Advance selling to strategic consumers

Abstract

Advance selling of goods and services is a form of separating purchase from consumption. It is often employed when consumers are uncertain about their consumption utilities until a short time period before consumption. A book to be released, a concert to attend, or a cruise to take are some examples. Invariably, in consumers’ mind inventory availability (of copies, seats, or rooms) is a concern. In this paper we study a retailer’s inventory and pricing decisions in an advance selling scenario that involves consumers who are strategic. Some consumers not only consider advance and spot prices, but also the uncertainty in future availability of the product (during the spot period) and in their consumption utility from it. We characterize the optimal inventory management and pricing policies, and discuss several interesting aspects of the solution. For example, it can be optimal for the retailer to limit advance sales even if there is more demand for it, and it can be optimal for the retailer to limit its inventory even though there is more capacity to keep it, but not both.

Appendices

Appendix A: Proofs

Lemma 6

$\Lambda _{Ds}(X|Q)$ is convex for $Q \le M - \alpha /4$.

Proof of Lemma 6

Note that,

$$\begin{aligned} \frac{\partial ^2 \Lambda _{D_s}(Q,X)}{\partial X^2} = [(1-\alpha )^2f(u)(2\sigma -\alpha (u-1))(2\sigma -\alpha (u+1))]/4 \sigma ^3. \end{aligned}$$

(29)

The terms $(1-\alpha )^2$, f(u), and $4\sigma ^3$ in expression 29 are obviously nonnegative. Substituting u in $(2\sigma -\alpha (u-1))$, we get $\alpha (M-Q)/\sigma + (\sigma + \alpha ) \ge 0$. The same substitution for $(2\sigma -\alpha (u+1))$ results in $\alpha (M-Q)/\sigma + (\sigma - \alpha )$, which may not always be nonnegative. Assuming that the condition holds, i.e. $\alpha (M-Q)/\sigma + (\sigma - \alpha ) \ge 0$, multiplying both sides with $\sigma$, and substituting $\sigma ^2$, we get $\alpha [(M-Q)+(M-X)(1-\alpha )-\sigma ]\ge 0$. Letting $q=M-Q$ and $p=(M-X)(1-\alpha )$, and dividing both sides by $\alpha$ we can rewrite this expression as $q+p \ge \sigma$. Squaring both sides and substituting $\sigma ^2$ results in, $p^2+(2q-\alpha )p+q^2 \ge 0,$ which has real roots if $\alpha \ge 4 q$. This implies that for $4 q = 4 (M-Q) \ge \alpha$, or equivalently $Q \le M-\alpha /4$, the inequality, and therefore our initial assumption $\alpha (M-Q)/\sigma + (\sigma - \alpha ) \ge 0$ hold, in which case $\frac{\partial ^2 \Lambda _{Ds}(Q,X)}{\partial X^2}$ becomes nonnegative. Note that $\alpha /4$ is a small number less than or equal to 1 / 4, so we can assume that $\Lambda _{D_s}(X|Q)$ is convex for almost any Q value except for the very small interval $(M-\alpha /4, M]$ at the boundary. $\square$

Proof of Theorem 1

Let $\mathbf {H}_L(Q, X)$ be the Hessian of $\Pi _L(Q, X)$, whose entries are,

$$\begin{aligned} \frac{\partial ^2 \Pi _L(Q,X)}{\partial X^2}= & {} 2[(H-L)\alpha (M-Q)]/(M-X)^3 \end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial ^2 \Pi _L(Q,X)}{\partial X \partial Q}= & {} -(H-L)\alpha /(M-X)^2 \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial ^2 \Pi _L(Q,X)}{\partial Q^2}= & {} 0. \end{aligned}$$

(32)

It is easy to show that $\mathbf {H}_L$ has a real (repeated) eigenvalue, $\lambda = \frac{(H-L)\alpha }{(M-Q)^2}$, only when $X=Q$, and, in which case $\Pi _L$ becomes linear in Q from expression 9.

Let $\sigma = \sigma _{D_s}$, and $u=\frac{(Q-X)-\mu _{Ds}}{\sigma _{D_s}}$ be the standardized value for $Q-X$. It can be shown that $\mathbf {H}_H= -H\mathbf {H}_\Lambda$, where $\mathbf {H}_H$ is the Hessian for $\Pi _H$, and $\mathbf {H}_{\Lambda }$ is the Hessian for the loss function $\Lambda _{D_s}$. The entries of $\mathbf {H}_{\Lambda }$ are as follows, where f is the standard normal distribution,

$$\begin{aligned} \frac{\partial ^2 \Lambda _{D_s}}{\partial X^2}= & {} [(1-\alpha )^2f(u)(2\sigma -\alpha (u-1))(2\sigma -\alpha (u+1))]/4 \sigma ^3 \end{aligned}$$

(33)

$$\begin{aligned} \frac{\partial ^2 \Lambda _{D_s}}{\partial X \partial Q}= & {} [(1-\alpha )f(u)(\alpha u -2\sigma )]/2\sigma ^2 \end{aligned}$$

(34)

$$\begin{aligned} \frac{\partial ^2 \Lambda _{D_s}}{\partial Q^2}= & {} f(u)/\sigma \end{aligned}$$

(35)

To check for convexity we look at the principal minors of k dimensions, where $k=1, 2$. This is equivalent to checking expressions 33 and 35 for $k=1$ and the determinant of $\mathbf {H}_{\Lambda }$ for $k=2$. It is clear that $\frac{\partial ^2 \Lambda _{D_s}(Q,X)}{\partial Q^2} = f(u)/\sigma \ge 0$. Lemma 6 presents the conditions for nonnegativity for expression 33, and completes the check for $k=1$. Checking for the case $k=2$ involves the determinant of $\mathbf {H}_{\Lambda }.$ From expressions 33–35, it can be shown that,

$$\begin{aligned} \det (\mathbf {H}_{\Lambda }) = -\frac{\alpha ^2 (1-\alpha )^2 f(u)^2}{4\sigma ^4} \le 0. \end{aligned}$$

(36)

For $0<\alpha <1$, inequality 36 becomes strict, hence makes $\Lambda _{D_s}$ indefinite. This implies that $\det (\mathbf {H}_H) = H^2 \det (\mathbf {H}_{\Lambda }) < 0$. $\square$

Proof of Lemma 2

Convexity follows from the second order condition,

$$\begin{aligned} \frac{d^2\Pi _L(X|Q)}{dX^2} = \frac{2(H-L)(M-Q)\alpha }{(M-X)^3} \ge 0. \end{aligned}$$

Let $\overline{X}_L(Q) = \min \{N_a, Q\}$, then, to find $X^*_L(Q)=\arg \max \{ \Pi _L(X|Q):0\le X \le \overline{X}_L(Q)\}$, we check the profit at the extremes, $\Pi _L(0|Q)$ and $\Pi _L(\overline{X}_L(Q)|Q)$, due to convexity. This implies that $X=\overline{X}_L(Q)$ is optimal if $((H-L)\alpha (M-Q))\frac{\overline{X}_L(Q)}{M-\overline{X}_L(Q)}-c_a \overline{X}_L(Q)+(L-c)Q \ge (L-c)Q$, which reduces to $\overline{X}_L(Q)((H-L)\alpha (M-Q)-(M-\overline{X}_L(Q))c_a)\ge 0$. Since $\overline{X}_L(Q) \ge 0$, then $(H-L)\alpha (M-Q)-(M-\overline{X}_L(Q))c_a \ge 0$, which can be rearranged as, $\overline{X}_L(Q) \ge \frac{(H-L)\alpha }{c_a}Q-\frac{(H-L)\alpha -c_a}{c_a}M.$ If this expression does not hold, then $X=0$ is optimal. $\square$

Proof of Lemma 3

Proof follows from the slope of the linear expression in 12. If the slope is nonnegative, that is, $L-c-\frac{X(H-L)\alpha }{M-X} \ge 0$, then $Q=K$ is the solution, otherwise, it is $Q=X$. This condition can be rearranged as $\frac{L-c}{L-c+(H-L)\alpha }M \ge X$. $\square$

Proof of Theorem 1

Lemmas 2 and 3 produce extreme point solutions for fixed values of Q and X. Thus, for the problem $\arg \max _{Q,X}\{\Pi _L(Q,X):0 \le X \le N_a \le K, X \le Q \le K\},$ the optimal solution should also be at the vertices of the feasible region, for otherwise it would contradict with Lemma 2 and/or Lemma 3.

Note that the vertices of the feasible region are determined by the bounding parameters, K and $N_a$, and the optimal solution becomes a function of these parameters with respect to the boundaries $N_a=\frac{(H-L)\alpha }{c_a}K-\frac{(H-L)\alpha -c_a}{c_a}M$, and $N_a=\frac{L-c}{L-c+(H-L)\alpha }M$ from Lemmas 2 and 3, respectively.

The possibility of advance sales is dependent on the existence of the region $\mathcal {R}_L^2$, for which $N_a \ge \frac{(H-L)\alpha }{c_a}K-\frac{(H-L)\alpha -c_a}{c_a}M$ must hold for any $K\ge N_a$. This implies that the intercept of this linear expression should be negative, that is, $-\frac{(H-L)\alpha -c_a}{c_a}M \le 0$, which implies $(H-L)\alpha \ge c_a$.

When $\frac{L-c}{L-c+(H-L)\alpha }M \le N_a \le \frac{(H-L)\alpha }{c_a}K-\frac{(H-L)\alpha -c_a}{c_a}M$, we compare $\Pi _L(K,0)$ to $\Pi _L(N_a,N_a)$, which yields a third boundary $N_a=\frac{L-c}{L-c+(H-L)\alpha -c_a}K$.

Three boundaries introduced above partition the cartesian plane of possible parameter pairs $(K,N_a)$ into three regions, $\mathcal {R}^L_i$, with the corresponding extreme solutions as functions of these parameters. $\square$

Proof of Lemma 4

Concavity follows from the second order condition, $\frac{d^2\Pi _H(Q | X)}{dQ^2} = -H f_{D_s}(Q-X)\le 0.$ and the optimal order quantity result follows from concavity and the first order condition $\frac{d\Pi _H(Q|X)}{dQ} = H(1-F_{D_s}(Q-X))-c = 0.$ The resulting expression can be written as a function of X using the equivalence $F_{D_s}^{-1}(\cdot )=F^{-1}(\cdot ) \sigma _{D_s}+\mu _{D_s}.$

Note that as c approaches to 0, $F_{D_s}^{-1}(\frac{H - c}{H})$ approaches to infinity, but $Q \le M$, hence we take the minimum of the two values. $\square$

Proof of Lemma 5

It is easy to show that $\frac{\partial ^2 \Pi _H(Q, X)}{\partial X^2} = -H \frac{\partial ^2 \Lambda _{D_s}(Q,X)}{\partial X^2}.$ Concavity of $\Pi _H(X|Q)$ follows from Lemma 6 since $\frac{d^2\Pi _H(X|Q)}{d^2 X} \le 0$. From expression 18 and concavity,

$$\begin{aligned} \frac{\partial \Pi _H(Q, X)}{\partial X} = L(1-\alpha )-c_a-H(1-\alpha )[(1-F(u))-\alpha f(u)/(2\sigma )] = 0, \end{aligned}$$

(37)

for $X^*(Q)$, where $u=((Q-X)-(M-X)\alpha )/\sigma$, and $\sigma =\sqrt{(M-X)\alpha (1-\alpha )}$. Note that $\alpha f(u)/(2\sigma ) \approx 0$ for any practical case, and therefore, can be ignored. Then, from expression 37, we can write, $F(u) = \frac{(1-\alpha )(H-L) + c_a}{(1-\alpha )H},$ which implies, $u=F^{-1}\left( \frac{(1-\alpha )(H-L) + c_a}{(1-\alpha )H} \right) =\mathcal {Z}_Q$, and can be rearranged as,

$$\begin{aligned} Q = M \alpha + (1-\alpha ) X + \mathcal {Z}_Q \sqrt{(M-X) \alpha (1-\alpha )}. \end{aligned}$$

(38)

After squaring both sides, this expression can further be rearranged in X as,

$$\begin{aligned}&(1-\alpha )^2 X^2 - (2(1-\alpha )(Q-M\alpha )-\alpha (1-\alpha )\mathcal {Z}_Q^2)X\\&\quad +(Q-M\alpha )^2-\alpha (1-\alpha ) M \mathcal {Z}_Q^2=0, \end{aligned}$$

whose appropriate roots results in expression 20. $\square$

Proof of Corollary 1

Proof follows from expression 38 in the proof of Lemma 5. $\square$

Lemma 7

Let $Q(X) = M\alpha + (1-\alpha )X+\mathcal {Z}\sqrt{(M-X)\alpha (1-\alpha )}$, then, $\Pi _H(Q(X),X)$ is convex.

Proof of Lemma 7

Using expression 5 and 19, and substituting Q(X) in expression 18 yields,

$$\begin{aligned} \Pi _H(Q(X),X)= & {} (L(1-\alpha )-c_a) X \nonumber \\&- H \sigma _{D_s} \Lambda (\mathcal {Z}) - c (\mathcal {Z}\sigma _{D_s}+\mu _{D_s}+X) + H M \alpha , \end{aligned}$$

(39)

whose second derivative is, $\frac{d^2 \Pi _H(Q(X),X)}{d X^2} = \frac{\alpha ^2 (1-\alpha )^2 (H \Lambda (\mathcal {Z})+cz)}{4 \sigma _{D_s}^3}.$ It can be shown that $f(\mathcal {Z})+ZF(\mathcal {Z}) \ge 0$ for all $\mathcal {Z}\in \mathbb {R}$. Using expression 5,

$$\begin{aligned} H \Lambda (\mathcal {Z})+c\mathcal {Z}\ge c(\Lambda (\mathcal {Z})+\mathcal {Z}) = c(f(\mathcal {Z})+zF(\mathcal {Z})) \ge 0, \end{aligned}$$

hence the second order condition, $\frac{d^2 \Pi _H(Q(X),X)}{d X^2} \ge 0$. $\square$

Proof of Corollary 2

Let $(L-c)(1-\alpha ) > c_a$, $(L-c)(1-\alpha ) < c_a$ and $(L-c)(1-\alpha )=c_a$ be conditions 1, 2, and 3, respectively. Each condition can be converted into an equivalent condition, in which the left hand side is $\frac{H-c}{H}$, and the right hand side is $\frac{(H-L)(1-\alpha )+c_a}{H(1-\alpha )}$, with the corresponding sign from each condition. From expressions 19 and 21, this implies that $\mathcal {Z}_X > \mathcal {Z}_Q$ ($X^*(Q)$ is to the left of, or above $Q^*(X)$), $\mathcal {Z}_X < \mathcal {Z}_Q$ ($X^*(Q)$ is to the right of, or below $Q^*(X)$), and $\mathcal {Z}_Z=\mathcal {Z}_Q$ ($X^*(Q)$ is overlapping with $Q^*(X)$), under conditions 1,2, and 3, respectively.

Since both $\Pi _H(Q|X)$ and $\Pi _H(X|Q)$ are concave, then, under condition 1, for any X and $X^\prime =X+\delta$ for small $\delta >0$, we have, $\Pi _H(Q^*(X),X) \le \Pi _H(Q^*(X),X^\prime ) \le \Pi _H(Q^*(X^\prime ),X^\prime )$, and for any $Q \ge Q_0$ and $Q^\prime =Q+\delta$ for small $\delta$, we have, $\Pi _H(Q,X^*(Q)) \le \Pi _H(Q^\prime ,X^*(Q)) \le \Pi _H(Q^\prime ,X^*(Q^\prime ))$, where $Q_0=\arg \max _Q\{Q:X^*(Q)=0\}$ proving that $\Pi _H(Q,X^*(Q))$ is nondecreasing. For condition 2, a similar argument can be constructed for any X and $X^\prime =X-\delta$, and any $Q\ge Q_0+\delta$ and $Q^\prime =Q-\delta$ for small $\delta >0$, where $Q_0=Q^*(0)$ and the inequalities change direction, proving that $\Pi _H(Q,X^*(Q))$ is monotonically decreasing. For condition 3, $\Pi _H(Q^*(X),X)$ is convex from Lemma 7. Let us assume that $X^*>0$ minimizes $\Pi _H(Q^*(X),X)$. In order to find $X^*$, we can write the first order condition

$$\begin{aligned} \frac{d \; \Pi _H(Q^*(X),X)}{d \; X}=((L-c)(1-\alpha )-c_a)+\frac{\alpha (1-\alpha )(H\Lambda (\mathcal {Z}_X)+c \mathcal {Z}_X}{2 \sigma _{D_s}}=0, \end{aligned}$$

which implies $\sigma _{D_s}=\sqrt{(M-X)\alpha (1-\alpha )}=-\frac{\alpha (1-\alpha ) (H\Lambda (\mathcal {Z}_X)+c \mathcal {Z}_X)}{2((L-c)(1-\alpha )-c_a)}$. Note that the numerator of the foregoing expression is always positive for $\alpha >0$, which means that a positive solution $X^*>0$ exists only under condition 2, therefore contradiction. Moreover, from expression 39, $\Pi _H(X^*(M),M) - \Pi _H(X^*(0),0)=((L-c)(1-\alpha )-c_a)+\sqrt{M\alpha (1-\alpha )}(H\Lambda (\mathcal {Z}_X)+c \mathcal {Z}_X)>0$ under condition 3, which shows that $\Pi _H(Q,X^*(Q))$ is a nondecreasing convex function. $\square$

Proof of Theorem 2

In $\mathcal {R}^H_1$, $Q^*_H=K$ for $X \le N_a$ since $K < Q^*_H(X)$ and due to concavity of $Q^*_H(X)$. Similarly, $X^*_H=N_a$ for $Q_0 \le Q \le K$ since $N_a < X^*_H(Q)$ and due to concavity of $X^*_H(Q)$.

In $\mathcal {R}^H_2$, $K\ge Q^*_H(X)$, therefore $Q^*_H=Q^*_H(X)$ for $0 \le X \le N_a$ due to concavity of $Q^*_H(X)$. Same argument goes for $X^*_H$ as in $\mathcal {R}^H_1$ case.

In $\mathcal {R}^H_3$, $N_a\ge X^*_H(Q)$, therefore $X^*_H=X^*_H(Q)$ for $Q_0 \le Q \le K$ due to concavity of $X^*_H(Q)$. Same argument goes for $Q^*_H$ as in $\mathcal {R}^H_1$ case.

In $\mathcal {R}^H_4$, $X^*_H(Q) < 0$ for $K<Q_0$, therefore $X^*_H=0$ due to concavity of $X^*_H(Q)$. Same argument goes for $Q^*_H$ as in $\mathcal {R}^H_1$. $\square$

Proof of Theorem 3

First we note that we are interested in region $\mathcal {R}_L^1$ regarding curve $K_2(N_a)$, and therefore, we only consider the interval $0 \le N_a \le \frac{L-c}{L-c+(H-L)\alpha }M$, which implies that the denominator of the second term in $K_2(N_a)$ is nonnegative. Then, $K_2(N_a) \le K_1(N_a)$ if the numerator of the second term in $K_2(N_a)$ is nonnegative, that is,

$$\begin{aligned} (H-L)\alpha N_a(M-K_1(N_a))-c_a(M-N_a)N_a \ge 0. \end{aligned}$$

Plugging $K_1(N_a)$ from expressions 24 and 39, and simplifying we get,

$$\begin{aligned} \overline{N}_a=\left[ 1-\frac{((H-L)\alpha -c_a)-\sigma \frac{H\Lambda (\mathcal {Z}_x)+c\mathcal {Z}_x}{M}}{((L-c)(1-\alpha )-c_a)-\frac{c_a}{(H-L)\alpha }(L-c)}\right] M. \end{aligned}$$

It is hard to isolate $N_a$ in this expression since on the right-hand-side of the expression we still have $\sigma =\sqrt{(M-\overline{N}_a) \alpha (1-\alpha )}$. However, we can replace $\sigma$ with $\sigma _0$ since the term $\frac{H \Lambda (\mathcal {Z}_x)+c \mathcal {Z}_x}{M}$ is small, and the inequality in expression 27 still holds since $\sigma _0 \ge \sigma$. $\square$

Appendix B: Additional plot and details

1.1 Appendix B.1: Profit surface plots (Fig. 6)

Fig. 6

Profit as a function of Q and X for low spot price (left) and high spot price (right)

1.2 Appendix B.2: Approximation plot for $Q^*_H(X)$

To demonstrate how tight our approximation is, we include here the results of a parameter sweep for a small market size of 400 buyers. Figure 7 shows the difference between the profit obtained using the optimum values and the approximated values of $X_H^* (Q)$ as a percentage of the high spot price H. Optimum values of $X_H^* (Q)$ are obtained numerically using binary search (using first order conditions in Lemma 4) at a high precision level of $10^{-6}.$ The most significant contributor of difference is the $\alpha$ parameter. Therefore we include the plots for the range $\alpha =[0.1,0.9]$, with increments of 0.05. In order to produce this plot, the maximum, average, and standard deviation of the profit differences as a percentage of the high spot price, H, of a single commodity are computed for each level of $\alpha$ over all other combinations of the full parameter sweep.

Fig. 7

Difference between the optimal and approximated profit values as a percentage of high spot price H for the solution pairs $(Q, X^*_H(Q))$ when Q is fixed

In the following plot, the solid line shows the average difference values together with error bars and the dashed line shows the maximum difference values. It is very clear from this plot that the approximated profit is extremely close to the optimal profit. The tightness of this approximation can be understood better with an example: if we consider that the product is an expensive concert ticket for $100 at the high price H for a market size of 400 people, then the average difference between the optimal and the approximate total profit for the entire market is less than $1, the maximum difference is less than $1 for most cases and only $15 for the most extreme case of $\alpha$ in the plot.

We note that we use the continuous versions of the functions we derived, whereas in real life, the quantities are integers, which may further reduce the gap between the optimal and approximated results. For larger market sizes the difference between optimal and approximated results further decreases.

Fig. 8

Minimum values of percent difference in quantity and profit as a function of $\alpha$ for the rare cases where $K(N_a) < Q_H^*(N_a)$

1.3 Appendix B.3: Approximation plots for solution boundaries

In Fig. 8, $\Delta ^{\%}_Q(\alpha )=\frac{\Delta _Q(\alpha )}{M}$, the minimum difference between $K(N_a)$ and $Q^*_H(N_a)$ as a percent of the market size M is depicted as a function of $\alpha$, where,

$$\begin{aligned} \Delta _Q(\alpha )=\min _{L,c,c_a} \left\{ \min _{N_a}\{K(N_a)-Q^*(N_a)\}: \min \{(H-L)\alpha , (L-c)(1-\alpha )\} \ge c_a \right\} . \end{aligned}$$

It can be observed that in most cases the difference is less than 2 %. Moreover, for such cases,

$$\begin{aligned} \Delta ^{\%}_\Pi (\alpha )=\frac{\Pi _H(Q^*_H(N_a),N_a)-\Pi _H(K(N_a),N_a)}{\Pi _H(Q^*_H(N_a),N_a)}, \end{aligned}$$

Fig. 9

Left functions $K_1(N_a)$ and $K_2(N_a)$; right for fixed $N_a$, the case where $Q^*_H(N_a)< K_2(N_a) < K_1(N_a)$

which is the corresponding percent difference between the profit values is also given in Fig. 8. This difference is less than 1 % for most cases. It is also worth noting that $K(N_a) \ge \overline{Q}_H(N_a)$ in all iterations of the parameter sweep, where $\overline{Q}_H(N_a)$ is the leftmost boundary of the narrow region $\mathcal {R}^H_1$. Curves $K_1(N_a)$ and $K_2(N_a)$ are given in Fig. 9, whose minimum constitute the function $K(N_a)$ which approximates the general boundary between optimal low spot and high spot prices. An example of the intersections among the functions $\Pi _H(Q^*_H(N_a),N_a)$, $\Pi _L(K,N_a)$, and $\Pi _L(K, 0)$ is also depicted in Fig. 9.