Sunday, October 07, 2012

A Simple, But Perverse, Neoclassical Model

 Figure 1: Less Plentiful Supply of Capital Lowers the Interest Rate
1.0 Introduction

I claim that capital reversing can be a source of instability and interesting dynamics in neoclassical models. I am interested in, for example, the convergence or not of equilibrium paths in models of intertemporal and temporary equilibrium to steady states, but not in tâtonnement dynamics. The ill-behaved nature of many neoclassical models is a challenge in demonstrating this claim.

This post is a start on revisiting these issues. I here outline a simple model of overlapping generations with a simple production model that cannot exhibit reswitching, capital reversing, or even price Wicksell effects. Yet, in this model, a greater willingness among the households to save is associated with a higher interest rate. This is inconsistent with the supposedly intuitive stories told in outdated and exploded neoclassical textbooks.

2.0 The Model

The model describes an economy in which a single commodity, corn, is produced. In this model, corn functions as both the consumption good and as the only capital good. In production, all (seed) corn is used up in producing the harvest; that is, all capital is circulating capital. For my purposes in this post, I want to consider an economy in a stationary state.

The point of these assumptions is not to describe any actually existing capitalist economy. Rather, the point is to demonstrate that neoclassical theory does not justify conclusions commonly made. I suppose you can say that these types of models raise the following empirical question: why do mainstream economists continue to teach, both in the classroom and in policy work, conclusions long exposed as nonsense by their own theory?

2.1 Utility-Maximizing Agents

Suppose the population consists of overlapping generations, as in Figure 2. Each generation lives for two years. In a given year, all members of the generation born at the start of that year work a full year. They are paid their wages at the end of the year. Out of their wages, they consume some and they save the remainder at the going interest rate. They are retired during the second year of their life. At the end of their second year, they consume the remainder of their income and die.

 Figure 2: Lifespans of Overlapping Generations

Furthermore, assume that each generation consists of a single individual, also known as an agent. Furthermore, suppose all generations are identically characterized by the following Cobb-Douglas utility function:

U(c0, c1) = (c0)γ(c1)(1 - γ)

where c0 is the bushels corn the agent consumes at the end of the first year of their life, c1 is the bushels corn consumed at the end of the second year, and

0 < γ < 1

A higher value of γ indicates a lesser willingness to defer consumption and a smaller supply of savings. Let w be the wage, and r the interest rate. Under these assumptions, the agent born in each generation solves the following utility-maximization problem:

Given w, r
Choose c0, c1
To Maximize U(c0, c1)
Such that c0(1 + r) + c1 = w(1 + r)
ci ≥ 0; i = 0, 1.

The constraint states that the total value of consumption, evaluated at a single point in time, equals the income of the agent, also evaluated at the same point in time. The solution to this mathematical programming problem is:

c0 = γ w
c1 = (1 - γ) w(1 + r)
S = (1 - γ) w

where S is the bushels corn saved at the end of each year.

2.2 Production

For simplicity, I assume a Leontief, fixed coefficients production function. Let L be the person-years of labor employed during the year, K be the bushels corn used as capital during the year, and q be the bushels corn produced during the year. The production function is:

q = min( L/a0, K/a1)

where:

a0 > 0
0 < a1 < (1/2)

(Productivity has to exceed a certain threshold for an equilibrium to exist in this model.)

Only consider cases where both constraints bind. In a stationary state, the corn available at the end of the year is divided up into a1/a0 bushels to use as capital next year and (1 - a1)/a0 corn to consume, per person-year employed.

Given this technology, the wage-rate of profits frontier is easily expressed:

a1(1 + r) + a0w = 1

Hence, one can solve for the wage as a function of the interest rate and the coefficients of production:

w = [(1 - a1)/a0] - (a1/a0)r

When the interest rate is zero, the wage is (1 - a1)/a0, that is, the total surplus of corn, after subtracting the seed corn needed to sustain production at the same level. When the wage is zero, the interest rate is (1 - a1)/a1.

2.3 Equilibrium

This model is completed by assuming that the households want to hold the capital stock at the end of every year. since only one generation is saving for retirement at the end of this year, this equilibrium condition is:

S = a1/a0

I might as well make an aside on marginal productivity. In models in which the firms choose the cost-minimizing technique, marginal productivity conditions are used to specify the coefficients of production. The price of each commodity used as a capital good is equal, in equilibrium, to the present value of the marginal product of that commodity. In models in which the technology is specified as a set of fixed-coefficient techniques, the value of marginal product, as I understand it, is an interval in which left-hand and right-hand derivatives enter. In any case, since prices and the quantities of capital goods are both found by solving the model, one cannot say that the (rental) price of a capital good is determined by its marginal product. Furthermore, wages are not determined by the marginal product of labor. A fortiori, the rate of profits is not determined by the marginal product of finance capital, even if one can concoct some equation involving the return on capital, some measure of the value of capital goods, and its marginal product.

Anyways, one can solve the above model to find the following closed-form expression for the interest rate in a stationary state:

r = [(1 - a1)/a1] - [1/(1 - γ)]

Figure 1 above graphs this function. And one can see that, in this model, a stationary state in which households are less willing to save is associated with a lower interest rate. If the interest rate were the price of capital and prices were indices of relative scarcities, this example could not be created. But equilibrium prices are not scarcity indices and neoclassical economics, as taught by most university professors, is nonsensical poppycock.

3.0 Conclusion

This post has presented a simple neoclassical model, a limit point, in some sense, of the kind of model that neoclassical economists advocated as a resolution of the Cambridge Capital Controversies. And this simple model shows that much of mainstream teaching and policy work is theoretically unfounded, by their own logic.

Anonymous said...

Professors teaching economics usually start with assumptions that explicitly rule out production function assumed here. You have shown that if we use different assumptions (like Leontieff instead of Cobb-Douglas), we may get different results. And the point is?

Besides, it's already well known that OLG models may lead to all kinds of pathologies (multiple equilibria, failure of welfare theorems, bubbles, etc.), see e.g. New Palgrave entry by Geanakoplos. For example, in your model there exists another (more efficient) equilibrium with zero interest rate, independent of parameters.

Robert Vienneau said...

Me: "The ill-behaved nature of many neoclassical models is a challenge".

You: "Besides, it's already well known that OLG models may lead to all kinds of pathologies (multiple equilibria, failure of welfare theorems, bubbles, etc.)"

Thank you for the agreement.

Anony2 said...

Robert,

Do you believe that a capital reversal model is more realistic than the cobb douglas model?

Note Pol Antras at Harvard is not too keen on the play cobb-douglas gets: http://www.economics.harvard.edu/faculty/antras/papers/CESPublished.pdf

Anonymous said...

Robert Vienneau: You are in agreement with him that if you use assumptions that economics scholars generally rule out, you can tweak a susceptible model to produce whatever results you want?

I agree, that if you insist on thinking in the past before things like Cobb-Douglas production functions (which are the true workhorse "neoclassical" functions) you can believe whatever cockamamie results you get.

It's nothing new nor impressive.