Silviase/jgov_v1.4_full_ann huggingface dataset:
$ \geq p_{2}^{N} $ and $ V_{i}^{N}-p_{2}^{N}\geqV_{i}^{U}-p_{2}^{U} $ . Combining these results, we obtain the reservation price for a used (new) good by type i in period 2, $ R_{i2}^{U}(R_{i2}^{N}) $ , to be

$$ \mathbf{R}_{\mathrm{i}2}^{\mathrm{U}}=\min\left[\mathbf{V}_{\mathrm{i}}^{\mathrm{U}},\mathbf{V}_{\mathrm{i}}^{\mathrm{U}}-\mathbf{V}_{\mathrm{i}}^{\mathrm{N}}+\mathbf{p}_{2}^{\mathrm{N}}\right]\qquad\mathbf{R}_{\mathrm{i}2}^{\mathrm{N}}=\min\left[\mathbf{V}_{\mathrm{i}}^{\mathrm{N}},\mathbf{V}_{\mathrm{i}}^{\mathrm{N}}-\mathbf{V}_{\mathrm{i}}^{\mathrm{U}}+\mathbf{p}_{2}^{\mathrm{U}}\right] $$

We will impose two restrictions on the taste parameters of the two types. The first gives type K the greatest valuation on new goods and type S the greatest valuation on used goods:

$$ \left(\mathrm{A}.1\right)\mathrm{V}_{\mathrm{K}}^{\mathrm{~N}}>\mathrm{V}_{\mathrm{S}}^{\mathrm{~N}}+\alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{~U}}\mathrm{a n d}\mathrm{V}_{\mathrm{S}}^{\mathrm{~U}}>\mathrm{V}_{\mathrm{K}}^{\mathrm{~U}}\geq0 $$

Note that the value of a new good to S buyers includes the expected return from being able to sell the good to a type S buyer in the used market in the future. Assumption (A.1) ensures that type S will always be the high valuation buyers in the used market, since it implies $ R_{S2}^{U} > R_{K2}^{U} $ for all $ p_{2}^{N} $ . This assumption seems plausible for the case of textbooks, since a student planning to hold on to the book will place a higher valuation on having a high quality copy and making careful use of the book. On the other hand, a student wanting the book only for the purpose of getting through the course is likely to find the used book to be a much better substitute for the new good.

The second assumption relates the taste parameters of the S buyers to the marginal cost of production

$$ \left(\mathrm{A}.2\right)\mathrm{V}_{\mathrm{S}}^{\mathrm{~N}>}\mathrm{\boldmath~c>V_{\mathrm{S}}^{\mathrm{~N}}-V_{\mathrm{S}}^{\mathrm{~U}}\:} $$

The left hand inequality restricts attention to cases in which type S buyers have a sufficiently large willingness to pay for the services of the good that they would purchase a new good in the second period if it was priced at marginal cost. Utilizing the right hand inequality in (A.2), we have $ R_{S_{2}}^{U} > 0 $ for all $ p_{2}^{N} \geq c $ . Since sellers will only choose prices that are no less than marginal cost, there will
be a positive price in the second period as long as supply is less than $ N_{s} $ . An implication of this assumption is that it is not socially efficient to kill off the used market, since the used market would exist under a seller pricing at marginal cost.

Letting $ X_{2} $ denote the stock of used goods in the second period, this analysis yields

Lemma 1: If $ X_{2} \leq N_{S} $ , $ p_{2}^{N} \geq c $ , and assumptions (A.1) - (A.2) hold, $ p_{2}^{U} = R_{2S}^{U} = \min\left[V_{S}^{U}, V_{S}^{U} - V_{S}^{N} + p_{2}^{N}\right] > 0 $ .

Under the assumption that type K buyers always keep their goods and that the number of consumers is constant in each period, this is the relevant case for consideration here since the maximum possible second period supply is $ \alpha N_{s} $ . For the benchmark case of a durable being supplied by perfectly competitive sellers, $ N_{K} + N_{s} $ units of the good would be sold in the first period and $ \mathrm{N}_{\mathrm{K}} + \mathrm{N}_{\mathrm{s}}(1 - \alpha) $ units in the second period. Used market sales in period 2 would be $ \alpha N_{s} $ units at a price of $ V_{s}^{U} - V_{s}^{N} + c > 0 $ .

### I -A. Monopoly Pricing of the Durable Good

We now turn to the analysis of the optimal pricing policy for a monopoly seller of the durable good. Since the first period valuations of type S buyers depend on the seller's second period pricing policy, we solve the problem using backward induction. In period 2, the seller chooses the price for a new good given a stock $ X_{2} $ of used goods resulting from sales in the previous period. If $ p_{2}^{N} \leq V_{S}^{N} $ , then it follows from (1) and Lemma 1 that type S buyers will buy both new and used goods and type K buyers will purchase new goods. If $ p_{2}^{N} \in (V_{S}^{N}, V_{K}^{N}] $ , then type K buyers will receive non-negative surplus from new goods and type S buyers will purchase only used goods. This indicates that the monopolist will choose between two pricing strategies: selling to only type K buyers at a price of $ V_{K}^{N} $ and selling to both types at a price of $ V_{S}^{N} $ . The former strategy yields a profit of $ (V_{K}^{N} - c)N_{K} $ , while
the latter yields a profit of $ (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(\mathrm{N}_{\mathrm{K}} + \mathrm{N}_{\mathrm{S}} - \mathrm{X}_{2}) $ . Letting $ \lambda_{K} \equiv N_{K}/N_{S} $ and $ x_{2} \equiv X_{2}/N_{S} $ , the strategy of selling only to type K will be more profitable if $ \lambda_{\mathrm{K}} \geq (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c})(1 - \mathrm{x}_{2})/(\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) $ .

This comparison yields the following result:

Lemma 2: Under (A.1) and (A.2), the optimal policy of the monopolist in the last period of the editions is:

$$ (a)\boldsymbol{p}_{2}^{N}=\boldsymbol{V}_{K}^{N}and\boldsymbol{\pi}_{2}^{*}(\boldsymbol{x}_{2})=(\boldsymbol{V}_{K}^{N}-c)\boldsymbol{\lambda}_{K}for\boldsymbol{x}_{2}\geq\tilde{\mathbf{x}} $$

$$ (b)\boldsymbol{p}_{2}^{^{N}}=\boldsymbol{V}_{S}^{^{N}}and\boldsymbol{\pi}_{2}^{*}(\boldsymbol{x}_{2})=(\boldsymbol{V}_{S}^{^{N}}\cdot\boldsymbol{c})(\boldsymbol{1}+\boldsymbol{\lambda}_{K}\cdot\boldsymbol{x}_{2})for\boldsymbol{x}_{2}<\tilde{\mathbf{x}}. $$

$$ where\quad\tilde{\mathbf{x}}\equiv\max\left[1-\lambda_{\mathrm{K}}\left(\frac{\mathbf{V}_{\mathrm{K}}^{\mathrm{N}}-\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}}{\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}-\mathbf{c}}\right),0\right] $$

In either case, $ p_{2}^{U}=V_{S}^{U} $

The seller will sell only to type K buyers if the normalized stock of used goods exceeds $ \tilde{x} $ . This result establishes a non-decreasing relationship between the stock of used goods and the price that a monopoly seller charges for new goods in period 2, since a larger used stock makes it more attractive to sell only to the high valuation buyers.

The profit function and used price solution from Lemma 2 can now be used to solve for the first period optimization problem for the seller. Since the monopolist's pricing strategy will depend only on the relative abundance of type K buyers, as in Lemma 2, we can simplify the notation by normalizing all quantities by the number of type S buyers. The discounted normalized profits of the firm over the life of the good will be

$$ \boldsymbol{\Pi}=(\mathbf{p}_{1}^{\mathrm{~N~}}-\mathbf{c})\mathbf{y}_{1}(\mathbf{p}_{1}^{\mathrm{~N~}})+\boldsymbol{\beta}\boldsymbol{\pi}_{2}^{*}(\mathbf{x}_{2}(\mathbf{p}_{1}^{\mathrm{~N~}})) $$

where $ y(p_{1}^{\mathrm{N}}) $ is the normalized first period demand and $ x_{2}(p_{1}^{\mathrm{N}}) $ is the second period stock resulting
from the firm’s first period price choice. The normalized second period stock will equal $ \alpha $ if type S consumers buy in period 1 and 0 otherwise, which from Lemma 2 yields

$$ \mathbf{0}\qquad\quad\mathbf{for}\mathbf{p}_{1}^{\mathrm{N}}>\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}+\alpha\beta\mathbf{V}_{\mathrm{S}}^{\mathrm{U}} $$

$$ \mathbf{x}_{2}(\mathbf{p}_{1}^{\mathrm{~N~}}) $$

$$ \alpha\qquad\text{otherwise} $$

The firm’s first period choice can be simplified to choosing between setting a price of $ V_{K}^{N} $ and selling $ \lambda_{K} $ units or setting a price of $ V_{S}^{N} + \alpha\beta V_{S}^{U} $ and selling $ (1 + \lambda_{K}) $ units. $ ^{9} $ If the seller chooses to serve only high valuation buyers in the first period, there will be no second period stock of used goods and normalized profits will be $ (V_{K}^{\mathrm{N}} - c)\lambda_{K} + \beta\pi_{2}^{*}(0) $ . If the seller chooses to sell lower the price to sell to both types, the profits will be $ (V_{S}^{\mathrm{N}} + \alpha\beta V_{S}^{\mathrm{U}} - c)(1 + \lambda_{K}) + \beta\pi_{2}^{*}(\alpha) $ .

Comparing the payoffs from these strategies, we can obtain the critical values of $ \lambda_{K} $ for which the respective strategies will be optimal.

Proposition 1: Under $ (A.1) $ , the optimal strategy for the monopolist will be:

a. sell to type K consumers only in both periods at $ p_{l}^{N} = p_{2}^{N} = V_{K}^{N} $ if

$$

\lambda_{\mathrm{K}} > \frac{\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}_{2}^{\mathrm{U}}}

$$
b. sell to both types in period 1 but sell only to type K in period 2, $ p_{I}^{N} = V_{S}^{N} + \alpha \beta V_{S}^{U} < p_{2}^{N} $

$$ =\boldsymbol{V}_{K}^{^{N}}i f\lambda_{\mathrm{K}}\in\left(\frac{(\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}-\mathbf{c})(1-\alpha)}{\mathbf{V}_{\mathrm{K}}^{\mathrm{N}}-\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}},\frac{\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}+\alpha\beta\mathbf{V}_{\mathrm{S}}^{\mathrm{U}}-\mathbf{c}}{\mathbf{V}_{\mathrm{K}}^{\mathrm{N}}-\mathbf{V}_{\mathrm{S}}^{\mathrm{N}}-\alpha\beta\mathbf{V}_{\mathrm{S}}^{\mathrm{U}}},\right). $$

c. sell to both types in both periods, $ p_{I}^{N}=V_{S}^{N}+\alpha\beta V_{S}^{U}>p_{2}^{N}=V_{S}^{N} $ , if $ \lambda_{K}<\frac{V_{S}^{N}-c}{V_{K}^{N}-V_{S}^{U}} $

If the population consists of a sufficiently large fraction of type K buyers, the optimal pricing policy is to set a constant price for new goods in each period that extracts all surplus from type K buyers. In this case there will be no used market and type S obtain zero surplus. When the fraction of type K is sufficiently low, the optimal price for new goods declines over time to extract all surplus from type S buyers. Used markets are active in each case, and type K buyers earn a positive surplus in each period on purchases of new goods. The interesting case arises for intermediate values of $ \lambda_{K} $ , in which the price of new goods rises in the last period of the good's life. In this case, a sufficiently large fraction of type S buyers purchase goods in the used market that the seller writes off that segment of the market and sells only to type K buyers.

### I -B. Extensions of the Model

Proposition 1 established conditions under which the price of a new durable good would be higher in the second period than in the first period of an edition under the assumption that (A.1) holds. A natural question to ask is to how this possibility is affected by extending the life of the durable good to three periods. Is the rising price necessarily a “last period effect,” or could it arise in the second period of the life of the good as well? It is straightforward to show that either of these possibilities
can arise in the three period model. A rising price of new goods between the first and second periods arises when the monopolist's optimal policy is to sell to both types in period 1, but to sell only to K types in subsequent periods. The price rises in the last period when the optimal policy results in sales to both types in the first two periods, but only to the K types in the last period. Since S type buyers will have a higher reservation for new goods the greater is the remaining life of the good, the threshold stock for selling only to type K buyers in the second period is higher than that in the last period.

A second extension is to consider the case in which type K buyers place a higher valuation on both new and used goods. Such a case might arise if type K are high income, high transaction cost buyers. A high income would be associated with a relatively higher valuation for all goods, but a high time cost might prevent them from selling goods in the used market. It can be shown in this case a rising price of new goods might arise if we modify (A.1) to

$$ \mathrm{V}_{\mathrm{K}}^{\mathrm{~N}>}\mathrm{V}_{\mathrm{S}}^{\mathrm{~N}+}\alpha\beta\mathrm{V}_{\mathrm{S}}^{\mathrm{~U}},\mathrm{V}_{\mathrm{K}}^{\mathrm{~U}>}\mathrm{V}_{\mathrm{S}}^{\mathrm{~U}},\mathrm{a n d}\mathrm{~V}_{\mathrm{K}}^{\mathrm{~N}-}\mathrm{V}_{\mathrm{K}}^{\mathrm{~U}>}\mathrm{V}_{\mathrm{S}}^{\mathrm{~N}}+(\alpha\beta-1)\mathrm{V}_{\mathrm{S}}^{\mathrm{~U}}. $$

Assumption (A.1') captures the case in which the K buyers value both new and used goods more highly, but have a relative preference for new goods.

As in the case where (A.1) held, a rising price of new goods over the life of the edition will be observed for parameter values such that the monopolist finds it optimal to sell new goods to both types in the first period, but only to type K in the second period. Establishing this possibility under (A.1') is slightly more complicated because of the possibility that type K buyers can outbid type S buyers for used goods in period 2. It is shown in Appendix I that if $ \lambda_{\kappa} < (\mathrm{V}_{\mathrm{s}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{s}}^{\mathrm{N}}) $ . 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. It is shown in Appendix I that the monopolist will choose to sell only to type K in the second period if $ \lambda_{\mathrm{K}} < (\mathrm{V}_{\mathrm{S}}^{\mathrm{N}} - \mathrm{c}) / (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) $ and $ x_{2} > 1 - (\mathrm{V}_{\mathrm{K}}^{\mathrm{N}} - \mathrm{V}_{\mathrm{K}}^{\mathrm{U}} + \mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{V}_{\mathrm{S}}^{\mathrm{N}}) / (\mathrm{V}_{\mathrm{S}}^{\mathrm{U}} - \mathrm{c}) $ . Since the former condition also ensures that it will be optimal to sell to both types in the first period, we obtain the following result:

Proposition 2: If $ (A.1') $ and $ (A.2) $ hold and $ \lambda_{K} < (V_{S}^{N} - c)/(V_{K}^{N} - V_{S}^{N}) $ , the optimal first period policy is to sell to both consumer types. The optimal second period policy is to sell only to type K if $ \alpha > 1 - (V_{K}^{N} - V_{K}^{U} + V_{S}^{U} - V_{S}^{N})/(V_{S}^{U} - c) $ , which results in a rising price of new goods over the life of the edition.

Proposition 2 shows that the threshold effect of the used stock and the rising price of new goods over time can both be observed in the model in which type K buyers have an absolute preference for new and used goods, although the upper bound on $ \lambda_{K} $ for which this holds is lower than that in Proposition 1b. $ ^{10} $

### I I. An Empirical Analysis of the Textbook Market

In this section, we use data from the sales of new economics textbooks to test hypotheses regarding the pricing of textbooks over the life of an edition. We first describe the data and then discuss the empirical model to examine the firm’s price setting behavior.
### I I-A. Data

Our textbook pricing data come from college bookshops and are collected by Monument Information Resource (MIR). Each semester, MIR collects textbook sales and adoption information for each school. Our sales data are aggregated at the national level and contain the number of textbooks sold (new vs. used), average prices (new vs. used), edition number, year and month of publication, author name, textbook category, publisher name, and the ISBN code. In contrast, adoption data are at the school level and contain the name, location, and ID of the school, the course name, estimated enrollments, and detailed information on the adopted textbook, including the ISBN code. If multiple sections or courses are offered in a school in the same semester, we observe the textbook adoption information separately. For this study, we focus our attention on economics textbooks that appear in the MIR database between 1997 and 2003. MIR collects data twice a year, and this gives us 14 semester of observations, at most, for each textbook. $ ^{11} $ We refer to these semesters as fall and spring semesters. MIR estimates that their 2003 data covered approximately 50.4 per cent of the total college textbook market in the U.S. $ ^{12} $

The main limitation of the data is that the MIR only covers the transactions that take place through college bookstores. For example, buying and selling textbooks through online bookstores are not captured in the data. We are fortunate, however, that during the data period (1997 to 2003), the online sales of textbooks appear to be relatively small. According to the estimate by the National Association of College Stores, online textbook sales accounts for only 7% in 2002. Nonetheless, the
results should be viewed with care because the data also do not contain other transactions, such as transactions among students.

For the 14 semesters between 1997 and 2003, we have a total of 4,708 observations for the sales data. An observation is a title-edition-semester. Publishers commonly revise textbook editions over time with almost identical names. A “title” refers to the name of a textbook. There are 483 unique textbook titles in the data set, and each title has an average of 1.6 editions; we observe a total of 440 edition revisions. We limit our attention to the textbooks that cost more than $40 in 2003 dollars, averaged across the semesters in the data. We exclude textbooks that did not revise editions during our data period and textbooks with no edition information assigned by MIR, Amazon, or Barnes & Noble. Therefore, we do not include popular press books that are sometimes assigned to economics classes, for which publishers are likely to have different pricing problems. Study guides, custom textbooks, government publications, and Canadian editions are excluded from the analysis.

As a measure of textbook price, we focus our analysis on the price of the textbook as a standalone textbook. That is, we do not use the “package” price of a textbook in our estimation, which contains supplementary materials, such as study guides, software, and CD-ROMs. Various types of these supplementary materials exist especially for popular textbooks, and controlling for these product attributes is difficult. Thus, comparing the prices of stand-alone textbooks is more reliable for examining the textbook pricing cycles. $ ^{13} $ In our preliminary analysis, however, we also estimated the model with average textbook prices of packages and stand-alone textbooks. The qualitative results changed little for this sample.
To examine how prices change over the life of an edition, we need to know when an edition was launched. If we observe multiple editions of the same title, we identify the entry of a new edition when we observe a new edition of a textbook in the MIR data. $ ^{14} $ For those textbook-editions that appear in the MIR data for the first time, the month of the new edition entry is identified using the MIR data, Amazon.com, and BarnesandNoble.com. Sometimes, an old edition of a title is sold even after a new edition of the same title is introduced. These “overlapping” observations were dropped from the data set.

Table I presents summary statistics. An average price for a new textbook in our sample is 91.5 in 2003 dollars, and the average quantity share of used textbooks (UsedRatio) is 40%. There is substantial variation in both textbook prices and used market shares, as shown in the table (see Figure 1 for the distribution of textbook prices in the data.) The average age of the textbooks is 6.9 semesters. In Figure 2, we look at the textbooks that experienced a revision between 1997 and 2003 and show the distribution of the age of the textbook in semester when a new edition is introduced. Although textbooks are thought to follow a pre-determined “3-year revision cycle,” the data show that there is substantial variation in the timing of textbook revision; while 35% of the textbooks in this sample revise editions within 6 semesters (or 3 years), as indicated by the highest bar in the graph, 33% of the textbooks introduce new editions in 7 semesters or longer. Additionally, 32% of the textbooks revise editions in 5 or fewer semesters. $ ^{15} $
### I I-B. Empirical Model

We now turn to the estimation of the firm’s price setting behavior derived from the theoretical model. The theoretical model in the previous section suggests estimating a relationship of the following form:

$$ \mathbf{p}_{\tau}^{\mathrm{~N~}}=\mathbf{f}(\mathbf{X}_{\tau},\mathbf{T}-\tau) $$

where $ p_{\tau}^{N} $ is the price of a new textbook in the $ \tau $ th period of the current edition, $ X_{\tau} $ is the stock of used goods for the edition at age $ \tau $ , and T is the expected life of the current edition. If the textbook market consists of homogeneous buyers who plan to sell the books, such as the type-S buyers of the theoretical model, then the primary determinant of the price of the book should be the remaining life of the edition, i.e., T- $ \tau $ . As in Proposition 1c, the price of the book will fall as the remaining life of the edition becomes shorter because the expected value of the good in resale will be lower. However, the price discrimination model with heterogeneous buyers in Proposition 1b suggests a positive relationship between the stock of used goods and the price of new goods; the policy of raising the price of new goods to sell only to the high valuation buyers of new goods becomes more attractive as the number of low valuation buyers purchasing in the used market rises. In addition, raising the price will be more attractive with the shorter remaining life of the edition.

To examine the textbook publisher’s price setting behavior, we estimate the following price equation:

$$ \begin{array}{r}{\ln p_{\dot{\mathbf{x}}}=\mathbf{A}_{\mathbf{\Gamma}}+\mathbf{B}_{\mathbf{\Gamma}}+\alpha\mathit{U s e d R a t i o}_{i t}+\sum_{\mathcal{M}=1}^{4}\beta_{\mathcal{M}}R m n L i f e N_{i t}+\gamma\ln Q_{i t}+\varepsilon_{i t}}\end{array} $$

where the dependent variable is the natural logarithm of the new price of a textbook title-edition i during semester t. $ A_{t} $ and $ B_{i} $ are semester and textbook title-edition dummy variables, respectively. As noted, one of the key factors that would affect the pricing policy is the stock of used goods. To adjust