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?
