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^{0.5}L^{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)^{0.5}L^{0.25} = 300

(6/5)^{0.5} x L^{0.5}L^{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

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