Question

3. This question is adapted from our textbook. Anne and Bill live in an island economy and consume only two goods. Let x? = (x1, xi) denote the consumption bundle for i = A, B. Their endowments are wa = (WA,WA) = (2,5) and wb = (wp,w?) = (10, a). Both have identical Cobb-Douglas utility functions ui(x) = xix, for i = A, B. Normalizing the price of good 2 to be p2 = 1, we just write pı = p > 0. The individual demand functions for Anne and Bill are then: xi = 2p where m' refers to the value of i’s endowment at the current market price p. (a) Draw the set of interior PE allocations in an Edgeworth box for this economy. (b) Calculate the Walrasian equilibrium price p and the Walrasian equilibrium allo- cation (C2A, 2A), (zB,2))

Answer 1

Anne and bill have endowments of Good 1 and Good 2 and their utility functions are given as: w* = (2,5), u*(174, 27) = 471, 47 W3 = (10,a), u® (,2,)=x, 1, The social endowment of Good 1 is 12 and Good 2 is 5+a. The set of interior PE allocations in an Edgeworth box for this economy is given below: Good Y. 5+a ICA Set of Interior PE allocations ICE 12 Good X

= x. Here, the diagonal line The set of pareto efficient allocation are those set of point where x shows the set of pareto efficient allocations. b) The price of Good 2 is normalized that is P = 1 And, P=P>0. Now get the value of equilibrium allocations for Anne: H2+5) -(2039)

Similarly, for Bill: C+d P 1 (10P+a 1+11 P

(otarot- (6+201) (10 4201) -

Demand = Supply x1 + x = wľ + w [2P+5] [10P+a] = 2+10 | 2P ] | 2P ] 2+ 2P + 5+10P+a=12x2P 12P+5+a= 24P 12 P= 5+ a p=Sta 12 Thus, the Walrasian equilibrium price P= 5+a 12 Substitute the value of P in above equation to get equilibrium allocations: 5 + IX- (10+2a+6012 | 10+2a J 12 70+ 2a 10+2 a

(10 + 2a +60 - 24 T - 70+ 2a) 24 Similarly, for Bill:

지음 [50+10 +12 ][ 12 [ 10 +20+12a JL 12 ] [50+22 ai 10+14 ] 10P+a (10 ) [ 50+10+12a] | 24