Economics 713: Notes on the Assignments

Assignment 1: Problems 2, 12, 71

12. The statistician's utility function is the power of the test, and the budget constraint is the size of the test. Since utility is linear, and there is only one unit of each good available, the goods are ordered according to marginal utility per dollar, and the consumer keeps buying until all the money is gone. If the budget (size) runs out when the next item is too expensive, the consumer will jump to another item that has lower MU per dollar, with a lower price tag. But Neyman-Pearson randomizes on whether to include this point in the rejection region: it does not skip to another sample point with a smaller likelihood ratio. And the consumer problem can be solved this way as well, with the interpretation that the consumer offers a suitable bet to the seller.

Assignment 2: Problems 11, 16, 70, 14.B.4 (Mas-Colell, page 507)

• Answer for Problem 11

16. The surprise is that the new Walrasian equilibrium is worse for the first person than the alternative of starting from the old equilibrium and consuming the extra endowment. This just means that the person could do better by acting as a monopolist.

Assignment 3: Problems 6, 55, 90, 14.BB.1 (Mas-Colell, page 510)

Assignment 4: Problems 69, 82, 21C2 (Mas-Colell, page 812), 21D6 (Mas-Colell, page 813)

Assignment 5: Problems 27, 74, 83, 13C5 (Mas-Colell, page 475),

27. Quasiconvexity can be stated as F(x) <= max [F(x'),F(x")], where x is a convex combination of x' and x".
Proof of Quasiconvexity of v(p,y) in p. Let p and p' be two price vectors, and let p" = tp+(1-t)p' be a convex combination of these, with 0 <= t <= 1. v" = v(p",y) = u(x") for some x" with p"<=x" <= y. Say p<=x" <= y. Then v" <= max(v,v'), since either x" or x (perhaps both) must be available at the price vector p".
NOTE: this does not require that u is quasiconcave (although it does require that a maximum exists). Let p" be Houston, p is LA and p' is Miami. Then Houston can't be strictly better than both LA and Miami.

74. See Excel Worksheet

Assignment 6: Problems 72, 84, 85, 87

85. The offers should be two (p,x) combinations, one for the high type and one for the low type. If the low-type's offer is accepted, there is an incentive to renegotiate.
There is a full-commitment solution and a no-commitment solution and an in-between case with commitment not to renegotiate.
Gross profit is 2a² - (a-[x-3])², so the optimal choice of x is a+3. This means the full commitment solution is either the separating value 32/2 (x = 7 with R = 16) or the pooling value 2 (x = 4 with R = 2), and the separating value is obviously better. But if the vineyard can commit to no renegotiation, then it's possible to offer a distorted contract that would be accepted by both types, with an expected payoff of 11.
A complication: if x is zero, both winery types make negative profits: (a-3)² - 18. So perhaps this can be used as a threat. More plausibly, the winery has the option to close down.

87. There isn't enough information to determine factor prices after trade opens up. It should be obvious that the conditions for factor price equalization are not met: both technologies are symmetric in K and L, so if (v,w)=(a,b) is an equilibrium factor price pair, then (v,w)=(b,a) is also an equilibrium pair. Thus it is possible to have (v,w)=(a,b) in one country, and (v,w)=(b,a) in the other.
The cost functions are c = 2sqrt(vw) for the Cobb-Douglas, and sqrt(c)=sqrt(v)+sqrt(w) for the CES technology. Then if v=1 and w=4, the unit cost is 4 for the Cobb-Douglas, and 9 for the CES, so if these are the world prices, there are two equilibrium factor price combinations for each country. But the endowments might not sustain an interior equilibrium [check this].

Assignment 7: Problems 58, 59, 10.E.3 (Mas-Colell, page 347),
Due: noon, Monday, May 3