Suppose your firm needs to produce 40 units if computers at the minimum cost. The function for production equates to Q=10L^2/3 K^1/3 with prices, L=50 and K=200.
Compute for the Minimum Total Cost and optimal combination to produce the required units
To find the minimum total cost and the optimal combination to produce the required units, you can follow these steps:
Step 1: Understand the given information:
The given production function is Q = 10L^(2/3)K^(1/3). This equation represents the relationship between the inputs (labor, L, and capital, K) and the output (number of units, Q).
The prices of labor and capital are L = 50 and K = 200.
Step 2: Determine the objective function:
The objective is to minimize the total cost of production. In this case, the total cost is a function of labor (L) and capital (K), which can be represented as C(L,K). So, our objective function is C(L,K).
Step 3: Convert the production function into the cost function:
To find the cost function, we need to determine the cost of labor and the cost of capital. The cost of labor is given by CL = wL, where w is the price of labor. The cost of capital is given by CK = rK, where r is the price of capital.
Given that L = 50 and K = 200, and the prices L=50 and K=200, the cost function can be computed as follows:
CL = 50 * L = 50 * 50 = 2500
CK = 200 * K = 200 * 200 = 40000
Now, we can write the cost function as C(L, K) = 2500 + 40000.
Step 4: Minimize the cost function:
To find the optimal combination that minimizes the cost function, you need to differentiate the cost function with respect to labor (L) and capital (K), and set the derivatives to zero.
dC/dL = 0 (partial derivative of the cost function with respect to L)
dC/dK = 0 (partial derivative of the cost function with respect to K)
Step 5: Solve the equations:
Differentiating the cost function with respect to L:
dC/dL = 0
0 + 0 = 10 * (2/3) * L^(-1/3) * K^(1/3)
Simplifying the equation:
2/3 * L^(-1/3) * K^(1/3) = 0
Since the value of L cannot be negative, we can conclude that L = 0.
Differentiating the cost function with respect to K:
dC/dK = 0
0 + 0 = 10 * (1/3) * L^(2/3) * K^(-2/3)
Simplifying the equation:
1/3 * L^(2/3) * K^(-2/3) = 0
Similarly, since the value of K cannot be negative, we can conclude that K = 0.
Step 6: Compute the minimum total cost:
To compute the minimum total cost, substitute the values of L and K into the cost function.
C(L,K) = 2500 + 40000
= 42500
Therefore, the minimum total cost to produce 40 units of computers is 42500. However, the optimal combination of labor and capital is not obtained from the given information and equations.