Mathematical methods for economic theory

Martin J. Osborne
Previous section Previous page Contents Text Exercises Search Next page Next section

8.5 Differential equations: phase diagrams for autonomous equations

We are often interested not in the exact form of the solution of a differential equation, but only in the qualitative properties of this solution. In economics, in fact, the differential equations that arise usually contain functions whose forms are not specified explicitly, so there is no question of finding explicit solutions. One way of studying the qualitative properties of the solutions of a differential equation is to construct a “phase diagram”. I discuss this technique for the class of “autonomous” first-order differential equations.

Autonomous equations

An autonomous first-order ordinary differential equation is a first-order ordinary differential equation in which the value of x'(t) depends only on x(t), not independently on the value of t.
Definition
An autonomous first-order ordinary differential equation is a first-order ordinary differential equation that may be written in the form
x'(t) = F(x(t))
for a function F of a single variable.
An equilibrium of such an equation is a value of x for which F(x) = 0 (because if F(x) = 0 then x'(t) = 0, so that the value of x does not change).

A phase diagram indicates the sign of x'(t) for a representative collection of values of x. To construct such a diagram, plot the function F, which gives the value of x'. For values of x at which the graph of F is above the x-axis we have x'(t) > 0, so that x is increasing; for values of x at which the graph is below the x-axis we have x'(t) < 0, so that x is decreasing. A value of x for which F(x) = 0 is an equilibrium state.

If x* is an equilibrium and F'(x*) < 0, then if x is slightly less than x* it increases, whereas if x is slightly greater than x* it decreases. If, on the other hand, F'(x*) > 0, then if x is slightly less than x* it decreases, moving further from the equilibrium, and if x is slightly greater than x* it increases, again moving away from the equilibrium. That is, we have the following result.

Proposition 8.5.1
For an autonomous first-order ordinary differential equation x'(t) = F(x(t)), x* is an equilibrium if and only if F(x*) = 0. An equilibrium x* is locally stable if F'(x*) < 0 and unstable if F'(x*) > 0.
The example in the following figure has three equilibria, a, b, and c. The arrows on the x-axis indicate the direction in which x is changing (given by the sign of x'(t)) for each possible value of x.

a b c x → x'

We see that the equilibrium b is locally stable, whereas the equilibria a and c are unstable.

Example 8.5.1 (Solow's model of economic growth)
The following model generalizes the one in an earlier example. Suppose that the production function is an increasing and strictly concave function G that is homogeneous of degree 1 (i.e. has “constant returns to scale”), rather than taking the specific form assumed in the earlier example. Assume also that G(0, L) = 0 for all L (no output can be produced without any capital).

We now have

K'(t) = sG(K(t), L(t)).
As before, the labor force grows at the constant rate λ, so that
L'(t)/L(t) = λ.
We may study the behavior of the capital-labor ratio K(t)/L(t) as follows. Let k = K/L and define the function f of a single variable by
f(k) = G(k, 1) for all k.
Since G is increasing and concave we have f' > 0 and f" < 0, and since it is homogeneous of degree 1 we have
G(K, L) = LG(K/L, 1) = Lf(k).
Thus
K'(t) = sL(t)f(k(t)).
Now,
k'(t)  =  [K'(t)L(t) − K(t)L'(t)]/(L(t))2
 =  [K'(t) − k(t)L'(t)]/L(t)
so
k'(t) = sf(k(t)) − λk(t).
Assuming that sf'(0) > λ and sf'(k) < λ for large values of k, so that the function sf(k) lies above the function λk for small values of k and below it for large values of k, the phase diagram of this equation is shown in the following figure.

λk k* k → sf(k) sf(k) − λk

We see that under these assumptions the equation has two equilibria, k* = 0, which is unstable, and the value of k* for which sf(k*) = λk*, which is locally stable.

The stable equilibrium value of k depends on s: the equation

sf(k*) = λk*
implicitly defines k* as a function of s. Which value of s maximizes the equilibrium value of per capita consumption? Per capita consumption is equal to (1 − s)G(K, L)/L, or, given our previous calculations, (1 − s)f(k). Thus in an equilibrium per capita consumption is equal to (1 − s)f(k*(s)). A stationary point s* of this function satisfies
(1 − s*)f'(k*(s*))k*'(s*) = f(k*(s*)).
Now, differentiating the equation defining k* we obtain
f(k*(s)) + sf'(k*(s))k*'(s) = λk*'(s) for all s.
Using this equation for s = s* reduces the first-order condition for the savings ratio s* that maximizes per capita consumption to
f'(k*(s*)) = λ.
The homogeneity of degree 1 of the production function G implies that that the derivatives of G are homogeneous of degree zero, which in turn implies that the left-hand side of this equation is equal to the derivative of the production function G with respect to capital at the optimal amounts of capital and labor. In economic jargon, that's the marginal product of capital. Thus the result says that the marginal product of capital at the equilibrium capital-labor ratio for the optimal savings rate is equal to the rate of population growth.