Calculating optimal level of negative externality

Question

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 $.

1. Compute the pareto optimal level of $h$, $h^o$.

2. Compute the firm's optimal level of $h$, $h^f$.

3. 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:

1. To find $h^o$ I maximize the aggregate surplus/welfare of h. $$\max_{h \geq0} \pi(h) + \phi(h)$$ $$ W(h)=120-2(h-10)^2 - 2h^2 = 0 $$ $$ \frac{dW}{dh}= -4(h-10)-4h = -8h + 40 = 0 $$ $$ 8h=40 \rightarrow h^o=5$$

2. To find $h^f$ $$FOC\ of\ \pi(h): \frac{d\pi}{dh} =-4(h-10) = -4h+40 = 0 \rightarrow h^f=10 $$

3. To find $h^c$ $$ FOC\ of\ \phi(h): \frac{d\phi(h)}{dh} = -4h = 0 \rightarrow h^c=0 $$

Is this correct?

Answer 1

Almost correct. Setting $W(h)=0$ is wrong (but inconsequential for the solutions). Checking the SOC for completeness should be included, but is somewhat obvious here. Correctness of 3. holds only under the assumption that the consumer owns no shares of the firm, which seems to hold in this exercise.