Economics 713: Assignment 1
Q1. An inventor has discovered a new method of producing a precious stone, using spring water found only in Venice and Tipton. The process is patented and manufacturing plants are set up in both places. The product is sold only in Europe and America. Trade laws are such that the price must be uniform within Europe and America, but the European and American prices may differ. Transport costs are negligible, and there is no second-hand market in the stones because of the risk of forgeries. From the production and marketing data given below, determine the profit-maximizing production and sales plans. In particular, determine the output in Venice and Tipton, sales in America and Europe, quantity shipped from America to Europe or vice versa, and prices in America and Europe.
Demand: America, p = 1500 - ½ Q; Europe, p = 1000 - Q
Average Cost: Tipton, AC = 150 + .375 Q; Venice, AC = 100 + ½ Q
Q2. The Likelihood Ratio Test (Neyman-Pearson Lemma)
A statistician expects to obtain a vector of data generated by one of two probability distributions, p or q. The vector will lie in a finite set X = {x¹, x2, ,...,xi ..., xS}, called the sample space, and the probabilities associated with the points in this set are either p1,p2,...,pi,...,pS or q1,q2,...,qi,...,qS, according to whether p or q is the true distribution.
The statistician wishes to design a test of the hypothesis H0 that the true distribution is p. The test involves dividing X into two parts, R and A, rejecting H0 if the observed data lie in R and accepting H0 if the data lie in A. Two types of error are possible here: the test might reject H0 when it is actually true, or it might accept H0 when it's false.
The statistician is primarily concerned about errors of the first type. Provided that these can be held to an acceptable level it is also desirable to avoid errors of the second type. Specifically, the aim is to maximize the probability that H0 will be rejected when it's false (this is called the power of the test), subject to the constraint that the probability of type I error should not exceed some number (called the size of the test).
(a) Translate this problem into the language of consumer theory. (What is the utility function? What is the budget constraint?).
(b) Solve the problem.
(c) Translate the solution back into language that the statistician can understand. (Note the title of the problem).
Q3. Consider a duopoly model in which firm 1 can choose a technology in period 1, but firm 2's technology is fixed. Firm 1 can pay nothing in period 1 and have period 2 production costs of 2 per unit. Alternatively, it can pay 2q* and have capacity q*; production in period 2 is costless up to q* and infinitely costly thereafter. Firm 2's production costs in period 2 are 2 per unit.
Demand in period 2 is given by P = 10 + e - Q, where Q is industry production and e is a stochastic demand shock. The firms play Cournot in period 2, then the game ends. The expectation of e is zero.
(a) Show that if demand is nonstochastic, (the distribution of e degenerates to a spike at 0) firm 1 will choose the fixed capacity technology.
(b) Now suppose that e is distributed as follows:
e is equal to -1 with probability a
e is equal to 1 with probability a
e is equal to 0 with probability 1 - 2a.
Firms learn the value of e between periods 1 and 2, so that they know e when they play the Cournot game. However, firm 1 does not know e when choosing whether or not to invest in the fixed capacity technology. For what values of a will the firm choose the fixed capacity technology?
(c) Explain the economics of the trade-off driving this model.