Public Housing and Housing Subsidies: In exercise 2.14, you considered two different public housing programs in parts A(a) and (b) —one where a family is simply offered a particular apartment for a below-market rent and another where the government provides a housing price subsidy that the family can use anywhere in the private rental market.
A. Suppose we consider a family that earns $1500 per month and either pays 50 cents per square foot in monthly rent for an apartment in the private market or accepts a 1500 square foot government public housing unit at the government’s price of $500 per month.
(a) On a graph with square feet of housing and “dollars of other consumption”, illustrate two cases where the family accepts the public housing unit—one where this leads them to consume less housing than they otherwise would, another where it leads them to consume more housing than they otherwise would.
(b) If we use the household’s own judgment about its well-being, is it always the case that the option of public housing makes the households who choose to participate better off?
(c) If the policy goal behind public housing is to increase the housing consumption of the poor, is it more or less likely to succeed the less substitutable housing and other goods are?
(d)What is the government’s opportunity cost of owning a public housing unit of 1500 square feet? How much does it therefore cost the government to provide the public housing unit to this family?
(e) Now consider instead a housing price subsidy under which the government tells qualified families that it will pay some fraction of their rental bills in the private housing market. If this rental subsidy is set so as to make the household just as well off as it was under public housing, will it lead to more or less consumption of housing than if the household chooses public housing?
(f) Will giving such a rental subsidy cost more or less than providing the public housing unit? What does your answer depend on?
(g) Suppose instead that the government simply gave cash to the household. If it gave sufficient cash to make the household as well off as it is under the public housing program, would it cost the government more or less than $250? Can you tell whether under such a cash subsidy the household consumes more or less housing than under public housing?
B. Suppose that household tastes over square feet of housing (x1) and dollars of other consumption (x2) can be represented by u(x1,x2) = αlnx1 +(1−α) lnx2.
(a) Suppose that empirical studies show that we spend about a quarter of our income on housing. What does that imply about α?
(b) Consider a family with income of $1,500 per month facing a per square foot price of p1 = 0.50. For what value of α would the family not change its housing consumption when offered the 1500 square foot public housing apartment for $500?
(c) Suppose that this family has α as derived in B(a). How much of a rental price subsidy would the government have to give to this family in order to make it as well off as the family is with the public housing unit?
(d) How much housing will the family rent under this subsidy? How much will it cost the government to provide this subsidy?
(e) Suppose the government instead gave the family cash (without changing the price of housing). How much cash would it have to give the family in order to make it as happy?
(f) If you are a policy maker whose aim is to make this household happier at the least cost to the taxpayer, how would you rank the three policies? What if your goal was to increase housing consumption by the household?