## Appendix I

Proof of Proposition 1: First suppose that $ \tilde{x} \leq 0 $ , which from (1) requires $ \lambda_{\mathrm{K}} \geq (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}1}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) $ . With $ \tilde{x} \leq 0 $ the monopolist sells only to type K in period 2 regardless of the first period sales policy, the desirability of selling to type S in the first period depends only on first period profits. Selling to type K only in the first period gives $ (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{c})\lambda_{\mathrm{K}} $ and selling to both types in the first period gives $ (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} + \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c})(1 + \lambda_{\mathrm{K}}) $ . It follows that selling to type K only is preferred if $ \lambda_{\mathrm{K}} > (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} + \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}}) $ , which establishes part (a) of the proposition. Selling to both types will be optimal for $ \lambda_{\mathrm{K}} \in [(\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}1}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}), (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} + \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}})) $ . Now suppose that $ \tilde{x} \in (0, \alpha) $ , which requires that $ \lambda_{\mathrm{K}} \in [(\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 - \alpha) / (\mathrm{V}_{\mathrm{K}1}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}), (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}1}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}})) $ . If the firm sells to type S buyers in period 1, the supply of used goods in period 2 is $ \alpha N_{2} $ . This exceeds the critical value for period 2, so that the firm sells only to type K buyers in period 2 and earns a discounted profit of $ (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} + \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c})(1 + \lambda_{\mathrm{K}}) + \beta(\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{c})\lambda_{\mathrm{K}} $ . If the firm sells only to type K buyers in period 1, the second period stock will be 0 and the firm will sell to both types in period 2. This yields a discounted profit of $ (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{c})\lambda_{\mathrm{K}} + \beta(\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 + \lambda_{\mathrm{K}}) $ . Comparing these expressions, the monopolist will find it profitable to sell only to type K buyers in the first period if $ [(\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}})(1 - \beta) - \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}}]\lambda_{\mathrm{K}} > ((\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 - \beta) + \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}}) $ . However, this inequality cannot be satisfied for any feasible values of $ \lambda_{K} $ . If $ (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}})(1 - \beta) - \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} < 0 $ , this inequality cannot be satisfied for $ N_{1}, N_{2} \geq 0 $ . If $ (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}})(1 - \beta) - \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} > 0 $ , the inequality requires $ \lambda_{K} > (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) $ which lies outside the required range. Therefore, the seller will sell to both types in period 1 and only to type K in period 2 for $ \lambda_{K} $ in this range. Combining this result with that of the previous paragraph yields part b.

Finally, consider the case where $ \tilde{x} > \alpha $ , which requires that $ \lambda_{\mathrm{K}} < (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 - \alpha)/(\mathrm{V}_{\mathrm{K}1}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) $ . If the firm sells to type S in the first period it will also sell to both types in the second period, yielding a return of $ (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} + \alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c})(1 + \lambda_{\mathrm{K}}) + \beta(\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 + \lambda_{\mathrm{K}} - \alpha) $ . If the firm sells only to type K in the first period, it $ (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{c})\lambda_{\mathrm{K}} + \beta(\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 + \lambda_{\mathrm{K}}) $ . Selling to both types in period 1 is preferred if $ (\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}_{2} - \mathrm{c})(1 - \beta) + \alpha\beta\theta_{2})\mathrm{N}_{2} $ . This condition will always be satisfied when Assumption 1 holds, which yields part d of the Lemma.

Proof of Proposition 2: Under Assumption (A.1'), Lemma 1 no longer applies because $ R_{2K}^{U} > R_{2S}^{U} $ for $ p_{2}^{N} > V_{K}^{N} - V_{K}^{U} + V_{S}^{U} $ . Therefore, the demand curve for new goods has three regions in period 2. For $ p_{2}^{N} \in (V_{K}^{N} - V_{K}^{U} + V_{S}^{U}, V_{K}^{N}) $ , type S are priced out of the used market and the sales of new goods will be $ (\lambda_{K} - x_{2}) $ . The maximum profit in this region is $ (V_{K}^{N} - c)(\lambda_{K} - x_{2}) $ . For $ p_{2}^{N} \in (V_{S}^{N}, V_{K}^{N} - V_{K}^{U} + V_{S}^{U}) $ , type K buy new goods and S buy used goods, yielding new sales of $ \lambda_{K} $ . The maximum profit in this region is $ (V_{K}^{N} - V_{K}^{U} + V_{S}^{U} - c)\lambda_{K} $ . For $ p_{2}^{N} \leq V_{S}^{N} $ , both types buy new goods and the monopolist sells $ (\lambda_{K} + 1 - x_{2}) $ units. The maximum