# Question on profit maximization with government taxation

(I) A monopolist has a cost function $c(q)=q$. It faces the following demand function $q(p)=100/p$ for $p\le 20$ and $q(p)=0$ for $p\ge 20$. What are the profit maximizing price and output.

(ii) a monopolist has a cost function $c(q)=\alpha q$ where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is $-3$. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?

My solutions are as follows:

I am not sure whether it is correct or not. Please tell me and show me my mistakes.

(I) A monopolist has a cost function $c(q)=q$. It faces the following demand function $q(p)=100/p$ for $p\le 20$ and $q(p)=0$ for $p\ge 20$. What are the profit maximizing price and output.

Given the demand function \begin{eqnarray*} q(p) = \begin{cases} \frac{100}{p} & \text{if } p \leq 20 \\ 0 & \text{if } p > 20\end{cases} \end{eqnarray*}

So, revenue as a function of quantity $(q)$ is \begin{eqnarray*} r(q) = \begin{cases} 100 & \text{if } q \geq 5 \\ 0 & \text{if } q < 5\end{cases} \end{eqnarray*}

Given that the cost is $c(q) = q$, profit as a function of quantity $(q)$ is \begin{eqnarray*} \pi(q) = r(q) - c(q) = \begin{cases} 100 - q & \text{if } q \geq 5 \\ -q & \text{if } q < 5\end{cases} \end{eqnarray*}

We simply solve the following problem to find the optimal quantity of the monopolist: \begin{eqnarray*} \max_{q \geq 5} \ 100 - q \end{eqnarray*}

Solving it we get the optimal quantity $(q^m)$ and the corresponding price $(p^m)$ chosen by the monopolist as \begin{eqnarray*} q^m & = & 5 \\ p^m & = & 95\end{eqnarray*}

(ii) a monopolist has a cost function $c(q)=\alpha q$ where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is $-3$. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?

Given that the demand function has a constant price elasticity whose value is $-3$, it is of the form: $$q(p) = \beta p^{-3}$$, where $\beta > 0$ is a constant.

Now profit maximization problem of the monopolist in terms of price $(p)$ can be written as : \begin{eqnarray*} \max_{p} \ \ pq(p) - c(q(p)) \end{eqnarray*}

Given that the cost is $c(q)=\alpha q$ and the demand is $q(p) = \beta p^{-3}$, the profit maximization problem is : \begin{eqnarray*} \max_{p} \beta p^{-2} - \alpha\beta p^{-3} \end{eqnarray*} Solving it we get the optimal price as: \begin{eqnarray*} p^m = \frac{3\alpha}{2} \end{eqnarray*}

With tax of 6 per unit, the profit maximization problem is \begin{eqnarray*} \max_{p} \beta p^{-2} - \alpha\beta p^{-3} - 6\beta p^{-3}\end{eqnarray*}

Solving it we get the optimal price as: \begin{eqnarray*} p^t = \frac{3(\alpha+6)}{2} \end{eqnarray*}

Therefore, increase in price due to tax equals : \begin{eqnarray*} \Delta = p^t - p^m = 9 \end{eqnarray*}

ii.

I don't think the demand function in (i.) applies for this question. Here's what I did:

By Lerner Index, $\displaystyle{\frac{P - MC}{P} = \frac{-1}{E_d}}$

Since $E_d = -3$, we get $P = \frac{3}{2} MC$

We know before tax MC is $c'(q) = \alpha$, and after tax MC is $\alpha + 6$

In this sense, MC increased by 6, so P will probably increase by 9