Microeconomics Help

May 2010
Hey all, first post, 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 guys.
May 2010
You have 2 unknowns so you need to find 2 simultaneous equations and solve them.

There are 2 ways to do this, the long way and the short way.
The long way to do this is to set up a lagrangian. You want to minimise the total cost, subject to the constraint that you have produced 121000 airframes.

Total cost: K + 10L
Constraint: \(\displaystyle 121000 = (\sqrt{L} + \sqrt{K})^{2}\)

\(\displaystyle Z = K + 10L -\lambda ((\sqrt{L} + \sqrt{K})^{2} -121000)\)

If you solve this in the usual way, you will get 2 simultaneous equations:
\(\displaystyle 121000 = (\sqrt{L} + \sqrt{K})^{2}\)
\(\displaystyle \frac{MPL}{10} = \frac{MPK}{1}\)

Solve those to get your answers of L and K.

short cut
You only need the ratio L/K. You can get this from the second simultaneous equation on its own
\(\displaystyle \frac{MPL}{10} = \frac{MPK}{1}\)

\(\displaystyle \frac{MPL}{MPK} = \frac{10}{1}\)

\(\displaystyle \frac{MPL}{MPK} = 10\)

\(\displaystyle \frac{(K^{0.5} + L^{0.5})L^{-0.5}}{(K^{0.5} + L^{0.5})K^{-0.5}} = 10\)

\(\displaystyle \frac{L^{-0.5}}{K^{-0.5}} = 10\)

\(\displaystyle \frac{K^{0.5}}{L^{0.5}} = 10\)

\(\displaystyle \frac{K}{L} = 100\)

Interestingly, for this production function, the ratio of K/L is constant at all levels of output. So you didn't need to know that there were 121000 airframes after all.