I am trying to solve the following question(s):
Let $h \geq 0$ represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility function and attaches a utility of $\phi(h) = -2h^2$ to the externality. The firm's profit function is $\pi(h)=120-2(h-10)^2 $.
Compute the pareto optimal level of $h$, $h^o$.
Compute the firm's optimal level of $h$, $h^f$.
Suppose the consumer rather than the firm can determine the specific externality levelh with which the firm operates. Compute the optimal level of $h$, $h^c$, from the consumer's perspective.
My approach so far:
To find $h^o$ I maximize the aggregate surplus/welfare of h. $$\max_{h \geq0} \pi(h) + \phi(h)$$ $$ W(h)=120-2(h-10)^2 - 2h^2 = 0 $$ $$ \frac{dW}{dh}= -4(h-10)-4h = -8h + 40 = 0 $$ $$ 8h=40 \rightarrow h^o=5$$
To find $h^f$ $$FOC\ of\ \pi(h): \frac{d\pi}{dh} =-4(h-10) = -4h+40 = 0 \rightarrow h^f=10 $$
To find $h^c$ $$ FOC\ of\ \phi(h): \frac{d\phi(h)}{dh} = -4h = 0 \rightarrow h^c=0 $$
Is this correct?