Microeconomics Help

May 2010
2
0
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
1,034
272
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: \(\displaystyle 121000 = (\sqrt{L} + \sqrt{K})^{2}\)

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

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.