Economics Stack Exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. It only takes a minute to sign up.
Sign up to join this community
A city has a single electricity supplier. Electricity production cost is $c$ per unit. Utility function $U(q,t)=a\ln(1+q) - t$, where $q$ is electricity consumption and $t$ is electricity tariff. What is the profit maximizing tariff?
I have figured out $\text{MC}=c$ and the supplier is monopolist so $\text{MR}=\text{MC}$ condition have to be used but I can't figure out how to get MR from the utility function.
I assume $t=pq$ is meant. $p$ is the price per unit.
We want to know when
$$U(q,p)=a \ln(1+q) - pq$$
is maximal, given $a>0$ and $p>0$, since this is the utility function and the quantity $q$ the customer buys is the quantity where the utility function is maximal.
The derivative with respect to $q$ is
$$\frac{d}{dq} U(q,p) = \frac{a}{1+q} - p$$
Set derivative to zero gives $q =\frac{a-p}{p}$.
Now we want to know when $(p-c)q$ is maximal, since that gives maximal profit.
Insert $q =\frac{a-p}{p}$ to get $\frac{(a-p)(p-c)}{p}$ is maximal. Rewrite this to $-p+a+c-\frac{ac}{p}$. Take the derivative with respect to $p$ and set to zero to get
$$-1+\frac{ac}{p^2}=0$$
$$-p^2+ac=0$$
$$p^2=ac$$
$$p=\sqrt{ac}$$
Thus we get $t=pq=a-p=a-\sqrt{ac}$.
