4. Consider a two-consumer, two-good exchange economy. Utility functions and endowments are u'(x1, x2) = (x1x2)² and e' = (18, 4),u²x1. x2) = In(x1) + 2 In(x2) and e² = (3, 6). (a) Characterize the set of Pareto-efficient allocations as completely as possible. (b) Characterize the core of this economy. (c) Find a Walrasian equilibrium and compute the WEA. (d) Verify that the WEA you found in part (c) is in the core.

4. Consider a two-consumer, two-good exchange economy. Utility functions and endowments are u'(x1, x2) = (x1x2)² and e' = (18, 4),u²x1. x2) = In(x1) + 2 In(x2) and e² = (3, 6). (a) Characterize the set of Pareto-efficient allocations as completely as possible. (b) Characterize the core of this economy. (c) Find a Walrasian equilibrium and compute the WEA. (d) Verify that the WEA you found in part (c) is in the core.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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2(b)There are three people and two goods in an economy. The utility functions and the initial bundles are given below: U = X11X 12 U2 = 2x 21+ X 22 U3 = X 31 X 3z+X 32 W; = (0,8) W;= (2,2) W3= (8,0) Suppose it is suggested that the three traders move to the allocation x given by x= (1,2), x2= (8, 4), X3= (1,4) C. Show that x is not in the core.
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Find the marginal rate of substitution for Household A.
Consider a pure exchange economy with two goods k = 1,2 and two households i = A, B. The total available amount of good 1 and 2 is 9 and 12 units, respectively. Utility functions for households A and B are UA (TA1, A2) = TA1TA2 and UB(B1, B2) = XB1T32. (a) Find an expression for the Pareto optimal set. (b) Assume e A = (4,4) and eg (5,8) are A and B endowments, respectively. Find the competitive equilibrium. (c) Is the competitive equilibrium allocation in the Pareto Optimal set? Why this result should not surprise us?
2(b)There are three people and two goods in an economy. The utility functions and the initial bundles are given below: U1 = X11X 12 U = 2x 21+ X 22 U3 = X 31 X 3z+X 32 W; = (0,8) W;= (2,2) Wy= (8,0) Suppose it is suggested that the three traders move to the allocation x given by x;= (1,2), x2= (8,4), X3= (1,4) b. Show that x makes no one worse off than the original allocation.
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5. Consider a two-consumer one-firm economy with a single input good and a single output good. Consumers' utility functions are u4(x4) Suppose that the input good is divisible and that wA = wB = 1/2. Suppose that the output good is indivisible (i.e., that xg, q = 0, 1, 2, ...), and that the firm's production technology is such that it requires 1 unit of input to produce one unit of output. rA + 3r and uB(xB) 4x? + x. (a) Derive the Pareto set of this economy. (b) For all Pareto-efficient allocations from (a), derive a competitive equilibrium for this economy that generates these allocations.
9. Consider an Edgeworth box economy with two consumers, whose utility func- tions and endowments are u'(x},x}) = (x}Xx})} (5,5) In the following, use the normalization p2 = 1. (a) Find the competitive equilibrium price. (b) State the first fundamental theorem of welfare and verify that it holds in this economy. (e) Consider the allocation ĩ = (x',) = (2,3), (8, 7). Show whether this allo- cation can supported as an equilibrium with transfers. (d) State the second fundamental theorem of welfare, and briefly discuss whether the result in part (c) conform with or violate this theorem. 3.