Question

A bakery produces cheesecakes using bakers (labor) and specialized baking ovens (cap- ital). Let q be the number of cheesecakes produced per week, K be the number of hours of oven use, and L be the number of hours of labor use. The bakery’s production function is given by q = 6K^0.5L^0.25. The current hourly wage for bakers is w = $15, and the per hour user cost of capital (ovens) is r = $25.

(a) Derivethemarginalproductoflaborandmarginalproductofcapitalforthebakery.

(b) What is the optimal ratio of capital to labor at the cost-minimizing combination of inputs?

(c) How many labor hours (L) and oven hours (K) will the bakery use to minimize the cost of producing q = 300 cheesecakes per week? Hint: You can allow for fractions of hours to be used. Keep any fractions at up to 5 decimal places when making your calculations.

Answer 1

q = 6K^0.5L^0.25

(a)

MPL = q/L = 6 x 0.25 x (K^0.5 / L^0.75) = 1.5 x (K^0.5 / L^0.75)

MPK = q/K = 6 x 0.5 x (L^0.25 / K^0.5) = 3 x (L^0.25 / K^0.5)

(b)

Cost is minimized when MPL/MPK = w/r = 15/25 = 3/5

MPL/MPK = [1.5 x (K^0.5 / L^0.75)] / [3 x (L^0.25 / K^0.5)] = K/2L = 3/5

K/L = 6/5 [Optimal K/L ratio]

(c)

K = 6L/5

When q = 300,

6 x (6L/5)^0.5L^0.25 = 300

(6/5)^0.5 x L^0.5L^0.25 = 50

1.09545 x L^0.75 = 50

L^0.75 = 45.64334

L = (45.64334)^(1/0.75) = 163.11852

K = (6 x 163.11852) / 5 = 195.74222