You run a cost-minimizing firm with production function f(L,K) = [min{L, K}], where L is labor and K is capital. Assume that you are a price-taker in the input markets: you pay w for each unit of labor you hire and r for each unit of capital (where w and r are set exogenously), and face no costs other than those from labor and capital. (a) (15 points) Assuming that you can freely choose both labor and capital (i.e.. the long run prob- lem"), derive expressions for your cost-minimizing conditional input demands, L*(r, w, Q) and K*(r, w, Q). Confirm that the conditional input demand functions are "homogeneous of degree zero" in w and r; that is, [L* (tr, tw,Q) = L*(r, w.Q) K*(tr, tw.Q)= K*(r, w. Q) for all t> 0 F(L.K) = {min L. K} is a perfect complements production function, so we know that we will always be producing at L = K. Using this. Q = F(L. K) = {min L, K} = {min L. L} = L L* = Q and K* = Q. That is, L* (r, w, Q) = K*(r, w, Q) = Q are independent of the input prices r and w. In particular, we verify homogeneity of degree zero: L* (tr, tw,Q) = Q = L*(r, w, Q) and same for K*(r, w, Q). (b) (8 points) What will happen to your conditional demand for labor if there is an increase in the wage rate, assuming that r and Q remain the same? Explain in one sentence why your answer makes intuitive sense. = As can be seen both in the picture and by plugging into the formulas, since L*(r, w, Q) K*(r, w,Q) = Q are independent of the prices r and w, there will be no change in the in- puts for a change in prices (given the perfect complement production function, producers cannot substitute away from labor when its price w increases). (c) (5 points) Use your answers from (a) to write down an expression for your total cost function TC(r, w. Q). Is this function "homogeneous of degree one" in w and r; that is, does TC(tr, tw, Q) = t-TC(r, w, Q)? The total cost function TC(r, w, Q) states how much it costs to produce quantity Q. when prices are r and w (and the producer behaves optimally). We have: TC (r, w,Q) = wL* (r. w,Q) +rK* (r,w.Q)=wQ +rQ = (r+w)Q TC(r, w,Q) is indeed homogenous of degree one: TC(tr, tw,Q) = (tr + tw)Q=t(r+w)Q = t-TC(r.w,Q)