The production function is given by $Y = f(L, K) = 4LK$, where $Y$ is the output, $L$ is labor, and $K$ is capital. The goal is to minimize the cost while producing 64 units of output, i.e., $Y = 64$.

Step 1: Find the constraint based on the output

Since $Y = 4LK$, we can set $Y = 64$ to find the constraint for $L$ and $K$: $64 = 4LK$ $LK = 16$ Thus, the product of $L$ and $K$ must equal 16.

Step 2: Set up the cost function

The cost function is the total cost of labor and capital. Given that the price of $L$ (labor) is £40 and the price of $K$ (capital) is £10, the total cost $C$ is: $C = 40L + 10K$ We need to minimize this cost subject to the constraint $LK = 16$.

Step 3: Test the options

Option a: $L = 16$ and $K = 1$

• $LK = 16 \times 1 = 16$ (satisfies the constraint)
• Cost: $C = 40(16) + 10(1) = 640 + 10 = 650$

Option b: $L = 2$ and $K = 8$

• $LK = 2 \times 8 = 16$ (satisfies the constraint)
• Cost: $C = 40(2) + 10(8) = 80 + 80 = 160$

Option c: $L = 2$ and $K = 2$

• $LK = 2 \times 2 = 4$ (does not satisfy the constraint)

Option d: $L = 32$ and $K = 32$

• $LK = 32 \times 32 = 1024$ (does not satisfy the constraint)

Conclusion:

The cost-minimizing combination that satisfies the constraint is Option b: $L = 2$ and $K = 8$, with a cost of £160.

Would you like more details or have any questions? Here are some relative questions for further practice:

1. How do you derive the cost function in production problems?
2. What is the method to minimize cost when given a production function?
3. How would the cost change if the price of $K$ were £20 instead of £10?
4. What is the importance of the marginal product of labor and capital in production?
5. How does changing the production function affect the optimal labor-capital combination?

Tip: Always verify that the solution satisfies the production constraint before checking the cost.