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Description

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 c 1 and c 2 denoting consumption in periods 1 and 2, utility

is equal to c 1 + δ c 2 , 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 fi rst-period consumption.

Everyone is a renter in the fi rst 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 fi rst 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:

c 1 = income – rent – down payment

260 Exercises

and

c 2 = income – owner-occupier ’ s user cost + down payment.

For a renter,

c 1 = income – rent

and

c 2 = 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 = 0.1 V . For simplicity, let the house

size be fi xed 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 benefi t from this lower second-period housing

cost, the consumer must save funds for a down payment in the fi rst

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 c 1 and c 2 for an owner-occupier.

(c) Compute c 1 and c 2 for a renter.

(d) Plug the results of parts (b) and (c) into the utility formula c 1 + δ c 2

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

Exercises 261

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 fi rst

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) Suppose instead that consumer incurs $5,000 worth of default costs

(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?