Exercise 7

Exercise Session 7, November 15th , 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua Chen (Cissy), Natalia Guseva 1. Consider an economy with 100 units of labor. It can produce chocolates x or watches y: To produce x chocolates, it takes x2 units of labor, likewise y 2 units of labor are needed to produce y units of watches. Economy has the following objective function F (x; y) = ax + by: Solve for the optimal amounts of x and y. 2. Consider the state planning where the production of the various goods is already known and the only remaining question is that of distributing them among the consumers. There are C consumers labeled c = 1; 2; :::; C and G goods, labeled g = 1; 2; :::; G: Let X be the …xed total amount of good g and xcg the amount allocated to consumer c: Each consumer’s utility is a function only of its own allocation, uc = U c (xc1 ; :::; xcG ) and has income Ic : Social welfare is a function of these utility levels w = W (u1 ; u2 ; :::; uC ): The constraints are that for each good, its allocation to the individuals should add up to no more than the total amount available. Solve the maximization problem of the social planner. Now, assume that each individual may choose freely the demand for each type of good, look for conditions in the distribution of income (Ic ) such that the quantities chosen optimally by the planner coincide with the (optimal) quantities of the individual. 3. A person can buy various brands of liquor at his home-town store, or at the duty free stores of the various airports he travels through. The duty free stores have cheaper prices, but the total quantity he can buy there is restricted by his home country customs regulations. There are n brands. Let p be the row vector of home-town prices and q that of duty free prices. The duty free prices are uniformly lower: q p: Let x be the column vector of his home town purchases and y that of the duty free. Suppose all variables are continuous. The amount of income he has decided to spend on liquor during the year is …xed at I: The budget constraint is then px + qy I: The number of bottles allowed in the customs is K; y1 + ::: + yn K: Solve the agent’s problem, assuming that his utility U (c) depends only on total consumption c = x + y; and x and y are restricted to be non negative. 4. Production function Q of a …rm is given by a combination of labor(L) and 1= capital(K), Q = (KL) : Let w be the wage rate and r the user cost of capital. Solve the cost minimization problem faced by the …rm and obtain the long-run cost function, C(w; r; Q). Now, de…ne the short-run as the period in which the capital cannot be chosen optimally (it’s …xed), …nd short-run cost function C(w; r; Q; K):

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