Assume the following production function, where a (0,1) and # > 0 are parameters, while A > 0 is productivity: 8 Y=A [QK% +(1- a)L]\"" (a) Calculate the marginal product of capital and labor. (b) Verify whether the function satisfies the 4 properties we discussed in class. () Assume a firm hires labor at a wage rate W and capital at a rental rate R. Find the firm's labor and capital demand functions. (d) Use the demand functions above to calculate the elasticity of substitution between capital and labor, that is, o dlog(L/K) lasticity = S\\ ) elasticity dlog(W/R) [Hint: Remember that this function is called CES - constant elasticity of substitution] (e) Under what conditions on @ does this production function imply that the share of payments to capital and labor is constant (i.e, it does not change with A, W, or R)? (f) Prove that the production function above converges to the Cobb-Douglas production function as ! 1. To do so, first write down the logarithm of both functions, simplify the algebra by letting v = 3%91 and use L'Hopital's rule to calculate the limit as v 0. Properties of production functions 1. Monotonicity 2. Constant Returns to Scale 3. Diminishing Marginal Product of Capital and Labor 4. Constant Factor Shares