Exercise $11\mathrm{.}1$

In this question, you will carry out the algebraic equivalent to the dia$-$ grammatic analysis investigating the effect of amenities on incomes and real$-$estate prices. To start, let the consumer utility function be given by q$1/2$c$1/2$a$1/2,$ where c is consumption of $$bread$($a catch$-$all commodity$),$ q is real estate $($housing$),$ and a is amenities, which are valued by the consumer given that a$$s exponent is positive. Letting y denote income, it can be shown that the consumer demand functions for bread and housing are given by c $=$ y$/2$ and q $=$ y$/(2$p$),$ where p is the price per unit of real estate.

$($a$)$ Substitute the above demand functions into the utility function to get what is known as the $$indirect$$ utility function, which gives utility as a function of income, prices, and amenities.

$($b$)$ Using your answer from $($a$),$ how does utility change when income y rises? When the real$-$estate price p rises? How does utility change when amenities increase?

With free mobility, everyone must enjoy the same utility level regard$-$ less of where they live. Let this constant utility level be denoted by u$.$

$($c$)$ Set the utility expression from $($a$)$ equal to u$.$ The resulting equa$-$ tion shows how y and p must vary with amenities a in order for

Exercises $271$

everyone to enjoy utility u$.$ To see one implication of the equation, solve it to yield p as a function of the other variables. According to your solution, how must p change when amenities rise, with y held constant? Given an intuitive explanation of your answer. How must p change if y were to rise, with amenities held constant? Again, explain your answer.

As was explained in the chapter, another condition is needed to pin

down an explicit solution that tells how y and p vary as amenities

change. That condition comes from requiring that the production cost

of firms be constant across locations. To generate this condition, let the

production function for bread be given by Dq$1/2$L$1/2$a$,$ where q now

represents the firm$$s real$-$estate input, L is labor input and a again is

amenities $($D is a constant$).$ The exponent could be either positive or

negative, indicating that an increase in a could either raise or lower

output. Recalling that p is the price of real estate and y is the price of

labor, it can be shown that the cost per unit of bread output is equal to p$1/2$y$1/2$a$$ $.$