15.3: Effect of Taxes

Learning Objectives

• 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 $π=p(q)q−c(q)−tq.$\)

The first-order condition for profit maximization yields $0= ∂π ∂q =p( q m )+ q m p ′ ( q m )− c ′ ( q m )−t.$

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

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

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.

Key Takeaways

• 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.

EXERCISES

1. 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.
2. 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).