Question

In this question, you will carry out the algebraic equivalent to the diagrammatic analysis investigating the effect of amenities on incomes and real-estate prices. To start, let the consumer utility function be given by q^1/2c^1/2a^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/(2p), where p is the price per unit of real estate.  PART D ONWARDS ONLY

(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. (PART D ONWARDS ONLY)

solved: U = (y/2)(sqrt(a/p))

(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? (PART D ONWARDS ONLY)

With free mobility, everyone must enjoy the same utility level regardless of where they live. Let this constant utility level be denoted by ū.

Solved: U = (y/2)(a/p)^(1/2)

(c) Set the utility expression from (a) equal to ū. The resulting equation shows how y and p must vary with amenities a in order for everyone to enjoy utility ū. 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. (PART D ONWARDS ONLY)

solved: ū = (y/2) (a/p)^(1/2)

P= (ay^2)/4(u^2)

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^1/2L^1/2a^θ, 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 θ 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^1/2y^1/2a^{– θ}.

(d) This function above shows that an increase in p or y raises unit cost, but that an increase in a  could either raise or lower costs. Give an example for each possibility, identifying an amenity a  valued by consumers that could alternatively raise, or lower, production costs for particular goods.  

(e) The condition ensuring that costs are constant across locations can be written as p^1/2y^1/2a^{– θ}  = 1. Suppose that θ >  0, so that higher amenities reduce costs. What must happen to p as a  increases to keep costs constant, with y held fixed? Give an intuitive explanation for your answer.

(f) To generate an explicit solution for y in terms of amenities, take the p solution from (c) and substitute it into the constant-cost condition from (e). The resulting equation just involves y, and use it to solve for y as a function of a.

(g) Suppose that θ  is negative. Using your solution from (f), how does y change when a increases? Suppose instead that θ is positive but that its magnitude is unknown. Can you say how y responds to an increase in a? How about if θ is positive and small? How about if θ is positive and large? How about if θ is zero?

(h) Now take the y solution from (f) and use it to eliminate y from the p solution in (c). Solve the resulting equation for p as a function of a. Suppose that θ is positive. How does p change when a increases? Suppose instead that θ is negative but that its magnitude is unknown. Can you say how p responds to an increase in a? How about if θ is negative but close to zero? How about if θ is negative and far from zero? How about if θ equals zero?

(i) Summarize your conclusions about how amenities affect incomes and real-estate prices. Although some conclusions are ambiguous, it is possible to offer a clear-cut statement when the effect of amenities on production is “ small, ” either positive or negative (with θ close to zero). What conclusion can be stated in this case?

(j) Relate your answer from (i) to the diagrammatic analysis from the chapter.

Answer 1