Wednesday, September 23, 2015

For Technical Discussions Of Cavalry Tactics At The Battle Of Austerlitz?

Figure 1: Steady States As Function Of Effective Return On Savings

1.0 Introduction

I have previously said I am not thrilled about arguments about whether or not assumptions are realistic. In this post, I describe some analysis 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:

"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
2.0 Generalization of Hahn and Solow's Model of Overlapping Generations

I have previously outlined 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.

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:

g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]

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 g.

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 t, ht, grow at the rate G:

ht = Gt,


G ≥ 1.

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:

g(Q) = G [ξ/(ξ - 1)] [β/(1 - β)]

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.

3.0 Other Generalizations

Here are some other possible generalizations and explorations one might make to the model:

  • Household lives more than two years.
  • Endogenous supply of labor, with leisure entering the utility function.
  • Introduction of a bequest motive.
  • Heterogeneous households.
  • Non-homothetic preferences.
  • Various specific forms of utility functions.
  • Multiple sectors in production, instead of the production of a single good.
  • Introduction of fixed capital (with radioactive depreciation), instead of only circulating capital.
  • Various specific forms of production functions.
  • Introduction of stochastic noise.
  • Analysis of reactions to different kind of shocks.
  • Introduction of government, foreign trade.
  • More detailed analysis of money, finance, and banks.

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.

4.0 Empirical Applicability of Generalized Model?

This program entails lots of work, some of it empirical. How could an outsider have standing to criticize this approach?

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 here 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 bifurcation diagram for some models, I will consider myself to be approaching advanced mathematics.) Some conventional concepts from economics (marginal conditions, excess demand functions, Walras' law, steady states) help organize the approach.

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.

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?

4.1 How Are Dynamic Equilibrium Paths Found?

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?

4.2 No Keynesian Uncertainty

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.

5.0 Conclusions

I intend to pursue some generalizations suggested above. (I could be distracted by trying to develop a bifurcation diagram 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?

Monday, September 14, 2015

Paul Krugman Stumbles

In his editorial in the New York Times 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:

"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

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, "The Second Coming". I have in mind the following lines:

"The best lack all conviction, while the worst
Are full of passionate intensity." -- W. B. Yeats

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.

I do like Krugman's previous allusions to Talking Heads lyrics.

Thursday, September 03, 2015

Failure To Replicate Hahn And Solow (1995), Figure 2.1

Figure 1: Stationary States As Function Of Effective Return On Savings

1.0 Introduction

In Chapter 2 of their Critical Essay, Frank Hahn and Robert Solow present an overlapping generations model1. 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.

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.

I have looked at other models of overlapping generations 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.

2.0 Overlapping Generations Model

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.

2.1 Households

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 labor2 during their second year. At the end of the second year, they consume all of their savings, and then die.

Households can save their income in the form of two assets:

  • Money, which earns a real return only if prices decline while a household holds it3.
  • Corporate bonds, which at the end of each year are paid off with the full (accounting) profits earned by firms.

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 constraint4.

Formally, the household born at the start of year t must choose decision variables to solve the following non-linear program:

Maximize u(ct,t, ct,t + 1)

such that:

ct,t + stwt
ct,t + 1Qξ(Rt) st
ct,t + 1 ≤ ξ mt pt/pt + 1

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, Qξ(Rt). The notation for the effective rate of return reflects the dependence of that rate on the real rate of return, R, 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 period5, where

ξ > 1

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.

2.1.1 Hahn and Solow's First Example

To be a bit more concrete, Hahn and Solow gives two examples of possible forms of the utility function. The first is:

u(ct,t, ct,t + 1) = (1/α)(ct,t)α + (1/α)(ct,t + 1)α


α < 1

Sometimes it is more convenient to express the solution of the household's program in terms of the parameter ε:

ε = α/(α - 1)
2.2 An Aggregate Cobb-Douglas Production Function

The firms are characterized by an aggregate production function6. To be concrete, they specify a Cobb-Douglas form:

yt = (kt - 1)β (lt)β + 1


0 < β < 1

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.

3.0 Stationary States

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.

Stationary states are conveniently expressed in terms of the following function.

g(Q) = Q s(Q)

where s(Q) is the stationary state savings found by solving the household's constrained maximization problem and substituting in a wage of unity in the solution7.

Exactly one real rate of return, R, corresponds to each each stationary state value of Q, 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:

g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]

Figure 1 graphs g(Q) 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 Q = 1.

Anyways, Hahn and Solow present a figure showing possible shapes and locations of g(Q). 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.

Table 1: Number of Stationary States
Hahn and Solow
Example 1
  • None.
  • No PIS, Exactly one LCS.
  • Exactly one PIS, No LCS.
  • Exactly one PIS, two LCS.
  • None.
  • No PIS, Exactly one LCS.
  • Exactly one PIS, No LCS.

4.0 Conclusion

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 example8.

Update (10 September 2015): 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 g(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.)

  1. 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.
  2. In other models of overlapping generations, how much labor a household supplies each year is a decision variable.
  3. In a stationary state, prices are stationary and money earns a real return of unity.
  4. 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.
  5. Hahn and Solow suggest this unrealistic approach to the transactions demand for money can be justified by a deeper analysis.
  6. 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.
  7. This definition only works for homothetic utility functions, another unrealistic assumption justified here by the critical intent of the model.
  8. I like that their second household has a parameter for time-discounting for households, anyways.
  • Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press