profit from this strategy is $ \left(\mathrm{V}_{\mathrm{s}}^{\mathrm{N}} - \mathrm{c}\right)\left(1 + \lambda_{\mathrm{K}} - \mathrm{x}_{2}\right) $ . Comparing the profits from the first option (K indifferent between new and used) with the third (S indifferent between new and used), it follows that the latter option will be preferred for all $ x_{2} \in [0,1] $ if $ \lambda_{\mathrm{K}} < \left(\mathrm{V}_{\mathrm{s}}^{\mathrm{N}} - \mathrm{c}\right)/\left(\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{s}}^{\mathrm{N}}\right) $ . This restriction requires that the type S be sufficiently abundant in the market that it never pays for the monopolist to set the price of new goods so high that type S are driven out of both the new and used markets. Comparing the profits from the second option (selling only to K in the second period) with that of the third option, we again obtain a threshold result that the monopolist sells only to type K in the second period if $ x_{2} > X_{2}^{\prime} = 1 - \left(V_{\mathrm{K}}^{\mathrm{N}} - V_{\mathrm{K}}^{\mathrm{U}} + V_{\mathrm{S}}^{\mathrm{U}} - V_{\mathrm{S}}^{\mathrm{N}}\right)/\left(V_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c}\right) $ . This result is similar to that in Lemma 2, in that the firm will switch from selling to both types to selling only to the high valuation buyers in period 2 if the stock of used goods is sufficiently high. One difference in this case is that the monopolist must leave some surplus to the high valuation buyers of new goods when $ x_{2} > x_{2}^{\prime} $ because they are also the high valuation buyers of new goods.

The optimal second period profits can then be expressed as $ \pi_{2}^{*}(\mathbf{x}_{2}) = (\mathbf{V}_{\mathbf{K}}^{\mathbf{N}} - \mathbf{V}_{\mathbf{K}}^{\mathbf{U}} + \mathbf{V}_{\mathbf{S}}^{\mathbf{U}} - \mathbf{c})\lambda $ for $ x_{2} < \hat{x} $ and $ \pi_{2}^{*}(\mathbf{x}_{2}) = (\mathbf{V}_{\mathbf{S}}^{\mathbf{N}} - \mathbf{c})(1 + \lambda_{\mathbf{K}} - \mathbf{x}_{2}) $ for $ x_{2} > \tilde{x}_{2}^{'} $ . We can then define the profit differential between selling only to type K in period 1 and selling to both types in period 1 (assuming $ V_{K}^{N} > V_{S}^{N} + \alpha\beta V_{S}^{U} $ ) to be

$$ \Gamma(\alpha)=(\mathrm{V_{K}}^{^{\mathrm{N}}}-\mathbf{c})\lambda_{_{\mathrm{K}}}+\beta\pi_{_{2}}^{^{*}}(0)-(\mathrm{V_{S}}^{^{\mathrm{N}}}+\alpha\beta\mathrm{V_{S}}^{^{\mathrm{U}}}-\mathbf{c})(1+\lambda_{_{\mathrm{K}}})-\beta\pi_{_{2}}^{^{*}}(\alpha) $$

Using the definition of $ \pi_{2}^{*} $ , it can be shown that this expression is strictly decreasing in $ \alpha $ . Evaluating at $ \alpha = 0 $ , we have $ \Gamma(0) = (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \alpha \beta \mathrm{V}_{\mathrm{S}}^{\mathrm{U}}) \lambda_{\mathrm{K}} - (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} + \alpha \beta \mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c}) < -\alpha \beta \mathrm{V}_{\mathrm{S}}^{\mathrm{U}} (1 + \lambda_{\mathrm{K}}) < 0 $ for $ \lambda_{\mathrm{K}} < (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) $ . This establishes that the optimal policy will be to sell to both types in period 1 for all $ \alpha $ , establishing Proposition 2.

## Appendix II

To examine how textbook quantities change over the life of an edition, we estimate simple regression models with textbook title-edition fixed effects. The number of semesters since the introduction of an edition and its squared and cubic terms are also included as regressors. Table A-1 reports the results. The first two columns indicate that new textbook quantity decreases by an average of 12.7%, or 53 units, in each semester after the introduction of a new edition. The next two columns add the squared and cubic terms of the time trend variable and indicate that new textbook quantity sharply declines, especially in the initial periods after the introduction of a new edition.

Columns 5 through 8 of the same table show used textbook quantity. Columns 5 and 6 show that used textbook quantity increases by an average of 7.3%, or 57 units, per semester after the