# Profit maximization question

Two firms are in a market together. They produce a product.

Total revenue from sales is $y=K+L$

K is the amount of capital

L is the amount of labor

These two firms each specialized in supplying one of inputs and the cost of the input supplied is incurred by the person providing it.

For capital, the cost is $K^2/2$ and for labor, the cost is $L^2/2$

And they share revenue from their business equally.

I need to find firstly that if eac is interested in maximizing his own profit, then how much of each input will they use, and how much profit will each make. And secondly, I need to find the profit maximizing level of inputs for whole business.

As a result, I need to compare what I have found in both parts.

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my solution

For firm 1,

$$max [1/2(K+L)-K^2/2]$$

FOCs:

$1/2-K=0$

So $K=1/2$. He use 1/2 unit of capital.

And his profit is $\pi= 1/2*(1/2+L)-1/4=1/2*L\ge 0$ for $L>0$

Similarly,

For firm 2,

$$max [1/2(K+L)-L^2/2]$$

FOCs:

$1/2-L=0$

So $L=1/2$. He use 1/2 unit of labor.

And his profit is $\pi= 1/2*(1/2+K)-1/4\ge 0$ for $L\ge 0$

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And I have tried to find the profit maximization for entire business

$$max[(K+L)- K^2/2-L^2/2]$$

FOCs;

$1-K=0$ so K=1

and

$1-L=0$ so L=1

And profit $= (1+1)-(1/2+1/2)=1>0$

Now, all parts of my solution are correct?

You make a mistake in setting up your individual profit functions. Given that the total revenues from sales are $y=K+L$ and they share revenues equally their revenues are $1/2*(K+L)$