EXAMPLES OF COMPARATIVE STATICS

To illustrate the preceding principles, let us consider three alternative hypotheses about the behavior of firms. Specifically, suppose we were to postulate that:

1.95 Firms maximize profits n, where ix equals total revenue minus

cost.

1.96 Firms maximize some utility function of profits U(n), where

U'(ji)> 0, so that

higher profits mean higher utility. Thus, profits are desired not for their own sake,

but rather for the utility they provide the firm owner.

1.97 Firms maximize total sales, i.e., total revenue only.

By what means shall these three theories be tested and compared? It is not possible to test theories by introspection. Contemplating whether these postulates sound to us like "reasonable" behavior is not an empirically reliable test. Also, asking firm owners if they behave in these particular ways is similarly unreliable. The only way to test such postulates is to derive from them potentially refutable hypotheses and ultimately to see if actual firms conform to the predictions of the theory.

What sorts of refutable hypotheses emerge from these behavioral assertions? Among the logical implications of profit maximization is the refutable hypothesis that if a per-unit tax is applied to a firm's output, the amount of goods offered for sale will decrease. This hypothesis is refutable because the reverse can be true. We therefore begin our first example by asserting that firms maximize profits in order to derive this implication.

Example 1. Let

R(x) = total revenue function (depending on output*) C(x) = total cost function

tx = total tax revenue collected, where the per-unit tax rate t

is a parameter determined by forces beyond the firm's control

If the firm sells its output in a perfectly competitive market, i.e., it is a. price taker, then

R(x) = px

where p is the parametrically determined market price of x. If the firm is not a perfect competitor, then p is determined, along with x, via the demand curve, and revenue is simply some function of output, R(x).

In the general case, the tax rate t represents the only parameter, or test condition, of the model. The first model thus becomes

maximize

3)

By simple calculus, the first-order condition for a maximum is

R'(x) - C'(x) -t = 0 (1-

4)

COMPARATIVE STATICS AND THE PARADIGM OF ECONOMICS 17 For a maximum, the sufficient second-order condition is

R" - C" < 0 (1-

5)

Condition (1 -4) is the choice function for this firm in implicit form. It states that the firm will choose that level of output such that marginal revenue (MR) equals marginal cost (MC) plus the tax (t). If the firm is a perfect competitor, then R'(x) — p, and R"(x) = 0. Equations (1-4) and (1-5) then become, respectively,

p-C'(x)-t = 0 (1-

4')

-C"(x) < 0 (1-

5')

We shall pursue the model from the standpoint of a firm with an unspecified revenue function R(x). Application of the model to the perfectly competitive case will be left as a problem for the student.

Equation (1-4) is a well-known application of "marginal" reasoning. Equation (1-4) states that a firm will produce at a level such that the incremental (marginal) gain in revenues is exactly offset by the incremental cost (including, of course, the tax). This condition, however, does not guarantee a maximum of profits. It is also perfectly consistent with minimizing profits with the same cost and revenue functions, since the same first-order conditions are implied. What we mean to express is that as long as marginal receipts exceed marginal cost, the firm will produce at a higher rate, and if marginal receipts are less than marginal costs, the output will be reduced. This idea is given a precise statement by Eq. (1-5), which says that receipts are increasing at a slower rate than costs. Or, in terms of the marginal revenue and marginal cost curves, Eq. (1-5) says that the marginal cost curve cuts the marginal revenue curve from below.

Notice that we do not assert that the "optimum" output for a firm is where marginal revenue equals marginal cost; this is a value judgment, not a statement about behavior. Likewise, Eq. (1- 4) does not represent what this firm does in equilibrium. Equation (1-4) is a necessary event, logically deduced from the assertion of maximization of profits. If Eq. (1-4) is not observed, it constitutes a refutation of the model, not disequilibrium or nonoptimal behavior. Thus, we assert that firms act as if they are obeying Eqs. (1-4) and (1-5), and on that account we make predictions about their behavior.

To simply assert MR = MC +1, however, is not likely to be useful. One is not likely to observe these marginal relationships. Just as tastes are difficult to observe, the total revenue and total cost functions and, hence, their derivatives, will likely not be known. However, a prediction about the response of the firm to a change in the economic environment, i.e., some test condition —in this case, a change in the tax rate—is, nonetheless, possible. Even if profit maximization, marginal revenue, and marginal cost are not directly observable, tax rates and quantities sold are potentially observable. And profit maximization contains implications about these observable quantities.

about marginal responses? Upon closer observation we notice that Eq. (1-4) is an implicit relationship between x and t. Under certain mathematical conditions this implicit relationship be- tween the variable x and the parameter t can be solved for the explicit choice function:

x=x*(t) (1-

18 THE STRUCTURE OF ECONOMICS

That is, if we knew the equations of the MR and MC curves, then as long as the firm can be counted on to always obey the appropriate marginal relations, no matter what tax rate prevails, we can, in principle, solve for the explicit relationship that states how much output will be produced at each tax rate. Again, although it would be desirable to know the exact form of Eq. (1- 6), the economist will not typically have this much information. Hence, predictions about total quantities will not generally be forthcoming. We can, nonetheless, make predictions about marginal quantities. If Eq. (1-6) is substituted into Eq. (1-4), the identity

R'{x*(t))-C(x*(t))-t = 0 (1-

7)

results. This is an identity because the left-hand side is 0 for all values of /. It is 0 for all values of t precisely because x*(t) is that level of output that the firm chooses in order to make the left-hand side of (1-7) always equal 0. That is, the firm, by always equating MR to MC plus the tax, for any tax rate, transforms the Eq. (1-4) into the identity (1-7). Because we are interested in what happens to x as t changes, the indicated mathematical operation is the differentiation of identity (1-7) with respect to t, keeping Eq. (1-6) in mind. The student must observe that this differentiation makes sense only if x is a function of t. Otherwise, the symbol dx/dt has no meaning. It is premature to simply differentiate Eq. (1-4) with respect to t until such functional dependence is formally implied. It is the assertion that the firm will always equate at the margin, i.e., obey Eq. (1-4)/or any tax rate that allows the specification of Eq. (1-6): the functional dependence of x upon t. The resulting identity, (1-7), can be validly differentiated on both sides; Eq. (1-4) cannot be. This step is often left out, yet it is critical from the standpoint of clearly understanding the implied economic relationships as well as mathematical validity^

Performing the indicated differentiation of identity (1-7),

^ ^ = Q (1- 8) R \ x ) ^ C { x ) ^_{dt dt} Equivalently, assuming (R" — C") ^ 0, dx* 1 * K--C- ( 19 )

Since R" — C" < 0 by the sufficient second-order condition for profit maximization, this implies

dx*_-r <0 dt

Note well what has been accomplished here. The postulate of profit maximization (not observable), as specified in Eq. (1-3), has led to the refutable proposition that output will decline as the tax rate the firm faces increases. In addition, nothing has been assumed as to the specific functional form of the demand or cost curves,

t As an example of the latter, differentiation of both sides of the identity

(x + 1 )2 = x2 + 2x + 1 is valid; differentiation of both sides of the

equation 2x = 6 yields nonsense. The difference is that the former holds for all x, whereas the latter holds only for x = 3.

COMPARATIVE STATICS AND THE PARADIGM OF ECONOMICS 19 and hence the result holds for all specifications of those functions. A prediction about changes in the choice variable, that is, marginal adjustment of output when the parameter facing the decision maker changes, has been rather easily derived, i.e., shown to be implied by a single behavioral assertion. This is the goal of comparative statics; the limitations and abilities of the methodology to accomplish that goal are the subject of this book.

Example 2. Consider now the second previously mentioned behavioral postulate. Let us suppose that profits are desired not for their own sake, but rather for the utility derived from them. Thus, let us now assert that the firm owner maximizes U(n), where U'(n)> 0, so that increased profits mean increased utility. The function U(rc) is some unspecified ordinal measure of the "satisfaction" that this firm owner gains from earning profits. It might seem that since we have replaced a potentially observable quantity, profits, with an unobservable variable, utility, that this theory will be devoid of refutable implications. Let us see. The objective function is now

maximize

U(R(x)-C{x)-tx) = U{n) (1-

10)

The firm's choice function, as before, is found by setting the derivative of U(TT) with respect to x equal to 0. Using the chain rule, dU dn _ dn dx or U'(n)[R'{x) - C'{x) - t] = 0 (1- 11)

Since U'irc) > 0, the choice function (1-11) is equivalent to the previous one for simple profit maximization:

R'(x)-C'(x)-t = 0 (1-4)

Since the implicit functions (1-4) and (1-11) are equivalent, their solutions

x=x\t) (1-

12)

are identical. Thus, these firms will act identically; they have the same explicit choice functions (1-6) and (1-12) governing the response of output to tax rates. One technicality must not be overlooked, however. We must check that the point of maximum profits is also maximum, rather than minimum, utility; i.e., we have to check the second-order conditions for this problem. Otherwise we might be discussing two entirely different points, and the derivatives dx/dt at those points would in general differ. The second-order conditions for the two problems are, however, identical: We have, for the first-order condition,

^ 2 (*)] =0

Thus, using the product rule,

20 THE STRUCTURE OF ECONOMICS