Question

Suppose a consumer has quasi-linear utility: u(x1, x2) = 3.01 + x2. The marginal utilities are MU(X) = 2x7"! and MU2:) = 1. Throughout this problem, assume P2 = 1. (a) Sketch an indifference curve for these preferences (label axes and intercepts). (b) Compute the marginal rate of substitution. (c) Assume w> . Find the optimal bundle (this will be a function of pı and w). Why do we need the assumption w> (d) Sketch the demand function for good 1 (holding w and p2 constant). (e) Suppose pı = 2. Compute the consumer's surplus. (f) Now suppose the government imposes a tax on good 1, so that the price of good 1 becomes 2 + t for some t > 0. Compute the consumer's surplus (this will be a function of t).

Answer 1

a. Plotting an indifference curve for a given satisfaction level:

b. The marginal rate of substitution is:
$\dpi{80} \fn_phv \small MRS=\frac{MU_{x_1}}{MU_{x_2}}=\frac{2x_1^{-1/3}}{1}=\frac{2}{x_1^{1/3}}$

c. At equilibrium, marginal rate of substitution is equal to the price ratio:
$\dpi{80} \fn_phv \small \frac{2}{x_1^{1/3}}=p_1\rightarrow x_1=(\frac{2}{p_1})^3$
Substituting this into the consumer's budget equation:
$\dpi{80} \fn_phv \small p_1*(\frac{2}{p_1})^3+x_2=w\rightarrow x_2=w-p_1*(\frac{2}{p_1})^3$
The assumption guarantees an interior solution to the consumer's problem.

d. Plotting demand for good one:

with demand for good one on the vertical axis and price of good one on the horizontal axis.