Using a two-period model, this problem investigates tenure choice in the presence of a down-payment requirement along with the incentives for mortgage default. Consumer utility depends on non-housing 1

Using a two-period model, this problem investigates tenure choice in the presence of a down-payment requirement along with the incentives for mortgage default. Consumer utility depends on non-housing con- sumption in each period, which equals what is left after paying housing costs. With c1 and c2 denoting consumption in periods 1 and 2, utility is equal to c1 + δc2, where δ is the discount factor. A high value of δ indicates that the consumer is “patient,” placing a relatively high value on second-period consumption relative to first-period consumption. Everyone is a renter in the first period. To become an owner-occu- pier, which happens in the second period, the consumer must accumu- late a down payment D while renting. At the end of the first period, the consumer purchases a house, which costs V, using the down payment D along with a mortgage equal to M = V – D. The consumer moves in at the beginning of the second period, paying the user cost during that period, and the house is sold at the end of the period. When the house is sold, the mortgage is paid off, and the consumer gets back the down payment. If the consumer instead remains a renter in the second period, there’s no need to accumulate a down payment, and housing cost in the second period just equals rent. Using the previous information, the non-housing consumption levels for an owner-occupier are as follows: c1 = income – rent – down payment and c2 = income – owner-occupier’s user cost + down payment. For a renter, c1 = income – rent and c2 = income – rent. Suppose that the simple model of subsection 6.3.5 (where e = 0) applies, and that property taxes, depreciation, and capital gains are all 0 (h = d = g = 0). But the mortgage interest rate equals 5 percent, so that i = 0.05, and the consumer’s income tax rate is τ = 0.3. In addition, V = 200 and income = 40 (dollar amounts are measured in thousands, so that the house’s value is $200,000). The required downpayment equals 10 percent of the house’s value, so D = 0.1V. For simplicity, let the house size be fixed at q = 1, so that V = v (house value and value per unit are then the same). With this assumption, V can be used in place of v in the user-cost and rent formulas in subsection 6.3.5. (a) Using this information, compute D along with rent R and the owner-occupier’s user cost. Note that the user cost is given by the usual formula, even though a down payment is present. Your answer should show that the owner-occupier’s user cost is less than rent. Note that, to benefit from this lower second-period housing cost, the consumer must save funds for a down payment in the first period. Whether the lower housing cost makes it worthwhile to under- take this saving depends on the consumer’s patience, as you will see below. (b) Using the formulas above, compute c1 and c2 for an owner-occupier. (c) Compute c1 and c2 for a renter. (d) Plug the results of parts (b) and (c) into the utility formula c1 + δc2 to get the utilities of an owner-occupier and a renter as functions of the discount factor δ. (e) Compute the value of δ that makes the consumer indifferent between being a renter and an owner-occupier. Let this value be denoted by δ*. (f) Pick a δ value larger than your δ* (but less than 1) and compare the utilities of the renter and the owner-occupier for this value. Then pick a δ value smaller than your δ* (but greater than 0) and compare the utilities of the renter and the owner-occupier. (g) What do your results say about the effect of consumer patience on the rent-own decision? Recall that a higher δ means greater patience. Give an intuitive explanation of your conclusion. Suppose now that, once the second period is reached, the value of the house drops unexpectedly. Instead of staying at 200, V drops to 190. This drop occurs after the owner-occupier has made the mortgage- interest payment (so that the user cost is already paid). (h) Under the previous numerical assumptions, what is the size (M) of the consumer’s mortgage? Are the proceeds from sale of the house enough to payoff the mortgage? Does the consumer get all of his down payment back? (i) Suppose that V were instead to drop to 170. Are the proceeds from sale of the house enough to pay off the mortgage? If the answer to part (i) is No, the consumer has two options. The first is to default on the mortgage, which means handing the house over to the bank rather than paying off the loan. The second option is to pay off the mortgage, which means paying the bank an additional amount beyond the proceeds from selling the house in order to retire the loan. (j) Suppose that default is costless, so that the default cost C is 0. In the situation from part (i), which option (default or paying off the mort- gage) is better? In other words, which option imposes a lower cost on the consumer? (k) Supposeinsteadthatconsumerincurs$5,000worthofdefaultcosts (the future cost of impaired credit, the psychic cost of guilt, and so on), so that C = 5. Which of the options from part (j) is best? (l) Suppose instead that C = 12. Which option is best?

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