Use The Tangency Rule To Determine The Cost-Minimizing Bundles Of Labor And Capital For A Japanese Synthetic Rubber Firm’s Production Function, Q = LO.5K0.5 (Flath, 2011), Where W= $20 And $10 At The Cost-Minimizing Bundle As A Function Of L, K= 2L How Does Your Answer Change If W= $40 And R= $5? At The Cost-Minimizing Bundle As A Function Of L
The Correct Answer and Explanation is :
To find the cost-minimizing bundle of labor (L) and capital (K) for a firm’s production function, we can apply the Tangency Rule. This rule tells us that at the cost-minimizing point, the ratio of the marginal products of labor (MPL) and capital (MPK) should equal the ratio of the input prices (wage rate (W) for labor and rental rate (R) for capital).
Step 1: Determine the Marginal Products of Labor and Capital
Given the production function:
[
Q = L^{0.5} K^{0.5}
]
The marginal product of labor (MPL) is the partial derivative of (Q) with respect to (L):
[
MPL = \frac{\partial Q}{\partial L} = 0.5L^{-0.5}K^{0.5}
]
Similarly, the marginal product of capital (MPK) is:
[
MPK = \frac{\partial Q}{\partial K} = 0.5L^{0.5}K^{-0.5}
]
Step 2: Apply the Tangency Rule
The Tangency Rule states:
[
\frac{MPL}{MPK} = \frac{W}{R}
]
Substitute the expressions for MPL and MPK:
[
\frac{0.5L^{-0.5}K^{0.5}}{0.5L^{0.5}K^{-0.5}} = \frac{W}{R}
]
Simplifying:
[
\frac{K}{L} = \frac{W}{R}
]
Now substitute the relationship (K = 2L) (as provided in the problem):
[
\frac{2L}{L} = \frac{W}{R}
]
Simplifying further:
[
2 = \frac{W}{R}
]
Thus, (W) must be twice the value of (R).
Step 3: Solve for the Cost-Minimizing Bundle
Now we solve for the cost-minimizing quantities of (L) and (K) under different wage and rental rates.
- When (W = 20) and (R = 10):
The tangency condition (W = 2R) holds true, so the relationship (K = 2L) remains valid. The cost is minimized by substituting (K = 2L) into the cost function (C = WL + RK):
[
C = 20L + 10(2L) = 40L
] - When (W = 40) and (R = 5):
Here, the new tangency condition is still valid because (W = 2R). Again, using (K = 2L), the cost function is:
[
C = 40L + 5(2L) = 50L
]
Explanation of Changes
The cost-minimizing bundles change as the price of labor (W) and the price of capital (R) change. The tangency rule ensures that at the cost-minimizing point, the ratio of the prices of labor and capital is equal to the ratio of their marginal products. As (W) increases and (R) decreases, the overall cost of employing labor and capital increases, as evidenced by the higher total cost when (W = 40) and (R = 5) compared to when (W = 20) and (R = 10). The relationship (K = 2L) remains unchanged, but the cost of employing these inputs increases, leading to higher total costs.