Question:

1. Suppose a consumer's utility function is given by:

U=x^(3/5)y^(2/5)

This is an example of a Cobb-Douglas model. The Cobb-Douglas model is used extensively in economics.

a) Set y = 1 and graph the marginal utility of x; put marginal utility on the vertical axis and x on the horizontal axis. Your graph does not have to be perfect, but it should have the correct shape.

b) What is the MRS for this consumer? Explain in words what the MRS is.

DRAW ONE FIGURE FOR C, D, E, F, G

c) Draw three examples of indifference curves for this consumer and give the slope (you can just find the formula for the slope; you don't have to report a specific value).

d) Suppose the consumer has $1,000 of income, the price of x is $2, and the price of y is $1. Add the budget line for this consumer to your graph for c). Label the slope and intercepts of the budget line.

e) Add an indifference curve to your graph that is tangent to the budget line and label the optimal consumption bundle (you do not need to give values for x and y in the optimal bundle; you need only label the point). What is true about the MRS and the ratio of the price of x to the price of y at the optimal bundle?

f) Suppose the consumer's budget increases to $2,000. Add a new budget line to your graph showing this higher income level. Label the slope and intercepts of the new budget line.

g) Suppose the consumer's budget is still just $1,000 and the price of y increases to $4. Add a new budget line to your graph that shows this price change. Label the slope and intercepts of the new budget line.

h) Suppose that the consumer with preferences given by the utility function above has herMRS equal to 1 and that the ratio of the price of x to the price of y is equal to 2. Why can this not be the optimal bundle? Note that answering "because the MRS and price ratio are not equal" is not sufficient. You must describe how she could improve upon this bundle.

i) Graph the consumer's inverse demand curve for x when her income is equal to $800.That is, graph her demand curve with the price of x on the vertical axis and the quantity of x on the horizontal axis. Label one point on your demand curve. HINT: Start with the MRS equal to the price ratio. Solve for p_yy as a function of p_xx ,and then write the budget line as a function of p_xx only (i.e., not including p_yy).