Question

Suppose there are 2 consumers, A and B. The utility functions of each consumer are given by: UA(X, Y ) = X^1/2 Y^1/2 UB(X, Y ) = 3X + 2Y The initial endowments are: WXA = 10, WYA = 10, WXB = 6, WYB = 6

a) (20 points) Using an Edgeworth Box, graph the initial allocation (label it W) and draw the indifference curve for each consumer that runs through the initial allocation. Be sure to label your graph carefully and accurately.

b) (4 points) What is the marginal rate of substitution for consumer A at the initial allocation?

c) (4 points) What is the MRS for consumer B at the intial allocation?

d) (4 points) Is the initial allocation Pareto efficient? How do you know

Answer 1

a) Total units of good X = WXA + WXB = 10+6 = 16

Total units of good Y = WYA + WYB = 10+6 = 16

The utility of both the consumers are given below:

U_A(X,Y) = X^1/2Y^1/2

U_B(X,Y) = 3X+2Y

At the endowment point of consumer A, U_A = 10^1/210^1/2 = 10

Therefore, IC of consumer A through his endowment : 10 = X^1/2Y^1/2. This IC has been drawn in figure 1 as IC_A1 in red color.

At the endowment point of consumer B, U_B = 3*6+2*6 =30

Therefore, IC of consumer A through his endowment: 30=3X+2Y. This IC has been drawn in figure 1 as IC_B1 in green color.

Figure 1

b)  marginal rate of substitution for consumer A at the initial allocation:

c)  marginal rate of substitution for consumer B at the initial allocation:

d) Since MRS of conusmer A is not equal to MRS of consumer B at the initial location, this means atleast one of them can be made better off without making the other one worse off. As see from figure 1, as we move to the area highlighted in blue, both the consumers are better off. Therefore, the initial allocation is not pareto efficient.