Problem 1 A rm produces a single output using labor (L) and capital (K ) according to the production function Q = L%K%. The output sells at price p = 9 per unit. A unit of labor costs 11; = 1 and a unit of capital costs 7' : 1. (1) Does the production technology exhibit increasing, constant, or decreasing returns to scale? (2) Find the optimal capital and labor input choices for a given output level, Q. Denote the optimal input choices by L(Q) and K(Q). Derive the long run cost function C(Q) : wL + TX. (3) Solve for the optimal output choice. How much labor and capital does the rm demand? (4) Another way to obtain the rm optimal choices is to directly maximize prot pQ (wL + TK) with respect to L and K. Verify that this approach leads to the same solution as part (3). Now, suppose that labor costs change from w = 1 to 3. The following parts of the question decompose the effect on labor and capital into the substitution eect and the scale effect. (5) From part (3), when (11),?) = (1,1), the rm optimally produces Q = 9, and uses labor L = 27' and capital K = 27. To calculate the substitution effect, we hold constant the production level Q = 9, and solve for the optimal input choices given the new input prices ("10,7") : (3, 1). The change in the input choices will be the substitution effect. Is the substitution effect on labor positive or negative? (6) To solve for the scale effect, we derive the optimal production level under the new input prices (10, T) : (3, 1). The new production level leads to a further change (from part (5)) in the input choices. This change is the the scale effect. Is the scale effect on labor positive or negative