
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
illbehaved 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stories told in outdated and exploded neoclassical textbooks.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 UtilityMaximizing 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 CobbDouglas utility function:
U(c_{0}, c_{1}) = (c_{0})^{γ}(c_{1})^{(1  γ)}
where c_{0} is the bushels corn the agent consumes
at the end of the first year of their life, c_{1} 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 utilitymaximization
problem:
Given w, r
Choose c_{0}, c_{1}
To Maximize U(c_{0}, c_{1})
Such that c_{0}(1 + r) + c_{1} = w(1 + r)
c_{i} ≥ 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:
c_{0} = γ w
c_{1} = (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 personyears 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/a_{0}, K/a_{1})
where:
a_{0} > 0
0 < a_{1} < (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 a_{1}/a_{0} bushels
to use as capital next year and
(1  a_{1})/a_{0} corn to
consume, per personyear employed.
Given this technology, the wagerate of profits frontier
is easily expressed:
a_{1}(1 + r) + a_{0}w = 1
Hence, one can solve for the wage as a function of the
interest rate and the coefficients of production:
w = [(1  a_{1})/a_{0}]  (a_{1}/a_{0})r
When the interest rate is zero, the wage is (1  a_{1})/a_{0}, 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  a_{1})/a_{1}.
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 = a_{1}/a_{0}
I might as well make an aside on marginal productivity.
In models in which the firms choose the costminimizing
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 fixedcoefficient
techniques, the value of marginal product, as I understand
it, is an interval in which lefthand and righthand
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 closedform expression for the interest
rate in a stationary state:
r = [(1  a_{1})/a_{1}]  [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.