# Final Assignment There are 3 questions and 40 points in total You need to show a

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Final Assignment
There are 3 questions and 40 points in total
You need to show all steps of your work, only writing the final answer is not enough.
Question 1 [13 points] A seller of a good chooses its price (p ≥ 0) and quality (q ≥ 0). The cost of quality q for the seller is C(q) = q2 .
A buyer can be either of type 1 or type 2. If a type 1 buyer purchases the good of quality q at price p, its net utility is 2q − p. If a type 2 buyer purchases the good of quality q at price p, its net
utility is 3q − p. Any buyer who does not purchase the good gets zero net utility. The seller knows the fraction 1/2 of buyers is type 1 while the remaining 1/2 is type 2.
(a) [4 points] Suppose the seller offers a menu of price-quality pairs ((p1, q1), (p2, q2)) where (pt, qt) is intended for type t for t = 1,2. For the two types, write down the individual rationality
constraints IR1,IR2 and the incentive compatibility constraints IC1,IC2 .
(b) [4 points] From the constraints above, show that q2 ≥ q1 .
(c) [5 points] We know that at any menu that maximizes profit of the seller: IR1, IC2 hold with equality and IR2, IC1 can be ignored (you don’t have to prove these results). Using these results, determine menu that maximizes profit of the seller.
Question 2 [17 points] A credit market has two types of borrowers: s (safe) and r (risky); each has proportion 1/2. Any borrower borrows 1 unit of capital to invest in a project. A project can result in either one of the two outcomes: good or bad. Under bad outcome, the return is 0. Under good outcome, the return is xs = 108 for type s and xr = 111 for type r. The probability of good outcome is ps = 2/9 for type s and pr = 1/6 for type r.
A credit contract is given by interest i (which includes both principal and interest). Under this contract, a borrower pays back i to lender if the outcome is good and pays back nothing if the outcome is bad. The opportunity cost of a borrower is B0 = 12. The opportunity cost of a lender is L0 = 7. Assume the credit market is competitive, so a lender makes zero net profit. Showing all steps of your work, answer the following questions.
(a) [3 points] Find the maximum acceptable rate of interest for each type.
(b) [5 points] Consider the full information case where a lender knows types of individual borrowers. Determine interest rates offered, which type gets loan and the aggregate income.
(c) [9 points] Consider the asymmetric information case where a lender does not know types of individual borrowers and only knows there is proportion 1/2 of each type. Determine interest rate offered, which type gets loan and the aggregate income. Then determine if there is a problem of underinvestment or overinvestment.
Question 3 [10 points] Consider an economy with n = 3 goods. The representative consumer of the economy equally weights the goods. The economy has 7 units of labour that the representative consumer supplies.
There are three sectors 1, 2, 3 in the economy, with sector i producing good i. Each sector has a competitive fringe of firms. Initially firms in all sectors use the existing technology (traditional mode) in which 1 unit of labour can produce 1 unit of good and the wage is 1.
There is a new technology (modern mode) in which 1 unit oflabour can produce t > 1 units of the good. The modern mode pays higher wage 1 + v. It is known that 7 = 400, t = 4 and 1 + v = 2 (so that v = 1).
Suppose sectors 2,3 are non-industrialized (that is, all firms in sectors 2,3 use traditional mode) and sector 1 is industrialized (that is, all firms in sector 1 use modern mode). Showing all steps of your work, determine (i) the income of economy and (ii) the demand for each of the goods 1, 2, 3.

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