I. Consider the following static optimization problem. Suppose that a consumer has financial wealth W and owns the house H¯ . She has utility over housing H and nonhousing consumption C. Suppose that P is the price of housing in terms of non-housing consumption, so the budget constraint is W + P . H¯ = C + P. H . Finally, assume that nonhousing wealth happens to equal housing wealth, so that W = H¯ .
Suppose that utility is log(C) + log(H).
Note: Housing is continuous variable. The consumer can just live in her house with H = H¯ , or you might choose to live in a smaller ( H < H¯ ) or larger (H > H¯ ) house. In those cases, as the budget constraint makes clear, the consumer buys or sells some extra house.
A. Prove that if P=1, the solution to the consumers problem is C* = H* H¯ =W.
B. Show the solution in a chart in (C,H)-space, i.e., with the indifference curve tangent to the budget constraint.
C. Now consider the situation where P>1.
i. Can the consumer still consume (C*, H¯) ? That is, is the previous allocation still feasible?
ii. How will the optimal solution change from (C*, H¯) when P>1? Solve for the optimal values of C and H in terms of P and H¯ .
iii. Show the new optimum in the chart.
D. Now consider the situation where P<1.
i. Show the new optimum in the chart.
E. Calculate optimized utility in terms of P (i.e., the indirect utility function). Rank optimized utility in the cases P=1, P>1, and P<1.
F. What does the result in E tell you about the dual role of housing as an asset and as a consumption good?
II. Suppose all that happens in this economy is that that household consumes housing and non-housing consumption. (This question is a bit tricky, but is relevant for understanding the housing bubble.)
A. What is GDP in this economy (in terms of the variables in Part I)?
B. How does GDP in this economy change with P?
C. But we learned that asset prices do not directly affect GDP. So why does GDP increase with P in this economy.