I. Consider the subsequent static optimization problem. Assume that a consumer has financial wealth W and owns house H. She has utility over housing H and non-housing consumption C. Assume that P is the price of housing in terms of non-housing consumption, so the budget constraint is W + P .H = C + P .H. At last, suppose that non-housing wealth happens to equivalent housing wealth, so that W = H.
Assume 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 may select to live in a smaller (HH) house. In those cases, as the budget constraint makes clear, the consumer purchases 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 budget constraint.
C. Now consider the situation where P>1.
i. Can the consumer silent consume (C*, H)? That is, is the previous allocation still feasible?
ii. How will optimal solution change from (C*, H) when P>1? describe for the optimal values of C and H in terms of P and H.
iii. Show new optimum in chart.
D. Now consider situation where P<1.
i. Show the new optimum in the chart.
E. Compute optimized utility in the terms of P (i.e., the indirect utility function). Rank optimized utility in cases P=1, P>1, and P<1.
F. What does result in E tell you about dual role of housing as an asset and as consumption good?
I. assumes all that happens in this economy is that household consumes housing and non-housing consumption. (This problem 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 don’t directly affect GDP. So why does GDP raise with P in this economy?