## Exercise

1T let the equilibrium condition for national income be S(Y) + T(V) -/(/) + Co (St T\ V > 0; V + V > V) where S, Yt Ttlt and C stand for saving, national income, taxes, investment, and government expenditure, respectively. Al! derivatives are continuous.

(a) interpret the economic meanings of the derivatives 5', 7"', and /'.

(b) Check whether the conditions of the implicit-function theorem are satisfied. If so, write the equilibrium identity.

(c) Find (rf/*/dCo) and discuss its economic implications,

2. Let the demand and supply functions for a commodity be

Y0) (Dp< 0; DYg> 0) Qs = S(PrT0) (Sp > 0; Stq < 0)

where Yo is income and Tq is the tax on the commodity Ad derivatives are continuous.

(a) Write the equilibrium condition in a single equation,

(b) Check whether the implicit-function theorem is applicable. If so, write the equilibrium identity.

(c) Find (tiPVaVo) and Q)P*ftiTo), and discuss their economic implications.

(id) Using a procedure similar to (8.37), find (a Q'fd Yo) from the supply function and (9Q737o) from the demand function. (Why not use the demand function for the former, and the supply function for the latter?)

3. Solve Prob. 2 by the simultaneous-equation approach.

4. Let the demand and supply functions for a commodity be Qd =D(P, to) and Q, = Q5o where £0 is consumers' taste for the commodity, and where both partial derivatives are continuous.

(a) What ¡5 the meaning of the - and + signs beneath the independent variables P and

(b) Write the equilibrium condition as a single equation.

(c) Is the implicit-function theorem applicable?

(d) How would the equilibrium price vary with consumers' taste?

5. Consider the following national-income model (with taxes ignored): Y-C(Y)- /{/) - Co = 0 (0 < C' < 1; /' < 0)

kY + i{i) - = 0 (k = positive constant; L' < 0) (a) Is the first equation in the nature of an equilibrium condition? (ib) What is the total quantity demanded for money in this model? (c) Analyze the comparative statics of the model when money supply changes {monetary policy) and when government expenditure changes (fiscal policy).

6, In Prob, 5, suppose that while the demand for money \$till depends on Y as specified, it is now no longer affected by the interest rate.

(a) How should the model statement be revised?

(b) Write the new Jacobian, call it [/ ['. Is \j numerically (in absolute value) larger or smaller than \) |?

(c) Would the implicit-function rule still apply?

(d) Find the new comparative-static derivatives,

(e) Comparing the new (3Y*/dCo) with that in Prob, 5, what can you conclude about the effectiveness of fiscal poficy in the new model where /is independent of i?

(0 Comparing the new (-9r79Ko) ^at 'n ^ w'iat can y°u 5aV about the effectiveness of monetary policy in the new model?

8.7 Limitations of Comparative Statics_

Comparative .statics is a useful area of study, because in economics we are often interested in finding out how a disequilibrating change in a parameter will affect the equilibrium state of a model. It is important to realize, however, that by its very nature comparative statics ignores the process of adjustment from the old equilibrium to the new and also neglects the length of time required in that adjustment process. As a consequence, it must of necessity also disregard the possibility that, because of the inherent instability of the model, the new equilibrium may not be attainable ever. The study of the process of adjustment per se belongs to the field of economic dynamics. When we come to that, particular attention will be directed toward the manner in which a variable will change over lime, and explicit consideration will be given to the question of stability ofequilibrium.

The important topic of dynamics, however, must wait its turn. Meanwhile, in Part 4. we shall undertake to study the problem of optimization, an exceedingly important special variety of equilibrium analysis with attendant comparative-staiic implications (and complications) of its own. Chapter 