Monday, August 30, 2010

Stephen Williamson, Fool or Knave?

Stephen Williamson quotes Narayana Kocherlakota, apparently a very stupid person:
"Kocherlakota says this...:
'But over the long run, money is, as we economists like to say, neutral. This means that no matter what the inflation rate is and no matter what the FOMC does, the real return on safe short-term investments averages about 1-2 percent over the long run.'
Again, uncontroversial." -- Stephen Willaimson
This, of course, is false. Communities of economists exist who set their theories in historical time and dispute that money is neutral in any run. I prefer to point to Post Keynesians, but Austrian School economists satisfy these criteria also. Furthermore, economists within such schools surpassed mainstream economists in the current historical conjuncture by having pointed out the possibility of the global financial crisis before its occurrence.

I think economists should strive not to tell untruths abouts what economists believe.

Friday, August 27, 2010

Why Income Inequality Leads To Recessionary Conditions

1.0 Introduction
Apparently, some have been discussing whether the gross increased inequality in the USA is connected with the depressionary conditions we are in. So I thought I would climb on my bicycle and do some arithmetic.

I take it as a stylized fact that an increase in inequality is associated with an increase in the average and marginal propensity to save.

There's something called the Harrod-Domar model of growth. I'm not sure I've ever read Domar. I've certainly read more of Harrod than I have of Domar. So in the sequel, I refer exclusively to Harrod.

Harrod defined three rates of growth: the actual rate, the warranted rate, and the natural rate. Increased inequality can result in the warranted rate exceeding the natural rate. Since the warranted rate is unstable and the actual rate cannot long exceed the natural rate, increased inequality is likely to lead to the actual rate of growth falling below and away from the warranted rate, that is, to depressions.

2.0 Harrod's Model
Harrod's model is fairly simple, but it raises deep questions.

2.1 The Actual Rate
Along a steady state growth path, the ratio, v, of the value of capital to the value of net income is constant:
v = K/Y,
where K is the value of the capital stock, and Y is the value of net income. v is known as the capital-output ratio. Thus:
dY/dt = (1/v) dK/dt
Investment, I, is defined to be the change in the value of capital with time. Hence,
(1/Y) dY/dt = (1/v) (I/Y)
The left-hand-side of of the above equation is, by definition, the rate of growth, g, of the economy. The equality of investment and savings is an accounting definition in a model with no foreign trade and no government. Therefore,
g = (1/v) (S/Y)
Define the savings rate, s:
s = S/Y
Then, a steady state growth ratio is the ratio of the savings rate to the capital-output ratio:
g = s/v
That is, the (actual) rate of growth is the quotient of the savings rate and the capital-output ratio.

2.2 The Warranted Rate
Suppose the savings rate and the capital-output ratio are as desired by income recipients (consumers) and firms, respectively. This defines Harrod's warranted rate of growth:
gw = sd/vd
where the subscripts on the right hand side stand for "desired". The warranted rate of growth is being achieved when expectations are being realized and current actions are not setting up forces to disturb current expectations.

The warranted rate of growth extends Keynes' analysis to the long period. Consider the stability of a warranted growth path. If the actual rate of growth exceeds the warranted rate, capacity will be utilized at a greater rate than firms expected. They will increase investment faster than the warranted rate, and the rate of growth will deviate from the warranted rate even more. Likewise, if the actual rate falls below the warranted rate, firms will cut back on investment since the plans upon which their investment was made are not being realized. Hence, the warranted rate is unstable.

Harrod suggested that this instability of the warranted rate is more like an inverted flat-bottomed bowl than a knife-edge.

2.3 The Natural Rate
Suppose the labor force is initially fully employed. Let n be the rate of growth of the labor force:
n = (dL/dt)/L
Define the value of output produced per employed worker:
f = Y/L
Harrod-neutral technical change occurs when the value of output per worker grows at a constant rate, m, while the rate of profit stays unchanged:
m = (1/f) df/dt
Harrod-neutral technical progress implies that the productivity of labor is growing at the same rate in all industries.

Anyways, the following equation follows:
dY/dt = f dL/dt + L df/dt
Some algebra yields:
(1/Y) (dY/dt) = ( 1/L) (dL/dt) + (1/f) (df/dt)
The left hand side of the above equation is the rate of growth that keeps the labor force fully employed (or a constant percentage unemployed). Harrod calls this the natural rate of growth. Hence, assuming Harrod-neutral technological progress, the natural rate of growth is the sum of the rate of growth of the labor force and the rate of growth of labor productivity.
gn = n + m

3.0 Conclusions
Notice that the determinants of the warranted rate of growth - the savings rate and the desired capital-output ratio - are taken as exogeneous constants. The determinants of the natural rate of growth - the growth of the labor force and Harrod-neutral technological progress - are also given. Hence, the warranted and natural rates can only be equal by a fluke.

Solow, following up on some work by Pivlin, suggested that the desired equality between the warranted and natural rates can be brought about by considering the capital-output ratio as a well-behaved function of the rate of interest. Divergences between the two rates can be corrected by variations in the distribution of income. This approach of neoclassical macroeconomics is exemplified in Solow's eponymous growth model, but it has been shown to be not well-founded in the Cambridge Capital Controversy.

If the warranted rate is below the natural rate, a moderate increase in the saving rate is desirable if the economy is exhibiting boom-like conditions. This would bring the warranted rate towards the actual rate of growth while still keeping it below the natural rate of growth.

Notice that when the warranted rate exceeds the natural rate, the economy must sometime fall below the warranted rate. The natural rate sets a limit which the economy cannot long exceed. Because of the instability of the warranted rate, such an economy will experience frequent and perhaps prolonged recessionary conditions. Since increased savings intensify the discrepancies between the warranted and natural growth rates under these conditions, increased savings intensify the frequency and severity of recessions. That is, increased inequality can intensify the frequency and severity of recessions.

References
  • A. Asimakopulos (1991) Keynes's General Theory and Accumulation, Cambridge.
    1991
  • Roy F. Harrod (1948) Towards a Dynamic Economics, Macmillan.
  • Joan Robinson (1962) Essays in the Theory of Economic Growth, Macmillan.

Wednesday, August 25, 2010

Barnett's Fried Apples


Ingredients

4 Tablespoons butter
1 #2 can sliced apples or 2 1/2 cups fresh apple
1/8 teaspoon salt
1/4 cup sugar
Cinnamon to taste

1) Peel and core apples.

2) Melt butter in iron skillet. Add apples, salt, sugar, and cinnamon. (I'm generous with the cinnamon.)

3) Fry until soft, between low and medium heat about 1/2 hour. (Do not fry dry.)

Makes approximately 3 servings. (I like them served with pork chops.)

Tuesday, August 24, 2010

That You Should Listen To Mainstream Economists...

... seems often to me to be the main point of many mainstream economists these days. I deliberately don't write, "Why you should listen..." Somebody as stupid as Kartik Athreya, a PhD. with the research department of the Federal Reserve Bank of Richmond, appears to be doesn't deal in arguments. I also see this sort of babble in recent posts by Frances Woolley, and Mike Moffat. (See also Nick Rowe's comments to those posts.)

(I, of course, have read papers making points along Colander's line.)

Sunday, August 22, 2010

Jeffrey Miron And Propertarian Advocacy Taught At Harvard

Jeffrey Miron teaches EC1017 at Harvard. "A Libertarian Perspective on Economic and Social Policy" is the course title, and PDFs for the lectures are available for download.

Based on the notes for the three lectures I looked at, Miron supposedly derives propertarian policy from intermediate principles (e.g., "efficiency"), with little to no data on relative magnitudes. I don't care for this approach myself, never mind the policy conclusions. He seems to mention no names. The reading list (from Spring 2009) does not include his book (which I haven't read). Perhaps Miron's experience is that Harvard students can be counted on to bring up Rawls, Karl Popper's piecemeal social engineering, Alan Haworth, and even Nozick.

Thursday, August 19, 2010

"When Adam Delved And Eve Span, Who Was Then The Gentleman?"

I had associated the title of this post with the 17th century and the period of the English Civil War. I think it occurs somewhere in Christoper Hill's The World Turned Upside Down: Radical Ideas During the English Revolution. Hill's book is an account of Anabaptists, Diggers, Levellers, Muggletonians, the New Model Army, Ranters, and Quakers - a very heady and confusing mix.

So I was startled yesterday to read the phrase in Crispin: The Cross of Lead. This is a Newberry-prize winning children's book, by Avi. It is set in England in the 14th century. I think it conveys a good idea of the drudgery and isolation of village life at the time; the seemingly unchangable hierarchy; and the bustle, confusion, and filth of a city before modern plumbing. I also like that Christianity is presented as a form of life, a language that all we see cannot but help using.

So is Avi's use of the phrase an anachronism? Hill may reference it, but, if so, the people of his time were harking back to a previous one. Apparently, the phrase is associated with John Ball, the leader of the 1381 Peasants’ Revolt. I know nothing about the Peasants’ Revolt, although Hill does refer to it in one line. But John Ball does appear in Avi’s book. Crispin, our thirteen-year old hero, overhears him conspiring. John Ball says:
"...that no man, or woman either, shall be enslaved, but stand free and equal to one another. That all fees, obligations, and manorial rights be abolished immediately. That land must be given freely to all with a rent of no more than four pennies per acre per year. Unfair taxes must be abolished. Instead of petty tyrants, all laws shall be made by consent of a general commons of all true and righteous men.

Above all persons, our lawful king shall truly reign, but no privileged or corrupt parliaments or councilors.

The church, as it exists, should be allowed to wither. Corrupt priests and bishops must be expelled from our churches.. In their place will stand true and holy priests who shall have no wealth or rights above the common man..."


Update: I've learned a new vocabulary word: A Jacquerie is a peasants' revolt, named after the French peasants' revolt of 1358.

Friday, August 13, 2010

Infinities Of Infinities

1.0 Introduction

This is mathematics, not economics. It is meant to be an introduction to how abstract mathematicians can be.

2.0 Some Definitions for Set Theory

Two sets are the same size if and only if they can be put into a one-to-one correspondence with each other.

A set S1 is bigger than the set S2 if and only if:
  • A subset of S1 can be put into one-to-one correspondence with S2, and
  • S2 cannot be put into one-to-one correspondence with S1.
A set is countably infinite if and only if it can be put into one-to-one correspondence with the set of natural numbers N = {0, 1, 2, ...}. (The integers and the rational numbers are both countably infinite.)

The power set P(S) formed from the set S is the set of all subsets of S. For example, the power set for the set {a, b} contains four elements:
P( {a,b} ) = {S | S ⊂ {a, b}.} = { ∅, {a}, {b}, {a, b} }

3.0 A Theorem

Theorem For all sets S, the power set P(S) is bigger than the set S.

Proof: First, show that a subset of P(S) can be put into one-to-one correspondence with S. Consider the set of singletons:
{ {a} | a is an element of S }.
Since each singleton {a} is a subset of S, the set of all singletons is a subset of P(S). And the set of all singletons maps one-to-one to S.

Next, show, by a proof by contradiction, that P(S) cannot be put into one-to-one correspondence with S. Suppose that there exists a one-to-one function f that maps S into P(S).

Notice that, for all a ∈ S, f(a) is a subset of S. For any given a in S, either
a ∈ f(a)
or
a ∉ f(a).
Define the set T to be the set of all elements in S that map under f to a set not containing themselves:
T = { a | a ∈ S and a ∉ f(a)}
Since f is one-to-one and T is a (possibly empty) subset of S, there exists, by hypothesis, an element b in S such that
f(b) = T.
Now consider whether or not
b ∈ T.
Suppose true. But, by the definition of T as the set of elements of S that are not elements of the subset of S that they map to, b cannot be in f(b), that is, T. But, if b is not in f(b), by the definition of T, b must be in T. So either way yields a contradiction. Thus, no such b can exist.

So I have shown that there does not exist an element b in S that maps under f to T. Yet T is in P(S). Thus, f cannot be one-to-one. Which was to be demonstrated.

4.0 Applying the Theorem to the Set of Natural Numbers

An interesting property of the above proof is that it applies to both finite sets and infinite sets. So start with N, the set of natural numbers. N contains an infinite number of elements. But, by the theorem, P(N), the set of all subsets of the natural numbers, is a set containing a bigger infinity. One can go on to form a set of infinite sets, each with a bigger size infinity:
U0 = { N, P(N), P(P(N)), ..., Pn(N), ...}
(Under the Zermelo Frankel axioms for set theory, the elements of a set do not need to all be of the same "type".) One can repeat the process of forming a sequence of power sets:
U1 = { U0, P(U0), P(P(U0)), ..., Pn(U0), ...}.
One can even imagine constructing a power set of all these difference size infinite sets in this sequence of sequences of sets:
P( { U0, U1, U2), ...} )
The definitions of infinite sets need not stop here.

4.0 Conclusion
I don't find the above hard to follow if I think of it as merely a matter of syntactic manipulation of symbols. Do I have a clear idea of these infinities of different size infinities after every point in this sequence of definitions? Does anybody? This is not so clear to me.

Reference
  • Paul R. Halmos (1960) Naive Set Theory, Springer Verlag

Tuesday, August 10, 2010

Onieda-Like Community Near Stroud, In Gloucestershire?

Martin Gardner once received a letter referring to "an Oneida-like community near Stroud, in Gloucestershire". The topic of the letter was something else, on visualizing four-dimensional space. Can anybody provide me with more information on this community?

Saturday, August 07, 2010

Full Unemployment

I find amusing the political slogan with which I title this post. We want the engineers to do their job in applying control theory to stepping motors, in creating Artificial Intelligences, in developing techniques of information management, etc. such that nobody need work out of necessity. Maybe in some future day, machinery will produce all we need, including more machinery. When the economic problem is solved:
"Man will be faced with his real, his permanent problem - how to use his freedom from pressing economic cares, how to occupy the leisure, which science and compound interest will have won for him, to live wisely and agreeably and well." -- John Myanard Keynes (1930)
Marx and Engels envision a post-capitalist society:
"Where nobody has one exclusive sphere of activity, but each can become accomplished in any branch he wishes, society regulates the general production and thus makes it possible for me to do one thing today and another tomorrow, to hunt in the morning, fish in the afternoon, rear cattle in the evening, criticize after dinner, just as I have a mind, without ever becoming hunter, fisherman, shepherd or critic. -- Karl Marx (1947, p. 22)
Bruce Sterling (1989) imagines that, in such a world, one will cultivate ones taste for "The Beautiful and the Sublime". At any rate, in this pleasant world of tomorrow, all will be able to devote themselves to great cooking, fostering social relationships, art, or whatever one may choose.

Curiously enough, the classical tradition in economics, as exemplified, for example, in Sraffa or Von Neumann, provides tools for analyzing how prices might be formed in a post-scarcity world. For example, Joan Robinson, in her first essay in (Robinson 1962) has a section titled "A model for the future" with a subsection on "The Robots". This is a model of a (maybe impossible) capitalist economy. In my version, all production is carried out in automated factories, and these factories are owned by firms traded on a stock exchange. Everybody owns shares, and the trading of these shares sets up a tendency torwards a uniform rate of profits.

I have described before some formulation of a price system consistent with this institutional set up. For now, I want to describe prices when the managers of each firm in an industry have chosen a process for producing the firm's output. As usual, I assume, for simplicity that all processes require the same time to operate, say, a year. Inputs must be purchased at the beginning of the year, and outputs become available at the end of the year. A reference set of prices satisfies the following system of equations:
p A β = p B
where
  • A is a square matrix; ai,j is the quantity of the ith commodity used as input when the jth process is operated at a unit level.
  • B is a square matrix; bi,j is the quantity of the ith commodity produced as output when the jth process is operated at a unit level.
  • p is a row vector of prices; pi is the price of the ith commodity.
  • (β - 1) is the rate of profits.
This formulation allows for robots to last for more than one year. The quantity of a dated robot enters as an input, and the output includes a robot one year older, as well as whatever other outputs are produced by that process. The use of such robots is a special case of the more general model of joint production encapsulated in the above system of equations.

Various conditions must be imposed on the coefficients of production A and B to ensure a solution simultaneously exists for prices and the dual problem of the choice of technique. Von Neumann, in fact, assumes that each commodity is either used as an input or produced as an output in a poisitive amount in each process. Joan Robinson assumes the existence of "some standard physical elements (say, nuts and bolts) that enter into the production both of robots and of salable goods." But I do not want to discuss more of the mathematics in this post.

References
  • D. G. Champernowne (1945-1946) "A Note on J. v. Neumann's Article on 'A Model of Economic Equilibrium'", Review of Economic Studies, V. 13, N. 1: pp. 10-18.
  • John Maynard Keynes (1930) "Economic Possibilities for our Grandchildren", in Essays in Persuasion, W. W. Norton & Company
  • Karl Marx and Frederick Engels (1947) The German Ideology: Parts I & III, International Publishers
  • Joan Robinson (1962) Essays in the Theory of Economic Growth, Macmillan.
  • Piero Sraffa (1960) , Cambridge University Press.
  • J. v. Neumann (1945-1946) "A Model of General Economic Equilibrium", Review of Economic Studies, V. 13, N. 1: pp. 1-9.
  • Bruce Sterling (1989) Crystal Express, Ace Books

Nortz's Johnny Cake


Ingredients

1/4 cup sugar
1/3 cup shortening
1 beated egg
1 cup sour milk
1 teaspoon baking soda
1 teaspoon baking powder
1 cup flour
1 1/2 cup cornmeal
1/2 teaspoon salt

1) Mix in above order, stirring thoroughly after adding each ingredient. Bake about 1/2 hour at 400 F.

2) Serve sliced with applesauce or maple syrup.

Makes approximately 10 servings. I usually make a double recipe when making my great-grandmother's Johnny cake.

Wednesday, August 04, 2010

Phenomenology

"One of the embarrassing dirty little secrets of economics is that there is no such thing as economic theory properly so-called. There is simply no set of foundational bedrock principles on which one can base calculations that illuminate situations in the real world." -- Brad DeLong

My title does not refer to an approach in continental philosophy associated with Husserl and Heidegger. Rather, I refer to a term used in physics and engineering by practitioners who know they are not trying to develop models derived from fundamental laws, but only modeling the phenomena.

I find it of interest that Brad DeLong has recently described economics as phenomenology. A noted "rocket scientist" on Wall Street came to the same conclusion:
"The techniques of physics hardly ever produce more than the most approximate truth in finance because 'true' financial value is itself a suspect notion. In physics, a model is right when it correctly predicts the future trajectories of planets or the existence and properties of new particles, such as Gell-Mann's Omega Minus. In finance, you cannot easily prove a model right by such observation. Data are scarce and, more importantly, markets are arenas of action and reaction, dialectics of thesis, antithesis, and synthesis. People learn from past mistakes and go on to make new ones. What's right in one regime is wrong in the next.

As a result, physicists turned quants don't expect too much from their theories, though many economists naively do. Perhaps this is because physicists, raised on theories capable of superb divination, know the difference between a fundamental theory and a phenomenological toy, useful though the latter may be. Trained economists have never seen a really first-class model. It's not that physics is 'better', but rather that finance is harder. In physics you're playing against God, and He doesn't change his laws very often. When you've checkmated Him, He'll concede. In finance, you're playing against God's creatures, agents who value assets based on their ephemeral opinions. They don't know when they've lost, so they keep trying." -- Emanuel Derman (2004) My Life as a Quant: Reflections on Physics and Finance, John Wiley & Sons.
I think one can read intimations of Soros' reflexitivity or Joan Robinson's historical time in the above quote. Derman is even more direct about a Post Keynesian concept elsewhere:
"Slowly it began to dawn on me that what we faced was not so much risk as uncertainty. Risk is what you bear when you own, for example, 100 shares of Microsoft - you know exactly what those shares are worth because you can sell them in a second at something very close to the last traded price. There is no uncertainty about their current value, only the risk that their value will change in the next instant. But when you own an exotic illiquid option, uncertainty precedes its risk - you don't even know exactly what the option is currently worth because you don't know whether the model you are using is right or wrong. Or, more accurately, you know that the model you are using is both naive and wrong - the only question is how naive and how wrong." -- Emanuel Derman (2004)

Sunday, August 01, 2010

Jacob Schwartz (9 January 1930 - 2 March 2009)

Jacob T. Schwartz was a mathematician at the Courant Institute of Mathematical Sciences at New York University. He once gave a series of lectures on mathematical economics, published as Lectures on the Mathematical Method in Analytical Economics, Gordon and Breach (1961). This book, coming out a year after Sraffa's work, seems to very little known. Its findings parallel Sraffa's in many ways:
"Our interest...will...be...in the use of the input-output model as a framework for ...abstract economic analysis." (p. 8)
"If each time labor appears as an input in production we replace this input by the corresponding real wage bill, we come to a hypothetical situation in which the only inputs required for the production of commodities are other (non-labor) commodities. Thus we may, if it is convenient for one or another theoretical purpose, consider our model to refer to a self-enclosed world of material commodities, produced out of each other with no additional input." (p. 10)
"The proper conclusion at this point is that the rate of profit ρ is not successfully determined by the Walrasian theories from considerations of production coefficients, utility functions, and so forth. What our analysis shows, in fact, is that the determination of the rate of profit is not purely a question of economics at all, but is rather a social-political question involving, among other things, union-management relations, pressures, and counterpressures, etc. Thus an initial skepticism about classical equilibrium analysis is justified. What this analysis aims to give us is a set of prices. But all the price-ratios are already determined by a small part of the theory, to wit by the competitive equality of profit rates. All that remains to be determined on the score of prices, is the rate of profit - but, as we have just seen, the Walrasian determination of this rate is questionable... What are determined more successfully are the amounts of production - but this is more a humble matter of consumption habits at given prices than a highly recondite matter of consumption schedules at a variety of hypothetical prices." (pp. 196-197)
"As our analysis in Lecture 16 shows, as long as we assume a fixed scheme of production the Keynesian conclusion that wage cuts by lowering wage-generated commodity demand must lower demand for labor is inescapable. The neoclassical contention thus depends on the possibility of shifts in the production scheme; a conclusion which the neoclassicist would be the first to emphasize, since the whole apparatus of neoclassical theory, revolving about the notion of marginal product accruing to an increment of each input facor, does in fact center on an analysis of variations in production. This means that the equilibrium analysis of Lecture 16 has come to such distinctly Keynesian conclusions as it has only by assuming away the basis for the neoclassicist's argument. At the present point, therefore, we shall attempt to generalize the analysis of Lecture 16 to include the possibility of shifts in the production scheme, hoping to estimate the extent to which such shifts are likely to affect our earlier conclusions." (p. 239)
"We may at this point remark once more that our analysis of prices shows that even in the framework of the present general model [with a choice of technique] price ratios are determined up to a single parameter from the conditions of production. As we have emphasized in the final paragraph of Section 1, Lecture 3, this conclusion constitutes strong presumptive evidence against theories which attempt to tie prices to consumer demand. More generally, we see that by allowing variation in the scheme of production, we in fact introduce no changes in the fixed-matrix Leontief model other than to make the Leontief matrices dependent on the [rate of profits]." (p. 248)
I prefer Sraffa's book partly on the basis of style.