Solve for the household’s optimal bundle (c*, h*). Your answer will depend on the parameter α. Suppose the government subsidizes housing by reimbursing a household for a fraction t of their spending h. The household only pays (1 − t)h out of their own pocket.

Problem Set 5 Due April 27th before Class Begins
You may submit typed or scanned .pdf’s on the NYU classes website or, if necessary, photos of your work. All submissions must be strictly prior to the start of class (4:55pm EST).

1. A household has Cobb-Douglass utility over housing h and other goods c: u(c,h) = α lnc + (1 − α)lnh. The household’s budget constraint is
c + h ≤ y , where y is total income and we are normalizing the prices of housing and other goods to one. Assume there is no labor supply decision, so y is fixed. You can think of h as capturing housing quality or expenditure, where higher quality housing is more expensive.

(a) Solve for the household’s optimal bundle (c*, h*). Your answer will depend on the parameter α.
Suppose the government subsidizes housing by reimbursing a household for a fraction t of their spending h. The household only pays (1 − t)h out of their own pocket.

(b) Solve for the optimal bundle (c(t), h(t)) as a function of the subsidy rate. How does it compare to (c*, h*)? (c) Graphically illustrate how the housing subsidy affects the household’s chosen bundle. Decompose the income and substitution effects (you may draw multiple graphs).

Now suppose that the subsidy is a fixed amount H that can be spent on housing. If a household spends less than H on housing, they pay nothing out of their own pocket. If they spend more than H, they pay h − H and the government pays the rest.
(d) Calculate the optimal bundle (c(H), h(H)) as a function of H. For which values of H does h(H) = H? (e) Graphically, decompose the income and substitution effects of a fixed subsidy H in the two cases of (i) h(H) = H, and (ii) h(H) > H. (f) In which case is the housing subsidy equivalent to an unrestricted cash transfer?

2. You earn \$40,000 per year at your current job. With probability p, you will lose your job next year and earn only \$10,000. With probability 1 − p you will keep your job and continue to earn \$40,000. Your utility function over your income y is
u(y) = √y
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