For the following two-person three-commodity pure exchange economy, the price of

Question

For the following two-person three-commodity pure exchange economy, the price of good z is normalized to $1. Calculate the Walrus equilibrium prices, (p_x, p_y), as well as the Walrus allocation for this economy, ((x^a, y^a, z^a), (x^b, y^b, z^b)). Check that the Walrus allocation is Pareto efficient. Pick the market for good z. Set the total demand for good z equal to the total supply of z of unit(s), obtaining the equation ()px + ()py = 72. Similarly, set the total demand for good x equal to the total supply of x of unit(s), obtaining another equation ()px - ()py = 24. Solving both equations simultaneously, we obtain px = $ and py = $. Therefore, the Walrus prices are (p_x, p_y, p_z) = (). Then at the Walrus allocation, consumer a consumes the bundle (), and consumer b consumes (). From the given utility functions, the marginal rates of substitution between x and y for each consumer is MRS^a_xy =. and MRS^b_xy =. Similarly, the marginal rates of substitution between y and z for each consumer is MRS^a_yz = and MRS^b_yz =, and between x and z is MRS^a_xz = and MRS^b_xz =. At the Walrus allocation, MRS^a_xy = and MRS^b_xy =; MRS^a_yz = and MRS^b_yz =; and MRS^a_xz = and MRS^b_xz =. Therefore, the Walrus allocation Select Pareto efficient.

Explanation / Answer

This is 2 person and 3 good exchange economy.

Total supply of z=8 ( given from the net endowment)

Individulal demand of the goods are given. ma= 6px similarly mb=16py+8 as pz=1( normalized)

total demand of z= total supply of z

6px/3 + (16py+8)/4 = 8 .............eq 1

solving eq 1

24px+48py=72 -----eq 2

repeating the same thing for+ good x demand = supply

6px/3px+(16py+8)/2px=6 -----------eq 3

solving eq 3

24px-48py=24 ------eq 4

solving eq 3 and 4 simultaneously

px=2 , py=1/2, pz=1

now insert these prices in individual demands and simplify

consumer a consumes the bundle ( 2, 8, 4)

consumer b consumes the bundle (4, 8, 4)

now calculating individual MRS using the utility function partial derivatives ratio and demand bundles

MRSaxy= axa-1yaza/axaya-1za = y/x = 4

similarly MRSbxy= 2y/x = 4

MRSayz=z/y = 1/2

MRSbyz=z/y = 1/2

MRSaxz=z/x = 2

MRSbxz=2z/x = 2

MRSaxy = MRSbxy =4

MRSayz=MRSbyz= 1/2

MRSaxz=MRSbxz= 2

all MRS are equal hence the allocation is pareto efficient

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