Saturday, December 31, 2016
Welcome
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.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
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.
Comments Policy: 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.
Friday, January 23, 2015
Approximating a Continuous Time Markov Process
Figure 1: Rate of Transitions Between States in a Three-State Markov Chain |
This post, about Markov processes, does not have much to do with economics. I here define how to approximate a continuous time Markov chain with a discrete time Markov chain. This mathematics is useful for one way of implementing computer simulations involving Markov chains. That is, I want to consider how to start with a continuous time model and synthesize a realization with a small, constant time step.
2.0 Continuous Time Markov ChainsConsider a stochastic process that, at any non-negative time t is in one of N states. Assume this process satisfies the Markov process: the future history of the process after time t depends only on the state of the process at time t, independently of how the process arrived at that state. I consider here only processes with stationary probability distributions for state transitions and for times between transitions. A continuous time Markov chain is specified by a state transition matrix. In this section, I define such a matrix as well as specifying two additional assumptions.
Formally, let P_{i, J} denote the conditional probability that the next transition will be into state j, given that the process is in state i at time zero. (As seen below, in the notation adopted here it matters that these conditional probabilities are not a function of time.) Assume that for each state, the next transition when the process is in that state is into a different state:
P_{i, i} = 0; i = 0, 1, ..., N - 1
Further, assume that for each state, the time to the next transition is from an exponential distribution with the rate of occurrence of state transitions dependent only on the initial state:
F_{i, j}(t) = 1 - e^{- λi t}; i, j = 0, 1, ..., N - 1;
where F_{i, j}(t) is the conditional probability that the next transition will be before time t, given that the chain is in state i at time zero and that the next transition will be into state j. In other words, F_{i, j}(t) is the Cumulative Distribution Function (CDF) for the specified random variable. Under the above definitions, the stochastic process is a continuous time, finite state Markov chain.
Let P_{i, j}(t) be the conditional probability that the chain is in state j at time t, given that the chain is in state i at time zero. These conditional probabilities satisfy Kolmogorov's forward equation:
,
where the transition rate matrix Q is defined to be:
The elements in each row of the transition rate matrix sum to zero. Kolmogorov's forward equation can be expressed in scalar form:
The above equation applies to continuous time Markov chains with a countably infinite number of states only under certain special conditions.
Steady state probabilities of this Markov chain satisfy:
p Q = 0,
where p is a row vector in which each element is the steady-state probability that the chain is in the corresponding state.
3.0 Discrete Time ApproximationA discrete time Markov chain is specified by a state transition matrix A, where a_{i, j} is the probability that the chain will transition in a time step from state i to state j, given that the chain is in state i at the start of the time step. Steady state probabilities for a discrete time Markov chain satisfy:
p A = p
The above equation compares and contrasts with how steady state probabilities relate to the transition rate matrix in a continuous time Markov chain.
Let the time step h be small enough that the probability of the continuous time Markov chain undergoing two or more transitions in a single time step is negligible. In other words, the following probability, calculated from a Poisson distribution, is close to unity for all states i:
P(0 or 1 transitions in time h | Chain in state i at time 0) =(1 + λ_{i} h) e^{- λi h}
The probability that the chain remains in a given state for a time step is the probability that no transitions occur during that time step, given the state of the chain at the start of the time step. This probability is also found from a Poisson distribution:
a_{i, i} = e^{- λi h} = e^{- qi, i h}; i = 0, 1, ..., N - 1
The probability that the chain transitions to state j, given the chain is in state i at the start of the time step, is the product of:
- The probability that a transition occurs during that time step, and
- The conditional probability that the next transition will be into state j, given the chain is in state i at the start of the time step.
The following equation specifies this probability:
a_{i, j} = (1 - a_{i, i})P_{i, j} = (1 - a_{i, i}) q_{i, j}/(- q_{i, i}); i ≠ j
These equations allow one to write a computer program to synthesize a realization from a finite state Markov chain, given the parameters of a continuous time, finite state Markov chain. Such a program will be based on a discrete time approximation.
4.0 An ExampleConsider a three-state, continuous time Markov chain. Figure 1 shows the rate of transitions between the various states. The transition rate matrix is:
To discretize time, choose a small time step h such that, for all states i, the following probabilities are approximately unity:
P(0 or 1 transitions in time h | Chain in state 0 at time 0) =[1 + (λ_{0, 1} + λ_{0, 2})h] e^{-(λ0, 1 + λ0, 2)h}
P(0 or 1 transitions in time h | Chain in state 1 at time 0) =[1 + (λ_{1, 0} + λ_{1, 2})h] e^{-(λ1, 0 + λ1, 2)h}
P(0 or 1 transitions in time h | Chain in state 2 at time 0) =[1 + (λ_{2, 0} + λ_{2, 1})h] e^{-(λ2, 0 + λ2, 1)h}
The state transition matrix A for the discrete-time Markov chain is:
I have not tested the above with concrete values for a continuous time Markov chain.
Reference- S. M. Ross (1970). Applied Probability Models with Optimization Applications. San Francisco: Holden-Day
Friday, January 16, 2015
Laughing At Neoclassical Economists, Elsewhere
- Matthew Yglesias lists "Nine Things Only Neoclassical Economists Will Understand". Strangely, his twitter announcement of this article is about a tenth.
- Noah Smith purports to explain each thing in only a couple of sentences. Stranegly, only for the Modiliani-Miller theorem does he note, "Obviously this doesn't work in the real world".
- Tyler Cowen attempts to clarify the Heckscher-Ohlin theorem, but fails to note that "capital" cannot be a factor of production in the Heckscher-Ohlin-Samuelson model. (He does note Leontief's empirical demonstration that the theory fails.)
Saturday, January 10, 2015
Because Something Is Happening Here/But You Don’t Know What It Is/Do You, Mister Jones?
Strangely, some prominent, somewhat liberal, economics bloggers have decided simultaneously to complain about (unnamed) left-leaning heterodox economists:
- Noah Smith, at Bloomberg View and at his blog.
- Simon Wren-Lewis, at his blog.
- Paul Krugman, at his New York Times blog.
All three, incorrectly in my view, think the heterodox economists who they object to are only arguing politics. As far as I know, many, including me, do not take issue with Krugman's short-term policy views. Smith, in his trollish approach, raises a side comment about Austrian economists and the Mont Pelerin society. (I will state the proper label for Friedrich Hayek and Ludwig Von Mises is "economist", not "quasi-economist", as Smith would have it. But I've seen for some time that I am more well-informed on Austrian economics than Smith is.)
I think more pertinent issues center around modeling approaches, the image the profession projects in the public sphere, and the sociology of the profession. How is it than so many rightists have been able to push the view that their politics is good economics, while simultaneously insisting that economics is a positive science? The involvement of economists with neoliberal politics is not confined to some fringe. Consider, for example, the Chicago school, the lack of a strong ethics policy in the American Economic Association, the Washington consensus, and even Paul Samuelson's 1960s research that led to to Efficient Market Hypothesis.
There is probably also a personnel element here. Non-mainstream, heterodox economists would like more acknowledgement by mainstream economists. If your knowledge of heterodox economics is confined to what you can get off the Internet, aside from what professional literature is now available there, you might not know what you are talking about when you talk about heterodox economics. (And this includes the Austrian school.) Furthermore, when you develop parallel ideas, or draw on heterodox economics, you should acknowledge it. In the linked post above, Krugman makes the point that "a country that borrows in its own currency" cannot easily become like Greece, under attack from "bond vigilantes", without saying anything about endogenous money or the economists at the University of Missouri Kansas City. (I could also say something about the research for which Krugman won the "Nobel Prize".) If you know where to look, you can find Joseph Stiglitz acknowledging that he learned a lot from such Cambridge economists as Nicky Kaldor and Joan Robinson.
Maybe economics would be a better place if the center of gravity in economics in the United States were arguments between mainstream economists and, say, economists at the New School and the University of Massachusetts at Amherst. If the profession were to move in this direction, young doctorates would need to be socialized to not dismiss viewpoints because of the rankings of the universities and the journals in which they were advanced. Methodology would continue to need to be broadened to include more than mathematical models of optimizing agents.
Update: Reactions from Chris Dillow, Peter Dorman, and Alex Marsh.
Friday, January 09, 2015
Greg Mankiw, Fool Or Knave?
Greg Mankiw seems determined to continually attempt to bring his supposed profession into disrepute. Last week, at the annual meeting of American economists (the Allied Social Science Associations), Greg Mankiw chaired a session on Thomas Piketty's Capital in the 21st Century. In his draft of his prepared remarks, Mankiw writes:
"Equation (3) says that capital earns its marginal product." -- Greg Mankiw, "Yes, r > g. So what?" (24 November 2014).
Because of price Wicksell effects, the marginal product of finance capital is generally unequal, in equilibrium, to the rate of profits. Even Champernowne's chain index, which abstracts from price Wicksell effects, cannot generally be used to defend the equality in aggregate models of the rate of profits and the marginal product of capital. Economic theory imposes no restriction on the direction of price and real Wicksell effects, and the chain index is not well-defined in the presence of positive Wicksell effects. Neoclassical theory claims, at best, that the price of each capital good is equal, in equilibrium, to its marginal product. But marginal productivity is not a theory of the functional income distribution, since every point on the wage-rate of profits frontier is compatible with all valid marginal productivity conditions. Even if the returns to capital could be explained by marginal productivity, this would not justify any particular size of the tolls that capitalists are able to impose. A conceptual distinction can and should be made between the cost of capital goods and the returns to capitalists.
As far as I am concerned, the above is just good economics, agreed to by all non-ignorant economists, neoclassical or otherwise. But the confusion and general muddleheadness promoted by such as Mankiw, seems to serve a functional purpose in the sublunary world.
Monday, December 29, 2014
On "Privatized Keynesianism"
I have been reading Colin Crouch's The Strange Non-Death of Neoliberalism^{1}. A major theme is that an ideological divide between more reliance on markets and on government misses issues raised by the existence of large - including multinational - corporations. The neoliberal assault on government has been increasing the strength of corporations, not competitive markets. Furthermore, corporations have been taking on the role of government. Crouch mentions, for example, the "seconding" of corporate executives to various ministries; the likelihood that internal policies of a Multi-National Corporation on, say, child labor may be more restrictive than laws in many third world countries; and the role of corporations in setting international standards, where organizations with nation-states may be weak.
But my point in this post is to note Crouch's introduction(?) of a new technical term, Privatized Keynesianism. A contrast between the post-World War II golden age and the later neoliberal era^{2} is needed to make sense of this term. After the war, in the United States - and, I gather, in other advanced industrial capitalist economies - wages rose with average productivity. Furthermore, governments, under a somewhat Keynesian ideology, saw it as their responsibility to maintain aggregate demand. These conventions came undone in the 1970s. Productivity increased (at a slower pace), but wages failed to keep up, and governments came to emphasize fighting inflation, not unemployment.
Increased inequality, however, did not eliminate the need to manage aggregate demand. Neither consumer spending from wages nor an abdication from fiscal polity by government could fill this lacuna. This period saw the increased availability of debt, the creation of secondary markets for the trading of bets on bets on bundles of debts (derivatives), and the capture of credit rating agencies by sellers of debts. This institutional structure led to the collective, but private, macroeconomic regulation of aggregate demand^{3}. This institutional structure is what Crouch calls privatized Keynesianism^{4}. The irresponsibility of banks, in some sense, produced a (temporary, unsustainable) positive externality.
Footnotes- I might as well note two mistakes I found irritating. Somewhere in one of the early chapters, Crouch, who I gather is British, refers to Eugene McCarthy when he means Joe McCarthy. I also thought that Crouch's account of the role of Fanny Mae and Freddy Mac in subprime mortages reflected too much credence for right-wing liars.
- I date the start of the neoliberal era with Nixon ending the fixed exchange rate between the United States dollar and gold, a major element of the Bretton Woods system.
- Is this a non-microfounded, functionalist account?
- From this perspective, the accumulation of private debt was a symptom, not the ultimate cause of the recent Global Financial Crisis, a cause that has yet to be addressed. These ideas seem to me to be close to Thomas Palley's Structural Keynesianism. Has anybody read James K. Galbraith's The End of Normal: The Great Crisis and the Future of Growth?
Thursday, December 18, 2014
Slaves Identifying With Their Masters
Marx's attempt to describe how capitalism creates objective illusions, so to speak, is one aspect of Capital that I like. In this comment on a long-ago Crooked Timber post, "Ted" draws an analogy to J. S. Mill's Subjection of Women, which I have never read. Apparently, Mill explains how women can come to identify with their oppressors.
I happen to currently be reading the autobiography of local Rochester hero, Frederick Douglass. This passage identifies a curious phenomenon:
"Moreover, slaves are like other people, and imbibe prejudices quite common to others. They think their own better than that of others. Many, under the influence of this prejudice, think their own masters are better than the masters of other slaves; and this, too, in some cases, when the very reverse is true. Indeed, it is not uncommon for slaves even to fall out and quarrel among themselves about the relative goodness of their masters, each contending for the superior goodness of his own over that of the others. At the very same time, they mutually execrate their masters when viewed separately. It was so on our plantation. When Colonel Lloyd's slaves met the slaves of Jacob Jepson, they seldom parted without a quarrel about their masters; Colonel Lloyd's slaves contending that he was the richest, and Mr. Jepson's slaves that he was the smartest, and most of a man. Colonel Lloyd's slaves would boast his ability to buy and sell Jacob Jepson. Mr. Jepson's slaves would boast his ability to whip Colonel Lloyd. These quarrels would almost always end in a fight between the parties, and those that whipped were supposed to have gained the point at issue. They seemed to think that the greatness of their masters was transferable to themselves. It was considered as being bad enough to be a slave; but to be a poor man's slave was deemed a disgrace indeed." -- Frederick Douglas, Narrative of the Life of Frederick Douglas
Maybe some day I'll read Hegel's Phenomenology of Spirit - it is on my shelf - to learn some ideas about the master-slave dialetic. Mayhaps the above is analogous to the opinions of many wage-slaves. There seem to be many ways to be unfree, and many ways to deny this.