Question

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" consumption and q denote housing consumption in square feet of floor 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 a 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 the minimum housing-consumption constraint says that q must be 500 swuare 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 dwellings, 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+α 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)/α.

C) Assume that α=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 the 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 α=61. Repeat (C) for this case.

F)Repeat (D).

G) Give an intuitive explanation for why the outcomes in the two cases are different.

Answer 1

a) the budget line is given as the constraint of income . so the fesible indifference curve would lie in the budget set.

slope of buget line is the price ratio ie the = price of bread / price of floor space.

b)

the shaded area is the unaccessible part of the budget set since the minimum q is 500. the level of bread at q=500 would also be 500 since total value of y is 1000. so maximum bread consumption would be 500.

c)given the slope of the indiffernce curve(MRS), we find the point of tangency at which within the budget constraint the consumer gets the utility maximised so

equating absolute values of MRS and slope of BL

|-(q+1)/α| = |price of bread / price of floor space |(at α=101)

(q+1)/101 = 1

so q= 100 . so at q=100 c would be 900

yes the chosen q is smaller than 500.

substituting value in utility function we get u(c,q)= 900 + 101ln (101)

= 900 + 101*4.615 = 1366.115

d) when q=0 the utility would be derived from only by consumption of c which is 1000.

In case q=500 the utility is derived from both q and c which is = 500 + 101ln(501)= 1127.877

so utility in case when both are equal to 500 is more than when q=0.

when there is no housing constraint utility is 1366.115

so loss in case of constraint is 238.23 ie (1366.115-1127.87)

diagram for d) part.

e) with α=61 q=60 and c = 1000-60=1940

utility would be 1940+ 61ln(61) = 2190.76

f) loss in utility =2190.76-1127.877=1062.89

g) in both the cases values are different because of α becuse less the value of α more will be the value of c because q would reduce and we know that any value within ln would give less value than the number itself . so more c more will be the utility in case of the given utility function.