Consider a firm with a production function given by y = a ln x1 + ß lIn x2 with a, ß E (0,1) which can sell its product for price p and by its inputs at input costs w1 and w2. Find the conditional input demand functions and the cost function for some given level of output y.

(a) For the cost function C(w1, w2, y) = 2y²w} w, calculate the Allen elasticity of substitution between the two inputs at the cost-minimizing input point (xf(w1, w2, y), a(w1, w2, y)). (b) Consider the production function f(r, y, z) = Vry + rz+ yz. Find the scale elasticity SE at (x, y, z) = (1,2, 3), (5, 1,6), (6, 6, 6) and determine if the pro- duction function is IRTS, CRTS, or DRTS locally at each point. (c) A profit maximizing firm in the market operates where the production exhibits decreasing return to scale (DRTS). Is this market in its long-run equilibrium? Justify your answer. (d) Suppose that there are the infinite number of potential firms that produce the identical output good y under the cost function C(y) = + 3. Assume free entry and exit. Find the long-run equilibrium output price p, the amount of the output that each firm in the market produces in the long-run equilibrium, and the value of profit that each firm earns in the equilibrium.

Assume that the firm’s operation is subject to the following production function and price data: Q=3X+5Y-XY Px= 3 $ Py= 6 $ where X and Y are two variable input factors employed in the production of Q. A. In the unconstrained case, what levels of X and Y will maximize Q? B. It is possible to express the cost function associated with the use of X and Y in the production of Q as TC = 3X + 6Y subject to the firm’s constraint of an operating budget of $25. Use the Lagrange multiplier technique to determine the optimal levels of X and Y. What is the firm’s total output at these levels of input usage? What is the value of Lagrangian multiplier? C. What will happen to the firm’s output from a marginal increase in the operating budget? (Hint: This is another way of asking you to interpret the Lagrangian multiplier you found in B above).