Question

4. General Equilibrium An economy consists of two consumers, indexed by j = A, B, who consume two goods x, and x2. The first consumer's endowment of the two goods is (W1,W4) = (2,4), and the second consumer's endowment is (w,w) = (5,1), where w/ denotes consumer j's endowment of good i. a. Suppose the preferences of the two consumers are described by the utility functions U,(x) = (x^)(x4)4 and U2(x) = xPx, where x denotes consumer j's consumption of good i. I. Given prices p1 and P2, what is each consumer's demand for x2? ii. Find prices such that the aggregate demand for X2 equals the economy's endowment of x2. b. Suppose, instead, that the preferences of the two consumers are described by the utility functions U$(x) = x1 + x4 and Up(x) = x x Suggest a trade that would make both consumers better off compared to their endowment points.

Answer 1

Be , Y=_BM 2P2 . 84) VA= X1oxzt, uß = XjX2, A=2,4), CB=(5,1) a endownent incomes MA= QP1 +4P2, MB = 59, + P2 of coble Douglas utility funcn, u=X&YB, then with Income M X = XM (Q+B) Px so for Good 2, X2A = 4MA X2B = MB 5 P2 x + = 4 (@fit pe) , x9B = (69+P2) 1) Now x 2 + x 23 = X2= 4+1=5.. 8pit 16 P2 + (5 PitP2 = 5 БР. Ро : 16P1 +32P2 + 25P, + 5P2 = 5 41P, +37P2 = 50 P2 = 13 P2 8o. Poices, Let If Pi=1, then SP2 212 10P2 41P, PRE 4 Ang

6.) UA (8,4) = 2+4= 6 UB(5, 1) = 5. - utility Levels at initial endowments . # now of after pade COA = (3. 5,3) WB= (3.5,2) then uA'= 3.5+3 u Bl= 3.5 x 2 So both individuals are better off than at initial Allocations.