Let production be given by p = blαk1−α where b and α are positive and α < 1. if the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production p is subject to the constraint ml + nk = p. show that the maximum production occurs when l=αp/m and k=(1-α)p/n.

The proof is completed below

Step-by-step explanation:

1) Definition of info given

We have the function that we want to maximize given by (1)

  (1)

And the constraint is given by

2) Methodology to solve the problem

On this case in order to maximize the function on equation (1) we need to calculate the partial derivates respect to L and K, since we have two variables.

Then we can use the method of Lagrange multipliers and solve a system of equations. Since that is the appropiate method when we want to maximize a function with more than 1 variable.

The final step will be obtain the values K and L that maximizes the function

3) Calculate the partial derivates

Computing the derivates respect to L and K produce this:

4) Apply the method of lagrange multipliers

Using this method we have this system of equations:

And replacing what we got for the partial derivates we got:

  (2)

  (3)

  (4)

Now we can cancel the Lagrange multiplier with equations (2) and (3), dividing these equations:

  (4)

And simplyfing equation (4) we got:

  (5)

4) Solve for L and K

We can cross multiply equation (5) and we got

And we can set up this last equation equal to 0

  (6)

Now we can set up the following system of equations:

  (a)

  (b)

We can mutltiply the equation (a) by on both sides and add the result to equation (b) and we got:

And we can solve for L on this case:

And now in order to obtain K we can replace the result obtained for L into equations (a) or (b), replacing into equation (a)

With this we have completed the proof.

answer: for each 3 girls there is 2 boys . 3: 2

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