Econ 600 Schroeter Fall 2007 Homework 1. Given differentiable functions: ()()()ℜ→ℜℜℜ→ℜℜℜ→ℜℜmnmnmnxxhandxxgxxF :;,:;,:;ααα, consider the following optimization problems: (i) ()()0;;max,...=αααxgtosubjectxFgivenxtrw (ii) ()()()0;0;;max,...==ααααxhandxgtosubjectxFgivenxtrw In these problems, (where n > 2) is a vector of choice variables and nx ℜ∈A∈α, an open subset of , is a vector of parameters. Assume that each problem has a strict global solution for each mℜA∈α and denote the optimal values of the choice variables, the equilibrium values for the problems' Lagrange multipliers, and the value functions (all assumed differentiable functions of α) as follows: (i) () ()()()()ααααλα;,,****xFFx ≡ (ii) () () ()()()()ααααμαλα;ˆˆ,ˆ,ˆ,ˆxFFx ≡ For a given value of the parameter vector, A∈0α, assume that ()()0;00*=ααxh. (That is, the additional constraint in problem (ii) is satisfied at problem (i)'s solution for 0αα= .) Further assume: ()()00*;ααxxg∂∂ and ()()00*;ααxxh∂∂ are linearly independent. a.) It is obvious that and () ()0*0ˆααxx =()()0*0ˆααFF =. Show that: () ()00*ˆαααα∂∂=∂∂ FF. (Hint: Use the FONC for the two problems to argue that ()0ˆ0=αμ. Then use the envelope theorem.)2b.) Prove that () (02*2022ˆα)ααα∂∂−∂∂ FF is negative semi-definite. (Hint: Argue that for all () ()αα*ˆFF ≤0αα≠ and then use a Taylor series expansion.) (Note: The same result could be established for the more general case in which the numbers of constraints in problems (i) and (ii) are and , respectively, where and , and the additional constraints in problem (ii) are satisfied at problem (i)'s solution for 1r2r201−≤≤ nr nrr <<210αα= .) 2. A profit maximizing firm uses inputs, and , to produce output, y, via the production function . The firm faces parametric output and input prices: . 1x2x(21, xxfy =)21,, wandwp a.) In the "long run," the firm can respond to price changes by choosing profit-maximizing input quantities without constraint. In the "short run," the firm is required to maintain the same ratio between and that characterizes its current equilibrium. 1x2x (For example, suppose that is unionized labor and is the number of machines. The union, concerned about a trend toward declining employment, has negotiated a contract mandating that firms hire at least two workers for every machine, for "safety" reasons. In this case, the "short run" corresponds to the period for which the current contract is in force. Actually, this hypothetical is similar to a famous real-world instance of "featherbedding," a labor union practice designed to stimulate employment by imposing inefficient work rules. "The classic example of featherbedding in the United States is the railroad union requirement that railroad owners hire firemen, men who shoveled coal, to work on diesel locomotives, which don't use coal." This effectively required that there be two workers (engineer and fireman) for every machine (diesel locomotive), when only one was needed. 1x2xhttp://en.wikipedia.org/wiki/Featherbedding) Use the result of problem 1 to show that short-run factor demands are no more elastic than long-run factor demands. b.) As an alternative, suppose that the short-run constraint is that the firm maintains the same total expenditure on factors as in its current equilibrium. Is it true for this case that short-run factor demands are necessarily no more elastic than long-run factor demands?