5 Cobb-Douglas Production Function - Fixed Capital The firm produces using capital (K) and labor (N) given a production technology represented with the following production function: ZF (K, N) = zK"NI-& The firm has a fixed amount of capital K and can hire labor at a market (given) wage (w) (per unit of labor hired). Then the profits of the firm are: 7 = max zF (K, N) - WN N = max zK N1-4 - wN N 1. Derive and interpret the firm's first order condition with respect to labor. 2. The elasticity of the labor demand to wages is defined as: aln N Sw = aln w Find the elasticity of the firm. 3. Interpret the elasticity of labor demand. 4. Repeat the exercise for the elasticity with respect to capital and productivity: aln N aln N SK = aln K $2 = alnz 5. Solve for the optimal profits in terms of parameters and exogenous variables (z, K, w)