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# Profit Maximizing Welfare

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Question 1

1. Consider a firm that can produce x at a cost c(x)=x^2. This output can be sold in a competitive market at a price p=12 per unit. The profit of this firm is this ?(x) = px - c(x) = 12x - x^2.
a. How much x will this produce to maximize its profit? (Hint: simply max the profit function w.r.t. x to find the optimal level.) How much profit does the firm make in this case?

Differentiate the profit function w.r.t. x and equate it to zero
d ?(x)/dx = 12-2x
Equating it to zero, we get 12-2x=0
Solving for x, we get x=6
Thus, firm should produce 6 units of x to maximize profit.

Maximum profit = ?(6) = 12*6- 6^2=36

2. Now assume that production causes a negative externality that amounts to d(x)=2x^2. The efficient output level is the one that maximized welfare, which is the firms profit minus the damage caused, W(x) = ...

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