This is “Effect of Taxes”, section 15.3 from the book Beginning Economic Analysis (v. 1.0).
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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
The first-order condition for profit maximization yields
Viewing the monopoly quantity as a function of t, we obtain with the sign following from the second-order condition for profit maximization. In addition, the change in price satisfies
Thus, a tax causes a monopoly to increase its price. In addition, the monopoly price rises by less than the tax if or
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.