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15.3: Effect of Taxes

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    43838
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    LEARNING OBJECTIVE

    1. 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

    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 qbqa.
    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).
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