Market Model

First let us consider again flie simple one-commodity market model of (3.1). That model can be written in the form of two equations:

Q — a — bP (a, h > 0) [demand] Q = ~c + dP (c,d> 0) [supply]

with solutions b + d

These solutions will be referred to as being m the reduced form; The two endogenous variables have been reduced to explicit expressions of the four mutually independent parameters a, b, c, and d.

To find how an infinitesimal change in one of the parameters will affect the value of P*, one has only to differentiate (7,14) partially with respect to each of the parameters. If the sign of a. partial derivative, say, dP*/da, can be determined from the given information about the parameters, we shall know the direction in which P* will move when the parameter a changes; this constitutes a qualitative conclusion. If the magnitude of HP*/8a can be ascertained, it will constitute a quantitative conclusion.

Similarly, we can draw qualitative or quantitative conclusions from the partial derivatives of Q* with respect to each parameter, such as HQ*/da. To avoid misunderstanding, however, a clear distinction should be made between the two derivatives dQ*/()a and <)Q/da. The latter derivative is a concept appropriate to the demand function taken alone, and without regard to the supply function. The derivative 3Q*/da pertains, on the other hand, to the equilibrium quantity in (7.15) which, being in the nature of a solution of the model, takes into account the interaction of demand and supply together. To emphasize this distinction, we shall refer to the partial derivatives of P* and Q* with respect to the parameters as comparative-static derivatives. The possibility of confusion between 3Q*/da and dQ/da is precisely the reason why we have chosen to use the asterisk notation, as in Q* to denote the equilibrium value.

Concentrating on P* for the time being, we can get the following four partial derivatives from (7.14):

parameter a has the coefficient b + d

dP* _ 0(6 + d) - 1 (a + c) _ ~(a + c) ~ (b + d)1 ~ (b + df

Since all the parameters are restricted to being positive in the present model, we can conclude that dP* 'dP* a P* BP"

da dc 8b 3d

For a fuller appreciation of the results in (7.16), let us look at Fig. 7.5, where each diagram shows a change in one of the parameters. As before, we are plotting Q (rather than P) on the vertical axis.

Figure 7.5a pictures an increase in the parameter a (to ar). This means a higher vertical intercept for the demand curve, and inasmuch as the parameter h (the slope parameter) is unchanged, the increase in a results in a parallel upward shift of the demand curve from D

FIGURE 7.5

(Increase in a)

(Increase in a)

FIGURE 7.5

(lucrcasc in h)

(lucrcasc in h)

Q

(Increase in d)

S. / j / / ^Ss/ /

s

/ S 1 1

V P*' p* /

p

r <

to D1. The intersection of D! and the supply curve S determines an equilibrium price P*\ which is greater than the old equilibrium price P*. This corroborates the result that dP*/'da > 0, although for the sake of exposition we have shown in Fig. 7.5a a much larger change in the parameter a than what the concept of derivative implies.

The situation in Tig. 7.5c has a similar interpretation; but since the increase takes place in the parameter c\ the result is a parallel shift of the supply curve instead. Note that this shift is downward because the supply curve has a vertical intercept of -c: thus an increase in c would mean a change in the intercept, say, from —2 to -4, The graphical comparative-static result, that P** exceeds P*. again conforms to what the positive sign of the derivative HP*/3c would lead us to expect.

Figures 1.5b and 1.5d illustrate the effects of changes in the slope parameters h and d of the two functions in the model. An increase in h means that the slope of the demand eurvc will assume a larger numerical (absolute) value; i.e., it will become steeper. In accordance with the result HP*/db < 0, we find a decrease in P* in this diagram. The increase in d that makes the supply curve steeper also results in a decrease in the equilibrium price. This is, of course, again in line with the negative sign of the comparative-static derivative 3P*/'dd.

Thus far, all the results in (7.16) seem to have been obtainable graphically. If so, why should we bother to use differentiation at all? The answer is that the differentiation approach has at least two major advantages, First, the graphical technique is subject to a dimensional restriction, but differentiation is not, Even when the number of endogenous variables and parameters is such that the equilibrium state cannot be shown graphically, we can nevertheless apply the differentiation techniques to the problem. Second, the differentiation method can yield results that are on a higher level of generality. The results in (7.16) will remain valid, regardless of the specific values that the parameters a, A, c. and d take, as long as they satisfy the sign restrictions. So the comparative-static conclusions of this model are, in effect, applicable to an infinite number of combinations of (linear) demand and supply functions. In contrast, the graphical approach deals only with some specific members of the family of demand and supply curves, and the analytical result derived therefrom is applicable, strictly speaking, only to the specific functions depicted.

This discussion serves to illustrate the application of partial differentiation to comparative-static analysis of the simple market model, but only half of the task has actually been accomplished, for we can also find the comparative-static derivatives pertaining to Q\This we shall leave to you as an exercise.

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