Question

General Equilibrium:

Problem 4 Consider a pure exchange economy with two goods and two consumers, Rand J with utility functions UR(x,y) = x²y and U,(x,y) = x4y respectively, and endowments of wR = (2,1) and wj = (1,2). Compute the competitive equilibrium for this economy. Calculate the transfers ta and t, needed to support the allocation (XR, YR) = (1,1.5) and (xj. y.) = (2,1.5) as an equilibrium with transfers. %3D

Answer 1

3p Let the utility function of consumer R is Up (22. Yp) = XRye Since consumer R has a units of a and 1 unit of y, the income side of budget constraint is ap+py and the expenditure side of the budger constraint is Pr + PyYR - 2 P + Py = Px HR + PyYR. - 2 p + 1 = PRR +YR [Dividing both sides by py and p = Pr o Consumer Rs utility maximizing conditions : mor Por = p Given UR (Up, GR) = YR, MU= 2X RJR. and Mug = → 22 UR =P => 2 YR = >> X = 2 XR Replacing Xx in its budget constraint, we get : 2p+1 = p. 27R + Y ap+1 = 34 ) YR = gpt) Replacing@ene, *= tem ap#! - 4p#2 - 3P Consumer R's alemand for a is 4p+2 and demand for Y is 25 Similarly, uz (Xy, Yy) am yg and budget constraint for consumer Jus: Px +2Py = Px Kg + Pg Yg 1+ 2( 3 ) = x2 + 2 / Ys (dividing both sicks by Bz) >> it § - Ast ist => x3 = 1+ $ - Yo- Consumer I maximizes utility when mux le = 1 Geven U ( 7 ) = x Is , mux= 4x I, and muy = x => 4344.-4 t >>ly — ® Replacing @ in 6, X = 1+ } - ++2) X3 = 1+ } >> N = 6*2.4 = 4p2+ 8- 0 4p2+1 Again replacing xg in 0, y = to rapish] - Auto => 45 - 4 Ch?) >[85 = het > Consumere Is demand for x and y are respectively, 47818 Pt2 P' Ap2+1 * Ap2+1 Since wp=(2,1), a trof (-1,0.5), and since Wy=(1,2), a tz of (1,-0.5). is required to support the allocation (XRYR) =(), 15) &C. Vy) (2,1:5). icota 2010 (15) ay. ie. tp = (1, 1.5-(2,1)=(1,015) and t = (a, 5)-(1, 2) = (-05). iz 2+1