2.1 Introduction to multivariate calculus

2.1 Introduction to multivariate calculus

Economic theory consists of models designed to improve our understanding of economic phenomena. Many of these models have the following structure: each member of a set of economic agents optimizesgiven some constraints and, given the optimal actions, variables adjust to reach some sort of equilibrium.

Consider, for example, a model in which the agents are profit-maximizing firms. Suppose that there is a single input that costs w per unit, and that a firm transforms input into output using a (differentiable) production function  f  and sells the output for the price p. A firm's profit when it uses the amount x of the input is then

p f (x) − wx.

As you know, if the optimal amount of the input is positive then it satisfies the "first-order condition"

p f '(x) − w = 0.

Further, under some conditions on  f  this equation has a single solution and this solution maximizes the firm's profit. Suppose these conditions are satisfied. For any pair (w, p), denote the solution of the equation by z(w, p). Then the condition

p f '(z(w, p)) − w = 0 for all (w, p)

defines z(w, p) implicitly as a function of w and p.

What can we say about the function z? Is it increasing or decreasing in w and p? How does the firm's maximized profit change as w and p change?

If we knew the exact form of  f  we could answer these questions by solving for z(w, p) explicitly and using simple calculus. But in economic theory we don't generally want to assume that functions take specific forms. We want our theory to apply to a broad range of situations, and thus want to obtain results that do not depend on a specific functional form. We might assume that the function  f  has some "sensible" properties---for example, that it is increasing---but we would like to impose as few conditions as possible. In these circumstances, in order to answer the questions about the dependence of the firm's behavior on w and p we need to find the derivatives of the implicitly-defined function z. Before we do so, we need to study the chain rule and derivatives of functions defined implicitly, the next two topics. (If we are interested only in the rate of change of the firm's maximal profit with respect to w and p, not in the behavior of its optimal input choice, then the envelope theorem, studied in a later section, is useful.)

Having studied the behavior of a single firm we may wish to build a model of an economy containing many firms and consumers that determines the prices of goods. In a "competitive" model, for example, the prices are determined by the equality of demand and supply for each good---that is, by a system of equations. In many other models, an equilibrium is the solution of a system of equations. To study the properties of such an equilibrium, another mathematical technique is useful.

I illustrate this technique with an example from macroeconomic theory. A simple macroeconomic model consists of the four equations

Y	=	C + I + G
C	=	f (Y − T)
I	=	h(r)
r	=	m(M)

where Y is national income, C is consumption, I is investment, T is total taxes, G is government spending, r is the rate of interest, and M is the money supply. We assume that M, T, and G are "parameters" determined outside the system (by a government, perhaps) and that the equilibrium values of Y, C, I and r satisfy the four equations, given M, T, and G.

We would like to impose as few conditions as possible on the functions  f , h, and m. We certainly don't want to assume specific functional forms, and thus cannot solve for an equilibrium explicitly. In these circumstances, how can we study how the equilibrium is affected by changes in the parameters? We may use the tool of differentials, another topic in this section.