ECON11: Problem Set 2

Maurizio Mazzocco

Due October 15 before 9:30 in class

1. Mark has a weekly endowment of 600 dollars that he spends on buying games (G) and digitalmusic (M). The price of each game is 20 dollars and the price of each digital music album is5 dollars. Mark’s utility is given by:

U (G,M) = 2G1/2 +M1/2

(a) Write down Mark’s budget constraint.

(b) Set up the Lagrangian and find his optimal consumption.

(c) What is his new consumption if the price of games becomes 10 dollars?

2. Maggie is the only person in an island, and she has an endowment of 14 units of X and 7units of Y. Her utility function is:

U (X, Y ) = ln (X) + 5 ln (Y )

(a) Find the marginal utility of X and Y .

(b) Find the marginal rate of substitution when Maggie consumes all her endowment.

3. The following utility function is known as CES (constant elasticity of substitution) function:

U(x, y) = (αxδ + βyδ)1/δ, where α > 0, β > 0

(a) Is this function homothetic?

(b) How does the MRSx,y depend on the ratio x/y? Specifically, show that the MRSx,yis strictly decreasing in the ratio x/y for all values δ < 1, increasing in the ratio x/yfor all values δ > 1 and constant for δ = 1. (Hint: treat the ratio x/y as one variable:z = x/y)

(c) Show that if x = y, the MRS of this function depends only on the relatively sizes of αand β.

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