2. For the following, assume a firm's production function is given by q = LaKb. a. Find an expression for the firm's MRTS. b. Assume a + b = 1, where a > 0 and b > 0. Set the MRTS from above equal to - w/r. This condition is equivalent to the utility maximizing condition from consumer theory equating MRS with the slope of the budget constraint. We can manipulate this equation and restate it as: MPL _ MPK W T In a few sentences, provide intuition for why this has to be true when a firm is using the cost minimizing amounts of labor and capital (Hint: imagine the opposite, that the additional output per dollar spent on labor is greater than the last dollar spent on capital. Could the firm change the amount of capital/labor and do better?) Use the equality from part b) to derive the amount of capital the firms should use as a function of labor, that is K(L). (Hint: recognizing b = 1 - a may help you to simplify your answer). Substitute K(L) into the production function and assume that the production level is fixed at some output level (7. You now have an equation with one unknown (L) that also incorporates the relationship that has to exist between K and L when minimizing costs, much like when you substituted the utility constraint into expenditure when analyzing consumer expenditure minimization. Solve this equation for L to obtain the cost minimizing level of labor denoted [L(w, r, (7)]. Plug that labor level back into your expression for K(L) to find the cost minimizing capital level [K(w,r, 17)]