Suppose you have $400,000 to spend on a house and “other goods” (denominated in dollars).
A. The price of 1 square foot of housing is $100 and you choose to purchase your optimally sized house at 2000 square feet. Assume throughout that you spend money on housing solely for its consumption value (and not as part of your investment strategy).
(a) On a graph with “square feet of housing” on the horizontal axis and “other goods” on the vertical, illustrate your budget constraint and your optimal bundle A.
(b) After you bought the house, the price of housing falls to $50 per square foot. Given that you can sell your house from bundle A if you want to, are you better or worse off?
(c) Assuming you can easily buy and sell houses; will you now buy a different house? If so, is your new house smaller or larger than your initial house?
(d) Does your answer to (c) differ depending on whether you assume tastes are quasilinear in housing or homothetic?
(e) How does your answer to (c) change if the price of housing went up to $200 per square foot rather than down to $50.
(f) What form would tastes have to take in order for you to not sell your $2000 square foot house when the price per square foot goes up or down?
(g) True or False: So long as housing and other consumption is at least somewhat substitutable, any change in the price per square foot of housing makes homeowners better off (assuming it is easy to buy and sell houses.)
(h) True or False: Renters are always better off when the rental price of housing goes down and worse off when it goes up.
B. Suppose your tastes for “square feet of housing” (x1) and “other goods” (x2) can be represented by the utility function u(x1,x2) = x1x2.
(a) Calculate your optimal housing consumption as a function of the price of housing (p1) and your exogenous income I (assuming of course that p2 is by definition equal to 1.)
(b) Using your answer, verify that you will purchase a 2000 square foot house when your income is $400,000 and the price per square foot is $100.
(c) Now suppose the price of housing falls to $50 per square foot and you choose to sell your 2000 square foot house. How big a house would you now buy?
(d) Calculate your utility (as measured by your utility function) at your initial 2000 square foot house and your new utility after you bought your new house? Did the price decline make you better off?
(e) How would your answers to B(c) and B(d) change if, instead of falling, the price of housing had increased to $200 per square foot?