A model of advance selling with consumer heterogeneity and limited capacity

Abstract

We study advance selling in a model with a capacity constraint for the seller and in the presence of both consumer heterogeneity and demand uncertainty. Buyers face different levels of uncertainty about their valuations in the advance selling period: one group of buyers (called informed buyers) know their individual valuations while the other group (called uninformed buyers) only know the distribution of their valuations. We find that the seller’s optimal pricing strategy depends on his capacity size as well as the size of informed buyers. For a small capacity size, the Constant Price strategy with the highest possible price is adopted. For sufficiently large capacity sizes, both Advance Purchase Discount and Advance Purchase Premium strategies may be optimal. In general, the larger the size of informed buyers the more likely an Advance Purchase Premium strategy is adopted.

Appendix: Proofs

1.1 Comparison of expected profits in (6) and (7)

From (6) and (7),

$$\begin{aligned} \Pi (P_{1i}, v_L)-\Pi (\bar{v}, v_L)&=\frac{h}{2}\left( v_H-(v_H-v_L)(1-R_i)\right) +\left( k-\frac{h}{2}\right) v_L-k\bar{v}\\&=\frac{h}{2}\left( v_H-v_L-(v_H-v_L)(1-R_i)\right) +kv_L-k\bar{v}\\&=\frac{h}{2}(v_H-v_L)R_i-k\frac{(v_H-v_L)}{2}\\&=\frac{(v_H-v_L)}{2(2-h)}\left( h-\frac{2k}{2-k}\right) . \end{aligned}$$

Hence, if $h\ge \frac{2k}{2-k}$, $\Pi (P_{1i}, v_L)\ge \Pi (\bar{v}, v_L)$; and if $h< \frac{2k}{2-k}$, $\Pi (P_{1i}, v_L)< \Pi (\bar{v}, v_L)$.

1.2 Comparison of expected profits in (6) and (8)

From (6) and (8),

$$\begin{aligned}&\Pi (P_{1i}, v_L)-\Pi (P_{1u}, v_L) \\&\quad =\frac{h}{2}P_{1i}+\left( k-\frac{h}{2}\right) v_L-\left( 1-\frac{h}{2}\right) P_{1u}-\left( k-\left( 1-\frac{h}{2}\right) \right) v_L\\&\quad =\frac{h}{2}(P_{1i}-v_L)-\left( 1-\frac{h}{2}\right) (P_{1u}-v_L)\\&\quad =\frac{h}{2}(v_H-v_L)R_i-\left( 1-\frac{h}{2}\right) \frac{v_H-v_L}{2}R_u\\&\quad =\frac{(v_H-v_L)(1-k)}{2h(2-h)}\left( h^2+4h-4\right) . \end{aligned}$$

The solution to the equation $h^2+4h-4=0$ is $\hat{h}\equiv 2(\sqrt{2}-1)>0$. Then, if $h \ge \hat{h}$, $\Pi (P_{1i}, v_L)\ge \Pi (P_{1u}, v_L)$; and if $h<\hat{h}$, $\Pi (P_{1i}, v_L)<\Pi (P_{1u}, v_L)$.

1.3 Proof of Lemma 4

We first pair up the pricing strategies and conduct the comparisons.

• (i) CP versus APD. It is obvious that the CP strategy ($P_1=v_L, P_2=v_L$) is dominated by the APD strategy ($P_1=\bar{v}, P_2=v_H$) because all capacity are sold under either strategy. Regarding the CP strategy ($P_1=v_H, P_2=v_H$), we have $\Pi (v_H, v_H)-\Pi (\bar{v}, v_H)= v_H/2-k\bar{v}$. Hence, if $k\le \frac{v_H}{2\bar{v}}$, $\Pi (v_H, v_H)\ge \Pi (\bar{v}, v_H)$; and if $k> \frac{v_H}{2\bar{v}}$, $\Pi (v_H, v_H)< \Pi (\bar{v}, v_H)$.

• (ii) APP versus APD. The APD strategy ($P_1=\bar{v}, P_2=v_H$) yields the same expected profit as the APP strategy ($P_1=\bar{v}, P_2=v_L$). Regarding the APP strategy ($P_1=P_{1i}, P_2=v_L$), from (10), if $h\ge \frac{2k}{2-k}$, $\Pi (P_{1i}, v_L)\ge \Pi (\bar{v}, v_H)$; and if $h<\frac{2k}{2-k}$, $\Pi (P_{1i}, v_L)< \Pi (\bar{v}, v_H)$.

• (iii) CP versus APP. We compare CP strategy ($P_1=v_H, P_2=v_H$) with APP strategy ($P_1=P_{1i}, P_2=v_L$) below.

  $$\begin{aligned} \Pi (v_H, v_H)-\Pi (P_{1i}, v_L)&=v_H/2-\left( (h/2)P_{1i}+(k-h/2)v_L\right) \\&=v_H/2-\left( \frac{h(v_H-v_L)(1-k)}{2-h}+kv_L \right) \\&=\frac{2v_H-(3v_H-2v_L)h-2(2v_L-hv_H)k}{2(2-h)}. \end{aligned}$$

  Hence, if $k\le \frac{2v_H-(3v_H-2v_L)h}{2(2v_L-hv_H)}$, $\Pi (v_H, v_H)\ge \Pi (P_{1i}, v_L)$; and if $k> \frac{2v_H-(3v_H-2v_L)h}{2(2v_L-hv_H)}$, $\Pi (v_H, v_H)< \Pi (P_{1i}, v_L)$.

Lemma 4(1) is obtained by combining (i) and (iii) above; Lemma 4(2) is obtained by combining (i) and (ii) above; and Lemma 4(3) is obtained accordingly.

1.4 Proof of Lemma 5

It is easy to see that the CP strategy ($P_1=v_L, P_2=v_L$) and the APD strategy ($P_1=v_L, P_2=v_H$) are dominated by APP strategies. We then compare the remaining strategies below.

• (i) CP versus APD. $\Pi (\bar{v}, v_H)-\Pi (v_H, v_H)=\left( 1-\frac{h}{2}\right) \bar{v}-\frac{1}{2}v_H=\frac{v_L}{2}-\frac{\bar{v}}{2}h$. Hence, if $h\le \frac{v_L}{\bar{v}}$, $\Pi (\bar{v}, v_H)\ge \Pi (v_H, v_H)$; and if $h> \frac{v_L}{\bar{v}}$, $\Pi (\bar{v}, v_H)\le \Pi (v_H, v_H)$.

• (ii) APP versus APD. We first compare the APD strategy ($P_1=\bar{v}, P_2=v_H$) with the APP strategy $(P_1=P_{1u}, P_2= v_L)$.

  $$\begin{aligned} \Pi (\bar{v}, v_H)-\Pi (P_{1u}, v_L)&=\left( 1-\frac{h}{2}\right) \bar{v}-\left( \left( 1-\frac{h}{2}\right) P_{1u}+\left( k-\left( 1-\frac{h}{2}\right) \right) v_L\right) \nonumber \\&=\left( 1-\frac{h}{2}\right) \bar{v}-\left( 1-\frac{h}{2}\right) \frac{(v_H-v_L)(1-k)}{h}-kv_L\nonumber \\&=\left( 1-\frac{h}{2}-k\right) \left( \bar{v}-\frac{v_H-v_L}{h}\right) . \end{aligned}$$

  Since $1-\frac{h}{2}-k<0$, if $h\le \frac{v_H-v_L}{\bar{v}}$, $\Pi (\bar{v}, v_H)\ge \Pi (P_{1u}, v_L)$; and if $h>\frac{v_H-v_L}{\bar{v}}$, $\Pi (\bar{v}, v_H)< \Pi (P_{1u}, v_L)$. Note that $\frac{v_H-v_L}{\bar{v}}<\hat{h}$. Hence, if $h\le \frac{v_H-v_L}{\bar{v}}$, APD strategy yields higher profit than APP strategies; and if $h>\frac{v_H-v_L}{\bar{v}}$, APD strategy yields lower profit than APP strategies.

• (iii) CP versus APP. From the Proof of Lemma 4, if $k\le \frac{2v_H-(3v_H-2v_L)h}{2(2v_L-hv_H)}$, $\Pi (v_H, v_H)\ge \Pi (P_{1i}, v_L)$; and if $k> \frac{2v_H-(3v_H-2v_L)h}{2(2v_L-hv_H)}$, $\Pi (v_H, v_H)< \Pi (P_{1i}, v_L)$. Next, we compare the CP strategy ($P_1=v_H, P_2=v_H$) with the APP strategy ($P_1=P_{1u}, P_2=v_L$).

  $$\begin{aligned} \Pi (v_H, v_H)-\Pi (P_{1u}, v_L)&=v_H/2-\left( \left( 1-\frac{h}{2}\right) P_{1u}+\left( k-\left( 1-\frac{h}{2}\right) \right) v_L\right) \\&=v_H/2-(1-\frac{h}{2})\frac{(v_H-v_L)(1-k)}{h}-kv_L\\&=v_H/2-\left( 1-\frac{h}{2}\right) \frac{v_H-v_L}{h}\\&\quad -\left( v_L-\left( 1-\frac{h}{2}\right) \frac{v_H-v_L}{h}\right) k . \end{aligned}$$

  Hence, if $h\le \frac{v_H-v_L}{\bar{v}}$, or if $h> \frac{v_H-v_L}{\bar{v}}$ and $k>\frac{v_Hh-(2-h)(v_H-v_L)}{2v_Lh-(2-h)(v_H-v_L)}$ then $\Pi (P_{1u}, v_L)> \Pi (v_H, v_H)$; otherwise, $\Pi (P_{1u}, v_L)\le \Pi (v_H, v_H)$.

Lemma 5(1) is obtained by combining (i) and (ii) above because $\frac{v_H-v_L}{\bar{v}}<\frac{v_L}{\bar{v}}$; Lemma 5(2) is obtained by combining (ii) and the last half of (iii); Lemma 5(3) is obtained by combining (ii) and the first half of (iii); and Lemma 5(4) is implied by the above conclusions.

1.5 Derivations of $\Pi$, CS and SS in each region

In region I, the seller’s profit $\Pi =v_H/2$ [see the second line in (12)]. Only high valuation consumers buy this product, and they will be able to get it at price $v_H$. Thus, $CS=(v_H-v_H)/2=0$, $SS=CS+\Pi =v_H/2$.

In region II, the first lines in (10) and (11) yield

$$\begin{aligned} \Pi&=(h/2)P_{1i}+(k-h/2)v_L \\&=\frac{h}{2}(v_H-v_L)R_i+kv_L \\&=\frac{h(1-k)}{2-h}v_H+\frac{2k-h}{2-h}v_L. \end{aligned}$$

Note that the stock-out probability for consumers who purchase in period 2 is $R_i$. Thus,

$$\begin{aligned} CS&=\frac{h}{2}(v_H-P_{1i})+\frac{1-h}{2}(v_H-v_L)(1-R_i) \\&=\frac{h}{2}(v_H-v_L)(1-R_i)+\frac{1-h}{2}(v_H-v_L)(1-R_i) \\&=\frac{(2k-h)(v_H-v_L)}{2(2-h)}, \end{aligned}$$

Then we have

$$\begin{aligned} SS=CS+\Pi =\frac{2k+h-2kh}{2(2-h)}v_H+\frac{2k-h}{2(2-h)}v_L. \end{aligned}$$

In region III, the second line in (11) yields

$$\begin{aligned} \Pi&=(1-h/2)P_{1u}+(k-(1-h/2))v_L \\&=\left( 1-\frac{h}{2}\right) \frac{v_H-v_L}{2}R_u+kv_L \\&=\frac{(2-h)(1-k)}{2h}v_H+\frac{hk+2k+h-2}{2h}v_L. \end{aligned}$$

All consumers except low valuation informed buyers pre-order in period 1. Thus,

$$\begin{aligned} CS&=\frac{h}{2}(v_H-P_{1u})+\frac{1-h}{2}(v_H-P_{1u})++\frac{1-h}{2}(v_L-P_{1u}) \\&=\frac{h}{2}(v_H-v_L)R_u+\frac{1}{2}(v_H-v_L)(1-R_u) \\&=\frac{(2h+2k-2-kh)(v_H-v_L)}{2h}, \end{aligned}$$

Then we have

$$\begin{aligned} SS=CS+\Pi =\frac{v_H}{2}+\frac{2k-1}{2}v_L. \end{aligned}$$

In region IV, the first two lines in (1) yield that $\Pi =k\bar{v}$ if $k\le 1-\frac{h}{2}$, and $\Pi =\left( 1-\frac{h}{2}\right) \bar{v}$ if $k> 1-\frac{h}{2}$. First, if $k> 1-\frac{h}{2}$, consumers who pre-order are able to get this product for sure. Thus,

$$\begin{aligned} CS=\frac{h}{2}(v_H-\bar{v})+(1-h)(\bar{v}-\bar{v})=\frac{h}{4}(v_H-v_L). \end{aligned}$$

Then

$$\begin{aligned} SS=CS+\Pi =\frac{v_H}{2}+\frac{1-h}{2}v_L. \end{aligned}$$

Second, if $k\le 1-\frac{h}{2}$, the capacity can not fulfill all pre-orders in period 1. Consumers who pre-order are able to get this product with probability $\frac{k}{1-h/2}.$ Thus,

$$\begin{aligned} CS=\frac{k}{1-h/2}\frac{h}{4}(v_H-v_L)=\frac{kh}{2(2-h)}(v_H-v_L). \end{aligned}$$

Then we have

$$\begin{aligned} SS=CS+\Pi =\frac{k}{2-h}v_H+\frac{k(1-h)}{2-h}v_L. \end{aligned}$$

1.6 Proof of Proposition 3

The results in region I are obvious. In region II,

$$\begin{aligned} \frac{\partial \Pi }{\partial k}= & {} \frac{2v_L-hv_H}{2-h}>0; \quad \frac{\partial \Pi }{\partial h}=\frac{2(1-k)(v_H-v_L)}{(2-h)^2}>0;\\ \frac{\partial CS}{\partial k}= & {} \frac{v_H-v_L}{2-h}>0; \quad \frac{\partial CS}{\partial h}=\frac{-(1-k)(v_H-v_L)}{(2-h)^2}<0;\\ \frac{\partial SS}{\partial k}= & {} \frac{\partial \Pi }{\partial k}+\frac{\partial CS}{\partial k}>0; \quad \frac{\partial SS}{\partial h}=\frac{\partial \Pi }{\partial h}+\frac{\partial CS}{\partial h}>0. \end{aligned}$$

In region III, note that $\frac{v_H-v_L}{\bar{v}}<h<\hat{h}$, so we have

$$\begin{aligned} \frac{\partial \Pi }{\partial k}= & {} \frac{(2+h)v_L-(2-h)v_H}{2h}>0; \quad \frac{\partial \Pi }{\partial h}=\frac{-(1-k)(v_H-v_L)}{(h^2}<0;\\ \frac{\partial CS}{\partial k}= & {} \frac{(2-h)(v_H-v_L)}{2h}>0; \quad \frac{\partial CS}{\partial h}=\frac{(1-k)(v_H-v_L)}{h^2}>0;\\ \frac{\partial SS}{\partial k}= & {} \frac{\partial \Pi }{\partial k}+\frac{\partial CS}{\partial k}>0; \quad \frac{\partial SS}{\partial h}=\frac{\partial \Pi }{\partial h}+\frac{\partial CS}{\partial h}=0. \end{aligned}$$

In region IV, if $k\le 1-h/2$, we have

$$\begin{aligned} \frac{\partial \Pi }{\partial k}= & {} \bar{v}>0; \quad \frac{\partial \Pi }{\partial h}=0;\\ \frac{\partial CS}{\partial k}= & {} \frac{h(v_H-v_L)}{2(2-h)}>0; \quad \frac{\partial CS}{\partial h}=\frac{k(v_H-v_L)}{(2-h)^2}>0;\\ \frac{\partial SS}{\partial k}= & {} \frac{\partial \Pi }{\partial k}+\frac{\partial CS}{\partial k}>0; \quad \frac{\partial SS}{\partial h}=\frac{\partial \Pi }{\partial h}+\frac{\partial CS}{\partial h}>0. \end{aligned}$$

If $k>1-h/2$, we have

$$\begin{aligned} \frac{\partial \Pi }{\partial k}= & {} 0; \quad \frac{\partial \Pi }{\partial h}=-\bar{v}/2<0;\\ \frac{\partial CS}{\partial k}= & {} 0; \quad \frac{\partial CS}{\partial h}=\frac{v_H-v_L}{4}>0;\\ \frac{\partial SS}{\partial k}= & {} \frac{\partial \Pi }{\partial k}+\frac{\partial CS}{\partial k}=0; \quad \frac{\partial SS}{\partial h}=\frac{\partial \Pi }{\partial h}+\frac{\partial CS}{\partial h}<0. \end{aligned}$$

1.7 Proof of Proposition 4

The comparison is between the Advance Purchase Discount strategy ($P_1=\bar{v}, P_2=v_H$), resulting in profit $\Pi ^1=(1-h/2)\bar{v}$, and selling exclusively at $v_L$, resulting in profit $\Pi ^2=v_L$. Since $\Pi ^1-\Pi ^2=\left( 1-\frac{h}{2}\right) \bar{v}-v_L=\frac{v_H-v_L-h\bar{v}}{2}$, it follows that $\Pi ^1<\Pi ^2$ when $h>\frac{v_H-v_L}{\bar{v}}$, and $\Pi ^1\ge \Pi ^2$ when $h\le \frac{v_H-v_L}{\bar{v}}$. Hence, we have Proposition 4.

1.8 Proof of Proposition 5

The seller chooses to expand the capacity to $k=1-h/2$ if and only if $\left( 1-\frac{h}{2}-\frac{1}{2}\right) c<\left( 1-\frac{h}{2}\right) \bar{v}-\frac{v_H}{2}$, which can be reduced to $c<\frac{v_L-h\bar{v}}{1-h}$. Furthermore, the seller continues to expand his capacity to $k=1$ if and only if $\left( 1-\frac{1}{2}\right) c<v_L-\frac{v_H}{2}$, which can be reduced to $c<2v_L-v_H.$ Next, we show that $2v_L-v_H<\frac{v_L-h\bar{v}}{1-h}$.

$$\begin{aligned} \frac{v_L-h\bar{v}}{1-h}- 2v_L+v_H&=\frac{1}{1-h}\left( v_L-h\bar{v}+(1-h)(v_H- 2v_L)\right) \\&=\frac{1}{1-h}\left( 1-\frac{3}{2}h\right) (v_H-v_L) \\&>0. \end{aligned}$$

Hence, we have Proposition 5.

1.9 Proof of Lemma 7

In this proof the seller’s possible optimal pricing schedules under Dynamic Pricing are identified by examining all possible price paths. This is done by considering two possible ranges for $P_{1i}$.

Suppose $P_{1i}>\bar{v}$ (case 1). In this case we have $P_{1u}<\bar{v}<P_{1i}$. The first-period price $P_1$ must be chosen from one of the following four regions.

• (i) $P_1 \in [v_L, P_{1u}]$. If the seller chooses $P_1 \in [v_L, P_{1u}]$, all high valuation informed buyers and all uninformed buyers pre-order in period 1 regardless of the price $P_2$. As a result, in period 2 the seller optimally charges $P_2=v_L$ to capture the demand from low valuation informed buyers. In period 1, the seller charges the highest possible in the interval $[v_L, P_{1u}]$. That is, $P_1=P_{1u}$.

• (ii) $P_1 \in (P_{1u}, \bar{v}]$. If the seller chooses $P_1 \in (P_{1u}, \bar{v}]$, all high valuation informed buyers pre-order in period 1 regardless of the price $P_2$. When the capacity is large, that is $k>1-h/2$, realizing that the seller will charge $P_2=v_L$, all uninformed buyers decide to wait until period 2. The seller therefore charges $(P_1=\bar{v}, P_2=v_L)$, which is inferior to the price strategy $(P_1=P_{1i}, P_2=v_L)$ in part (iii). When the capacity is small (i.e., $k<1-h/2$), uninformed buyers pre-order at $P_1$ to avoid a stock out in period 2. Sales in period 2 is zero. As a result, the optimal price is $P_1=\bar{v}$ and $P_2=v_H$.

• (iii) $P_1 \in (\bar{v}, P_{1i}]$. If the seller chooses $P_1 \in (\bar{v}, P_{1i}]$, all high valuation informed buyers surely pre-order in period 1 and all uninformed buyers surely wait until period 2. As a result, the seller charges $P_1=P_{1i}$. In the second period, the seller charges either $v_L$ or $v_H$. But the price strategy $(P_1=P_{1i}, P_2=v_H)$ is inferior to the strategy $(P_1=v_H, P_2=v_H)$ in part (iv).

• (iv) $P_1 \in (P_{1i}, v_H]$. If the seller chooses $P_1 \in (P_{1i}, v_H]$, only high valuation informed buyers may pre-order. Anticipating that $P_2=v_H$ high valuation informed buyers pre-order and otherwise they wait to buy in period 2. Thus, the seller will consider price strategy $(P_1=v_H, P_2=v_H)$ when the capacity is small.

To summarize, in case 1, the optimal price strategy candidates are ($P_1=P_{1u}, P_2=v_L$) in (i); ($P_1=\bar{v}, P_2=v_H$) in (ii); ($P_1=P_{1i}, P_2=v_L$) in (iii); and ($P_1=v_H, P_2=v_H$) in (iv).

Consider next $P_{1i}<\bar{v}$ (case 2). The first-period price $P_1$ must be chosen from one of the following four regions.

• (i) $P_1 \in [v_L, P_{1u}]$. If the seller chooses $P_1 \in [v_L, P_{1u}]$, as in part (i) of case 1 above the optimal prices are $P_1=P_{1u}$ and $P_2=v_L$.

• (ii) $P_1 \in (P_{1u}, P_{1i}]$. If the seller chooses $P_1 \in (P_{1u}, P_{1i}]$, all high valuation informed buyers pre-order in period 1 regardless of the price $P_2$. When the capacity is large ($k>1-h/2$), realizing that the seller will charge $P_2=v_L$, all uninformed buyers decide to wait until period 2. The seller therefore charges $(P_1=P_{1i}, P_2=v_L)$. When the capacity is small (i.e., $k<1-h/2$), uninformed buyers pre-order at $P_1$ to avoid a stock out in period 2. Sales in period 2 is zero. As a result, the optimal price is $(P_1=P_{1i},P_2=v_H)$. However, since $P_{1i}<\bar{v}$, the price strategy $(P_1=P_{1i},P_2=v_H)$ is inferior to strategy $(P_1=\bar{v}, P_2=v_H)$ in part (iii) under which the seller also sells out in period 1.

• (iii) $P_1 \in (P_{1i}, \bar{v}]$. If the seller chooses $P_1 \in (P_{1i}, \bar{v}]$, all high valuation informed buyers and all uninformed buyers pre-order in period 1 only when the capacity is small ($k<1-h/2$). It follows that the optimal price is $P_1=\bar{v}$ and $P_2=v_H$ in this case. Otherwise, when the capacity is large ($k>1-h/2$), no buyers pre-order anticipating that the seller sets $P_2=v_L$ in period 2, which yields a profit $kv_L$. As a result, the seller will never employ APP when $P_1 \in (P_{1i}, \bar{v}]$.

• (iv) $P_1 \in (\bar{v}, v_H]$. If the seller chooses $P_1 \in (\bar{v}, v_H]$, as in part (iv) of case 1 above, only high valuation informed buyers may pre-order. Thus, the seller will consider price strategy $(P_1=v_H, P_2=v_H)$ when the capacity is small.

To summarize, in case 2, the candidates for optimal prices are ($P_1=P_{1u}, P_2=v_L$) in (i); ($P_1=P_{1i}, P_2=v_L$) in (ii); ($P_1=\bar{v}, P_2=v_H$) in (iii); and ($P_1=v_H, P_2=v_H$) in (iv).²⁰

1.10 Proof of Proposition 6

It is straightforward that when $0<k\le 1/2$ the seller charges the highest possible price of $P=v_H$ and all his capacity will be sold out to high valuation buyers.

For $1/2<k<1$, the comparison is between the CP strategy ($P_1=v_H, P_2=v_H$), APD strategy ($P_1=\bar{v}, P_2=v_H$) and the APP strategies $(P_1=P_{1u}, P_2= v_L)$ and $(P_1=P_{1i}, P_2= v_L)$. Based on the Proofs for Lemmas 4 and 5, we obtain the results in Proposition 6.