Question

1 [75 points; Chapter 5] You are to solve the consumer choice problem for three different consumers. Each consumer has $150 to spend (income) and faces prices Px = $2 and Py = $3 for goods X and Y. Consumers I, II, and III have utility functions U'(X, Y) = X? + Y, U"(X, Y) = X12 + Yl2, and UTM(X, Y) = X´Y, respectively. For each consumer, do the following steps. a [15 points] Carefully express the consumer's choice problem, using the given information (this is where you write out the max operator, the choice variables, the objective function, and the budget constraint). b [30 points] Compute the absolute value of the consumer's marginal rate of substitution, and inspect it to determine the shape of the consumer's indifference curves: C-shaped, linear, )-shaped, or some other shape. To show your work, neatly use the arrow argument, increasing X (1) and decreasing Y (1) to see whether MRS(X, Y) is diminishing along an indifference curve. c[30 points. If the indifference curves are C-shaped write out the budget line and the equal slopes condition that characterize an interior solution to the consumer's choice problem. Use the particulars for the given consumer. Solve these conditions to find the interior solution. On the other hand, if the indifference curves are linear or )-shaped, look for a boundary solution where the consumer either buys all X or all Y. Determine which boundary is the solution by comparing the consumer's utility at (75, 0) and (0, 50), which are the corners of the budget line. 2 [25 points; Chapter 6] Determine the demand function for goods X and Y for a consumer with preferences U(X, Y) = X²Y?. This is where you solve for the consumer's optimal choice of X and Y in terms of the prices Px and Py and income M, leaving these prices and income as variables rather than setting them at specific values as in problem 1. Neatly show your work.

Answer 1

1.

a.

$\dpi{80} \fn_phv \small U^1=X^2+Y$

The consumer's problem is:

$\dpi{80} \fn_phv \small max \ X^2+Y \ s.t. \ 2X+3Y=150$

$\dpi{80} \fn_phv \small U^2=X^{1/2}+Y^{1/2}$

The consumer's problem is:

$\dpi{80} \fn_phv \small max \ X^{1/2}+Y^{1/2} \ s.t. \ 2X+3Y=150$

$\dpi{80} \fn_phv \small U^3=X^2Y^2$

The consumer's problem is:

$\dpi{80} \fn_phv \small max \ X^2Y^2 \ s.t. \ 2X+3Y=150$

b.

For U1:

$\dpi{80} \fn_phv \small MRS_1=2X$

As the quantity of X rises, the MRS increases. This implies that the indifference curves are not C shaped and the consumer's problem will have a corner solution.

For U2:

$\dpi{80} \fn_phv \small MRS_2=\frac{\frac{1}{2}X^{-1/2}}{\frac12 Y^{-1/2}}=(\frac{Y}{X})^{1/2}$

As the quantity of X rises, the MRS diminishes. This implies that the indifference curves are C shaped and the consumer's problem will have an interior solution.

For U3:

$\dpi{80} \fn_phv \small MRS_3=\frac{2XY^2}{2X^2Y}=\frac{Y}{X}$

As the quantity of X rises, the MRS diminishes.This implies that the indifference curves are C shaped and the consumer's problem will have an interior solution.

c. Solving problem 1:

Since the indifference curves are not C shaped and MRS increases as X increases, the problem will have a corner solution. The consumer will either consume X or Y. Upon consuming only X, the consumer can purchase 75 units and his utility is 5625. Upon consuming only Y, the consumer can consume 50 units and his utility is 50. Since the consumer attains a higher satisfaction from consuming only X, the consumer's optimal bundle is (75,0).

Solving problem 2:

Since MRS diminishes as X increases, the problem has an interior solution which can be computed by equating MRS and price ratio as follows:

$\dpi{80} \fn_phv \small (\frac{Y}{X})^{1/2}=\frac23 \rightarrow Y=\frac{4X}{9}$

Substituting this value in the consumer's budget line:

$\dpi{80} \fn_phv \small 2X+3*\frac{4X}{9}=150 \rightarrow X=45, Y=20$

Solving problem 3:

Since MRS diminishes as X increases, the problem has an interior solution which can be computed by equating MRS and price ratio as follows:

$\dpi{80} \fn_phv \small \frac{Y}{X}=\frac23 \rightarrow Y=\frac{2X}{3}$

Substituting this value in the consumer's budget line:

$\dpi{80} \fn_phv \small 2X+3*\frac{2X}{3} =150 \rightarrow X=37.5, Y=25$