The production function for Superlite Sailboats, Inc., is Q = 20(K^0.5)*(L^0.5)
ID: 1121169 • Letter: T
Question
The production function for Superlite Sailboats, Inc., is
Q = 20(K^0.5)*(L^0.5) with marginal product functions
MPk= 10 * ([L^0.5]/[K^0.5]) and MPl = 10 * ([K^0.5]/[L^0.5])
a. If the price of capital is $5 per unit and the price of labor is $4 per unit, determine the expansion path for the firm.
b. The firm currently is producting 200 units of output per period using input rates of L = 4 and K = 25. IS this an efficient input combination? Why or why not? If not, determine the efficient input combination for producing an output rate of 200. What is the capital-labor ratio?
c. IF the price of labor increases from $4 to $8 per unit, determine the efficient input combination for an output rate of 200. What is the capital labor ratio now? What input substitution has the firm made?
Explanation / Answer
(a) Cost is minimized when MPL / MPK = w/r = $4 / $5 = 4/5
MPL / MPK = K / L = 4/5 (= 0.8)
K = 4L / 5 [Equation of expansion path]
(b)
When L = 4 & K = 25,
MPL / MPK = K / L = 25/4 = 6.25 > 0.8
Therefore this combination is not optimal.
When Q = 200,
200 = 20K0.5L0.5
K0.5L0.5 = 10
KL = 100
(4L / 5) x L = 100
L2 = 125
L = 11.18
K = (4 x 11.18) / 5 = 8.94
Capital-labor ratio = K / L = 0.8
(c)
MPL / MPK = K / L = $8 / $5 = 8/5
K = 8L / 5
When Q = 200,
K0.5L0.5 = 10
KL = 100
(8L / 5) x L = 100
L2 = 62.5
L = 7.91
K = (8 x 7.91) / 5 = 12.66
Capital-labor ratio = 8/5 = 1.6
Frm has substituted (costlier) labor for (cheaper) capital.
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