Wednesday, December 31, 2014
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.
Thursday, April 17, 2014
Estimating Probability of Extreme Events
Figure 1: Distribution for Mixture Distribution |
What is the probability that the Dow Jones Industrial Average (DJIA) will rise by at least 5% tomorrow? By 10%? Very few samples can be found in the data for a large enough rise, and, eventually, you will be asking about a rise beyond all historical experience. Some have argued that Extreme Value Theory can be applied to financial data to extrapolate these sorts of tail probabilities. In this post, I attempt to explain this theory. For purposes of exposition, I here disregard the possibility of such rises as being associated with states that might be impossible to foresee from the past history of the data-generating process.
2.0 A Random Sample from a Mixture DistributionThis exposition includes an example. I need a probability distribution in which tails differ from the portion of the distribution clustered around the center, in some sense. Consider a random variable X which can take on any real number. The probability distribution for this random variable is defined by the Cumulative Distribution Function (CDF). The CDF specifies the probability that a realization of the random variable is less than or equal to a given value:
F(x) = Pr(X ≤ x)
where:
- x is the argument at which the CDF is evaluated.
- F is the CDF.
- F(x) is the indicated probability, that is, the value of the CDF evaluated for the argument.
(Conventionally, uppercase letters toward the end of the alphabet denote random variables. The corresponding lowercase letter denotes a realization of that random variable resulting from the outcome of conducting the underlying experiment.)
To obtain a distribution with heavy tails, I consider a mixture distribution. (Mixture distributions are often used in the theory for robust statistics. I would appreciate a reference arguing that robust statistics and Extreme Value Theory are complementary, in some sense.) Suppose F_{1} and F_{2} are CDFs for Gaussian (also known as normal or bell shaped) distributions with possibly different means and standard deviations. And let p be a real number between zero and one. F is the CDF for a mixture distribution if it is defined as follows:
F(x) = p F_{1}(x) + (1 - p) F_{2}(x)
For definiteness, let the parameters for this distribution be as in Table 1. The two Gaussian distributions have equal means. The distribution with the 90% weight also has the smaller standard deviation. In other words, the distribution that is selected less frequently will have realizations that tend towards the tails of the overall mixture distribution.
Parameter | Value |
Probability Variate from First Distribution | 90% |
First Gaussian Distribution | |
Mean | 0.0 |
Standard Deviation | 1.0 |
Second Gaussian Distribution | |
Mean | 0.0 |
Standard Deviation | 3.0 |
2.1 A Random Sample
Suppose X_{1}, X_{2}, ..., X_{n} are mutually stochastically independent random variables, each of which has the probability distribution with CDF F. Under these conditions, these random variables comprise a random sample. I wrote a computer program to generate a realization of such a random sample of size n. Table 2 shows some statistics for this realization. I use this realization of a random sample to illustrate the application of various statistical techniques below.
Value | |
Sample Size | 500 |
Realizations from 1st Distribution | 443 |
Realizations from 2nd Distribution | 57 |
Sample Mean | -0.0086 |
Standard Deviation | 1.3358 |
Minimum | -6.0125 |
Median | -0.0034 |
Maximum | 9.932 |
2.2 Goodness of Fit
It is difficult to determine that a realization of the random sample is not from the distribution F_{1}. In other words, the existence of sample values, often in the tails, from F_{2} is not readily apparent from a straightforward statistical test for the goodness-of-fit. Consider the order statistics found by sorting the random sample:
X_{(1)} ≤ X_{(2)} ≤ ... ≤ X_{(n)}
(By convention, a subscript for a random variable without parentheses denotes a random variable from a random sample. Parentheses denotes an order statistic.)
An empirical CDF can be constructed from the order statistics. The probability that a random variable from the distribution generating the random sample is less than or equal to x_{(i)} is estimated as i/n, the proportion of the sample less than or equal to the given order statistic. Figure 1, above, shows the empirical CDF for my realization of the random sample, as well as the CDFs for the two Gaussian distributions in the mixture distribution. Both Gaussian CDFs have a value of 1/2 for an argument of zero, since that is their mean. The Gaussian distribution with the smaller standard deviation has a CDF with a steeper slope around the mean, since more of its probability is clustered around zero. The empirical CDF, estimated from the data, is a step function, with equal size steps occurring at each realization of a random variable in the sample. One needs to sort the data to calculate the empirical CDF.
The maximum vertical distance between a theoretical distribution and an empirical CDF is known as the Kolmogorov-Smirnov statistic. Under the null hypothesis that the random sample is drawn from the theoretical distribution, the Kolmogorov-Smirnov statistic will be a small positive number. Table 3 shows the Kolmogorov-Smirnov statistics for the data. This statistic is not statistically significant for the first Gaussian distribution. The probability that one would observe such a large value for the Kolmogorov-Smirnov statistic for the second Gaussian distribution is less than 1%. Thus, one could conclude that this data was not generated from the second distribution, but (incorrectly) conclude that it was generated from the first.
1st Gaussian Distribution | 2nd Gaussian Distribution | |
Kolmogorov-Smirnov Statistic | 0.0354 | 0.228 |
p Value | 54.59% | 0.00% |
3.0 Distribution for the Tail
With the description of the data out of the way, tail probabilities can now be defined. I concentrate on the upper tail.
3.1 Definition of a TailThe upper tail is defined in terms of the lower bound u for the tail and the tail probability q. These parameters are related like so:
q = Pr(X > u) = 1 - F(u)
The upper tail is defined as those values of the random variable such that the probability of exceeding such a value is less than the given parameter:
{x | Pr(X > x) < q}
In other words, the tail consists of values of the random variable that lie above the lower bound on the tail. It is sometimes convenient to define a new random variable, Y, for outcomes that lie in the tail:
Y = X - u
This new random variable is the distance from the lower bound of the tail, given that a realization of X lies in the tail. One could give a symmetrical definition of the lower tail and a corresponding random variable. Table 4 shows how many samples in my realization of the random sample, defined above, happen to come from the Gaussian distribution with the larger standard deviation, where, the parameter q is taken to be 10%.
Value | |
Number in Lower Tail | 18 |
Number in Center | 23 |
Number in Upper Tail | 16 |
Percentage of Lower Tail from 2nd Distribution | 36.7% |
Percentage of Center from 2nd Distribution | 5.7% |
Percentage of Upper Tail from 2nd Distribution | 32.7% |
For y > 0, the conditional probability that X exceeds any given value in the tail, given that X lies in the tail is:
Pr(X > y + u | X > u) = Pr[(X > y + u) and (X > u)]/Pr(X > u)
The above formula simply follows from the definition of conditional probability. The second clause in the "and" expression is redundant. So the above can be rewritten as:
Pr(X > y + u | X > u) = Pr(X > y + u)/Pr(X > u), y > 0
Let G(y) be the CDF for the distribution for the random variable Y. One can then rewrite the above formula as follows"
1 - G(y) = [1 - F(y + u)]/[1 - F(u)], y > 0
Substituting for the definition of the parameter q, one obtains:
F(y + u) = (1 - q) + q G(y), y > 0
Or:
F(x) = (1 - q) + q G(x - u), x > u
The above two expressions relate the CDFs for the distributions of the random variables X and Y.
3.2 Generalized Pareto DistributionA theorem states that if X is a continuous random variable, the distribution of the tail is from a Generalized Pareto Distribution with the following CDF:
G(y) = 1 - [1 + (c/a)y]^{-1/c}
The parameter a is called the scale parameter, and it must be positive. The parameter c is the shape parameter. It can take on any real number. When the shape parameter is zero, the Generalized Pareto Distribution reduces, by a limit theorem, to the exponential distribution.
Below, I will need the following expression for the Probability Density Function (PDF) for the Generalized Pareto Distribution:
g(y, a, c) = (1/a)[1 + (c/a)y]^{-(1 + c)/c}
The PDF is the derivative of the CDF. For any set A to which a probability can be assigned, the probability that Y lies in A is the integral, over A, of the PDF for Y.
3.3 Parameter EstimatesThe parameters defining the upper tail are easily estimated. Let r be an exogenously specified number of variates in the tail. The lower bound on the upper tail is estimated as:
u_{estimate} = X_{(n - r)}
The corresponding tail probability is estimated as:
q_{estimate} = r/n
Several methods exist for estimating the scale and shape parameters for the Generalized Pareto Distribution. I chose to apply the method of maximum likelihood. Since the random variables in a random sample are stochastically independent, their joint PDF is merely the product of the their individual PDFs. The log-likelihood function is the natural logarithm of the joint PDF, considered as a function of the parameters of the PDF.
ln g(a, c) = ln g(y_{1}, a, c) + ... ln g(y_{r}, a, c)
Maximum likelihood estimates are the values of the parameters that maximize the log-likelihood function for the observed realization of the random sample. I found these estimates by applying the Nelder-Mead algorithm to the additive inverse of the log-likelihood function. Table 5 shows estimates for the example.
Parameter | Estimate |
Tail Probability (q) | 10% |
Lower Bound on Tail (u) | 1.368 |
Scale Parameter (a) | 0.7332 |
Shape Parameter (c) | 0.2144 |
The above has described how to estimate parameters for a distribution characterizing a tail of any continuous distribution. Given these estimates, one can calculate the conditional probability that Y lies above any value in the tail. Figure 2 plots this probability for the example. Notice that this probability is noticeably higher, for much of the tail, for the mixture distribution, as compared to the probability found from the Gaussian distribution with the smaller standard deviation in the mixture. And the Kolmogorov-Smirnov goodness-of-fit would not have led one to reject estimates from the first Gaussian distribution. But the estimates from Extreme Value Theory are closer to the higher (and correct) probabilities from the true theoretical distribution.
Figure 2: Tail Probabilities |
4.0 Conclusion
This post has illustrated:
- A probability distribution in which the central part of the distribution's support tends to behave differently from the tails.
- The difficultly in rejecting the hypothesis that data is drawn from the distribution characterizing the central tendency of the data, with no account being taken of heavy tails.
- A method, applicable to any continuous random variable, for estimating a tail distribution.
- Such estimation yielding an appreciably larger estimate for a tail probability than the distribution characterizing the central tendency.
References
- J. B. Broadwater and R, Chellappa (2010). Adaptive Threshold Estimation via Extreme Value Theory, IEEE Transactions on Signal Processing, V. 58, No. 2 (Feb.): pp. 490-500.
- Damon Levine (2009). Modeling Tail Behavior with Extreme Value Theory, Risk Management Issue. 17.
- R. V. Hogg and A. T. Craig (1978). Introduction to Mathematical Statistics, Fourth edition, Macmillan.
- A. Ozturk, P. R. Chakravarthi, and D. D. Weiner (). On Determining the Radar Threshold for Non-Gaussian Process from Experimental Data, IEEE Transactions on Information Theory, V. 42, No. 4 (July): pp. 1310-1316.
- James Pickands III (1975). Statistical Inference Using Extreme Order Statistics, Annals of Statistics, V. 3, No. 1: pp. 119-131.
Wednesday, April 09, 2014
Illusions Generated By Markets Like Those Created By Language On Holiday
I have been reading a book, edited by Gavin Kitching and Nigel Pleasants, comparing and contrasting Ludwig Wittgenstein and Karl Marx. This is the later Wittgenstein of the Philosophical Investigations, not of the Tractatus. The authors of the papers from the conference generating this work do not seem too concerned with arguments about the differences between the young Marx and the mature Marx, albeit many quote a passage from the German Ideology about language. (I think this post is more disorganized than many others here.)
Anyways, I want to first consider a reading of Capital, consonant with the approach of Friedrich Engels and the Second International, but at variance with an analogy to Wittgenstein's later philosophy. One might think of the labor theory of value as a scientific approach revealing hidden forces and structures that are at a deeper level than observed empirical reality. Think about how, for example, physicists have an atomic theory that explains why tables are hard and water is wet. Even though a table may be seem solid, we know, if we accept science, that it is mostly empty space. Somewhere Bertrand Russell writes something like, "Naive realism leads to physics, and physics shows naive realism is wrong. Hence naive realism is false". Similarly, you may think purchases and sales on markets under capitalism are made between equals, freely contracting. But the science of Marxism reveals an underlying reality in which the source of profits is the exploitation of the workers.
Wittgenstein, in rejecting his early approach to language, rejects the idea of a decontextualized analysis of the sentences of our languages into an ultimate underlying uniform atomic structure which explains their meaning. Rather, in his later philosophy, he gathers togethers descriptions of the use of language, to dispel and dissolve the illusions characteristic of traditional philosophy. He is hostile to ideal of an ultimate essence for meaning, and points out the multifarious uses to which language is put. Some of his famous aphorisms include, "Nothing is hidden" and his explanation of the point of his philosophical investigations as "To show the fly the way out of the fly bottle". Some of his descriptions are not from actually existing societies, but from imagined primitive societies. Some of these imagined societies are described near the beginning of the Philosophical Investigations, much as in the first chapter of Piero Sraffa's Production of Commodities by Means of Commodities.
Can Marx be read in an analogous manner, as attempting to dispel illusions, while claiming that no hidden essence or foundation underlies capitalist economies? Such a reading, I think, will emphasize Marx's remarks on commodity fetishism and "real illusions" that come with non-reflective participation in a market economy. It also makes sense of Marx's literary style. Both Marx and Wittgenstein are attempting to encourage a fundamental change so that our form of life will not generate these illusions.
Perhaps such a reading is in tension with the view of Marx's account of exploitation as descriptive, not normative. What about Wittgenstein's saying that philosophy "leaves everything as it is"? How can one read Wittgenstein and Marx as pursuing complementary projects when Marx writes, "Philosophers have hitherto only interpreted the world in various ways; the point is to change it"? Various essays in this book address these issues. I guess what concerns me more is Marx's Hegelian style, quite different from Wittgenstein. (I rely on English translations.)
This book also alerted me to some issues in Wittgenstein interpretation. When Wittgenstein writes of a form of life, is he writing of human life in general (in contrast, say, to the form of life of a lion)? Or would different human cultures and societies have different forms of life? Does Wittgenstein encourage a political quietism since he does not provide an external standpoint outside of language to criticize rules? (I think the last objection draws lines more firm than is compatible with Wittgenstein's comments on family resemblances.)
I also have two new books to look up, Gellner (1959) and Winch (1963). Gellner sounds like an unscholarly polemic that yet was influential in turning philosophy away from the linguistic philosophy of the later Wittgenstein, J. L. Austen, and Gilbert Ryle. Winch seems to argue those studying society must use the terms that members of a culture use, and with the same understanding. So perhaps this is a Wittgensteinian argument that social science is not possible, or at least must lower its aims. But I have not read it yet.
References- Ernest Gellner (1959). Words and Things: A Critical Account of Linguistic Philosophy and a Study in Ideology London: Gollancz.
- Gavin Kitching and Nigel Pleasants (editors) (2002). Marx and Wittgenstein: Knowledge, Morality and Politics, London: Routledge
- Peter Winch (1963). The Idea of a Social Science, London: Routledge and Kegan Paul.
Thursday, March 27, 2014
Analytical TOC For Athreya
I finally finished Kartik Athreya's book, Big Ideas in Economics: A Nontechnical View. I have already offered two comments on it. I do not expect it to be successful. Do not look here for a discussion of the theory of the second best, the aggregation of production functions, the distinction between risk and uncertainty, or the problems with microeconomics (despite its point being that macroeconomics, as the author understands it, is applied microeconomics). Athreya does select and address some theoretical objections, such as the Sonnenschein-Debreu-Mantel theorem, related difficulties with using a representative agent, and the folk theorem in game theory. I was disappointed not to see an informed discussion of the relationship of steady state models, such as the Solow growth model, to very short run models such as the Arrow-Debreu model. On the other hand, you will find a lot of rationalization of assumptions on the ground that they are needed (useful?) to get definite conclusions, independent of any discussion of whether or not models with those models work empirically.
Anyways, I read the book on my Kindle. I found it difficult to keep the thread. So I have prepared the following analytical table of contents for my own use, if I should reread sections. I think Athreya could have gone through a couple more edits, reconsidering this structure. For example, maybe the book would have been more understandable with shorter and more chapters.
- Acknowledgements
- I. Introduction
- I.1 Why do Macroeconomists Think What They Think and Do What They Do?
- I.2 Whom Do I Want to Reach?
- I.3 Some Key Features
- I.4 Pictures, Talk, and Homework
- 1. The Modern Macroeconomic Approach and the Arrow-Debreu-McKenzie Model
- 1.1 Introduction
- 1.2 What is a Macroeconomic Model?
- 1.2.1 Macroeconomics as Hyperorganized Narrative with Hard-Nosed Data and Logic Checks
- 1.2.1.1 Ensuring Internal Consistency
- 1.2.1.2 informed Criticism
- 1.3 How Do Macroeconomists Account for the Facts?
- 1.3.1 How Macroeconomists Argue with Each Other (or, How to Argue with a Macroeconomist, if You Must!)
- 1.3.1.1 Step 1: They Tell Each Other Who Is in Their Model Economy, and What Those Participants Want to Do: Household Preferences and Firm Profit Maximization
- 1.3.1.2 Step 2: They Tell Each Other What Their Model's Participants Have: Endowments and Technology
- 1.3.1.3 Step 3: They Tell Each Other How Model Participants Can Interact: Trading Arrangements
- 1.3.1.4 Step 4: They Tell Each Other How Participants Will Interact: Equilibrium as Prediction
- 1.3.1.5 It Takes a Model to Beat a Model
- 1.4 Macroeconomic "Equilibrium": What It Does and Does Not Imply
- 1.5 Payoffs from the Standard Macroeconomic Model Building Recipe
- 1.5.1 Making Logical Errors Easier to Spot
- 1.5.2 Disciplining Claims about Causal Relationships
- 1.5.3 Better Policy Analysis: Welfare Economics
- 1.5.4 Better Policy Analysis: The "Lucas Critique"
- 1.5.4.1. All Models Are Susceptible to the Lucas Critique, but Some More Than Others
- 1.5.5 Making the Tent Bigger
- 1.6 The Benchmark Macroeconomic Model: Arrow-Debreu-McKenzie
- 1.6.1 Understanding the Basic ADM Structure Is a Must
- 1.6.2 ADM Terminology
- 1.6.2.1 Households: Preferences and Endowments
- 1.6.2.2 Firms
- 1.6.2.3 Profit Maximization
- 1.6.2.4 Markets and Prices
- 1.6.2.5 Pareto Efficiency and the Core
- 1.6.2.6 Don't Misunderstand Pareto Efficiency
- 1.6.3 The ADM Model: An Example and a Picture
- 1.7 Concluding Remarks
- 2. Prices, Efficiency, and Macroeconomics
- 2.1 Introduction
- 2.2 A Fanciful Macroeconomic Trading Institution: The Walrasian Clearinghouse
- 2.3 Why Is This Trading Process Interesting?
- 2.3.1 The First Welfare Theorem
- 2.3.2 Why Are Walrasian Outcomes So "Coordinated"? Some Intuitions
- 2.3.3 The Incentival Role of Prices
- 2.3.4 The Informational Role of Prices
- 2.3.4.1 Prices as Aggregators of Information
- 2.3.4.2 Prices as Conveyers of Information
- 2.4 Walrasian Prices Will Exist
- 2.4.1 Time and Uncertainty
- 2.4.2 Convexity and Existence
- 2.5 Decentralized Outcomes and the First Welfare Theorem
- 2.5.1 Decentralized Trade Seems to Generate "Workable" Outcomes
- 2.5.2 Decentralized Trade Seems to Centralize (and Locate Ownership) Sensibly
- 2.5.3 "ADM Minus Some Markets" Seems Like a Useful Description of the Real World
- 2.5.3.1 Externalities as Missing Markets
- 2.6 Should the Real World Look Like One in Which Most Trading Is Run Via a WCH, and If So, Why? Theoretical Foundations for Walrasian Equilibria
- 2.6.1 The Axiomatic or "Cooperative Game Theory" Approach
- 2.6.1.1 The Equivalence Principle
- 2.6.2 The Noncooperative Approach
- 2.6.2.1 Nash Equilibrium: The Most Important Kind of Equilibrium in Social Science
- 2.6.2.2 Why Look at "Nash" Outcomes? Because "Not Nash" Means "Not Likely"
- 2.6.2.3 What If Interactions Are Repeated and Not Anonymous
- 2.6.2.4 When Should Households and Firms Take Prices as Given?
- 2.6.2.5 Market Games
- 2.6.2.6 Summary of the Noncooperative Approach
- 2.6.3 The Experimental Approach
- 2.6.3.1 Markets as Calculators
- 2.6.3.2 Experiments, the Invention of New Trading Institutions, and Mechanism Design
- 2.7 The ADM Model Does Not Require "Perfect Information" to Deliver Pareto-Optimal Outcomes; It Requires a Complete Set of Walrasian Prices
- 2.7.1 The Interpretation of Prices: What's at Stake?
- 2.8 Some Real-World Complications
- 2.8.1 Walrasian Prices Are Sufficient, but Not Necessary
- 2.8.2 Costless Enforcement
- 2.8.3 Market Power
- 2.8.4 Imperfect Monitoring
- 2.8.4.1 The Myerson-Satterthwaite Theorem
- 2.8.4.2 The Revelation Principle
- 2.8.4.3 Further Reading
- 2.9 The Observational Implications of the ADM Model
- 2.9.1 Sonnenschein-Mantel-Debreu...
- 2.9.2 ...and Boldrin-Montrucchio
- 2.9.2.1 Does It Mean That "Anything Will Happen"? No
- 2.10 A Macro-Hippocratic Moment
- 2.11 Concluding Remarks
- 3. Macroeconomists, Efficiency, and Inequality
- 3.1 Economists, Efficiency, and Inequality
- 3.1.1 Decentralized Trading and Inequality
- 3.1.2 Economists' Preoccupation with "Efficiency"
- 3.1.3 Deadweight Loss from Taxation
- 3.2 The Second Welfare Theorem
- 3.2.1 The Welfare Theorems Inspire a Form of Central Planning!
- 3.2.2 A General Lesson of the Second Welfare Theorem: Taxes Can Hurt
- 3.2.3 Caveat 1: What's an "Initial" Endowment, Anyway?
- 3.2.4 Caveat 2: Knowledge and the Limits to Lump-Sum Redistribution
- 3.2.5 Caveat 3: Lump-Sum Redistribution Might Require Surprising People
- 3.2.6 The Second Welfare Theorem Does Not Require More Assumptions than the First Welfare Theorem
- 3.3 What's Right with Non-Lump Sum Taxes? Or, Sometimes Lump-Sum Taxes Are Bad for "Insurance"
- 3.3.1 Jargon Digression" "Ex-Ante" and "Ex-Post" Pareto Efficiency
- 3.3.2 Back to Lump-Sum Taxes Being Bad for Insurance...
- 3.3.3 Why Shouldn't I Trade Ex-Ante Efficiency for Equity?
- 3.3.3.1 Why Efficiency Is Important
- 3.4 A General Approach to Thinking about Allocations and Trading Institutions: Mechanism Design
- 3.4.1 Limits on Mechanisms
- 3.4.1.1 Implementing Social Outcomes: Gibbard-Satterthwaite and the Importance of the "Solution Concept"
- 3.4.1.2 Why Do Macroeconomists Care about Mechanism Design, and Why Should Policymakers?
- 3.5 Concluding Remarks
- 4. Macroeconomic Shortcuts
- 4.1 Introduction
- 4.1.1 Our Four Sin: Aggregation, Rationality, Equilibrium, and Mathematics
- 4.2 Macroeconomic Compromises
- 4.2.1 Aggregation
- 4.2.1.1 Aggregation of Producers
- 4.2.1.2 Aggregation of Consumers
- 4.2.1.3 Aggregation of Commodities
- 4.2.1.4 Aggregation and Modeling Tradeoffs
- 4.2.1.5 An Example: The Breeden-Lucas "Fruit Tree"
- 4.2.2 Rationality
- 4.2.2.1 No Rationality, No Utility Function
- 4.2.2.2 Bounded Rationality
- 4.2.2.3 Rational Expectations
- 4.2.2.4 Expected Utility
- 4.2.2.5 A Provisional Summary
- 4.2.3 Equilibrium Analysis
- 4.2.3.1 Steady States and Transitions
- 4.2.3.2 An Interesting Criticism of Steady-State Analysis
- 4.2.3.3 Equilibrium Analysis: A Provisional Summary
- 4.2.3.4 Race as an Equilibrium Outcome: The Work of Glenn Loury
- 4.2.4 Mathematics, Practicality, and Some Examples
- 4.2.4.1 Mathematics and Forecasting
- 4.2.4.2 Mathematics as a Language to Protect the Public from Economists
- 4.2.4.3 Example: The Continuum Assumption
- 4.2.4.4 Example: Infinitely Lived Households
- 4.2.4.5 Example: "Social Planning Problems"
- 4.3 Concluding Remarks
- 5. Benchmark Macroeconomic Models
- 5.1 ADM and the Real World
- 5.2 Time, Uncertainty, and the ADM Model
- 5.2.1 The Long Arm Attached to the Invisible hand
- 5.2.1.1 The Impossibility of Literal Arrow-Debreu Market Completeness
- 5.3 The Radner Version of the ADM Economy
- 5.3.1 A Summary of Radner Trading
- 5.3.2 Spot Markets and IOU Markets: Radner and How Macroeconomists Think about Market Dysfunction
- 5.3.2.1 Spots Are OK
- 5.3.2.2 IOUs, Maybe Not So Much?
- 5.3.2.3 Radner and the Real World: A Brief Recap
- 5.4 Many Important Macroeconomic Models Are Mainly Versions of Radner Economies
- 5.5 Macroeconomic Policy: A Brief General Discussion
- 5.5.1 What Is a Policy?
- 5.5.2 Two Questions to Ask before "Doing Policy"
- 5.5.2.1 Question 1: How Are the Preconditions for the First Welfare Theorem Violated?
- 5.5.2.2 Question 2: Why Do You Think You Can Do Better?
- 5.5.2.3 One Reason to Think You Can Do Better: Coordination Failure
- 5.5.3 Coordination Failure and Macroeconomics
- 5.6 Important Macroeconomic Models and Policy Implications
- 5.7 The Mother of All Walrasian Macroeconomic Models: Neoclassical Growth Models
- 5.7.1 Step 1: The Malthusian Growth Model: No Capital
- 5.7.2 Step 2: The Solow Growth Model: No Fixed Inputs
- 5.7.2.1 Labor-Saving Devices
- 5.7.2.2 Balanced-Growth Steady States
- 5.7.2.3 The Role Savings Rates Play in Living Standards
- 5.7.2.4 The Solow Model as a First Unified Model of Growth and Fluctuations
- 5.7.3 Step 3: The Modern Neoclassical Growth Model: Enter the Consumer
- 5.7.4 What Happens When There Is Uncertainty? The Stochastic Neoclassical Growth Modek
- 5.7.4.1 Deterministic and Stochastic Steady States
- 5.7.5 What Payoffs Do Stochastic Neoclassical Growth Models Offer Us?
- 5.7.5.1 A Step Toward a Unified Theory of Growth and Fluctuations
- 5.7.5.2 They Operationalize the ADM Model
- 5.7.5.3 Stochastic Neoclassical Growth Provides a Benchmark
- 5.7.6 The Influence of Neoclassical Growth Models on How We Think about Some Key Macroeconomic Issues
- 5.7.6.1 Macroeconomics Can Be Stable
- 5.7.6.2 Technological Progress is the Gift Horse
- 5.7.6.3 The Lives of Indian and American Barbers
- 5.7.6.4 Higher Tax Rates Mean Lower Income Levels, but May Not Lower Long-Run Growth Rates
- 5.7.6.5 The ADM Model Is Silent on Innovation
- 5.8 How Do Macroeconomic Models Provide Quantitative Information? Calibration and Estimation
- 5.8.1 Calibration and Estimation: Taking a Model Very (Too?) Seriously
- 5.9 The SGM and Keynesian Macroeconomics
- 5.9.1 Keynesian Economics and the SGM I: Coordination Failures
- 5.9.2 Keynesian Economics and the SGM II: Sticky Prices
- 5.9.2.1 Is Monopolistic Competition a UFO?
- 5.9.2.2 Tensions, Tensions
- 5.10 Less-Than-Perfect Worlds: The Standard Search Model, the Standard Incomplete Markets Model, and the Overlapping Generations Model
- 5.10.1 Who Knew?
- 5.10.2 No Representative Agent: Heterogeneity Galore
- 5.10.2.1 Equilibrium Doesn't Mean "Good": Redux
- 5.11 The Reality of Decentralized-Decentralized Trade: The Search Model
- 5.11.1 Optimal Decisions and Stationary Equilibria
- 5.11.2 What Kinds of Questions Can We Address with Search Models?
- 5.11.3 Keynesian Economics and the Search Model
- 5.11.3.1 Search Is Not Really about Searching
- 5.11.3.2 Search Models and Voluntary versus Involuntary Unemployment
- 5.11.3.3 What, Exactly, Is Being Traded? Walrasian Economics and the Importance of Defining the "Commodity Space"
- 5.12 The Reality of Missing Markets: The Standard Incomplete-Market Model
- 5.12.1 The Income Fluctuation Problem (IFP): The Lynchpin of Modern Macroeconomics
- 5.12.1.1 SIM Models: "IFPs in GE"
- 5.12.1.2 Stationary Equilibria
- 5.12.1.3 SIM as a Macroeconomic Model of Bounded Rationality
- 5.12.1.4 What Search and IM Models Give Us (I): Insurance vs. Incentives: The First Quantitative Pass
- 5.12.1.5 What Search and IM Models Give Us (II): Competitive Theories of Inequality
- 5.12.1.6 What Search and IM Models Give Us (III): Maybe "Competition" Isn't All That Great?
- 5.12.1.7 How Incomplete Are Decentralize Trading Arrangements?
- 5.12.1.8 It's the IOU Markets
- 5.13 The Reality of Life and Death: The Overlapping-Generations Model
- 5.13.1 Economists Get Precise about Policy, Inequality, and Intergenerational Conflict
- 5.14 Concluding Remarks
- 6. Macroeconomic Theory and Recent Events
- 6.1 Introduction
- 6.2 The Financial Crisis of 2007-2008: What Are the Questions?
- 6.2.1 The Facts: A Crisis Reading List
- 6.2.2 Radner and Financial Intermediation
- 6.2.3 What (Good) Are Financial Markets, and How Does the ADM Model Influence How Macroeconomists View Them?
- 6.3 Models for Question 1: Why Did Asset Prices Rise So Much?
- 6.3.1 Demand and Supply
- 6.3.2 Principal-Agent Conflicts
- 6.3.3 Financial Markets and the Importance of Beliefs
- 6.3.4 Differences of Opinion
- 6.3.5 Bubble Detection
- 6.3.5.1 What "Efficient Financial Markets" Means (Hint: It Does Not Mean Pareto Efficiency)
- 6.3.5.2 The EMH and "Random Walks"
- 6.4 Models for Question 2: Why Did Initial Changes Get Amplified
- 6.4.1 Debt
- 6.4.2 Models of Banks and Bank Runs
- 6.5 Models for Question 3: Why Has the Recovery Been So Slow?
- 6.5.1 Labor and Asset Market Search Models
- 6.6 Macroeconomics and the Financial Crisis of 2007-2008 Implications for Policy
- 6.6.1 (Try to End) "Too Big to Fail"
- 6.6.2 Asset Prices and Policy
- 6.6.2.1 The Great Price Diagnosis Dilemma for PolicyMakers
- 6.6.3 Spillovers and Ronald Coase
- 6.6.4 Ronald Coase and Macroeconomics
- 6.6.5 Dynamic Games
- 6.6.5.1 Things "off the Equilibrium Path" Can Matter for Things on It
- 6.6.5.2 The Limited Commitment of Benevolent Policymakers: Time Inconsistency
- 6.6.5.3 Consumer and Sovereign Debt
- 6.6.5.4 Ex-Ante versus Ex-Post Efficiency...Again
- 6.7 Macroeconomics and the Financial Crisis of 2007-2008: Navel Gazing and a Response to Those Gazing at Our Navels
- 6.7.1 Does Modern Macroeconomics Favor Laissez-Faire?
- 6.7.2 Where Did We Fail?
- 6.7.3 Criticism of DSGE Models
- 6.7.4 Reforming Macroeconomics
- 6.7.5 Policy: Some Perspective and a Caution
- 6.7.5.1 Global Policy Coordination
- 6.7.5.2 A Caution
- 6.8 What Should Macroeconomists Be Doing?
- Notes
- References
- Index
Friday, March 14, 2014
Philip Mirowski And Adolph Reed, Jr.: Separated At Birth?
I want to highlight the similarity in conclusions in Mirowski's recent book and Reed's controversial essay (see references below). Their understanding of the current conjuncture is fairly dispiriting. The right is winning in mass consciousness, despite their ideas being incoherent and vicious from an intellectual perspective. And their ideas extend over the entirety of the political spectrum, at least if one restricts oneself to what is seen to be practical. Arguments over how to make existing markets work better or to address current problems by constructing new markets, for example, accept the inevitability of capitalism.
Both Mirowski and Reed have something to say about what must be done by the left now. What is needed is a collective development of a leftist alternative. Those developing such an alternative need to be part of a group, like the Mont Pelerin Society was for the development of neoliberalism. And those developing this alternative, at least in their role in such a group, should not be overly concerned with the vagaries of this or that election in this or that country. This is a long term project, which, if successful, will spawn other groups over decades more concerned with implementation in specific times and places.
Are these authors correct in arguing the left does not currently have an inspiring vision to put before the public? You can talk about social democracy, but is that a way forward now? Are there powerful institutionalized groups working to improve our societies based on an architectonic view of what is possible? It seems to me more of a rearguard movement in advanced industrialized countries. And what about further left? I am aware of various statements of ideals - for example, Davidson and Davidson (1996), Rorty (1999)- but, without being built upon by a movement, these seem kind of idiosyncratic and quixotic to me.
An aside: If Mirowski is going to read literature produced by well-known writers who taught at Syracuse University, I wish he would mix some Raymond Carver in with the David Foster Wallace he has been reading.
References- Greg Davidson and Paul Davidson (1996). Economics for a Civilized Society, M. E. Sharp. [I HAVEN'T READ THIS]
- Philip Mirowski (2013). Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown, Verso.
- Adolph Reed Jr. (2014). Nothing Left: The Long, Slow Surrender of American Liberals, Harper's (March).
- Richard Rorty (1999). Achieving Our Country: Leftist Thought in Twentieth-Century America.
Saturday, March 01, 2014
Athreya Untrustworthy On History Of Thought
I continue to read Kartik Athreya's supposedly popular account of contemporary macroeconomics. Today I focus on the misleading presentation of the theory of economic growth.
Athreya presents the Solow-Swan Neoclassical Growth Model (NGM) as a contrast to Malthus' model of economic growth. He briefly alludes to Real Business Cycle (RBC) theory as the result of appending random shocks to the Solow-Swan model. He then goes on to discuss what he calls the Ramsey-Cass-Koopmans model. There are two problems here. (I bracket off the grouping of the Ramsey model of a central planning authority determining an optimal savings rate with models of household savings decisions.)
First, Solow developed his model in the context of many other economists also developing growth models. This setting is totally missing from Athreya's book. Neither "Harrod" nor "Domar" appear anywhere in the book. Yet Solow's work was a neoclassical response to the Harrod-Domar model. The Post Keynesian approach to steady-state growth, associated with such economists as Richard Kahn, Nicholas Kaldor, and Joan Robinson provided an alternative at the time. (I might also mention Michal Kalecki and Frank Hahn's doctoral thesis, if I recall correctly.) Maybe this approach is missing because Athreya is not aware of its existence.
Second, Athreya does not even get classical growth theory correct, as presented by Malthus or others. According to Athreya, Malthus' theory abstracts from the existence of capital. I guess income is supposedly distributed only in the form of wages and rents. Athreya then claims to consider the effects of a technological innovation, namely, the introduction of a vaccine in Malthus' theory. Supposedly, the effect is to lower the death rate, while leaving birth rates unchanged. That is, population increases. Since the quantity of land is fixed, the theory exhibts diminishing marginal returns to labor. So Athreya misrepresents Malthus as claiming that improved technology, while increasing total output, ultimately leads to lower average income per worker.
In the classical theory of value, the natural wage is given by habit and custom. Malthus, building on his predecessors, argued that transitory wages higher than the natural wage might lead to changes in habits, through what we now might call hysteresis. This effect would be to increase the natural rate of wages. At any rate, population was expected to increase when wages exceeded the natural wage. But, maybe, the classical economists emphasized more reactions to opportunities for jobs than reactions to wages. They accepted that unemployment could be persistent and expected lower and higher periods of unemployment to encourage increases and decreases of the rate of growth of population. Anyways, Athreya is right, at least, about the response to increased productivity being an initial increase in the population of workers.
But he is mistaken about the ultimate effect. Suppose the market wage falls below the natural wage, in a period in which the accumulation of capital has declined. Then the classical economists, such as Malthus, expected the rate of increase in population to fall. Emigration would increase, birth rates would fall, and workers would form families later in their lives. (It is unclear to me how the classical economists envisioned such mechanisms to kick in fast enough for their theories. At any rate, I can quote Ricardo suggesting that the stationary state was far away.) The ultimate effect of declining population would be for workers to obtain their natural wage, with the level of employment and distribution between wages, profits, and rent being consistent with technological possibilities after a change. That is, the ultimate effect, in Malthus' theory, of an improvement is not lower real wages. (I am here bracketing out any consideration of whether Malthus presented a stylized theory consistent with the empirical experience in the centuries prior to his time or overlooked the effects of the ongoing industrial revolution.)I cannot recommend Athreya's book, either for the general reader curious about macroeconomics or for the advanced undergraduate or beginning graduate student. It is too misleading. The above is only one of many examples. I suppose some professional economists might find it of interest to catalog the misconceptions, mistakes, inconsistencies, tendentious statements, and occasional insights.
Update: I want to recall the comments of David Glasner, John Quiggin, Noah Smith, and Stephen Williamson.
References- Kartik B, Athreya (2014). Big Ideas in Macroeconomics: A Nontechnical View, MIT Press.
- Nicholas Kaldor (1956). Alternative Theories of Distribution, Review of Economic Studies, V. XXIII: pp. 83-100.
- Antonella Stirati (1994). The Theory of Wages in Classical Economics: A Study of Adam Smith, David Ricardo and their Contemporaries, Edward Elgar,
Wednesday, February 26, 2014
Post Keynesianism Contrasted With Neoclassical Economics
The following is reproduced from "An Essay on Post-Keynesian Theory: A New Paradigm in Economics", Al Eichner and Jan Kregel's 1975 Journal of Economic Literature article. Of course, the table being a summary, all entries are highly stylized.
Aspect | Post Keynesian Theory | Neoclassical Theory |
Dynamic properties | Assumes pronounced cyclical pattern on top of a clearly discernible growth path | Either no growth, or steady-state expansion with market mechanisms assumed to preclude any but a temporary deviation from that growth path |
Explanation of how income is distributed | Institutional factors determine a historical division of income between residual and non-residual shareholders, with changes in that distribution depending on changes in the growth rate | The distribution of income explained solely by variable factor inputs and the marginal productivity of those variable factor inputs |
Amount of information assumed to be available | Only the past is known, the future is uncertain | Complete foresight exists as to all possible events |
Conditions that must be met before the analysis is considered complete | Discretionary income must be equal to discretionary expenditures | All markets cleared with supply equal to demand in each of those markets |
Microeconomic base | Imperfect markets with significant monopolistic elements | Perfect markets with all micro units operating as price takers |
Purpose of the theory | To explain the real world as observed empirically | To demonstrate the social optimality if the real world were to resemble the model |