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1

Between 1945 and 1965, average household size in North America increased dramatically (with the emergence of the baby boomers) leading to an increase in the demand for housing for a typical household. How would such a change affect the slope of the housing price function in a city? Would this tend to encourage or discourage the suburbanization of population? Explain your reasoning.

The slope of the housing price function is -t/h , where h is housing consumption. Thus, an increase in
family size should make the housing price function flatter. This will encourage suburbanization, since
builders will have an incentive to convert land at the edge of the city from agriculture to urban uses to
supply part of the increase in housing demand.

2

Consider a monocentric city in which all houses contain 1,500 square feet of floor space, and in which monthly commuting cost is $150 per mile round trip. If the price of housing one mile from the city centre is $3 per square foot per month, then what is the cost of 1,500 square feet of floor space 11 miles from the city centre?

The slope of the housing price function is p/h = - t/h* = - 150/1,500 = -0.1. If the price of housing 1 mile from the city centre is $3 per square foot, then the price of housing 11 miles from the city centre must be $2 per square foot. Δp = -( t/h ) Δd = -0.1(10) = -1. Thus, the cost of 1,500 square feet of space 11 miles from the city centre must be 2(1,500) = 3,000 per month.

3

Compare the residential bid rent functions given in equations 6.3 and 10.8, and explain any differences between them.

The only difference is that in equation 10.8, the vertical intercept of the residential land rent function
is reduced by the costs of housing capital (pk)(k).

4

Consider a monocentric city in which the unit cost of commuting is $50 per mile per month round trip. A household located 10 miles from the city centre (d = 10) occupies a dwelling with 1,000 square feet of floor space at a monthly rent of $800. Non-land cost per dwelling is $400. Housing demand is perfectly inelastic. Residential developers are flexible (i.e., they engage in factor substitution), and the number of housing units per acre is given by 20 - (8/5)d.

(a) How many units per acre are built at d = 10?
(b) What is the price per square foot of housing at d = 10?
(c) What is the bid rent per acre for residential land at d = 10?
(d) What is the price of housing at the city centre?
(e) What is the bid rent for residential land at the city centre? Express your answer per unit or per acre.
(f) Derive and illustrate the housing price function.

(a) The number of units per acre at this distance is
20 - (8/5)10 = 4.

(b) p(10) = $800/1,000 = $0.80 ft2.

(c) The residual is r(10) = 4(800 - 400) = $1,600 per acre.

(d) The slope of the housing price function is
p/d = - t/h = - 50/1,000 = -0.05.
Thus p(0) = 0.80 + 10(0.05) = $1.3 ft2.

(e) There are 20 units per acre at d = 0(20 8/5d). The total cost per unit is $1,300 (1.3 x 1,000) per month. Non-land cost is $400 per month. Therefore, bid rent is $900/unit/month ($1,300 - $400) or $18,000 per acre
(900 x 20).

(f) From part d., p(d) = 1.3 - 0.05d.
Students should include a diagram in their answer.
Diagram Located in Review and discuss answers question 4

5

Define the concept of "filtering" and discuss its implications for housing policy toward low income families.

In the context of housing markets, “filtering” refers to the fact that as a house ages, both its quality (or
the number of units of housing services that it provides) and the income of its occupants tends to
decrease. The most important implications of filtering for low income housing policy are that it may
be better to let the market supply low income families with used housing than to try to construct new
housing for them, and that policies that encourage housing construction at higher quality levels may
actually benefit the poor by accelerating the filtering process.

6

You are considering purchasing a $250,000 house. Your down payment is $70,000, the opportunity cost of equity is 10% per year, the mortgage interest rate is 8% per year, the property tax rate is 1% per year, and annual maintenance expenditures and depreciation together amount to 0.25% of property value. To begin, assume that the house is not expected to appreciate.

(a) The market rent on the house is $2,000 per month. Are your housing costs lower if you rent or own?
(b) How might your answer to part (a) change if you now assume that the property is expected to appreciate at 1.5% per year?
(c) Suppose that you are expecting to be transferred to another city in 12 months. The cost of selling your house (in current dollars) will equal 6% of its current value, but it is expected to appreciate at 1.5% per year, as in part
(b). Are your housing costs lower if you rent or own under these conditions?

(a) By equation 10.1, the user cost is:
UC = 0.08(180,000) + 0.1(70,000) + (0.01 + 0.0025)(250,000) = $24,525 or $2,044 per month. Thus, your housing costs are lower if you rent.

(b) The user cost is now:
UC = 0.08(180,000) + 0.1(70,000) + (0.01 + 0.0025 - 0.015)(250,000) = 20,775 or 1,731 per month. Now, your housing costs are lower if you own.

(c) Now your annual user cost is UC = 20,775 + 0.06(250,000 x 1.015) = 36,000 or 3,000 per month. Your housing costs are much lower if you rent.

7

Assuming that consumers purchase two goods: housing services h and other goods x. Housing services is plotted on the horizontal axis and other goods is plotted on the vertical axis. Generally, where will a consumer choose to locate on a graph, given their budget constraint so (I = p × h + x ), and the indifference curves represent different amounts of utility?

(1) Consumers will choose to locate where the utility curve is equal to the supply curve.
(2) Consumers will choose to locate where the budget constraint is tangent to the highest possible indifference curve.
(3) Consumers will choose to locate where the demand curve crosses the indifference curve.
(4) Consumers will choose to locate where the quantity of housing services equals other goods.

Answer: (2)
The demand for housing comes from the utility maximizing choices of consumers. Utility maximization
will generally occur at a point where the budget constraint is tangent to the highest possible indifference
curve.

8

Why would Joanne, a consumer, continue to live in the same house even if her indifference curve changes?

(1) Because of the adjustment costs, a consumer may be better off staying in the same house even though her preferences for housing have changed.
(2) If she chooses to move, utility will be at a lower level than it would be if she did not move.
(3) Options (1) and (2) are both incorrect.
(4) Options (1) and (2) are both correct.

Answer: (4)
Because of adjustment costs, Joanne may be better off staying where she is, even though her preference
for housing has changed. If she stays in her current accommodation, her utility level will be higher than
the highest level she can achieve by moving.

9

In general, housing demand is inelastic with respect to both price and income. This means:

(1) that the absolute values of both elasticities are higher than one, so Eh,p > -1 and Eh,i > 1.
(2) that the absolute values of both elasticities are lower than one, so Eh,p > -1 and Eh,i

Answer: (2)
When we say that demand is inelastic with respect to both income and price, this means that the absolute
value of both elasticities is less than one, so Eh,p > -1 and Eh,i

10

Assume that housing services are produced from capital and land, and that the amount of land that a particular property occupies is fixed. Also assume that the production function for housing services in period t is H t = f(K ), where K is structural capital in period t. t t The level of new investment in period t is represented by K , and depreciation on the property in period t is represented by dt. The marginal t benefit of investing is given by the value of the marginal product of structural capital: MB = p × MP . k In each period, how much should the property owner invest in the property?

(1) The owner should choose an amount that makes the amount of new investment equal to the amount of depreciation.
(2) The owner should choose kt, so that the marginal benefit of investing is equal to the marginal cost of supplying.
(3) The owner should invest an amount that makes the marginal benefit of investing equal to the marginal cost of investing.
(4) The owner should invest a large amount of money making the owner's house more valuable.

Answer: (3)
Each time period, the property owner has to decide how much additional capital to invest in the
property. To maximize the value of his or her holdings, the owner should choose kt, so that the
marginal benefit of investing and the marginal cost of investing are equal. The marginal cost of
investing another unit of capital in the property is simply the opportunity cost. The marginal benefit
of investing is given by the value of the marginal product of structural capital: MB = p × MPk.

11

In the short-run, the general view of the relationship between housing supply and the price of housing is that the stock of housing is essentially fixed and the supply of housing is very inelastic. This implies that in the short-run, the price of housing will be determined by ________________, while the quantity of housing will be determined by _____________________.

(1) demand conditions, supply conditions
(2) supply conditions, economic conditions
(3) income conditions, demand conditions
(4) market price, supply conditions

Answer: (1)
This implies that in the short-run, the price of housing will be determined by demand conditions, while
the quantity of housing will be determined by supply conditions. Conversely, the general view is that
in the long-run the supply of housing is very elastic. This implies that in the long-run, the price of
housing is determined by cost factors while housing quantity is determined by demand factors.

12

Consider a monocentric city in which all houses contain 1,300 square feet of floor space, and where the cost of commuting is $65 per kilometre per month, round trip. The price of housing is $1 per square foot at a distance of 10 kilometres from the city centre. What is the price of housing (per square foot) at the city-residential border?

(1) $0.95
(2) $1.00
(3) $1.50
(4) $2.47

Answer: (3)
From the equation, change in p = -(t/h*) × change in d, thus, -(65/1,300)(-10) = 0.5. Thus the price of housing at the city border is $0.50 + $1.00 = $1.50 per square foot. The housing price function is a straight line with a slope of -t/h* = -65/1,300 = 0.05, so it has the form p(d) = a - 0.05d, where a is the vertical intercept. Then p(10) = 1, which implies 1 = a - 0.05(10), a = 1.50, and the housing
price function p(d) = 1.50 - 0.05d. When we substitute d = 0 into the housing price function, we get a price per square foot of p(d) = 1.5 - 0.05(0) = $1.50.

13

Housing is supplied by a competitive building industry. Builders construct housing at different locations and then sell or rent it to households. The selling price/rent they charge at a distance of 5 kilometres is p(5)= $1.85 per square foot. Every household in the city consists of 43 units of capital, 1,100 square feet of floor space, and 1,400 square feet of land. All buildings in this city are the same height. The price of capital is $5. In the long-run, how much would a builder be willing to pay in land rent, r(d), for a piece of land they will build on, 5 kilometre from the city centre?

(1) $1.00 per square foot
(2) $1.07 per square foot
(3) $1.21 per square foot
(4) $1.30 per square foot

Answer: (4)
The profit of a builder who supplies housing at location d is pie = p(d) × h* - pk × k* - r(d) × l*.
Competition forces profits to zero in the long-run and therefore 0= p(d) × h*1 pk × k*-! r(d) × l*.
If we substitute in all the variables [p(d) = 1.85, h* = 1,100, pk = 5, k* = 43, l* = 1,400] into the formula, we have
r(d) = ((1.85 × 1,100) - (5 × 43))/1,400, then r(d) = $1.30.

14

Why are residential bid rent functions convex?

(1) The residential bid rent function is convex because households can substitute housing for other goods.
(2) The residential bid rent function is convex because houses closer to the city centre rise in price much more quickly.
(3) The residential bid rent function is convex because land price rises faster than lot size.
(4) None of the above answers are correct.

Answer: (1)
If households can substitute housing for other goods, they will attempt to reduce their housing
consumption when the price of housing is high (i.e., at locations near the city centre). This will cause
the housing price function to be convex - to be steeper near the city centre than near the boundary of
the city. If builders can also substitute capital for land in housing production, then they will try to
economize on their use of land where the rent on land is high (i.e., at locations near the city centre).
This will cause the land rent function to become convex as well, and will lead to higher population and
structural densities near the city centre than near the boundary of the city.

15

Houses have complex bundles of characteristics. Individual characteristics are not traded on a market and therefore do not have market prices. Hedonic pricing isolates the contribution of each characteristic by:

(1) dividing the bundles by average income and then multiplying the number by the utility.
(2) estimating the marginal value of each characteristic. (3) making all bundles of housing and other goods equal to the same utility.
(4) All of the above statements are incorrect.

Answer: (2)
Hedonic pricing estimates the marginal value or "implicit price" of each characteristic of a property.
This information allows us to construct a measure of the price of housing in different areas or at
different points in time where the characteristics of the houses are held constant. Such "constant quality
price indices" can then be used to make correct inferences about how housing prices vary across space
or over time.

16

What are the implications of filtering for housing policy toward low-income families?

(1) It may be better to let the market supply low-income families with used housing than to try to construct new housing.
(2) Policies that encourage housing construction at higher levels may actually benefit the poor by accelerating the filtering process.
(3) Regulations that restrict the supply of housing in higher-income and higher-quality submarkets may have the effect of worsening housing conditions in lower-quality submarkets, since supply restrictions tend to raise prices, encourage housing maintenance and improvements, and thus discourage downward filtering. (4) All of the above statements are true.

Answer: (4)
Filtering has implications for housing policy. The most important implications of filtering for low income
housing policy are that it may be better to let the market supply low-income families with used
housing for them, and that policies that encourage housing construction at higher quality levels may
actually benefit the poor by accelerating the filtering process. Regulations that restrict the supply of
housing in higher-income and higher-quality submarkets may have the perverse effect of worsening
housing conditions in lower quality submarkets, since supply restrictions tend to raise prices, encourage
housing maintenance and improvements, and thus discourage downward filtering.