I><SUB>α</SUB>)<SUP>1/2</SUP></BLOCKQUOTE></BLOCKQUOTE><P>The indirect average cost function shows the average cost of producing each widget, when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm. </P><P>These indirect average cost functions are homogeneous of the first degree. For the indirect average cost function for land of type alpha, this property is expressed as: </P><BLOCKQUOTE><I>c</I><SUB>α</SUB>(<I>a</I><I>p</I><SUB>α</SUB>, <I>a</I><I>w</I><SUB>1</SUB>, <I>a</I><I>w</I><SUB>2</SUB>) = <I>a</I> <I>c</I><SUB>α</SUB>(<I>p</I><SUB>α</SUB>, <I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) </BLOCKQUOTE><P>This a traditional assumption for cost functions. </P><P>Consider the indirect average cost function for a specific type of land. That type of land, unskilled labor, and skilled labor are substitutes. No inputs are complements in this example. In other words, the off-diagonal elements of the Hessian matrices formed from each indirect average cost function are all positive. The elements along the principal diagonal of each Hessian matrix are negative. </P><B>3.0 The Wage-Wage Frontier</B><P>Consider a long run equilibrium of the firms in which pure economic profits have been competed away and no firm is making a loss. Perhaps, the prospect of firms entering or exiting the industry has caused this situation to arise. Furthermore, suppose rents for both types of land happen to be unity. (Without this assumption, this example would have two more degrees of freedom.) If firms are producing on a given type of land, the indirect average cost function for that type of land will be equal to unity. For alpha type land, one has: </P><BLOCKQUOTE>1 = <I>c</I><SUB>α</SUB>(1, <I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) </BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>w</I><SUB>1, α</SUB> = [(2 - <I>w</I><SUB>2</SUB><SUP>1/2</SUP>)/(1 + <I>w</I><SUB>2</SUB><SUP>1/2</SUP>)]<SUP>2</SUP></BLOCKQUOTE><P>As shown in Figure 2, given the type of land employed, the wage for unskilled labor is a declining function of the wage for skilled labor. The maximum wage for unskilled labor, 4 widgets per person-year, corresponds to skilled labor working for free. Symmetrically, the maximum wage for skilled labor likewise corresponds to unskilled labor working for free. </P><P>Equating the indirect average cost function for production on land of type beta yields another trade-off in long run equilibrium between the wages of unskilled and skilled labor. </P><BLOCKQUOTE><I>w</I><SUB>1, β</SUB> = [(20 - 11 <I>w</I><SUB>2</SUB><SUP>1/2</SUP>)/(12 + 6 <I>w</I><SUB>2</SUB><SUP>1/2</SUP>)]<SUP>2</SUP></BLOCKQUOTE><P>When land of type beta is used, the maximum wage for unskilled labor is 2 7/9. The maximum wage for skilled labor is 3 37/121. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-XgfhNeU7ts8/VbYDjEFBJWI/AAAAAAAAAmA/azBW5qzjrVM/s1600/WageWageFrontier.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-XgfhNeU7ts8/VbYDjEFBJWI/AAAAAAAAAmA/azBW5qzjrVM/s320/WageWageFrontier.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 2: Wage-Wage Curves and the Frontier</B></TD></TR></TABLE><P>For some combination of wages of skilled and unskilled labor, firms will be indifferent between producing widgets with land of type alpha and type beta. The cost-minimizing technique at these wages, on each type of land, is equally cheap. These combinations can be found by equating the wages of unskilled labor for the expressions above. After some manipulation, one obtains the equation: </P><BLOCKQUOTE>5 <I>w</I><SUB>2</SUB> - 9 <I>w</I><SUB>2</SUB><SUP>1/2</SUP> + 4 = 0 </BLOCKQUOTE><P>This equation can be factored: </P><BLOCKQUOTE>(<I>w</I><SUB>2</SUB><SUP>1/2</SUP> - 1)(5 <I>w</I><SUB>2</SUB><SUP>1/2</SUP> - 4) = 0 </BLOCKQUOTE><P>Firms will thus be indifferent to the type of land used in production for ordered pairs of wages of unskilled and skilled labor, (<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>), of (1/4, 1) and (4/9, 16/25). </P><P>Firms produce widgets on land of type alpha for wages for skilled labor between zero and 16/25, and for wages of skilled labor between one and four. For wages for skilled labor between 16/25 and four, firms produce widgets on land of type beta. The outer frontier allows one to determine the wage of unskilled labor for any feasible wage for skilled labor, given the model assumptions. As well soon be apparent, this is <I>not</I> an example of reswitching. The overall indirect average cost function is almost always differentiable. It is not differentiable only at the two points found by the construction of the outer frontier. </P><B>4.0 Land and Labor</B><P>We have seen that when rents are unity, long run equilibrium of the firm necessitates that the wages of unskilled labor is a declining function of the wages of skilled labor. Shepherd's lemma can be used to find the coefficients of production for each feasible combination of wages of unskilled and skilled labor. The quantity of each input the firm wants to hire per unit output is the derivative of the indirect average cost function with respect to the price of that input. Thus, when land of type alpha is used, the number of acres of land employed per unit output of widgets is: </P><BLOCKQUOTE><I>t</I><SUB>α</SUB>(<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (1/4)(<I>w</I><SUB>1</SUB><SUP>1/2</SUP> + <I>w</I><SUB>2</SUB><SUP>1/2</SUP>) </BLOCKQUOTE><P>The number of acres of land of type beta per unit output of widgets, when land of that type is used, is: </P><BLOCKQUOTE><I>t</I><SUB>β</SUB>(<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (1/40)(12 <I>w</I><SUB>1</SUB><SUP>1/2</SUP> + 11 <I>w</I><SUB>2</SUB><SUP>1/2</SUP>) </BLOCKQUOTE><P>In what I hope is obvious notation, person-years of unskilled labor employed per unit output of widgets is, depending on the type of land used: </P><BLOCKQUOTE><I>l</I><SUB>1, α</SUB>(<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (1/4)(1 + <I>w</I><SUB>2</SUB><SUP>1/2</SUP>)/(<I>w</I><SUB>1</SUB><SUP>1/2</SUP>) </BLOCKQUOTE><BLOCKQUOTE><I>l</I><SUB>1, β</SUB>(<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (3/20)(2 + <I>w</I><SUB>2</SUB><SUP>1/2</SUP>)/(<I>w</I><SUB>1</SUB><SUP>1/2</SUP>) </BLOCKQUOTE><P>Finally, person-years of skilled labor employed per unit output of widgets is given by one of the following two functions of wages: </P><BLOCKQUOTE><I>l</I><SUB>2, α</SUB>(<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (1/4)(1 + <I>w</I><SUB>1</SUB><SUP>1/2</SUP>)/(<I>w</I><SUB>2</SUB><SUP>1/2</SUP>) </BLOCKQUOTE><BLOCKQUOTE><I>l</I><SUB>2, β</SUB>(<I>w</I><SUB>1</SUB>, <I>w</I><SUB>2</SUB>) = (1/40)(6 <I>w</I><SUB>1</SUB><SUP>1/2</SUP> + 11)/(<I>w</I><SUB>2</SUB><SUP>1/2</SUP>) </BLOCKQUOTE><B>5.0 Bringing it all Together</B><P>The above algebra can be used to generate various graphs. Figure 1 shows person-years of skilled labor firms desire to hire per unit output. As one moves to the right in the figure, the wage of skilled labor rises and the wage of unskilled labor falls. But at every point in the figure, the wages of the two types of labor are such as to maintain wages as on the outer frontier in Figure 2. That is, firms are minimizing costs, and the output price and input prices are such as to enforce the equilibrium condition that no pure economic profits are available in this industry. </P><P>Figure 3 shows the analogous graph for unskilled labor. The point for wages of 4/9 widgets per person-years and 16/25 widgets per person-year for unskilled and skilled labor, respectively, is emphasized. At any point to the left, wages for unskilled labor are higher, and wages for skilled labor are lower. And an infinitesimal variation around this point is associated with firms wanting to employ unskilled labor more intensively when their wage is relatively higher. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-Htks5WW16yk/VbYDqJMa5cI/AAAAAAAAAmI/lVUi97dT378/s1600/UnskilledLaborDemand.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-Htks5WW16yk/VbYDqJMa5cI/AAAAAAAAAmI/lVUi97dT378/s320/UnskilledLaborDemand.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 3: Unskilled Labor Hired by Firms per Unit Output</B></TD></TR></TABLE><P>Reswitching of techniques arises when one technique of production is cost-minimizing at, say, a high and low wage but not at an intermediate wage. A technique of production is specified by four coefficients of production in this example. The amount of skilled labor and unskilled labor hired per unit output are two of these coefficients. The acres of land of each type rented per unit output are the other two. The latter two coefficients of production obviously vary, depending on which type of land can be used in a cost-minimizing technique. In fact, the coefficients of production for the type of land not employed is zero. As can be seen in Figures 1 and 3, the coefficient of productions for the two types of labor vary monotonically with relative wages, given the type of land employed. </P><P>At one of the two switch points highlighted in Figure 2, two techniques of production are cost-minimizing. (This is the definition of a switch point.) In one technique, one type of land is used. And in the other, the other type of land is used. But a different pair of techniques of production is cost-minimizing at the other switch point. The coefficients of production vary among, for example, the cost-minimizing techniques in which alpha-type land is used at a switch point. Hence, as noted, no reswitching of techniques exists in this example. </P><B>6.0 Conclusion</B><P>This example has cost-minimizing firms in equilibrium in a single industry. Price and quantity relationships among factors of production have been analyzed, where factors of production consist of land of two types and labor of two types. Quantity relationships have been presented in terms of inputs per unit output for a firm. For simplicity, only the case in which the interest rate is zero and rents of land per acre are unity has been considered. When beta type land is adopted, more acres are cultivated for alpha-type land, for the same level of output. Thus, land has a higher proportion of total unit cost when beta-type land is used. Both skilled and unskilled labor are a lower proportion of total unit cost (as seen in Figures 1 and 3) than they would be if alpha type land was employed. A wage has been found for unskilled labor in which a higher relative wage for unskilled labor is associated with firms desiring to hire more unskilled labor per unit output. And a different relative wage for skilled labor has been found with the analogous property. </P><P>I wonder whether an example can be found with a continuum of types of land in which the analog of Figures 1 and 2 come out as continuous U-shaped curves. </P><P>So much for explaining wages and employment by well-behaved supply and demand curves in competitive labor markets. </P><B>Reference</B><UL><LI>Opocher, Arrigo and Ian Steedman (2015). <I>Full Industry Equilibrium: A Theory of the Industrial Long Run</I>, Cambridge University Press</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0