The U.S. tax code subsidizes housing through a deduction of mortgage interest. For new homeowners, mortgage interest makes up the bulk of their housing payments which tend to make up about 25% of a household’s income. Assume throughout that housing is a normal good.
A: For purposes of this problem, we will assume that all housing payments made by a household represent mortgage interest payments. If a household is in a 25% tax bracket, allowing the household to deduct mortgage interest on their taxes then is equivalent to reducing the price of $1 worth of housing consumption to $0.75.
(a) Illustrate a demand curve for a consumer, indicating both the with- and without-deductibility housing price.
(b) On the same graph, illustrate the compensated (or MWTP) curve for this consumer assuming that housing costs are deductible.
(c) On your graph, indicate where you would locate the amount that a consumer would be willing to accept in cash instead of having the subsidy of housing through the tax code.
(d) On your graph, indicate the area of the deadweight loss.
(e) If you used the regular demand curve to estimate the deadweight loss, by how much would you over- or under-estimate it?
B: Suppose that a household earning $60,000 (after taxes) has utility function u(x, y) = x0.25 y0.75, where x represents dollars worth of housing and y represents dollars worth of other consumption. (Thus, we are implicitly setting the price of x and y to $1.)
(a) How much housing does the household consume in the absence of tax deductibility?
(b) If the household’s marginal tax rate is 25% (and if all housing payments are deductible), how much housing will the household consume?
(c) How much does the implicit housing subsidy cost the government for this consumer?
(d) Derive the expenditure function for this household (holding the price of other consumption at $1 but representing the price of housing as p.)
(e) Suppose the government contemplates eliminating the tax deductibility of housing expenditures. How much would it have to compensate this household for the household to agree to this?
(f) Can you derive the same amount as an integral on a compensated demand function?
(g) Suppose you only knew this household’s (uncompensated) demand curve and used it to estimate the change in consumer surplus from eliminating the tax deductibility of housing expenditures. How much would you estimate this to be?
(h) Are you over- or under-estimating the deadweight loss from the subsidy if you use the (uncompensated) demand curve?
(i) Suppose that all 50,000,000 home-owners in the U.S. are identical to the one you have just analyzed. What is the annual deadweight loss from the deductibility of housing expenses? By how much would you over- or under-estimate this amount if you used the aggregate demand curve for housing in this case?