The production function for Hamburger Heaven (HH) is q=10KL. Given that the marg
ID: 1195694 • Letter: T
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The production function for Hamburger Heaven (HH) is q=10KL. Given that the marginal product of labor, MP_L = 10K, marginal product of capital, MP_k = 10L, and marginal rate of technical substitution (MRTS = MP_L/MP_k) As a manager, your job is determine the long-run and short-run costs, given that wage rate w = $10 and rental rate r = $15. Suppose in the short run, the capital of grills used by HH is fixed at 4, answer part a and b. How much labor HH should employ to produce 100 units of output How much does it cost in the short run How much labor HH should employ to produce 200 output How much does it cost Now consider in the long run, and both K and L are variable inputs, answer the following questions. What is the cost-minimizing combination of labor and capital HH should use to produce 100 units of output in the long run How much does it cost in the long run .What is the cost-minimizing combination of labor and capital HH should use to produce 200 output How much does it cost Given the calculation from the short-run and long-run scenario, what makes the long-run costs are less than (or equal to) the short-runExplanation / Answer
a) Cost = Units of capital*Price per capital + Units of labor*price per labor
Capital is fixed in short run at 4 unit. and price of capital is $15. Wage rate = $10
Production function: q = 10KL
putting q = 100 and K = 4 we get
100 = 10*4*L
L = 2.5 units
hence 2.5 unist of labor would be required in short run to produce 100 units.
Cost = 4*15 + 2.5*10 = $85.
b) Now let put q = 200 and K = 4 in production function again
200 = 10*4*L
L = 5 units. 5 labors will be required to produce 200 units.
Cost = 4*15 + 5*10 = $110.
c) Let put q = 100 in production function,
100 = 10KL
K = 10/L or L = 10/K .. eq i
We know that at optimal level of combinations of input, MRTS = rental rate/wage rate
MRTS = MPK/MPL = dq/dK / dq/dL = 10L/10K = L/K
Equating MRTS with rental rate/wage rate
L/K = 15/10
L = 15K/10 .. eq ii
from eq i and ii
10/K = 15K/10
100/15 = K2
K = 2.58 approx
Putting K = 2.5 in eq 1 we get
L = 10/2.5 = 4
Hence to produce 100 units in long run it required to have 2.5 units of capital and 4 units of labor.
Cost = 2.5*15 + 4*10 = $77.5
d) Putting q = 200 in production function we get the following:
L = 20/K . eq iii
and from part c we have L = 15K/10 . eq 11
from eq ii and iii
20/K = 15K/10
K = 3.65 approx
putting above value of K in eq iii
L = 20/3.6 = 5.5
hence to produce 200 units it is required to have 3.6 units of capital and 5.5 units of labor.
Cost = 3.6*15 + 5.5*10 = $109.
e) the variability of capital input in long run makes the long run costs less than short run cost.
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