Answer: Let y 2 denote rm 2 s output of food and L 2 denote rm 2 s labor input (so

Transcription

1 The Ohio State University Department of Economics Econ 805 Extra Problems on Production and Uncertainty: Questions and Answers Winter 003 Prof. Peck () In the following economy, there are two consumers, two rms, and two goods (labor/leisure and food). For i =,, consumer i is endowed with zero units of food and unit of leisure,! i =(0; ). Letting x i denote consumer i s consumption of food and `i denote consumer i s consumption of leisure, the utility function is: log(x i ) + log(`i). Let y denote rm s output of food and L denote rm s labor input (so that L must be nonnegative). Then rm s production function, the frontier of its production set, is given by: y = AL, where the parameter A is a positive real number. Firm is owned by consumer. Let y denote rm s output of food and L denote rm s labor input (so that L must be nonnegative). Then rm s production function, the frontier of its production set, is given by: y =(L ) =.Firmisownedbyconsumer. (a) De ne a competitive equilibrium for this economy. (b) Calculate the competitive equilibrium price vector and allocation, as a function of the parameter, A. Assume that we have an interior solution, where both rms produce output. (c) For what values of the parameter, A, will we have a corner solution, where one of the rms produces zero output? Answer: (a) Normalizing the price of food to be and denoting the price of labor as p, a Competitive Equilibrium is a price vector, (;p), and an allocation, (x ;`;x ;`;y ;L ;y ;L ), such that: (i) (x ;`) solves: max log(x )+log(`) x + p` = p (x ;`) 0: This relies on the fact that utility is monotonic and rm has CRS and receives zero pro ts at the CE. (ii) (x ;`) solves max log(x )+log(`) x + p` = p + ¼ (x ;`) 0:

2 (iii) (y ;L ) solves y = AL L 0: max y pl (iv) (y ;L ) solves max y pl y = (L ) = L 0: (v) x + x = y + y ` + ` + L + L = : (Equalities follow from strict monotonicity of utility.) (b) Starting with the pro t maximization conditions, for an interior solution where rm produces a nite positive quantity, we must have zero pro ts: AL pl =0; or p = A. Plugging the constraint into the pro t expression for rm, we have the unconstrained problem: max(l ) = pl : L Setting the derivative equal to zero and solving yields: L = 4p ; y = p ; and ¼ = 4p : The rst order conditions for consumer s utility maximization problem are: Solving, we get: x = p` (MRS condition) x + p` = p (budget) x = p and ` = : The rst order conditions for consumer are: x = p` x + p` = p + 4p :

3 Solving, we get x = p + 8p and ` = + 8p : To get the nal allocation, substitute p = A into the demand functions. Market clearing comes in, not to allow us to solve for the price, but instead to allow us to determine rm s input and output. We get: x = A ;` = ;x = A + 8A ;` = + 8A ; L = 4A ;y = A : Market clearing for labor/leisure implies +( + 8A )+L + 4A =: Solving for L,wehave L = 3 8A, and therefore, y = A 3 8A : (c) Firm will shut down if the previous expression for L is negative, which occurs if A is su ciently small. The condition is, A<( 3 8 )=. If you are wondering what the CE would be in this case, we could recalculate the CE, assuming rm does not produce. The equilibrium wage will end up being p =( 3 8 )=, and indeed rm does best not to produce. () In the following economy, there are consumers, one good, x, and states of nature, a and b. The probabilities of the states are A and B, where A + B =. The utility functions and endowments of the consumers are given by V = A log(x a )+Blog(xb ) (! a ;! b ) = (; 0) V = Ax a + Bxb (! a ;! b ) = (; ) (a) Draw an Edgeworth box diagram, indicating the endowment point and the contract curve. (b) Calculate the C.E. of this economy, as a function of the parameters, A and B. Be careful to remember the constraint that consumption must be nonnegative. (c) Interpret the C.E. as an insurance market. In other words, who is o ering the insurance, what is the amount of the premium, what is the amount of the claim in the event of an accident, and how does the premium compare with the expected value of the claim? 3

4 Answer: (a) The contract curve follows the 45-degree line from (0,0) to (,), then follows the top of the box from (,) to (3,). On the at segment, marginal rates of substitution are not equal, but these points are Pareto optimal. 0.8 b a (b) To calculate the CE, we rst notice that the probabilities will determine whether we have an interior solution, with all consumptions positive, or a corner solution, with x b =0. Let us calculate the demand function for consumer. Normalizing p b =, the rst order conditions are Solving, we have Ax b Bx a = p a p a x a + x b = p a : Ax b B + xb = p a ; or x b = Bpa A + B =Bpa : x a = Axb Bp a =A: If we have an interior solution, consumer s marginal rate of substitution must equal the price ratio, so A B = pa : 4

5 The allocation is found by plugging the price to determine consumer s consumption, and using market clearing to determine consumer s consumption. We have x a = A; x b =A; x a = 3 A; x b = A: This interior equilibrium applies if consumption is nonnegative, A. If we have A>, then we have a corner solution. Consumer s demand is determined by the two equations, Solving, we have x b = 0; and p a x a + xb = p a +: x a = pa + p a : Using market clearing for state b, we have: The allocation is: Bp a +0 = ; so p a = B : x a = A; x b =; x a = +B =3 A; and x b =0: (c) For the interior equilibrium, consumer (who is risk neutral) winds up on the same indi erence curve as his initial endowment. He provides insurance to consumer, at fair odds. To see this, the insurance premium paid by consumer is her endowment in the good state (no accident), less her consumption: premium = A =B: In the bad state, she submits a claim, where her nal consumption equals her endowment plus her claim, minus her premium: A = 0+ claim B; so claim = : The premium is B, and the expected claim is multiplied by the probability of submitting the claim, B, or B. (3) Consider the following exchange economy with two von Neumann-Morgenstern expected utility maximizers, two states of nature, s = and s =, and one commodity per state. We have V i = X ¼ s u i (x s i ); s= ; 5

6 where V i is consumer i s utility function, ¼ s is the probability of state s, and x s i is the consumption of consumer i in state s. Assume that each u i is twice continuously di erentiable, and that we have u 0 i > 0 and u00 i < 0. Also assume that the initial endowments,! s i, are strictly positive for each consumer in each state. (a) If the following statement is true, then carefully argue why, and if it is false, then present a counterexample: If the initial endowments of statecontingent commodities are Pareto optimal for ¼ = ¼ =, then the endowments are Pareto optimal for any other speci cation of the probabilities. (b) De ne a competitive equilibrium for this economy. (c) Does this economy satisfy the assumptions required to apply the Second Welfare Theorem? Answer: (a) This statement is true. Because of our assumptions of di erentiability, etc., Pareto optimality at the original probabilities implies It follows that ¼ (! (! ) = = ¼ (! (! ) : holds, ) solves which implies that the allocation is Pareto optimal for the new endowments. (b) A CE is a price vector, (p ;p ), and an allocation, (x ;x ;x ;x ),such that (i) (x ;x ) solves max ¼ u (x )+¼ u (x ) p x + p x = p! + p! x 0; (ii) (x ;x max ¼ u (x )+¼ u (x ) p x + p x = p! + p! x 0; (iii) x + x =! +! x + x =! +! : 6

7 (c) Because we have u 0 i > 0 for each consumer, strict monotonicity is satis ed. Because we also have u 00 i < 0 for each consumer, the Bernoulli utility functions, and the overall utility functions V i, are strictly concave. Strict concavity implies strict quasiconcavity. Also, the initial endowments,! s i,are strictly positive for each consumer in each state. Thus, all of the assumptions required to apply the SFTWE are satis ed. 7