Joint pricing and inventory control for additive demand models with reference effects

Abstract

We study a periodic review joint inventory and pricing problem of a single item with stochastic demand subject to reference effects. The random demand is contingent on the current price and the reference price that acts a benchmark against which customers compare the price of a product. Randomness is introduced with an additive random term. The customers perceive the difference between the price and the reference price as a loss or a gain. Hence, they have different attitudes towards them, such as loss aversion, loss neutrality or loss seeking. We model the problem using safety stock as the decision variable and show that the problem can be decomposed into two subproblems for all demand models under mild conditions. Using the decomposition, we show that a steady state solution exists for the infinite horizon problem and we characterize the steady state solution. Defining the modified revenue as revenue less the production cost, we show that a state-dependent order-up-to policy is optimal for concave demand models with concave modified revenue functions and provide example demand models with absolute difference reference effects and loss-averse customers. We also show that the optimal inventory level increases with the reference price. All of our results hold for finite and infinite horizon problems.

Appendices

Appendix 1: Proof of Theorem 1

Fix a period $t'$. We show that given Assumption 1, if ($i$) $\Delta ^0_{t+1} \ge \Delta ^0_{t}$ for $t=t',\ldots ,T$ and $(ii)\bigl (p_{t'}^0(r_{t'}),\Delta _{t'}^0\bigr )$ is feasible, i.e., $d_{t'}^0(r_{t'}) + \Delta ^0_{t'} \ge x_{t'}$, then

• a. ${I^*}$ can mimic the decisions of ${I^0}$ in periods $t',\ldots ,T$.

• b. $\Phi _{t'}(x,r) = \Phi ^0_{t'}(r)$ and $(p_t^0(r_t),\Delta _t^0)$ are indeed optimal for ${I^*}$ for $t=t',\ldots ,T$.

In order to show the feasibility in (a), we need to show that $x_{t'+1} \le d^0_{t'+1}\bigl (r_{t'+1}\bigr ) + \Delta ^0_{t'+1}$. Since $x_{t'+1} \!=\! d^0_{t'} \bigl ( r_{t'} \bigr ) \!+\! \Delta ^0_{t'} - \Bigl ( d^0_{t'}\bigl (r_{t'}\bigr ) + \epsilon _{t'} \Bigr )$, we need to show the following:

$$\begin{aligned} \Delta ^0_{t'} - \epsilon _{t'} \le d^0_{t'+1}\bigl (r_{t'+1}\bigr ) + \Delta ^0_{t'+1}. \end{aligned}$$

(14)

$d^0_{t'+1}\bigl (r_{t'+1}\bigr ) + \epsilon _{t'} \ge 0$ from Assumption 1(ii) and $\Delta ^0_{t'} \le \Delta ^0_{t'+1}$ from the hypothesis of the theorem. Hence (14) holds and this completes the proof of $(a)$.

For $(b)$, observe that the profit of the ${I^*}$ in last period cannot be greater than that of ${I^0}$ since ${I^*}$ has the same profit function with ${I^0}$ but has a smaller feasible region. Hence $\Phi ^0_t(r)\ge \Phi _t(x,r)$ for $\forall x$ can be shown inductively for $t=1,\ldots ,T$. If the system ${I^*}$ can mimic the decisions of ${I^0}$ starting from period $t'$, then $\Phi _{t}(x,r) = \Phi ^0_{t}(r)$ for $t=t',\ldots ,T$. Therefore the decisions of $(p_t^0(r), \Delta ^0_{t})$ are optimal for ${I^*}$ as well. Since we prove this for arbitrary $t'$, the proof is complete. $\square$

Appendix 2: Proof of Theorem 3

We first show that the value function $\phi _{T}(\Delta ,p,r)$ and the optimal value function $\Phi _{T}(x,r)$ are concave. Then we show by induction that $\phi _t(\Delta ,p,r)$ and $\Phi _t(x,r)$ are concave for every $t$.

The revenue function $\Pi _t(p,r)$ is concave by Assumption 3 and expected inventory related cost is convex by Assumption 2(i). Since $\Phi _{0} = 0,\,\phi _1(\Delta ,p,r)$ is concave.

The set $\Bigl \{(\Delta ,p,r,x): \hat{d}_{T}(p,r) + \Delta - x\ge 0 \Bigr \}$ is a convex set since upper level set of a concave function is a convex set. Therefore the maximization over $\phi _{T}(\Delta ,p,r)$ is performed over a convex set. The concavity of $\Phi _{T}(x,r)$ follows from Theorem A.3 in Porteus (2002). For the induction step, we assume that $\Phi _{t+1}$ is concave and show that $\phi _t$ and $\Phi _t$ are concave. $\Pi _t(p,r)$ is concave by Assumption 3 and $G^{\gamma }_t$ is convex by Assumption 2, hence we need to show that $\Phi _{t+1} (\Delta - \epsilon , \alpha r + (1-\alpha )p)$ is concave in $(\Delta ,p,r)$ in order to show the concavity of $\phi _t$. For arbitrary $\Delta _1, \Delta _2,\, (p_1,r_1)$ and $(p_2,r_2)$, following can be written:

$$\begin{aligned}&\Phi _{t+1}\left( \frac{\Delta _{1} + \Delta _2}{2} - \epsilon , \alpha \frac{r_{1} + r_2}{2} + (1-\alpha ) \frac{p_{1} + p_2}{2}\right) \\&\quad = \Phi _{t+1}\left( \frac{(\Delta _{1} + \epsilon ) + (\Delta _2 + \epsilon )}{2} , \frac{(\alpha r_{1} + (1-\alpha ) p_{1}) + (\alpha r_2 + (1-\alpha ) p_2)}{2} \right) \\&\quad \ge \frac{\Phi _{t+1}\left( \Delta _{1} + \epsilon ,\alpha r_{1} + (1-\alpha )p_{1} \right) + \Phi _{t+1}\left( \Delta _2 + \epsilon ,\alpha r_2+(1-\alpha )p_2 \right) }{2} \end{aligned}$$

where the inequality follows from the concavity of $\Phi _{t+1}(x,r)$. This completes the proof for the concavity of $\phi _t(\Delta ,p,r)$. The concavity of $\Phi _{T}(x,r)$ follows from Theorem A.3 in Porteus (2002).

We now prove that an SDO policy is optimal in period $t$. Given $r$, let

$$\begin{aligned} (\Delta _t^*(r),p_t^*(r))=\text{ arg }\max \limits _{ \underline{p}\le p\le \overline{p}} \phi _t(\Delta ,p,r). \end{aligned}$$

If $\Delta _t^*(r) + d_t^*(r) \ge x$, then $(\Delta _t^*(r),p_t^*(r))$ are the optimal decisions for the period and the optimal order-up-to level is given by $y_t^*(r) = \Delta _t^*(r) + d_t^*(r)$.

If the initial inventories are high, i.e., $\Delta _t^*(r) + d_t^*(r) < x$, we argue that the firm does not give any order. We prove it by showing that the firm can make a better profit by not ordering. Assume that the firm decides on a price $p_t'(r)$ and a safety stock $\Delta _t'(r)$ which are feasible and this decision results in a positive order replenishment. Then we have $\Delta _t'(r) + d_t'(r) > x$ where $d_t'(r) = \hat{d}(p'_t(r),r)$ and $\underline{p}\le (p_t'(r),r) \le \overline{p}$. One can pick a pair $(\overline{\Delta }_t(r),\overline{p}_t(r))$ such that

$$\begin{aligned} \left( \begin{array}{l@{\quad }l} \overline{\Delta }_t(r)\\ \overline{p}_t(r) \end{array} \right) = \psi \left( \begin{array}{l@{\quad }l} \Delta _t^*(r)\\ p_t^*(r) \end{array} \right) + (1-\psi ) \left( \begin{array}{l@{\quad }l} \Delta '_t(r) \\ p'_t(r) \end{array} \right) \end{aligned}$$

(15)

and $\overline{\Delta }_t(r) + \overline{d}_t(r) = x$ where $\overline{d}_t(r) = \hat{d}_t(\overline{p},r)$ and $\psi \in (0,1)$. This is possible since the pairs $\bigl (p_t^*(r),\Delta _t^*(r)\bigr )$ and $\bigl (p_t'(r),\Delta _t'(r)\bigr )$ are in disjoint spaces and the line passing through those points crosses the boundary which is defined by the set $\Bigl \{ (\Delta ,p): \Delta + \hat{d}_t(p,r) = x\Bigr \}$. Note that since $\Bigl \{p_t^*(r),p_t'(r) \Bigr \} \in [{p^l},{p^u}]$ and $\overline{p}_t(r)$ is a convex combination of these two, $\overline{p}_t(r) \in [{p^l},{p^u}]$. Hence the decision $(\overline{\Delta }_t(r),\overline{p}_t(r))$ is a feasible one. Now assume that the profit generated by the decision $(p_t'(r),\Delta _t'(r))$ is greater than that of the decision $(\overline{\Delta }_t(r),\overline{p}_t(r))$. Then one can write the following:

$$\begin{aligned} \phi _t\Bigl (\overline{\Delta }_t(r),\overline{p}_t(r),r\Bigr ) \le \psi \ \phi _t\Bigl ({\Delta }^*_t(r),{p}^*_t(r),r\Bigr ) + (1-\psi ) \phi _t\Bigl ({\Delta }_t'(r),{p}_t'(r),r\Bigr ) \end{aligned}$$

(16)

since $(p^*_t(r),\Delta ^*_t(r))$ is the optimal solution for $\phi _t$. However this contradicts the concavity of the function $\phi _t$. Therefore it is optimal not to order if $\Delta _t^*(r) + d_t^*(r) < x$ and hence $\Delta = x-\hat{d}(p,r)$. The optimal price is the value that maximizes $\phi _t(x-\hat{d}(p,r),p,r)$ for a given $r$. $\square$

Appendix 3: Proof of Theorem 4

• (a) The transformed model has a non-positive single-period profit function:

  $$\begin{aligned} \bigl (p \hat{d}(p,r) - M\bigr ) - c \hat{d}(p,r) - \hat{G}^{\gamma }(\Delta ) \le 0 \end{aligned}$$

  since $\hat{G}^{\gamma }(\Delta ) = \hat{G}(\Delta ) + c(1-\gamma )\Delta \ge 0$ by definition. Hence $\Phi ^{M}_i\le 0$ for all $i=1,2,\ldots$. Using Proposition 9.17 in Bertsekas and Shreve (1996), it suffices to show that for all $i=1,2,\ldots$, and all $\beta$, the sets:

  $$\begin{aligned} U_i(x,r,\beta ) = \Bigl \{(\Delta ,p): \hat{d}(p,r) + \Delta \ge x, \underline{p}\le p \le \overline{p} \text{ and } \phi ^{M}_i(\Delta ,p,r) \ge - \beta \Bigl \} \end{aligned}$$

  are compact. From ($i$) $\Phi ^{M}_i\le 0$ and $(ii)$ the definition of $M,\, p\hat{d}(p,r) + \gamma E\Bigl [\Phi ^{M}_{i-1}\bigl (\Delta -\epsilon ,\alpha r + (1-\alpha )p\bigr )\Bigr ] \le M$ can be written. Subtracting $c\hat{d}(p,r)+\hat{G}^{\gamma }(\Delta )$ from both sides and rearranging yields the following:

  $$\begin{aligned} \phi ^{M}_i(\Delta ,p,r) \le - c \hat{d}(p,r) - \hat{G}^{\gamma }(\Delta ). \end{aligned}$$

  (17)

  From Assumption 2, $\lim _{|\Delta |\rightarrow \infty } \Bigl (c\hat{d}(p,r) + \hat{G}^{\gamma }(\Delta )\Bigr ) = \infty$ on $\underline{p}\le p \le \overline{p}$. Moreover $\hat{G}^{\gamma }(\Delta )$ is convex. Hence for a given $\beta$, there exists $\Delta ^{\beta }_l$ and $\Delta ^{\beta }_u$ such that $\Delta ^{\beta }_l \le \Delta \le \Delta ^{\beta }_u$ implies $\hat{G}(\Delta ) + c\bigl (\Delta + \hat{d}(p,r)\bigr ) \le \beta$. Define a new set $U^*=\Bigl \{(\Delta ,p): \underline{p}\le p\le \overline{p}, \Delta ^{\beta }_l\le \Delta \le \Delta ^{\beta }_u \Bigr \}$. Note that $U^*$ is a bounded set. From (17), $U_i(x,r,\beta ) \subseteq U^*$ and hence $U_i(x,r,\beta )$ is also bounded. $\hat{d}(p,r)$ is continuous, and hence $\phi _i$ and $\Phi _i$ are continuous. Therefore $U_i(x,r,\beta )$ is closed since levels sets of continuous functions are closed (Rudin 1976). Hence $U_i(x,r,\beta )$ is compact for all $i=1,2,\ldots$.

• (b) $\Phi ^{M}$ and $\phi ^{M}$ satisfy the optimaliy equations in (9) from Proposition 9.8 in Bertsekas and Shreve (1996). $\phi _t$ and $\Phi _t$ satisfy the optimality equation in the original model (the equation in (4)) since $\phi _t = \phi ^{M}_t + M\frac{1-\gamma ^{t+1}}{1-\gamma }$ and $\Phi _t = \Phi ^{M}_t + M\frac{1-\gamma ^{t+1}}{1-\gamma }$.

• (c) Concavity of $\phi$ and $\Phi$ follows from Lemma 8.4 in Heyman and Sobel (2004).

• (d) Follows from (c) and Theorem 3. $\square$

Appendix 4: Proof of Theorem 5

Proof

Let $S = \left\{ (x,r)| r\in [{p^l},{p^u}]\right\}$ denote the state space and let

$$\begin{aligned} S_{1} = \left\{ (x,r) \in S : d^0(r) + \Delta ^0 \ge x\right\} \end{aligned}$$

and $S_2 = S\backslash S_{1}$. For the ${I^*}$ system, we will show the following:

• (i) If $(x,r)\in S_{1}$ and ${I^*}$ chooses $(\Delta ^0, p^0(r))$, then the new state $(x',r')$ is in $S_{1}$.

• (ii) If $(x,r)\in S_{1}$, then $\Phi (x,r) = \Phi ^0(r)$ and $(p^0(r),\Delta ^0)$ is also optimal for ${I^*}$ as well.

• (iii) If $(x,r)\in S_2$, then it is optimal not to order for the system ${I^*}$.

$(i)$ and $(ii)$ are proven similar to Theorem 1. For $(iii)$, we show that $(\Delta ^0, p^0(r))$ is also a maximizer of the function $\phi$. In order to do this, we compare the value functions $\phi$ and $\phi ^0$ which are given in (4) and (5), respectively. Note that the only difference between $\phi$ and $\phi ^0$ is the terms $E[\Phi (\cdot )]$ and $E[\Phi ^0(\cdot )]$. Consider the following inequality:

$$\begin{aligned} \phi (\Delta ,p,r) \le \phi ^0(\Delta ,p,r) \le \phi ^0({\Delta }^0(r),{p}^0(r),r) = \phi ({\Delta }^0(r),{p}^0(r),r) \end{aligned}$$

The first inequality follows from the fact that $\Phi (x,r)\le \Phi ^0(r)$ for all $(x,r)$ as shown in the proof of Theorem 1. The second inequality is from the definition of $({\Delta }^0(r),{p}^0(r))$. For the equality, if the decision of the system ${I^*}$ is $\bigl (\Delta ^0(r), p^0(r)\bigr )$ in the current period, then the state in the next period remains in $S_{1}$. Once the state is in $S_{1}$, it remains in $S_{1}$ and $\Phi _t(x,r) = \Phi ^0_t(r)$ from (ii). Therefore $(\Delta ^0(r),p^0(r))$ is a maximizer of $\phi$. Therefore $p^*(r)=p^0(r)$ and $\Delta ^*(r) = \Delta ^0$ by the definition of $p^*(r)$ and $\Delta ^*(r)$. Also $y^*(r) = y^0(r)$ and $d^*(r) = d^0(r)$. $\square$

Appendix 5: Examples of concave demand functions

In this section we provide mean demand models with concave base demands that satisfy the concavity assumption (Assumption 3).

Güler et al. (2014) employ concave demand models from the literature (Chen et al. 2006) as their base demand, e.g. $a-b(p-k)^{n-1},\,a-be^{np}$ and $n \log (a-bp)$ with $a\ge 0,\, b\ge 0$ and $n\ge 0$. They show that the mean demand models with these base demand functions employing RDRE are concave and have concave modified revenue functions.

Now consider the demand model employing ADRE with concave base demand:

$$\begin{aligned} \hat{d}_t (p,r) = {{\mu }}_t(p) + a_t (r-p) \end{aligned}$$

(18)

This mean demand model is concave since ${{\mu }}_t(p)$ is concave and $r-p$ is linear.

In order to show the concavity of the revenue of the model in (18), we introduce a transformation which enhances our subsequent analysis. Consider the following representation of the problem in (4) which is given by Chen et al. (2014). First, define $\Phi ^{\omega }_t(x,r) = \Phi _t(x,r) - \omega _t r^2$. Note that using $\Phi ^{\omega }_t(x,r)$ instead of $\Phi _t(x,r)$ in the representation of the value function $\phi _t(\Delta ,p,r)$ does not affect the value of $\phi _t(\Delta ,p,r)$ and hence does not affect the solution. For example, consider the final period. Since $\Phi _T(x,r)=0$, we have:

$$\begin{aligned} \phi _T(\Delta ,p,r) = \Pi _T(p,r) - G^{\gamma }_T(\Delta ) \end{aligned}$$

(19)

Using $= \Phi _{T+1}(x,r) = \Phi ^{\omega }_{T+1}(x,r) + \omega r^2 ,\, \phi _T(\Delta ,p,r)$ can be written as:

$$\begin{aligned} \phi _T(\Delta ,p,r)&= \Pi _T(p,r) - G^{\gamma }_T(\Delta ) \\&+ \gamma E\Bigl [\Phi ^{\omega }_{T+1}\bigl (\Delta -\epsilon _T,\alpha r + (1-\alpha )p\bigr ) +\omega _{T+1} (\alpha r + (1-\alpha )p)^2\Bigr ] \\&= \Pi _T(p,r) - G^{\gamma }_T(\Delta ) +\omega _{T+1} (\alpha r + (1-\alpha )p)^2 \\&+ \gamma E\Bigl [\Phi ^{\omega }_{T+1}\bigl (\Delta -\epsilon _T,\alpha r + (1-\alpha )p\bigr ) \Bigr ] \end{aligned}$$

Now, since $\Phi _{T+1}(x,r)=0$, we have $\Phi ^{\omega }(x,r) = -w_{T+1}r^2$. Substituting $\Phi ^{\omega }(x,r)$ into the equation above, we have:

$$\begin{aligned} \phi _T(\Delta ,p,r)&= \Pi _T(p,r) - G^{\gamma }_T(\Delta ) +\gamma \omega _{T+1} (\alpha r + (1-\alpha )p)^2 \nonumber \\&+ \gamma E\Bigl [-\omega _{T+1} (\alpha r + (1-\alpha )p)^2 \Bigr ] \nonumber \\&= \Pi _T(p,r) - G^{\gamma }_T(\Delta ) \end{aligned}$$

(20)

It can be observed that (19) and (20) are the same. Therefore the value of $\phi _t(x\Delta ,p,r)$ is the same irrespective of its representation with $\Phi _t(x,r)$ or $\Phi ^{\omega }_t(x,r)$.

Second, we define a transformed value function $\phi ^{\omega }_t(\Delta ,p,r) = \phi _t(\Delta ,p,r) - \omega _t r^2$. Note that $\Delta$ and $p$ are the decision variables and $r$ is the state variable. By this transformation, the optimal decisions are not affected since the transformation does not include any term that has an interaction between the state variable $r$ and the action variables $\Delta$ and $p$.

We show that by choosing $\omega _t$ carefully, the transformed value function $\phi ^{\omega }_t$ can be shown to be concave. The recursion in (4) can be re-written as follows:

$$\begin{aligned} \phi ^{\omega }_t(\Delta ,p,r)&= (p-c_t){{\mu }}_t(p) + (p-c_t)\Bigl ( a_t(r-p)\Bigr ) - \omega _t r^2 \\&+ \gamma \omega _{t+1} (\alpha r + (1-\alpha )p)^2 -G^{\gamma }_t(\Delta ) + \gamma E\Bigl [\Phi ^{\omega }_{t+1}\bigl (\Delta -\epsilon ,\alpha r + (1-\alpha )p\bigr )\Bigr ] \\ \Phi ^{\omega }_t(x,r)&= \underset{{}\underline{p}\le p \le \overline{p}, \hat{d}_t(p,r)+ \Delta \ge x}{\underset{\Delta ,p}{\max }} \Biggl \{\phi ^{\omega }_t(\Delta ,p,r)\Biggr \} - \omega _t r^2 \end{aligned}$$

The term $(p-c_t){{\mu }}_t(p)$ is concave since ${p^l}\ge c_t$ by the modeling assumptions. $G^{\gamma }_t$ is concave from Assumption 2(i). For the concavity of the remaining part of the single period profit function, one must check the hessian matrix which is given in the following:

$$\begin{aligned} \left[ \begin{array}{l@{\quad }r} 2(\gamma (1-\alpha )^2\omega _{t+1} - a_{t}) &{} a_{t} + 2\gamma \alpha (1-\alpha ) \omega _{t+1} \\ a_{t} + 2\gamma \alpha (1-\alpha ) \omega _{t+1} &{} 2(\gamma \omega _{t+1} \alpha ^2 - \omega _t) \end{array}\right] \end{aligned}$$

The hessian matrix is negative semi definite if and only if the following conditions hold:

$$\begin{aligned} \frac{a_{t}}{\gamma (1-\alpha )^2}&\ge \omega _{t+1} \end{aligned}$$

(21)

$$\begin{aligned} \omega _t&\ge \frac{a_{2t}/4 + a_{t} \alpha \gamma \omega _{t+1}}{a_{t} - \gamma (1-\alpha )^2 \gamma _{t+1}} \end{aligned}$$

(22)

Let us construct $\omega _t$ using (22) as an equality: $\omega _{T+1} = 0,\, \omega _T = n_T/4,\,\ldots$. If these $\omega _t$ satisfy (21), then the hessian matrix is negative semi-definite.

In order to find a sufficient condition over the parameters, let $\omega _{t+1} = k_t\frac{a_{t}}{\gamma (1-\alpha )^2}$. Then the hessian matrix is negative semi-definite if $0\le k_t <1$ and

$$\begin{aligned} \frac{a_{t-1}}{\gamma a_{t}} \ge \frac{(1-\alpha )^2 + \alpha k_t}{4 (1-k_t) k_{t-1}} \end{aligned}$$

(23)

Hence if there exists $k_t$ which satisfy $0\le k_t<1$ and (23), then the single period profit function is concave in $(\Delta ,p,r)$. For example, if $\frac{a_{t-1}}{ a_{t}} \ge \gamma$, then $k_t=0.5$ satisfies the required conditions. For example if the demand is stationary, then $k_t=0.5$ satisfies (23).

Now we give examples with loss-averse customers where Assumption 3 holds. In particular, consider the mean demand model in (10). Assume that ${\mu }(p)$ is concave, and customers are loss-averse, i.e., ${a_1}< {a_2}$. Then model in (10) can be rewritten as:

$$\begin{aligned} \hat{d}_t (p,r) = \min \left( {\mu }_t(p) + a_{1t} \frac{(r-p)}{F(r)}, {\mu }_t(p) + a_{2t} \frac{(r-p)}{F(r)}\right) \end{aligned}$$

(24)

The minimum operator preserves concavity. Hence the mean demand defined in (24) is concave if ADRE or RDRE is employed.

Consider the mean demand model with RDRE and $c_t=0$. Then the revenue can be written as:

$$\begin{aligned} \Pi _t(p,r)= \min \left( p\left( {\mu }_t(p) + a_{1t} \frac{(r-p)}{r}\right) , p\left( {\mu }_t (p) + a_{2t} \frac{(r-p)}{r}\right) \right) \end{aligned}$$

(25)

Since the minimum operator preserves concavity the revenue function defined in (25) is concave.

Finally we show that the revenue of the mean demand models with ADRE is concave. Consider the transformation given for the mean demand model employing ADRE with loss-neutral customers:

$$\begin{aligned} \Pi _t(p,r|a_{it})= (p-c_t)\left( {{\mu }}_t(p) +a_{it}(r-p)\right) - \omega _t r^2 + \gamma \omega _{t+1} (\alpha r + (1-\alpha )p)^2 \end{aligned}$$

(26)

The transformed revenue for the mean demand function with loss-averse customers can be written as:

$$\begin{aligned} \Pi _t(p,r)&= \min \Bigl ((p-c_t)\left( {{\mu }}_t(p) +a_{1t}(r-p)\right) , (p-c_t)\left( {{\mu }}_t(p) +a_{2t}(r-p)\right) \Bigr ) \nonumber \\&- \omega _t r^2 + \gamma \omega _{t+1} (\alpha r + (1-\alpha )p)^2 \nonumber \\&= \min \left( \Pi _t(p,r|a_{1t}), \Pi _t(p,r|a_{2t})\right) \end{aligned}$$

(27)

since the term $- \omega _t r^2 + \gamma \omega _{t+1} (\alpha r + (1-\alpha )p)^2$ is the same for both side and hence can be taken inside the minimization. $\Pi _t(p,r|a_{it})$ is concave for $i=1,2$, hence the revenue given in (27) is concave because of the minimum operator.

Appendix 6: Proof of Theorem 6

Without loss of generality, we suppress the time indices. Let $\bigl \{(d_1,r_1), (d_2,r_2)\bigr \}\in \mathbb {D}$ and $p_{ij} = \hat{p}(d_i,r_j)$.

(i) In order to show that $\mathbb {D}$ is a lattice, We need to show that

$$\begin{aligned} \bigl (\max (d_1,d_2),\max (r_1,r_2)\bigr )\in \mathbb {D}, \, \, \bigl (\min (d_1,d_2),\min (r_1,r_2)\bigr )\in \mathbb {D}. \end{aligned}$$

The case $d_1\ge d_2$ and $r_1\ge r_2$ is trivial. Let $d_1\le d_2$ and $r_1\ge r_2$. we need to show ${p^l} \le p_{12}\le {p^u}$ and ${p^l} \le p_{21}\le {p^u}$. We have ${p^l}\le p_{22}\le p_{12}\le p_{11} \le {p^u}$ since since the price function is strictly decreasing with $d$ and increasing with $r$ (Proposition 1). Similarly it can be shown that ${p^l}\le p_{21}\le {p^u}$ which completes our proof of part (i).

(ii) We show that every term of $\hat{\phi }^0_t$ is supermodular. Let $d_2\ge d_1$ and $r_2\ge r_1$. We first show that the revenue term is supermodular. Following can be written:

$$\begin{aligned} d_2 \bigl ( p_{22} - p_{21} \bigr ) \ge d_1 \bigl (p_{22} - p_{21} \bigr ) \ge d_1 \bigl ( p_{12} - p_{11} \bigr ) \end{aligned}$$

(28)

where the first inequality follows from the fact that $\hat{p}$ is increasing with $r$ (Proposition 1) and $d_2\ge d_1$ and the second inequality follows from supermodularity of the price function (Assumption 5). This completes the proof of supermodularity of $d\hat{p}(d,r)$.

We now show that $\Phi ^0_{t+1}\bigl (\alpha r + (1-\alpha )\hat{p}(d,r)\bigr )$ is supermodular in $d$ and $r$. Without loss of generality, let $r_2\ge r_1$ and $d_2\ge d_1$. Let $p_{12} = p_{11} + \delta _p/(1-\alpha )$. Since $\hat{p}$ is increasing with $r$, we have $\delta _p \ge 0$. From supermodularity of the price function, we have $p_{21} - p_{11} \le p_{22} - p_{12}$. Let $p_{11} + p_{22} = p_{12} + p_{21} + \delta _s$. Note that $\delta _s\ge 0$. Rearranging yields $p_{12} = p_{11} + \bigl (p_{22}-p_{21} - \delta _s \bigr )$ and hence $p_{22}-p_{21} - \delta _s = \delta _p/(1-\alpha )$. Moreover let $r_2 = r_1 + \delta _r/\alpha$. Finally, it can be shown in a similar way to the proof of Theorem 3 that $\phi ^0_{t+1}$ is concave. Using these definitions and properties, following can be written:

$$\begin{aligned}&\Phi ^0_{t+1}\Bigl (\alpha r_2 + (1-\alpha ) p_{12}\Bigr ) - \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{11}\Bigr ) \nonumber \\&\quad =\Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{11} + \delta _r + \delta _p \Bigr ) - \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{11}\Bigr ) \nonumber \\&\qquad \quad \le \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{21} + \delta _r + \delta _p \Bigr ) -\Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{21}\Bigr ) \end{aligned}$$

(29)

where the second inequality follows from $p_{21}\le p_{11}$ and from concavity of $\Phi ^0_{t+1}$. Substituting $\delta _r$ and $\delta _s$:

$$\begin{aligned}&\Phi ^0_{t+1}\Bigl (\alpha r_1 \!+\! (1\!-\!\alpha ) p_{21} \!+\! \alpha (r_2-r_1) \!+\! (1-\alpha )(p_{22} \!-\! p_{21} \!-\! \delta _s) \Bigr ) \!-\! \Phi ^0_{t+1}\Bigl (\alpha r_1 \!+\! (1-\alpha ) p_{21}\Bigr ) \nonumber \\&\quad =\Phi ^0_{t+1}\Bigl (\alpha r_2 + (1-\alpha ) p_{22} - (1-\alpha ) \delta _s \Bigr ) - \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{21}\Bigr ) \nonumber \\&\qquad \quad \le \Phi ^0_{t+1}\Bigl (\alpha r_2 + (1-\alpha ) p_{22}\Bigr ) - \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{21}\Bigr ) \end{aligned}$$

(30)

can be written. The inequality follows from the fact that $\Phi ^0_{t+1}(r)$ is increasing with $r$ (Lemma 1). From (29) and (30), following can be written:

$$\begin{aligned}&\Phi ^0_{t+1}\Bigl (\alpha r_2 + (1-\alpha ) p_{12}\Bigr ) - \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{11}\Bigr ) \\&\quad \le \Phi ^0_{t+1}\Bigl (\alpha r_2 + (1-\alpha ) p_{22}\Bigr ) - \Phi ^0_{t+1}\Bigl (\alpha r_1 + (1-\alpha ) p_{21}\Bigr ) \end{aligned}$$

which is the definition of supermodularity.

Since $\mathbb {D}$ is a lattice and $\hat{\phi }^0_t(d,r)$ is supermodular, $d^0_t(r)$ is increasing with $r$. Hence $y^0_t(r)= d^0_t(r)+\Delta ^0_t$ is increasing with $r$.

Appendix 7: Supermodularity of $\hat{p}(d,r)$

Without loss of generality, we suppress the time indicies.

Proposition 2

Assume that the price function is twice differentiable. The price function $\hat{p}(d,r)$ is supermodular if and only if $\hat{d}_r \hat{d}_{pp} \le \hat{d}_p \hat{d}_{pr}$.

Proof

A differentiable function is supermodular if its cross derivate is non-negative. Here the price function is supermodular if $\hat{p}_{dr} \ge 0$. We will show the equivalent condition on the demand function which assures $\hat{p}_{dr} \ge 0$. Subscripts denoting the partial derivatives, $\hat{p}_d = \frac{1}{\hat{d}_p}$.

Then:

$$\begin{aligned} \hat{p}_{dr}&= \frac{\partial \hat{p}_d}{ \partial r} = \frac{\partial (\hat{d}_p)^{-1}}{ \partial r} = -\left( \frac{1}{d^2_p} \right) \left( \hat{d}_{pp}\hat{p}_r + \hat{d}_{pr}\right) . \end{aligned}$$

Also $\frac{\partial d}{\partial r} = \frac{\partial \hat{d}(p,r)}{\partial r}$ and hence $\hat{p}_r = -\frac{\hat{d}_r}{\hat{d}_p}$. Using these equations, $\hat{p}_{dr} = \frac{\hat{d}_r \hat{d}_{pp} -\hat{d}_p \hat{d}_{pr} }{d^3_p}$ can be written. Since $\hat{d}^3_p \le 0$ from Assumption 1, $\hat{d}_r \hat{d}_{pp} \le \hat{d}_p \hat{d}_{pr}$ is necessary and sufficient for $\hat{p}(d,r)$ to be non-negative and hence $\hat{p}(d,r)$ to be supermodular. $\square$

Note that $\hat{d}_{pp}\le 0$ for concave demand models and $\hat{d}_r\ge 0$ and $\hat{d}_p\le 0$ from Assumption 1(ii). Then if $\hat{d}_{pr}\le 0$, particularly if the mean demand has Note that $\hat{d}_{pp}\le 0$ for concave demand models and $\hat{d}_r\ge 0$ and $\hat{d}_p\le 0$ from Assumption 1(ii). Then if $\hat{d}_{pr}\le 0$, particularly if the mean demand has ADRE then the price function is supermodular. If the mean demand is supermodular in $(p,r)$, then one must impose explicitly the condition $\hat{p}_{dr}\ge 0$.