The quantity, q, of a product manufactured depends on the number of workers, W, and the amount of capital invested, K, and is given by q = 6W(3/4)K(1/4) . Labor costs are $10 per worker and capital costs are $420 per unit, and the budget is $3000.
(a) What are the optimum number of workers and the optimum number of units of capital?
(b) Show that at the optimum values of W and K, the ratio of the marginal productivity of labor (∂q/qW) to the marginal productivity of captial (∂q/qK) is the same as the ratio of the cost of a unit of labor to the cost of a unit of capital.
(c) Recompute the optimum values of W and K when the budget is increased by one dollar. Check that increasing the budget by $1 allows the production of λ is the Lagrange multiplier.