# 15.3: Effect of Taxes

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## LEARNING OBJECTIVE

- 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 \(\begin{equation}π=p(q)q−c(q)−tq.\end{equation}\)\)

The first-order condition for profit maximization yields \(\begin{equation}0= ∂π ∂q =p( q m )+ q m p ′ ( q m )− c ′ ( q m )−t.\end{equation}\)

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

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

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

- 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).