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ECON1003 S1 2023

PROBLEM SET 7

1. Consider a consumer with the following utility function: U(x1, x2) = x1 .x2.

(a) Find the consumer’s utility for the following points: (1,1), (2,4), (1,3).

(b) Fix U= 5. Plot x2 as a function of x1. This graph is called an indifference curve.

(c) Fix U= 10. Plot x2 as a function of x1. Compare with the graph in part b) and comment on the direction of increasing utility.

(d) The consumer’s marginal utility with respect to good 1 is defined as the rate of          change of utility with respect to a small increase in good 1, holding the quantity of good

2 fixed. Compute the marginal utility of good 1.

(e) The consumer’s marginal utility with respect to good 2 is defined as the rate of change of utility with respect to a small increase in good 2, holding the quantity of good 1 fixed. Compute the marginal utility of good 2.

2. Consider a firm with the following production function Y = F (K, L) = K1/3 L2/3 where Y is output, K is the quantity of capital and L is the quantity of labor.

(a) The firm’s marginal product of labor is defined as the rate of change of output with   respect to a small increase in labor, holding capital fixed. Compute the marginal product of labor.

(b) The firm’s marginal product of capital is defined as the rate of change of output with respect to a small increase in capital, holding labour fixed. Compute the marginal product of capital.

3. Find second derivatives of the following functions:

(a) Y = K1/3 L2/3

(b) Q = lnx + 3lny

(c) C = 120(1+e-0.8Q)