Question

Firm 1 produces output X with a cost function C_1(X)=\frac{X^2}{200}. Firm 2 produces output Y with a cost function C_2(X,Y)=\frac{Y^2}{100}-2X. Both firms face competitive markets. The competitive price of X is 6 and the competitive price of Y is \$ 5. There is no entry or exit into this market. What is the socially optimal production of X?

Answer 1

Answer:

800

Explanation:

The objective here is to determine the socially optimal production of X.

For this to occur ; it is crucial that both firm must merge together.

Therefore; the Profit will be = Total revenue - Total Cost

From the question; the total revenue = 6X + 5Y ;     &

The total cost is : $\dfrac{X^2}{200} + \dfrac{Y^2}{100} - 2X$

Now: The profit = $6X+5Y - \dfrac{X^2}{200}- \dfrac{Y^2}{100}-2X$

= $8X+5Y - \dfrac{X^2}{200}- \dfrac{Y^2}{100}$

If the socially optimal production of X is the differential of the equation $8X+5Y - \dfrac{X^2}{200}- \dfrac{Y^2}{100}$

(X) = $8-\frac{2X}{200} =0$

= $8-\frac{X}{100} =0$

= $\dfrac{X}{100}=8$

= 800

Thus the social optimal production of X = 800