Exercise 7.1 Although adjustment to the equilibrium may take a long time

Exercise 7.1

Although adjustment to the equilibrium may take a long time in a

stock-fl ow housing model, adjustment is fast under some circumstances,

which makes for an easy analysis. This problem considers such

a case and illustrates the effect of rent control. Suppose that the initial

demand curve for housing is given by p = 3 – H , where p is the rental

price per square foot of housing and H is the size of the stock in square

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feet. Note that this equation gives the height up to the demand curve

at any H . The fl ow supply curve for housing is given by p = ? H + 2,

where ? H is the change in the stock. Again, this equation gives the

height up to the fl ow supply curve at any value of ? H . Note that the

slopes of the two curves are – 1 and 1, respectively, a fact that allows

simple answers to be derived below.

(a) Compute the equilibrium price p e (the price at which ? H = 0).

(b) Suppose that prior to the demand shock, the housing market is in

equilibrium, with a stock of size H = 1. Verify that the price in the

market equals p e when the stock is this size.

After the demand shock (e.g., arrival of the Cuban refugees), demand

increases to p = 8 – H .

(c) With the new higher demand, the price in the market shoots up to

a higher value, denoted by p '. Compute p '.

(d) Next, compute the change in the housing stock that occurs as

developers respond to this new price (compute ? H ). Then, compute the

new size for the housing stock, which equals the original stock plus ? H .

(e) Compute the price that prevails in the market after this increase in

the housing stock. Is further adjustment of the stock required to reach

equilibrium? How many periods does it take for the market to reach

the new equilibrium?

Instead of following the sequence you have just analyzed, now

suppose that rent control is imposed immediately after the demand

shock, with the controlled price set at p c = 3.

(f) Compute H' , the stock size at which rent control ceases to have an

effect (in other words, the stock size where the equilibrium price is

equal to p c ). How many periods does it take for the stock to reach H ‘

under rent control?

(g) How many periods does it take for the market to reach the new

equilibrium, where p = p e ?

(h) Illustrate your entire analysis in a diagram.

(i) On the basis of your analysis, does rent control seem like a good

response to a demand shock?

Exercise 7.2

This problem illustrates a consumer ’ s decision to be homeless in the

presence of a minimum housing-consumption constraint, imposed

through misguided government regulation. Let c denote “ bread ” conExercises

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sumption and q denote housing consumption in square feet of fl oor

space. Suppose that a unit of bread costs $1 and that q rents for $1 per

square foot. The consumer ’ s budget constraint is then c + q = y , where

y is income, which equals $1,000 per month.

(a) Plot the budget line, putting q on the vertical axis and c on the

horizontal axis. What is the budget line ’ s slope?

(b) Suppose that minimum housing-consumption constraint says that

q must be 500 square feet or larger. Show the portion of the budget line

that is inaccessible to the consumer under this constraint. Assuming

the consumer rents the smallest possible dwelling, with q = 500, what

is the resulting level of bread consumption?

Assume that the consumer ’ s utility function is given by U( c , q ) = c

+ ? ln( q + 1), where ln is the natural log function (available on your

calculator). Using calculus, it can be shown that the slope of the indifference

curve at a given point ( c , q ) in the consumption space is equal

to – ( q + 1)/ ? .

(c) Assume that ? = 101. Supposing for a moment that the minimum

housing-consumption constraint were absent, how large a dwelling

would the consumer rent? The answer is found by setting the

indifference-curve slope expression equal to the slope of the budget

line from (a) and solving for q . Note that this solution gives the

tangency point between an indifference curve and the budget line. Is

the chosen q smaller than 500? Illustrate the solution graphically.

Compute the associated c value from the budget constraint, and substitute

c and q into the utility function to compute the consumer ’ s

utility level.

(d) Now reintroduce the housing-consumption constraint, and consider

the consumer ’ s choices. The consumer could choose either to be

homeless, setting q = 0, or to consume the smallest possible dwelling,

setting q = 500. Compute the utility level associated with each option,

and indicate which one the consumer chooses. Compute the utility loss

relative to the case with no housing-consumption constraint. Illustrate

the solution graphically, showing the indifference curves passing

through the two possible consumption points.

(e) Now assume that ? = 61. Repeat (c) for this case.

(f) Repeat (d).

(g) Give an intuitive explanation for why the outcomes in the two cases

are different.