Question

Imagine that you are moving to a new city after you graduate to start a job. You are deciding how much of your paycheck to use for housing (call this H) and how much to save for other spending (call this S). Your utility function is U ( H , S ) = H 0.5 ⋅ S. Housing is measured in square feet, and the cost per square foot is p h. Saving is measured in terms of dollars and therefore the price (p s) is $1. Just to be clear, H is the “x” good, and S is the “y” good in this problem. Your income is $6000.

a) (6 points) What is the marginal rate of substitution? What is the price ratio? What is the budget constraint?

b) (8 points) Find the utility-maximizing demand for housing, H*. It will be a function of p H. Find the utility-maximizing demand for savings, S*. Show your work using your answer from part a for full credit.

c) (10 points) Draw a graph in the (H, S) plane. Put 3 budget constraints on the graph: one with p H = 1, one with p H = 2, and one with p H = 4. Label the intercept points of these budgets. On each budget, add a dot at the (H*,S*) bundle and label those values on each axis. Put an indifference curve through each utility-maximizing point (these don’t need to be drawn to scale). Add the price-offer curve to the graph.

Answer 1