This is “Effect of Taxes”, section 15.3 from the book Beginning Economic Analysis (v. 1.0).

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- How does a monopoly respond to taxes?

A tax imposed on a seller with monopoly power performs differently than a tax imposed on a competitive industry. Ultimately, a perfectly competitive industry must pass on all of a tax to consumers because, in the long run, the competitive industry earns zero profits. In contrast, a monopolist might absorb some portion of a tax even in the long run.

To model the effect of taxes on a monopoly, consider a monopolist who faces a tax rate *t* per unit of sales. This monopolist earns
$\pi =p(q)q-c(q)-tq\text{.}$

The first-order condition for profit maximization yields $0=\frac{\partial \pi}{\partial q}=p({q}_{m})+{q}_{m}{p}^{\prime}({q}_{m})-{c}^{\prime}({q}_{m})-t\text{.}$

Viewing the monopoly quantity as a function of *t*, we obtain
$\frac{d{q}_{m}}{dt}=\frac{1}{2{p}^{\prime}({q}_{m})+{q}_{m}{p}^{\u2033}({q}_{m})-{c}^{\u2033}({q}_{m})}<0$
with the sign following from the second-order condition for profit maximization. In addition, the change in price satisfies
${p}^{\prime}({q}_{m})\frac{d{q}_{m}}{dt}=\frac{{p}^{\prime}({q}_{m})}{2{p}^{\prime}({q}_{m})+{q}_{m}{p}^{\u2033}({q}_{m})-{c}^{\u2033}({q}_{m})}>0\text{.}$

Thus, a tax causes a monopoly to increase its price. In addition, the monopoly price rises by less than the tax if ${p}^{\prime}({q}_{m})\frac{d{q}_{m}}{dt}<1\text{,}$ or ${p}^{\prime}({q}_{m})+{q}_{m}{p}^{\u2033}({q}_{m})-{c}^{\u2033}({q}_{m})<0\text{.}$

This condition need not be true but is a standard regularity condition imposed by assumption. It is true for linear demand and increasing marginal cost. It is false for constant elasticity of demand, *ε* > 1 (which is the relevant case, for otherwise the second-order conditions fail), and constant marginal cost. In the latter case (constant elasticity and marginal cost), a tax on a monopoly increases price by more than the amount of the tax.

- A perfectly competitive industry must pass on all of a tax to consumers because, in the long run, the competitive industry earns zero profits. A monopolist might absorb some portion of a tax even in the long run.
- A tax causes a monopoly to increase its price and reduce its quantity.
- A tax may or may not increase the monopoly markup.

- Use a revealed preference argument to show that a per-unit tax imposed on a monopoly causes the quantity to fall. That is, hypothesize quantities
*qb*before the tax and*qa*after the tax, and show that two facts—the before-tax monopoly preferred*qb*to*qa*, and the taxed monopoly made higher profits from*qb*—together imply that*qb*≤*qa*. - When both demand and supply have constant elasticity, use the results of 0 to compute the effect of a proportional tax (i.e., a portion of the price paid to the government).