A capacitated firm’s pricing strategies for strategic consumers with different search costs

Abstract

This paper studies pricing strategies of a seller with budget constraints facing two types of strategic consumers with different search costs, and proposes three pricing strategies to motivate all consumers to visit his shop. These are the basic price strategy, differentiated compensation strategy and an improved differentiated compensation strategy. Based on the rational expectations paradigm, we characterize the rational expectations equilibrium in the game and propose a basic pricing strategy. In order to address the interplay between price and demand, we further propose a differentiated compensation strategy to improve the basic model. We then compare the differentiated compensation strategy to the basic pricing strategy when both are feasible. We find that selection of the optimal strategy is independent of composition of consumers but is dependent on the seller’s budget level and the difference between the two search costs. If the budget is large enough and the difference between the search costs is small enough, a differentiated compensation strategy can further improve the seller’s profitability. In addition to these findings, we first propose an improved differentiated compensation strategy to further enhance the firm’s profit. We find that the optimal strategy is to implement the improved differentiated compensation strategy when all three strategies are feasible. Interestingly, the firm may benefit from paying a high compensation to the consumers.

Appendix

Proof of Proposition 1:

To implement the basic pricing strategy, the following three conditions need to be satisfied: (1) the quantity in equilibrium $q^{*}_{basic}$ exists, (2) $q^{*}_{basic}$ can be realized and (3) the profit in equilibrium is positive. Now we will prove them one by one.

First, we prove the existence of the quantity in equilibrium $q^{*}_{basic}$. According to the conditions under which the rational expectations equilibrium comes to exist, we know $q^{*}_{basic}$ is a solution of the following equation $(v-c/\bar{F}(q^{*}_{basic}))A(q^{*}_{basic})=h_{H}$. Define $J(q)=(v-c/\bar{F}(q))A(q)$, then one has that J(0)=0,J′(0)>0,J′(∞)=−∞, J″(q)<0. Further, one has that J(q) is concave with respect to q∈[0,∞). Let J _max denote the maximum value of J(q). Since one has that h _H ≤J _max from assumptions of this Proposition, $q^{*}_{basic}$ exist.

Second, we prove that $q^{*}_{basic}$ can be realized. Let q ₁≤q ₂ be the two solutions to J(q)=h _H . Whether the quantity can be realized or not depends on the seller’s budget. When the basic pricing strategy is feasible, its minimum requirement of the budget is cq ₁. So according to the budget condition that K≥cq ₁ and the definition of rational expectations equilibrium, one has that this rational expectations equilibrium exists.

Finally, we prove the profit in equilibrium is positive. Define G(q)=vE(X∧q)−cq. We know that the function G(q) is concave. We assume the maximum value of G(q) is achieved at q _G . Clearly, there must be $\bar{F}(q_{G})=c/v$. Because $\bar{F}(q^{*}_{basic})=c/p$ and p<v, then $\bar{F}(q^{*}_{basic})>\bar{F}(q_{G})$. Further, one has that $q^{*}_{basic}<q_{G}$. Hence the function G(q) increases in $[0, q^{*}_{basic}]$. When K≥cq ₁, there are two cases of the profit in equilibrium.

Case 1::

If cq ₁≤K<cq ₂, then $q^{*}_{basic}=q_{1}$ and the profit is $\prod^{*}_{basic}=G(q_{1})-h_{H}\mathrm {E}[X]$. Because G(q ₁)>h _H E[X], one has that $\prod^{*}_{basic}>0$.

Case 2::

If K≥cq ₂, then $q^{*}_{basic}=q_{2}$ and the profit is $\prod^{*}_{basic}= G(q_{2})-h_{H}\mathrm {E}[X]$. Since h _H ∈(0,J _max ], q ₂ decreases in h _H and G(q) increases in q ₂. Then function G(q) decreases in h _H . Then $\prod^{*}_{basic}$ decreases in h _H . If $\prod^{*}_{basic}>0$ for h _H =J _max , then it holds that $\prod^{*}_{basic}>0$ for h _H ∈(0,J _max ]. If $\prod^{*}_{basic}<0$ for h _H =J _max , since $\prod^{*}_{basic}$ is continuous, then there exists a $h^{'}_{1}>0$ such that $\prod^{*}_{basic}=0$ with $h_{H}=h^{'}_{1}$. So, $\prod^{*}_{basic}>0$ for $h_{H}<h_{1}=\min\{h^{'}_{1}, J_{max}\}$, which means that the seller earns a positive profit. Notice that both $h^{'}_{1}$ and J _max are lower than v−c and increase in v and decrease in c. So h ₁ increase in v and decreases in c.

In summary, under the condition that h _H ≤J _max , if cq ₁≤K<cq ₂ and G(q ₁)>h _H E[X], there exists a unique rational expectations equilibrium, whereas if K≥cq ₂, there exist some h ₁ satisfying 0<h ₁<v−c such that a unique rational expectations equilibrium exists with h _H <h ₁. □

Proof of Proposition 2:

To implement the differentiated compensation strategy, the following three conditions need to be satisfied: (1) the quantity in equilibrium $q^{*}_{dc}$ exists, (2) $q^{*}_{dc}$ can be realized and (3) the compensation $u^{*}_{dc}\leq h_{H}$ as well as the profit in equilibrium is positive. Now we prove them one by one.

• (1) First, we prove the existence of the quantity in equilibrium $q^{*}_{dc}$. It is similar to the proof of Proposition 1. Only if h _L ≤J _max , then $q^{*}_{dc}$ satisfied the following equation $(v-c/\bar{F}(q^{*}_{dc}))A(q^{*}_{dc})=h_{L}$. Hence, it further shows that $q^{*}_{dc}$ exists.

• (2) Second, we point out that $q^{*}_{dc}$ can be realized. Let Q ₁≤Q ₂ be the two solutions to J(q)=h _L . Similarly to the proof of Proposition 1, when the differentiated compensation strategy is feasible, its minimum requirement of the budget is cQ ₁. So the budget only satisfies K≥cQ ₁, this rational expectations equilibrium exists.

• (3) Finally, we prove the compensation $u^{*}_{dc}\leq h_{H}$ and the profit in equilibrium is positive. When K≥cQ ₁, there are two cases of the profit in equilibrium.

  Case 1::

  If cQ ₁≤K<cQ ₂, $q^{*}_{dc}=Q_{1}$.

  When Δh≤h _L (1/A(Q ₁)−1), we know that $u^{*}_{dc}\leq h_{H}$. The profit in equilibrium is $\prod^{*}_{dc}=G(Q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]$. Because G(Q ₁)>h _L E[X] and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$, one has that $\prod^{*}_{dc}>0$.

  Case 2::

  When K≥cQ ₂, there are two cases.

  Case 2.1::

  When h _L (1/A(Q ₂)−1)<Δh≤h _L (1/A(Q ₁)−1),

  If $q^{*}_{dc}=Q_{2}$, then $u^{*}_{dc}> h_{H}$. This makes this strategy unfeasible. So, it is the only choice that $q^{*}_{dc}=Q_{1}$. The profit is $\prod^{*}_{dc}=G(Q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]$. Because G(Q ₁)>h _L E[X] and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$, one has that $\prod^{*}_{dc}>0$.

  Case 2.2::

  When Δh≤h _L (1/A(Q ₂)−1),

  If $q^{*}_{dc}=Q_{2}$, then $u^{*}_{dc}\leq h_{H}$, which leads no moral hazard problem for the consumers. This makes this strategy feasible. Then the profit is $\prod^{*}_{dc}=G(Q_{2})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]$.

  Because G(Q ₂)>h _L E[X] and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{2})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$, one has that $\prod^{*}_{dc}>0$.

In summary, under the condition that h _L ≤J _max , one has the following results:

When cq ₁≤K<cq ₂, if G(Q ₁)>h _L E[X], $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$ and Δh≤h _L (1/A(Q ₁)−1), there exists a unique rational expectations equilibrium. When K≥cQ ₂, if G(Q ₁)>h _L E[X], $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$ and h _L (1/A(Q ₂)−1)<Δh≤h _L (1/A(Q ₁)−1), there exists a unique rational expectations equilibrium. When K≥cQ ₂, if G(Q ₂)>h _L E[X], $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{2})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$ and Δh≤h _L (1/A(Q ₂)−1), there exists a unique rational expectations equilibrium. □

Proof of Proposition 3:

In equilibrium, the total cost to motivate high-cost type consumer’s ex-ante is (1−θ)E[X]Δh. But the number of high-cost consumers who get the compensation ex-post is $(1-A(q^{*}_{dc}))(1-\theta)\mathrm {E}[X]$. The individual high-cost type consumer who doesn’t get the product is provided compensation amounting to $((1-\theta)\mathrm {E}[X] \varDelta h)/ ((1-A(q^{*}_{dc}))(1-\theta)\mathrm {E}[X])=\varDelta h/(1-A(q^{*}_{dc}))$. So, $u^{*}_{dc}=\varDelta h/(1-A(q^{*}_{dc}))$. Because $0<A(q^{*}_{dc})<1$, then $\varDelta h/(1-A(q^{*}_{dc}))> \varDelta h$, i.e., $u^{*}_{dc}>\varDelta h$. □

Proof of Proposition 4:

In order to prove this proposition 4, we consider the three cases:

CASE 1::

When G(q ₂)≤h _H E[X], the profit of the basic pricing strategy is $\prod^{*}_{basic}=G(q_{2})-h_{H}\mathrm {E}[X]<0$. So the basic strategy is not feasible.

In the following, we prove the differentiated compensation strategy is feasible and optimal. Because G(Q ₁)>h _L E[X] and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<\!1$, the profit from the differentiated compensation strategy is $\prod^{*}_{dc}=G(Q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X] >0$. Under this condition, the differentiated compensation strategy is feasible and realizes the theoretical suboptimal solution (because the differentiated compensation strategy can only realize the theoretical suboptimal solution when h _L (1/A(Q ₂−1))<Δh≤h _L (1/A(Q ₁−1)). Because G(Q ₁)≤G(Q ₂), when G(Q ₁)>h _L E[X] and $\max\{0,\frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$, it always has that G(Q ₂)>h _L E[X] and $\max\{0,\frac{h_{H}\mathrm {E}[X]-G(Q_{2})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$. Further, we know the profit from the differentiated compensation strategy is $\prod^{*}_{dc}=G(Q_{2})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]>0$. Under this condition, the differentiated compensation strategy is feasible and realizes the theoretical optimal solution (Because the differentiated compensation strategy can realize the theoretical optimal solution when Δh≤h _L (1/A(Q ₂−1)). Therefore, in this case, the basic pricing strategy is not feasible while the differentiated compensation strategy is feasible. The optimal option is the differentiated compensation strategy.

CASE 2::

Because G(q ₂)>h _H E[X], the profit from the basic pricing strategy is satisfied with $\prod^{*}_{basic}=G(q_{2})-h_{H}\mathrm {E}[X]>0$. So the basic strategy is feasible. Because Δh>h _L (1/A(Q ₂)−1), the differentiated compensation strategy cannot realize the theoretical optimal solution. Then we consider the following two situations:

Subcase 1: The differentiated compensation strategy is not feasible and the only feasible strategy is the basic pricing strategy. So the basic pricing strategy is optimal.

Subcase 2: The differentiated compensation strategy is feasible and realizes the theoretical suboptimal solution.

The profit from the differentiated compensation strategy is

$$\prod^{*}_{dc}=G(Q_{1})-\bigl(\theta h_{L}+(1-\theta)h_{H}\bigr)\mathrm {E}[X]. $$

The difference between the profits of the two strategies is

$$\prod^{*}_{basic}-\prod^{*}_{dc}=G(q_{2})-G(Q_{1})-\theta \varDelta h \mathrm {E}[X]. $$

According to the proofs of Proposition 1 and Proposition 2, we get that q ₂, Q ₁ satisfies $(v-c/\bar{F}(q_{2}))A(q_{2})=h_{H}$, $(v-c/\bar{F}(Q_{1}))A(Q_{1})=h_{L}$ respectively. So, one has that $G(q_{2})-G(Q_{1})-\theta \varDelta h \mathrm {E}[X]=c((Q_{1}-\mathrm {E}[X\wedge Q_{1}]/\bar{F}(Q_{1}))-(q_{2}-\mathrm {E}[X\wedge q_{2}]/\bar{F}(q_{2})))$. According to the proof of Proposition 1 and Proposition 2, we know that $q^{*}_{basic}$ is the solution of the equation J(q)=h _H and $q^{*}_{dc}$ is the solution of the equation J(q)=h _L . Further, one has that Q ₁<q ₁≤q ₂<Q ₂<q _G . Since the function $q-\mathrm {E}[X\wedge q]/\bar{F}(q)$ decreases in q, one has that $\prod^{*}_{basic}>\prod^{*}_{dc}$.

Thus, under this condition, the basic pricing strategy is feasible. No matter whether the differentiated compensation strategy is feasible, the optimal strategy is the basic pricing strategy.

CASE 3::

According to the above proof, we know that when G(q ₂)>h _H E[X], the basic pricing strategy is feasible and realizes the theoretical optimal solution. According to the proof of Proposition 4, further when Δh≤h _L (1/A(Q ₂)−1), the differentiated compensation strategy is feasible and realizes the theoretical optimal solution. Therefore, when the basic pricing strategy is adopted under this condition, the price is $p^{*}_{basic}=v-h_{H}/A(q_{2})$ and the profit is $\prod^{*}_{basic}=G(q_{2})-h_{H}\mathrm {E}[X]$.

When the differentiated compensation strategy is adopted, the price is $p^{*}_{dc}=v-h_{L}/A(Q_{2})$ and the profit is

$$\prod^{*}_{dc}=G(Q_{2})-\bigl(\theta h_{L}+(1-\theta)h_{H}\bigr)\mathrm {E}[X]. $$

The difference between the two prices is

$$p^{*}_{basic}-p^{*}_{dc}= \bigl(h_{L}A(q_{2})-h_{H}A(Q_{2})\bigr)/ \bigl(A(q_{2})A(Q_{2})\bigr). $$

The difference between the two profits is

$$\prod^{*}_{basic}-\prod^{*}_{dc}=G(q_{2})-G(Q_{2})-\theta \varDelta h\mathrm {E}[X]. $$

Since A(q ₂)<A(Q ₂), one has that $p^{*}_{basic}-p^{*}_{dc}<0$, i.e., $p^{*}_{basic}<p^{*}_{dc}$. Further since G(q ₂)<G(Q ₂), one has that $\prod^{*}_{basic}-\prod^{*}_{dc}<0$, i.e., $\prod^{*}_{basic}<\prod^{*}_{dc}$. Based on the above analysis, we know that the optimal strategy is the differentiated compensation strategy when both of the strategies are feasible and both realize the theoretical optimal solution.

 □

Proof of Proposition 5:

The budget only has an impact on the realization of the quantities of the two strategies. When cQ ₂≤K<∞, if the differentiated compensation strategy is feasible, then $q^{*}_{dc}=Q_{1}$ or $q^{*}_{dc}=Q_{2}$; if the basic pricing strategy is feasible, then $q^{*}_{basic}=q_{2}$. When K=∞, it is the same with the above case in which cQ ₂≤K<∞. Therefore, when cQ ₂≤K<∞, the seller can be seen as having no constraint of budget. □

Proof of Proposition 6:

The minimum requirement of the budget is cQ ₁ when the seller chooses the differentiated compensation strategy which is lower than cq ₁, which is the minimum requirement of the budget when the seller chooses the basic pricing strategy. So, when cQ ₁≤K<cq ₁, only the differentiated compensation strategy can be implemented. Because G(Q ₁)>h _L E[X] and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$, then $\prod^{*}_{dc}=G(Q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]>0$. When Δh≤h _L (A(Q ₁)−1) is satisfied, the compensation $u^{*}_{dc}\leq h_{H}$ such that there is no moral hazard.

Therefore, the differentiated compensation strategy is the optimal strategy under this condition. □

Proof of Proposition 7:

In the following, we only show the proof the case cq ₁≤K<cq ₂. The case of cq ₂≤K<cQ ₂ can be proved similarly.

Both the basic pricing strategy and the differentiated compensation strategy have their own feasible regions. Under the condition cq ₁≤K<cq ₂,

• (1) If G(q ₁)<h _H E[X] is satisfied, then the profit from the basic pricing strategy is satisfied with $\prod^{*}_{basic}=G(q_{1})-h_{H}\mathrm {E}[X]<0$. Thus this strategy is not feasible. According to the proof of Proposition 2, when G(Q ₁)>h _L E[X], $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$ and Δh≤h _L (1/A(Q ₁)−1), the differentiated compensation strategy is feasible. So, the differentiated compensation strategy is the only feasible strategy in this case.

• (2) If G(q ₁)>h _H E[X] is satisfied, then the profit from the basic pricing strategy is satisfied with $\prod^{*}_{basic}=G(q_{1})-h_{H}\mathrm {E}[X]>0$. Thus this strategy is feasible.

  Because G(Q ₁)≤h _L E[X] or $0<\theta\leq\frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}$ is satisfied, the profit from the differentiated compensation strategy is satisfied with $\prod^{*}_{dc}=G(Q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]<0$. Thus this strategy is not feasible.

  Since Δh>h _L (1/A(Q ₁)−1), the compensation of a differentiated compensation strategy doesn’t satisfy $u^{*}_{dc}\leq h_{H}$. Then there will be a moral hazard. So this strategy is also not feasible under this condition.

  So, the basic pricing strategy is the only feasible strategy in this case.

• (3) According to the proof of the above (2), it is known that when G(q ₁)>h _H E[X], the basic pricing strategy is feasible. According to the proof of the above (1), it is known that when G(Q ₁)>h _L E[X], $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$ and Δh≤h _L (1/A(Q ₁)−1), the differentiated compensation strategy is feasible. Therefore, both of the strategies are feasible.

 □

Proof of Proposition 8:

We analyze the two strategies under the condition that both of them are feasible. Because the proof of the case that cq ₂≤K<cQ ₂ are similar to the proof of the case of cq ₁≤K<cq ₂, in the following, we only prove the case of cq ₁≤K<cq ₂.

When the basic pricing strategy is adopted, the price is $p^{*}_{basic}=v-h_{L}/A(q_{1})$, and the profit is $\prod^{*}_{basic}=G(q_{1})-h_{H}\mathrm {E}[X]$. And when the differentiated compensation strategy is adopted, the price is $p^{*}_{dc}=v-h_{H}/A(Q_{1})$, and the profit is $\prod^{*}_{dc}=\theta(G(Q_{1})-h_{L}\mathrm {E}[X])+(1-\theta)(G(Q_{1})-h_{H}\mathrm {E}[X])$. The difference between the two prices is $p^{*}_{basic}-p^{*}_{dc}=(h_{L}A(q_{1})-h_{H}A(Q_{1}))/(A(q_{1})A(Q_{1}))$. The difference between the two profits is $\prod^{*}_{basic}-\prod^{*}_{dc}=G(q_{1})-G(Q_{1})-\theta \varDelta h \mathrm {E}[X]$. According to the proof of proposition 4, we get that $(h_{L}A(q_{1})-h_{H}A(Q_{1}))/(A(q_{1})A(Q_{1}))=c/\bar{F}(q_{1})-c/\bar{F}(Q_{1})$, $G(q_{1})-G(Q_{1})-\theta \varDelta h \mathrm {E}[X]=c((Q_{1}-\mathrm {E}[X\wedge Q_{1}]/\bar{F}(Q_{1}))-(q_{1}-\mathrm {E}[X\wedge q_{1}]/\bar{F}(q_{1})))$. From the proof of Proposition 4, one has that q ₁>Q ₁. Further one has that $c/\bar{F}(q_{1})-c/\bar{F}(Q_{1})>0$, i.e., $p^{*}_{basic}>p^{*}_{dc}$. Since $q-\mathrm {E}[X\wedge q]/\bar{F}(q)$ decreases in q and q ₁>Q ₁, one has that G(q ₁)−G(Q ₁)−ΔhE[X]>0, i.e., $\prod^{*}_{basic}>\prod^{*}_{dc}$. Therefore, the optimal strategy is the basic pricing strategy in this case.

In summary, when cq ₁≤K<cQ ₂ and the two strategies are feasible, the optimal strategy is the basic pricing strategy. □

Proof of Proposition 9:

When the improved differentiated compensation strategy is adopted, the profit is

$$\begin{aligned} \hat{\prod}^{id} =& \hat{p}^{id}\mathrm {E}\bigl[X\wedge \hat{q}^{id}\bigr]-\bigl(\mathrm {E}[X] -\mathrm {E}\bigl[X\wedge \hat{q}^{id} \bigr]\bigr) \bigl(\theta \hat{u}^{id}_{L}+(1-\theta) \hat{u}^{id}_{H}\bigr) \\ =&\biggl(v-\frac{h_{L}-(1-A(\hat{q}^{id}))\hat{u}^{id}_{L}}{A(\hat{q}^{id})}\biggr) \mathrm {E}\bigl[X\wedge \hat{q}^{id}\bigr] \\ &{}-\bigl(\mathrm {E}[X]-\mathrm {E}\bigl[X\wedge \hat{q}^{id} \bigr]\bigr) \bigl(\theta \hat{u}^{id}_{L} +(1-\theta) \hat{u}^{id}_{H}\bigr) \\ =&v\mathrm {E}\bigl[X\wedge \hat{q}^{id}\bigr]-c \hat{q}^{id}-\bigl( \theta h_{L}+(1-\theta)h_{H}\bigr)\mathrm {E}[X]. \end{aligned}$$

From the conditions of equilibrium, we can get that

$$\begin{aligned} \bar{F}\bigl(\hat{q}^{id}\bigr) =&c/\bigl(\hat{p}^{id}+ \hat{u}^{id}_{H}\bigr) =\frac{c}{v-\frac{h_{L}-(1-A(\hat{q}^{id}))\hat{u}^{id}_{L}}{A(\hat{q}^{id})} +\hat{u}^{id}_{H}} =\frac{c}{v-\frac{h_{H}-\hat{u}^{id}_{H}}{A(\hat{q}^{id})}} \end{aligned}$$

Further, we know the profit reaches maximum only when $\hat{q}^{id}$ satisfies $\bar{F}(\hat{q}^{id})=c/v$. Further, one has that $v-\frac{h_{H}-\hat{u}^{id}_{H}}{A(\hat{q}^{id})}=v$, i.e., $\hat{u}^{id}_{H}=h_{H}$. Hence, one has that $\hat{u}^{id}_{L}= h_{H}-\frac{h_{H}-h_{L}}{(1-A(\hat{q}^{id}))}$. □

Proof of Proposition 10:

To implement the improved differentiated compensation strategy, the following three conditions need to be satisfied: (1) the quantity in equilibrium $\hat{q}^{id}$ exists, (2) $\hat{q}^{id}$ can be realized and (3) the compensation $\hat{u}^{id}_{H}\leq h_{H}$, $\hat{u}^{id}_{L}\leq h_{L}$ as well as the profit in equilibrium is positive. Now we prove them one by one.

• (1) First, we prove the existence of the quantity in equilibrium $\hat{q}^{id}$. Since one has that $\bar{F}(\hat{q}^{id})=c/v$. Hence, $\hat{q}^{id}$ must exist, and is a maximizer of G(q).

• (2) Second, we point out that $\hat{q}^{id}$ can be realized.

  From the proof of 1, we get that $\hat{q}=q_{G}=\bar{F}^{-1}(c/v)$. So one has that $\hat{q}^{id}=\hat{q}$. When the improved differentiated compensation strategy is feasible, its minimum requirement of the budget is $c\hat{q}$. So only if the budget satisfies $K\geq c\hat{q}$, this rational expectations equilibrium exists.

• (3) Finally, we prove the compensation $\hat{u}^{id}_{H}\leq h_{H}$, $\hat{u}^{id}_{L}\leq h_{L}$ as well as the profit in equilibrium is positive.

  According to Proposition 9, one has that $\hat{u}^{id}_{H}=h_{H}$ and $\hat{u}^{id}_{L}=h_{H}-\frac{h_{H}-h_{L}}{(1-A(\hat{q}^{id}))}$. Only when $0< A(\hat{q}^{id})<1$, we implement the improved differentiated compensation strategy. Thus one has that $\hat{u}^{id}_{L}< h_{H}-(h_{H}-h_{L})=h_{L}$. When product availability $A(q^{*}_{dc})=0$, it means that the seller leaves the market and no deals happen. When product availability $A(q^{*}_{dc})=1$, it means no stockouts take place and nobody gets the compensation. In sum, one has that $\hat{u}^{id}_{L}\leq h_{L}$. The profit in equilibrium is $\hat{\prod}^{id}=G(\hat{q})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]$. Since $G(\hat{q})>h_{L}\mathrm {E}[X]$ and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(\hat{q})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$, one has that $\hat{\prod}^{id}>0$.

Based on the above analysis, under the condition that $K\geq c\hat{q}$, there exists a unique rational expectations equilibrium if and only if $G(\hat{q})>h_{L}\mathrm {E}[X]$ and $\max\{0, \frac{h_{H}\mathrm {E}[X]-G(\hat{q})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$. □

Proof of Proposition 11:

The case that $K\geq c\hat{q}$ and G(q ₂)≤h _H E[X] can be proved similar to the proof of the case that $K\geq c\hat{q}$ and G(q ₂)>h _H E[X]. In the following, we only give a proof for the case that $K\geq c\hat{q}$ and G(q ₂)>h _H E[X].

First, we show that the basic pricing policy and the improved differentiated pricing policy are feasible under the condition of $K\geq c\hat{q}$.

• Under the condition that G(q ₂)>h _H E[X], the profit when adopting the basic pricing policy is $\prod^{*}_{basic}=G(q_{2})-h_{H}\mathrm {E}[X]>0$. So this policy is feasible.

• Under the condition that G(Q ₁)≤h _L E[X], or $0<\theta\leq\frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}$, the profit when adopting the differentiated pricing strategy is $\prod^{*}_{dc}=G(Q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]<0$. So this policy is not feasible.

• Under the condition that Δh>h _L (1/A(Q ₁)−1), the compensation may cause the moral hazard problem for the consumers when adopting the differentiated pricing strategy. This also results that the strategy is not feasible.

• From the proof of Proposition 4, one has that $q_{2}<q_{G}=\hat{q}$. Then one has further that $G(\hat{q})>G(q_{2})$, and the profit when adopting the improved differentiated pricing policy is $\hat{\prod}^{id}=G(\hat{q})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]>0$. So this policy is feasible.

Further, one has that $\hat{\prod}^{id}-\prod^{*}_{basic}=\theta (G(\hat{q})-G(q_{2})+\varDelta h\mathrm {E}[X])+(1-\theta)(G(\hat{q})-G(q_{2}))>0$.

In summary, under this condition, only the basic pricing policy and the improved differentiated pricing policy are feasible, and the latter is the optimal strategy. □

Proof of Proposition 12:

When the three strategies are feasible, there are two cases.

Case 1::

Under the condition of $K\geq c\hat{q}$ and G(q ₂)>h _H E[X], if G(Q ₁)>h _L E[X], $\max\{0,\frac{h_{H}\mathrm {E}[X]-G(Q_{1})}{\varDelta h\mathrm {E}[X]}\}<\theta<1$ and h _L (1/A(Q ₂)−1)<Δh≤h _L (1/A(Q ₁)−1), then we know from the proof of Proposition 1 that the basic pricing strategy can realize its theoretical optimal solution and the profit is $\prod^{*}_{basic}=G(q_{2})-h_{H}\mathrm {E}[X]$. From the proof of Proposition 2, we know that a differentiated compensation strategy can only realize its theoretical suboptimal solution and the profit is $\prod^{*}_{dc}=G(q_{1})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]$. From the proof of Proposition 10, we know that the profit is $\hat{\prod}^{id}=G(\hat{q})-(\theta h_{L}+(1-\theta)h_{H})\mathrm {E}[X]$ when the improved differentiated compensation strategy is adopted. According to Proposition 1, we know that $q^{*}_{basic}=q_{1}\ \mathrm{or} \ q_{2}$. According to the proof of Proposition 4, we know $Q_{1}<q_{1}\leq q_{2}<Q_{2}<q_{G}=\hat{q}$. Therefore, one has that $\hat{\prod}^{id}>\prod^{*}_{basic}$ and $\hat{\prod}^{id}>\prod^{*}_{dc}$. The optimal strategy is to implement the improved differentiated compensation strategy.

Case 2::

Under the condition of $K\geq c\hat{q}$, if Δh≤h _L (1/A(Q ₂)−1), all of the three strategies are feasible. Although both the basic pricing strategy and the differentiated compensation strategy can realize their theoretical optimal solutions, the improved differentiated compensation strategy is still the optimal one. Similarly to the proof proceeding of Case 1, this proof can be obtained.

From the two cases, we know that the optimal choice is to implement the improved differentiated compensation strategy when the three strategies are feasible. □