Hopefully someone can help me with this.

"Suppose the production of airframes is characterized by a CES production function: Q=(K^0.5 + L^0.5)^2 (MPL = (K^1/2 + L^1/2) /L^1/2), MPK = (K^1/2 + L^1/2) /K^1/2)

Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost minimizing combination of labor and input for an airframe manufacturer that wants to produce 121,000 airframes."

Thanks in advance everyone!

To find the cost-minimizing combination of labor and capital for an airframe manufacturer that wants to produce 121,000 airframes, we need to minimize the cost function while satisfying the production quantity.

The cost function C is given by the equation C = wL + rK, where w is the price of labor, L is the amount of labor, r is the price of capital, and K is the amount of capital.

In this case, the price of labor is $10 per unit, so w = 10, and the price of capital is $1 per unit, so r = 1.

We are given the production function Q = (K^0.5 + L^0.5)^2, where Q is the quantity of airframes produced. We want to produce 121,000 airframes, so Q = 121,000.

To find the cost-minimizing combination of labor and capital, we need to find the values of L and K that minimize the cost function, subject to the production quantity constraint.

To start, let's rewrite the cost function in terms of Q using the production function:

C = wL + rK
= 10L + K

Next, we substitute the production quantity constraint into the cost function:

Q = (K^0.5 + L^0.5)^2
121,000 = (K^0.5 + L^0.5)^2

Solving this equation, we find that K^0.5 + L^0.5 = √121,000 = 110.

Now, we can substitute K^0.5 + L^0.5 into the cost function:

C = 10L + K
= 10L + (K^0.5 + L^0.5)^2 - L^0.5

Simplifying further, we have:

C = 10L + (110)^2 - L^0.5

To minimize the cost function, we take the derivative of C with respect to L and set it equal to zero:

dC/dL = 10 - (0.5)L^(-0.5) = 0

Solving this equation, we find L = 400.

Once we have the value of L, we can substitute it back into the production quantity constraint to solve for K:

121,000 = (K^0.5 + 400^0.5)^2

Solving this equation, we find K = 81,000.

Therefore, the cost-minimizing combination of labor and capital for an airframe manufacturer that wants to produce 121,000 airframes is L = 400 and K = 81,000.