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Determine the optimal factor allocation for Long Vision Binoculars
1 .Given the production function y = k^1/2L 1/2 and the output target of 40 units, express the cost of labor and capital:
Labor Cost (LC): L * w = L
Capital Cost (KC): k * r = 4k
2.To minimize costs, Long Vision Binoculars will choose a factor combination that equalizes the marginal cost of labor and capital. Setting MCL = MCC, we get:
L^-1/2k^1/2 = k^-1/2L^1/2
L = k
3.Substituting L = k into the production function, we get:
40 = k^1/2 * k^1/2
40 = 2k^1/2
k = 400
4.Calculate the total cost per week for Long Vision Binoculars:
Total Cost (TC) = LC KC = L 4k
TC = 400 4(400)
TC = 1760
Long Vision Binoculars optimally allocates factors by equating marginal cost of labor to marginal cost of capital, leading to L=k. Substituting into the production function yields k=400. The total cost per week is 1760.
Determine the total cost per week for Excellent Binoculars and compare it to Long Vision Binoculars
Given the output target of 40 units and double the labor usage of Long Vision Binoculars, express the cost of labor and capital for Excellent Binoculars:
Labor Cost (LEC): 2L *w = 2L
Capital Cost (KEC): k *r = 4k
Substitute L = 2k into the production function and calculate the capital usage (k = 10) for Excellent Binoculars.
Calculate the total cost per week for Excellent Binoculars:
Total Cost (TEC) = LEC KEC = 2L 4k
TEC = 2(20) 4(10)
TEC = 120
Calculate the total cost ratio (TCR) comparing Excellent Binoculars to Long Vision Binoculars:
Total Cost Ratio (TCR) = TEC / TC
TCR = 120 / 1760
TCR ≈ 0.068
Therefore, the total cost per week of Excellent Binoculars is approximately 6.8% of the total cost per week of Long Vision Binoculars.
Excellent Binoculars (EB) incurs a weekly total cost of 120, with labor cost (LEC) of 40 and capital cost (KEC) of 80. The total cost ratio (TCR), comparing EB to Long Vision Binoculars, is approximately 6.8%, indicating EB’s cost is only a small fraction of LVB’s total cost.
By equating the marginal cost of labour to capital, Long Vision Binoculars allocates factors optimally, yielding (L = k). The production function returns (k = 400) when (L = k) is substituted, and the weekly total cost of Long Vision Binoculars is (1760). With twice as much labour used for Excellent Binoculars, the total cost is (120). When comparing Excellent to Long Vision Binoculars, the total cost ratio (TCR) is roughly (6.8%), showing that Excellent Binoculars are substantially less expensive.