PUBP720: Managerial Economics, Math Practice

s.t.         x + y = 6

Step 1: The Lagrangian function is set as:

Step 2: Take the first order partial derivatives of the Lagrangian and set them to 0 (corre­sponding to the maximum point)

What does the value of λ* tell you?

2.

Let's generalize the problem and solve again.

Max z =  xy s.t.        x + y = Q

Step 1: The Lagrangian function is set as:

Step 2: 2: Take the first order partial derivatives of the Lagrangian and set them to 0 (corre­sponding to the maximum point)

What does the value of λ* tell you?

2 Consumer Demand

Gilbert Grape gets enjoyment only from consuming Bordeaux wines and organic yogurt. Gilbert's preference are represented by the following utility function

U(x, y) = x^2/3y^1/3

where x is the consumption of Bordeaux and y is the consumption of yogurt.

a) Use the power function rule for taking a derivative (i.e. finding a slope!) to confirm that

Utility gain from small ↑ in x = marginal utility of x = ∂U/∂x = U_x = 2/3x^-1/3y^1/3

Utility gain from small ↑ in y = marginal utility in y = ∂U/∂y = U_y = 1/3x^2/3y^-2/3

b) Combine equations 1 and 2 to arrive at a general expression for the marignal rate of subtiution, that is

MRS = 0 - (∂U/∂x)/(∂U/∂y) = 2y/x

Gilbert's income M = $300. The prices of x and y are as follows:

p_x = $20/bottle

p_y = $4/container

c) What is the slope of the budget line? Does the slope of the budget line depend on consumer preferences? Why or why not?

d) Using your expression for the marginal rate of substitution, find the combination of x and y for which

• the slope of the indifference curve equals the slope of budget line and
• Gilbert spends his entire budget on x and y, that is

20x + 4y = 300

HINT: Answering this sub-question involves solving two equations for two unknowns. Make sure that you have Esteban Clavell's consumer theory handout, posted to the website. Find the the pair of demand functions at the bottom of page 2:

e) For the income and prices given above, use in 3 and 4 to solve for the optimum values of x* and y*.

f) Compare your answer from part (d) and part (e). Explain why they are the same or different.

g) Provide the Gilbert's choices of both Bordeaux and yogurt under the following cir­cumstances (consider each seperately):

i. Gilbert's income increases to $600.

ii. The price of Bordeaux increases to $40.

iii. Gilbert finds a new brand of yogurt that he absolutely loves. As a result, his preferences change, and are now represented by the following utility function

U(x,y) = x^1/4y^3/4