Suppose, as in end-of-chapter exercise 6.9, you have $400,000 to spend on “square feet of housing” and “all other goods”. Assume the same is true for me.
A. Suppose again that you initially face a $100 per square foot price for housing, and you choose to buy a 2000 square foot house.
(a) Illustrate this on a graph with square footage of housing on the horizontal axis and other consumption on the vertical. Then suppose, as you did in exercise 6.9, that the price of housing falls to $50 per square foot after you bought your 2000 square foot house. Label the square footage of the house you would switch to hB .
(b) Is hB smaller or larger than 2000 square feet? Does your answer depend on whether housing is normal, regular inferior or Giffen?
(c) Now suppose that the price of housing had fallen to $50 per square foot before you bought your initial 2000 square foot house. Denote the size of house you would have bought hC .
(d) Is hC larger than hB ? Is it larger than 2000 square feet? Does your answer depend on whether housing is a normal, regular inferior or Giffen good?
(e) Now consider me. I did not buy a house until the price of housing was $50 per square foot— at which time I bought a 4000 square foot house. Then the price of housing rises to $100 per square foot. Would I sell my house and buy a new one? If so, is the new house size hB′ larger or smaller than 4000 square feet? Does your answer depend on whether housing is normal, regular inferior or Giffen for me?
(f) AmI better or worse off?
(g) Suppose I had not purchased at the low price but rather purchased a house of size hC′ after the price had risen to $100 per square foot. Is hC′ larger or smaller than hB′ ? Is it larger or smaller than 4000 square feet? Does your answer depend on whether housing is normal, regular inferior or Giffen for me?
B. Suppose both you and I have tastes that can be represented by the utility function u(x1,x2) = x10.5 x20.5 , where x1 is square feet of housing and x2 is “dollars of other goods”.
(a) Calculate the optimal level of housing consumption x1 as a function of per square foot housing prices p1 and income I .
(b) Verify that your initial choice of a 2000 square foot house and my initial choice of a 4000 square foot house was optimal under the circumstances we faced (assuming we both started with $400,000.)
(c) Calculate the values of hB and hC as they are described in A (a) and (c).
(d) Calculate hB′ and hC′ as those are described in A(d) and (f ).
(e) Verify your answer to A (e).