Calculating deadweight loss from tax for a monopolist

Question

A monopolist has cost function $c(y) = y$ so that its marginal cost is constant at 1 per unit. It faces the following demand curve $D(p) = \begin{cases} \frac{100}{p}, &\text{if}&p ≤ 20 \\0,&\text{if} &p>20 \end{cases}$

Find the profit maximizing level of output if the government imposes a per unit tax of Re. 1 per unit, and also the deadweight loss from the tax.

$TR=100 \Rightarrow$ $MR=0$ while $MC=1$ before tax is imposed and $2$ after tax is imposed. So, I can't even understand how to calculate profit-maximising output from the usual $MR=MC$ condition.

I can maybe see the profit maximising problem being reduced to cost-minimisation since the total revenue is constant. In which case, firm minimises its cost while earning a positive profit for $q=5$ for both cases of $MC.$

But, I still don't understand how to calculate deadweight loss. Could someone please guide me with this problem?

Answer 1

If the govt. imposes a tax of $1$ per unit, then the new marginal cost becomes $2$ per unit.

Profit. Using $p \leq 20 \iff \frac{100}{q} \leq 20 \iff q \geq 5$, we have $\pi(q) = 100 - 2q$ when $q \geq 5$ $($or $p \leq 20)$ which is maximized at $q^{*} = 5$.

Deadweight loss. There's no tax-induced deadweight loss in this scenario. As the tax doesn't change the optimal price/quantity, the only thing that changes is a part of the producer's surplus going to the government as tax revenue.

Answer 2

The monopolists original problem without the tax was: $$\begin{aligned} \max_{p,y} \quad & py(p)-c(y)\\ \textrm{s.t.} \quad & y(p)=\begin{cases} \frac{100}{p} & \text{if } p \leq 20\\ 0 & \text{if } p>20\end{cases} \\ & c(y)=y \end{aligned}$$

Upon substitution the problem becomes: $$\begin{aligned} \max_{y\geq 5} \quad & 100-y\\ \end{aligned}$$

Clearly, the objective is decreasing output $y$, so we set output to the lowest value it can take i.e., $y^*=5$

Now let us consinder the case with the tax. The tax imposed is per unit so his new cost function is $c(y)=y+1\cdot y=2y$

The new problem of the monopolist is: $$\begin{aligned} \max_{p,y} \quad & py(p)-c(y)\\ \textrm{s.t.} \quad & y(p)=\begin{cases} \frac{100}{p} & \text{if } p \leq 20\\ 0 & \text{if } p>20\end{cases} \\ & c(y)=2y \end{aligned}$$

using substitution like before gives: $$\begin{aligned} \max_{y\geq 5} \quad & 100-2y\\ \end{aligned}$$ Which again gives $y^*=5$

Since the equilibrium quantity and price remains unchanged, it must be that the consumer surplus is still the same. Clearly the profit of the monopolist has reduced as compared to before due to paying taxes, but as Amit pointed out in the comment, the government's revenue increases by the difference in old and new profits of the monopolist. Therefore, the aggregate social surplus stays the same leading to no deadweight loss