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 model^{1}. 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 counterintuitive behavior, including bifurcations of steady states and interesting nonsteady 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 ModelThe 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 HouseholdsEvery year, one household is born. Households live two years. During the first year, they supply one personyear of labor, and they are paid their wages at the end of the year. At the end of the first year, they consume some of their wages and save the rest. They are retired and do not labor^{2} 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 it^{3}.
 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 cashinadvance constraint^{4}.
Formally, the household born at the start of year t must choose decision variables to solve the following nonlinear program:
Maximize u(c_{t,t}, c_{t,t + 1})
such that:
c_{t,t} + s_{t} ≤ w_{t}
c_{t,t + 1} ≤ Q_{ξ}(R_{t}) s_{t}
c_{t,t + 1} ≤ ξ m_{t} p_{t}/p_{t + 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, Q_{ξ}(R_{t}). 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 cashinadvance condition. The household must hold at least some given fraction (namely, 1/ξ) of the consumption planned at the end of the last period in the form of money during this period^{5}, 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 ExampleTo be a bit more concrete, Hahn and Solow gives two examples of possible forms of the utility function. The first is:
u(c_{t,t}, c_{t,t + 1}) = (1/α)(c_{t,t})^{α} + (1/α)(c_{t,t + 1})^{α}
where,
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 CobbDouglas Production Function
The firms are characterized by an aggregate production function^{6}. To be concrete, they specify a CobbDouglas form:
y_{t} = (k_{t  1})^{β} (l_{t})^{β + 1}
where:
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 profitmaximizing analysis. The demand for labor is constrained to match the households' supply of one personyear per year. That is, with flexible wages and prices, the labor market is assumed to clear.
3.0 Stationary StatesBy 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 solution^{7}.
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 upwardsloping 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.
Hahn and Solow Possibilities  Example 1 Possibilities 


I was hoping to find a model with multiple equilbria for some subset of the parameter space. Perhaps I have made some simple error in algebra, but I was disappointed to not find such. This post does not say that Hahn and Solow are in error. They do not claim multiple equilibrium can arise for every conventional form of the utility function in their problem. I guess I'll have to focus on their second example^{8}.
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 upwardsloping in both cases, unlike in Hahn and Solow's diagram for the case of three equilibria. (I do not see offhand why Hahn and Solow rule out a case of in which no PIS exists, but two LCS do.)
Footnotes 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 gametheoretic analysis of social norms, and with imperfect competition in product markets.
 In other models of overlapping generations, how much labor a household supplies each year is a decision variable.
 In a stationary state, prices are stationary and money earns a real return of unity.
 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.
 Hahn and Solow suggest this unrealistic approach to the transactions demand for money can be justified by a deeper analysis.
 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.
 This definition only works for homothetic utility functions, another unrealistic assumption justified here by the critical intent of the model.
 I like that their second household has a parameter for timediscounting for households, anyways.
 Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press
2 comments:
Do you have the source code for your program? Even a gist will do :)
So far, I have analytical results. That is, this is a matter of mathematical proof, not computer programming.
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