When only remanufactured products exist in the market, we first analyze the scenario when the government does not have a budget constraint, and then analyze the scenario when the government has a budget constraint.

#### 4.1.1. The Government Does Not Have a Budget Constraint

We begin our analysis by analyzing consumers’ purchasing behavior in the second period, since consumers are strategic. Note that consumers with valuation

$\theta \in \left[{\tau}_{1r},1\right]$ will purchase remanufactured products in the first period, and the demand in this period is

${D}_{1r}=1-{\tau}_{1r}$. A consumer remaining in the second period will purchase the remanufactured product if, and only if,

${u}_{2r}=\delta (\beta \theta -{p}_{2r}+\lambda {D}_{1r})>0\Rightarrow {\tau}_{2r}=\frac{{p}_{2r}-\lambda {D}_{1r}}{\beta}$. In this case, the demand in the second period is

${D}_{2r}={\tau}_{1r}-{\tau}_{2r}={\tau}_{1r}-\frac{{p}_{2r}-\lambda (1-{\tau}_{1r})}{\beta}$. The OEM maximizes its profit in the second period by the setting price,

${p}_{2r}$, as follows:

We obtain the optimal price, demand, and profit in the second period by using first-order condition with respect to

${p}_{2r}$.

Next, we investigate the OEM’s problem in the first period, where the OEM maximizes its total profits over the two periods by the setting price,

${p}_{1r}$. In the first period, a consumer will purchase a remanufactured product if, and only if, his or her net utility from buying is non-negative and is higher than that from purchasing in the second period, i.e.,

$\alpha \theta -{p}_{1r}+s>0$ and

$\alpha \theta -{p}_{1r}+s>\delta (\beta \theta -{p}_{2r}+\lambda {D}_{1r})$. By rational expectation equilibrium, a consumer with valuation

$\theta \in \left[{\tau}_{1r},1\right]$ will buy the remanufactured products in the first period, so that

$\alpha {\tau}_{1r}-{p}_{1r}+s=\delta (\beta {\tau}_{1r}-{p}_{2r}+\lambda {D}_{1r})$. By substituting

${{p}_{2r}}^{*}({\tau}_{1r})=\frac{{\tau}_{1r}(\beta -\lambda )+\lambda}{2}$, the equilibrium price

${p}_{1r}$ can be expressed as follows:

Hence, the OEM’s optimal problem in the first period can be formulated as follows:

By substituting

${p}_{1r}({\tau}_{1r})$ and

${\prod}_{2}^{B*}({\tau}_{1r})$, the OEM’s problem can be reformulated to the following:

To ensure the objective function is concave in ${\tau}_{1r}$, we impose the condition of $\alpha >\frac{(\beta -\lambda )(\beta +2\beta \delta -\lambda )}{4\beta}$. The optimal value of ${\tau}_{1r}$ can be obtained from the first-order conditions, which can in turn be used to solve the equilibrium outcomes for the OEM in both periods. We summarize the optimal results in the following proposition.

**Proposition** **1.** In Setting 1, the OEM only produces the remanufactured products, and there is no budget constraint. The optimal prices, demands, and total profit can be expressed as follows:${p}_{1r}^{S{1}_{N}*}=\frac{s+2\alpha}{2}+\frac{-4{\alpha}^{2}\beta +2\alpha \beta (\beta -(1+\delta )\lambda )+(\beta -\lambda )({\beta}^{2}{\delta}^{2}-s(\beta -\lambda ))}{8\alpha \beta -2(\beta -\lambda )(\beta +2\beta \delta -\lambda )}$, ${p}_{2r}^{S{1}_{N}*}=\frac{2\alpha \beta (\beta +\lambda )-(2s+\beta \delta )\beta (\beta -\lambda )}{8\alpha \beta -2(\beta -\lambda )(\beta +2\beta \delta -\lambda )}$, ${D}_{1r}^{S{1}_{N}*}=\frac{\beta (2s+2\alpha -\beta -\beta \delta +\lambda )}{4\alpha \beta -(\beta -\lambda )(\beta +2\beta \delta -\lambda )}$, ${D}_{2r}^{S{1}_{N}*}=\frac{2\alpha (\beta +\lambda )-(2s+\beta \delta )(\beta -\lambda )}{8\alpha \beta -2(\beta -\lambda )(\beta +2\beta \delta -\lambda )}$, ${\prod}^{S{1}_{N}*}=\frac{4{s}^{2}\beta +4\alpha \lambda \beta +{(2\alpha -\beta \delta )}^{2}\beta +4s(2\alpha -\beta (1+\delta )+\lambda )\beta}{16\alpha \beta -4(\beta -\lambda )(\beta +2\beta \delta -\lambda )}$.

Proposition 1 illustrates that the OEM’s optimal prices, demands, and profit are closely related to consumers’ valuation for remanufactured products, before and after the subsidy ($\alpha ,\beta $), consumers’ strategic behavior ($\delta $), network externality ($\lambda $), and government subsidy ($s$). For instance, consumers’ demand in the first period increases with the government subsidy, whereas it decreases with the government subsidy in the second period, though the government only provides the subsidy in the first period. This indicates that more consumers would make a purchase in the first period rather than delay and wait until more consumers purchase. We first analyze the structural properties when the government does not have a budget constraint, and then analyze the scenario when the government has a budget constraint.

The structural properties of the optimal results without a budget constraint, under Setting 1, are summarized in the following corollary.

**Corollary** **1.** The optimal results without a budget constraint, when only remanufactured products exist in the market have the following characteristics:

(a) The OEM implements a markup pricing (${p}_{1r}^{S{1}_{N}*}<{p}_{2r}^{S{1}_{N}*}$) when $0<s<{s}_{1}$, and implements a markdown pricing (${p}_{1r}^{S{1}_{N}*}>{p}_{2r}^{S{1}_{N}*}$) when ${s}_{1}<s<1$, where ${s}_{1}=\frac{\beta (2\alpha -\beta \delta +\delta \lambda )(\beta +\beta \delta -2\alpha )+2\alpha {\lambda}^{2}}{4\alpha \beta -2(\beta -\lambda )(\beta \delta -\lambda )}$;

(b) The optimal demands are so that${D}_{1r}^{S{1}_{N}*}<{D}_{2r}^{S{1}_{N}*}$when$0<s<{s}_{2}$, and${D}_{1r}^{S{1}_{N}*}>{D}_{2r}^{S{1}_{N}*}$when${s}_{2}<s<1$, where${s}_{2}=\frac{{\beta}^{2}(2+\delta )-\beta \lambda (2-\delta )-2\alpha (\beta -\lambda )}{6\beta -2\lambda}$;

(c) The OEM’s profit,${\prod}^{S{1}_{N}*}$, is always increasing with the government subsidy level,$s$.

When the government subsidy level, $s$, is low, the markup pricing strategy is an optimal strategy for the OEM, i.e., ${p}_{1r}^{S{1}_{N}*}<{p}_{2r}^{S{1}_{N}*}$. A low price ${p}_{1r}^{S{1}_{N}*}$ will cause more consumers to make the purchase in the first period, which would reduce remaining consumers’ uncertainty level about the quality of the remanufactured products in the second period. As a result, remaining consumers in the second period would have a higher valuation for the product, and hence, the OEM can charge a higher price in the second period. It is also interesting to note that many consumers would wait and delay their purchase when the government subsidy is low, even though the OEM offers a lower price in the first period. Consumers’ uncertainty about the quality of the remanufactured products discourages them to purchase in the first period. Consumers’ uncertainty and network externality can improve the OEM’s profit because it allows the firm to charge a higher price to the majority of consumers remaining in the market in the second period.

When the government subsidy level, $s$, is high, markdown pricing strategy is optimal for the OEM, i.e., ${p}_{1r}^{S{1}_{N}*}>{p}_{2r}^{S{1}_{N}*}$. Many consumers who formerly want to wait would purchase the product in the first period, when the government subsidy level is high. Consumers would have a higher utility for the product because of the subsidy, and therefore, the OEM can extract higher surplus from these consumers by charging a higher price. Fewer consumers delay their purchase to the second period, though the firm increases the prices in the first period. The higher subsidy can greatly improve consumers’ utility and ease their uncertainty about the quality of the remanufactured products.

Finally, the OEM’s profit increases with the government subsidy. The OEM can adjust its pricing strategy with respect to different levels of government subsidy, which influence the firm’s ability to extract surplus from consumers. In the presence of externality, consumers’ purchase decision directly affects consumers’ decision in the second period. The OEM can exploit this by implementing markup or markdown pricing strategy.

From Corollary 1, we know that, given consumers’ valuation for the remanufactured products over the two periods (

$\alpha $ and

$\beta $), consumers’ strategic behavior (

$\delta $) and externality effect (

$\lambda $) have an impact on a firm’s pricing strategy and consumers’ demands over the two periods. However, it is difficult to analyze their monotonicity. Therefore, we use a numerical example to show their impact. The impact of these parameters on a firm’ two periods pricing strategy or consumers’ demands is very similar. Hence, we only investigate their impact on a firm’s pricing. The numerical results show that the impact of consumers’ strategic behavior (

$\delta $) and externality effect (

$\lambda $) on a firm’s two periods’ pricing strategy are not significant. Here, we only present when

$\alpha $ and

$\beta $ are large (

$\beta =0.9,\alpha =0.8,\lambda =0.8,\delta =0.75$), and when

$\alpha $ and

$\beta $ are small (

$\beta =0.5,\alpha =0.4,\lambda =0.2,\delta =0.35$), respectively (

Figure 1).

Figure 1 shows that, when (1) consumers’ valuation for the remanufactured products over the two periods (2) consumers’ strategic behavior and (3) externality effect are high, the OEM’s pricing strategy over the two periods is consistent with Corollary 1. Namely, the OEM should employ a markup pricing strategy when the government subsidy level,

$s$, is small whereas the OEM should employ a markdown pricing strategy when

$s$ is large. However, when (1) consumers’ valuation for the remanufactured products over the two periods (2) consumers’ strategic behavior and (3) externality effect are small; therefore, the OEM should always implement a markdown pricing strategy. When consumers’ valuation for the remanufactured products is low, they are more reluctant to make a purchase. As a result, the OEM has to decrease selling price in the second period, to attract more consumers to purchase the remanufactured products.

#### 4.1.2. The Government Has a Budget Constraint

When the government has a budget constraint, the problem becomes binding. The government acts as the Stackelberg leader in the game, whereas the OEM is the follower. We solve the problem backward. Consumers’ purchase decisions and firms’ optimal results are the same with the results without a budget constraint. Therefore, we only need to consider the government optimal problem with a budget constraint. As stated in the previous section, the government’s optimal problem can be reformulated as follows, by substituting

${D}_{1r}^{S{1}_{N}*}$:

We summarize the optimal results with a budget constraint in the following proposition.

**Proposition** **2.** When the OEM produces only the remanufactured products, the government has a budget constraint, and the constraint is binding, so the optimal demand is${D}_{1r}^{S{1}_{B}*}=\frac{\beta (2{s}^{S{1}_{B}*}+2\alpha -\beta -\beta \delta +\lambda )}{4\alpha \beta -(\beta -\lambda )(\beta +2\beta \delta -\lambda )}$, and the optimal subsidy level is${s}^{S{1}_{B}*}=\frac{-{b}_{1}+\sqrt{{{b}_{1}}^{2}-4{a}_{1}{c}_{1}}}{2{a}_{1}}$, where${a}_{1}=2\beta $, ${b}_{1}=\beta (2\alpha -\beta -\beta \delta +\lambda )$, and${c}_{1}=-K\left(4\alpha \beta -(\beta -\lambda )(\beta +2\beta \delta -\lambda )\right)$.

The firm’s optimal prices, consumers’ demands, and the firms’ total profit can be obtained by substituting the optimal subsidy level, ${s}^{S{1}_{B}*}$. Proposition 2 suggests that the budget constraint is binding. The optimal subsidy level is increasing with the constraint, which indicates that the government can still increase consumers’ demand for remanufactured product with a limited budget. We analyze the structural properties in the following corollary.

**Corollary** **2.** The optimal results with a budget constraint, when only remanufactured products exist in the market, have the following characteristics:

(a) The OEM always implements a markup pricing (${p}_{1r}^{S{1}_{B}*}<{p}_{2r}^{S{1}_{B}*}$) when ${s}^{S{1}_{B}*}<{s}_{1}$, and it always implements a markdown pricing (${p}_{1r}^{S{1}_{B}*}>{p}_{2r}^{S{1}_{B}*}$) when ${s}_{1}<{s}^{S{1}_{B}*}$;

(b) The optimal demands are so that${D}_{1r}^{S{1}_{B}*}<{D}_{2r}^{S{1}_{B}*}$when${s}^{S{1}_{B}*}<{s}_{2}$, and${D}_{1r}^{S{1}_{B}*}>{D}_{2r}^{S{1}_{B}*}$when${s}_{2}<{s}^{S{1}_{B}*}$;

(c) The OEM’s profit,${\prod}^{S{1}_{B}*}$, is always increasing with the government subsidy level,${s}^{S{1}_{B}*}$.

Corollary 2 implies similar implications with Corollary 1. We note that there are several distinctions of the firm’s strategy when the government is with and without a budget constraint. The OEM may have to change from a markup pricing strategy to a markdown pricing strategy when the government increases its subsidy level or when there is no budget constraint. However, there is no pricing strategy change for the OEM when there is a budget constraint for the government. The OEM should always employ a markup pricing strategy when the government subsidy is low, whereas the OEM should always implement a markdown strategy when the government subsidy is high. The same applies to consumers’ demands for remanufactured products. The OEM’s total profit still increases with the government subsidy level, although there is an upper limit.