# Using a two-period model, this problem investigates tenure choice in the presence of a down-payment requirement along with the incentives for…

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 consumption 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-occupier, which happens in the second period, the consumer must accumulate 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 = δ = g = 0). But the mortgage interest rate equals 5 percent, so that i = 0.05, and the consumer’s income tax rate is t = 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 undertake 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

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 consumption 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-occupier, which happens in the second period, the consumer must accumulate 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 = δ = g = 0). But the mortgage interest rate equals 5 percent, so that i = 0.05, and the consumer’s income tax rate is t = 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 undertake 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.