J

pro Diem o

3. [Market Clearing) £.x* = (/, (z*},/7(zi)) and = (1,1).

l 1122 Lj j

Of course, Condition 2 can be replaced by the first-order condition p* =

c.iw?,wi) and z? ./z* - a. .(w*)/a_ .{w*h J i * i-J ¿J ¿J 2j

(b) Suppose now that we have two equilibria (p*,w*,x*fz*) and (p**,w",x**tz"j. Assume without loss of generality that p* = p** = 1 and

/L(z*) * fL{z**) and /2(z») * Note that w*/p* = fzU2)( w2 = ^l*2*5'

w**/p" - / (z"), w" = / fz") by the utility maximization and the market

* ^ W w W X i clearing. According to Exercise IS.D.Ub), the input allocations and 2** can be depicted as follows:  0 wi

Figure 15J).4(bJ2)

0 wi

Figure 15J).4(bJ2)

Hence w*/w* w*Vw*f. Thus, as we can see with the unit cost curves (and because of p* = p** = l)T we must have w* 2: w**, that isT the price of input 2 cannot increase even in the absolute value:

Hence /Jz*) * f.U"). Thus / fz?) = /.(z?*) and /0(z5) - /-(z"). Since i 1 i i i I I i ¿z ¿z <L

the /.(-) are strictlv c-uasiconcave, this implies that z* = z*"*. Hence, bv J

id Suppose row that we have two equilibria (p*,w*,x*,z*) and (p",w**,x",z"]. Assume without loss of generality that p* - p** = L We only show that it is not inconsistent to have /j(z*) < /jiz**), >

f2(Z**l w* = /^z*), W** - w*/p* - /2(z')t and w*«/p»* = £

the same time. In fact, again, we know from the unit cost curves that w* < w** and w* > w**. But, as we saw in the proof of Exercise 15.D.3, the profr maximization and the factor market clearing implies that w**/w* < p**/p*, th<

i is, w**/p*m < w£/p*. This *ast ineclua^ly ls nothing but f^iz> f0lz").

ls.D.5 Denote the initial factor allocation by z = (z^^) ^^ t*rie nsw r'actcr allocation bv z? = (zj,z')t after the endowment of inout 1 increases from zT

to z' Note on Figure 15,D.7 that z. and z'. are proportional for each j "1,2,

and that z; » zy In particular, z' > z . Since the endowment of input is

1 JL Wl CrX

fixed at the level of this implies that z^ < 222* ^ proportionality, z' < z__, Thus, 2! - zl, < z, ~ zt1, that is, z' - z.. >2! - Z-. Hence, ¿¿1 i Ii 1 11 li li 1 1

by dividing the left hand side by z and the right hand side by z. (and jkl «k because cf z., < zJf we obtain z!./z,, > z!/z,. By the homogeneity of degree

¿11 li li l l one and the orcDcrtionaiitv, we have /t(z')//T(z) > z!/z_.

15.D.6 (a) Writing w* = iw* w*), the eauilibrium conditions for w* and