## Xi

Define p° = (2,1) and p1 = (1,2). Then x(p°,2) = ((1 - e)/2, 1 + e), x(pl,2) = (1 - c, (1 + c)/2). Thus x(p°,2) >- x(p*,2). However, the area variation measure following the price-change path p° h> (1,1) p^ is

= 2eln2 > 0. Hence the area variation measure ranks p* over

3J.11 If (p* - > 0» then w > p^-x*. The local non-satiation implies that x° is preferred to x*. Hence the consumer must be worse off at (p\w). As for the interpretation in term of the first-order approximation, since e£p,u) is concave in p,

Since VeCp^uVip0 - p1) < 0, e(p0tu!} < eip^u1) = w. Thus u° = v(p°,w) >

Finally, (p1 - > 0 if and only if w > p°«x\ which, in turn, is equivalent to p^-tx1 - x°) < 0. This test is depicted in the picture below:

3.1.12 Let u° = v(p°,w°) and u^ = v(p\wV Then we define

0 0 1 L ,0 1, ,0 0. ,0 1, 0 EV[p ,w ;p ,w ) = e(p ,u ) - e(p ,u ) - eip ,u ) - w , n 0 0 1 "1 ' .11. ,10, 1 / 1 0. CV(p ,w ;p ,w ) = ,e(p ,u } - e(p ,u } - w - e(p ,u ].

The "partial information" test can be extended as follows: If < w\ then the consumer is better off at (p\w*). This can be proved in three ways.

The first one is the same revealed-preference argument as in the proof of

Proposition 3.1.1.

The second way is to use the indirect utility function. Since v(p,w) is quasiconvex, if

, 1 0, _ , 0 0. , 1 , 0 0W_ ^ _ (p - p }'VpV(p ,w ) + (w - w )<3v(p ,w J/ow > 0, then we can conclude that v(p\w*) > v(p^,w°). But, by Roy's identity, this sufficient condition is equal to

- (p1 - p°)"(5v(p0>w0)/aw)x(p0tw°) + (w1 - w°)(3v(p0(w°)/aw)

f 0 \( 1 0 0 1 0, = [dvip ,w )/aw)(- p *x + w 4- W - w )

Hence, if p^x^ < w\ then v(p*,w*) > v(p°,w°).

The third way is to use the expenditure function, vip^w*) > v(p°,w°) if

and only if e(p1,v(p\w1)) > e(p1fWp°,w0)). But eip^vfp^w1)) = w1 and etp^vtp^v/0)) ^ p^-x0. Hence, if < w\ then we can conclude that

3.J.1 [First pr intin.g errata: The difficulty level should probably be B. ] It follows immediately from the definition that if x(p,w) satisfies the strong axiom, then it satisfied the weak axiom. Conversely, if x{ p,w ) satisfies the weak axiom (in addition to the homogeneity of degree zero and Walras* law), then the Slutsky matrix is negative semidefinite and, by Exercise

2.F.11, symmetric. Hence x(p,w) is integrable, implying that there exists a preference relation that generates p,w ). Thus x{ptw) satisfies the strong

axiom as well.

3.AAJ If (p,w) = (1,1,1), then x(p,w) = (0,1). The locally cheaper condition is not satisfied since B = ix e (R^: x. + x^ = 1) and there is no p,w + i 2

y such that p*y < w, as depicted in the following figure.

To check that the demand function is not continuous at (1,1,1), consider the sequence (pnfwn) = (1 - 1/n, 1, 1 - 1/n). Then (pn,wn) (1,1,1) and x{pn,wn) = (1,0), but jc(1,1,1 j 2= (0,1). This discontinuous change in demands arises because the budget set B consists of (1,0) for every n, but B,. , =