Price–quality relationship in the presence of asymmetric dynamic reference quality effects

Abstract

The purpose of this study is to examine the relationship between price and reference quality and their combined effect on profits. An analytical modeling approach aimed at solving the optimal solution for the profit maximization problem under these conditions is developed, enabling the exact path of the optimal price and quality over time to be depicted. Based on separating the effects of price and reference quality on demand, this analysis also provides insight into the contribution of these two effects to the steady-state solution through elasticities. Our results show that a monotonic inverse relationship exists between price and quality, such that a steady-state level is obtained where the quality–price ratio is lower when reference quality effects exist than when such effects do not exist. In other words, consumers obtain higher quality for a higher price but with a lower price per unit of quality. Overall, accounting for reference quality effects will increase a firm’s profits.

Appendices

Appendix A: Proof of Proposition 1

The current value Hamiltonian is given by

$$ H=\left(p(t)-c\right)Q(t)-q^{2}(t)+\lambda \beta \left(q(t)-r_{q}(t)\right)~. $$

Therefore, the first-order conditions for optimality are

$$ \frac{\partial H}{\partial p} =a+\delta _{1}q-2\delta _{2}p+\gamma \left(q-r_{q}\right)+\delta _{2}c=0~, $$

(18)

$$\frac{\partial H}{\partial q} =(p-c)\left(\delta _{1}+\gamma \right)-2q+\lambda \beta =0~,$$

(19)

$$\begin{array}{rll}\frac{d\lambda }{\textrm{d}t} &=&\alpha \lambda -\frac{\partial H}{\partial r_{q}} ~=\left(\alpha +\beta \right)\lambda +\gamma (p-c),\\ \frac{\textrm{d}r_{q}}{\textrm{d}t} &=&\beta \left(q-r_{q}\right)~, \end{array} $$

with the initial condition $r_{q}(0)={r_{q}^{0}}$ and the assumption that lim _t→ ∞ p(t) < ∞. In steady-state $\frac{d\lambda }{\textrm{d}t}=0$ and thus, λ in steady-state $\lambda _{\rm ss}=-\frac{\gamma }{ \alpha +\beta }(p_{\rm optimal}^{\rm ss}-c)$. Furthermore, in steady-state $q_{\rm optimal}^{\rm ss}=r_{q}^{\rm ss}$. Substituting λ _ss in Eq. 19 and solving with Eq. 19 gives Eq. 11. The above equations are reduced to a linear differential equation that can be easily solved to get Eqs. 9 and 10. However, to avoid excessive detail, we note that substitution of Eqs. 9 and 10 in the above equations shows that Eqs. 9 and 10 is indeed the solution.

Appendix B: Proof of Proposition 4

This proof will show that under asymmetric conditions, optimal price and quality paths will not create a situation in which consumers’ evaluation of these variables will switch from gains to losses or vice versa. We use the technique introduced in Fibich et al. (2003) for that purpose. Throughout this proof, the notations q _optimal(t), p _optimal(t) represent the optimal solution for the symmetric problem given by Eqs. 9 and 10. To prove this proposition, we will need the following three lemmas.

Lemma 1

The steady-state quality, $q_{\rm optimal}^{\rm ss},$ given in Eq.  11 is a monotonic increasing function of γ .

Proof

Differentiating $q_{\rm optimal}^{\rm ss}(\gamma )$ with respect to γ gives

$$ \frac{d}{\textrm{d}\gamma }q_{\rm optimal}^{\rm ss}(\gamma )=\frac{4\left(a-\delta _{2}c\right)\frac{ \alpha }{\alpha +\beta }\delta _{2}}{\left[ 4\delta _{2}-\delta _{1}^{2}-\delta _{1}\gamma \frac{\alpha }{\alpha +\beta }\right] ^{2}}>0. $$

□

Lemma 2

$\left( q_{\rm optimal}(t)-r_{q}(t)\right)$ does not change this sign for any t .

Proof

From Eqs. 9 and 12, we have

$$ q_{optimal}(t)-r_{q}(t)=-\frac{m}{\beta }\left(r_{q}^{0}-q^{\rm ss}\right)e^{-mt} $$

which has a constant sign.□

Let us denote the profits under any quality–price strategy q(t), p(t) by

$$ g(\gamma _{\rm gain},\gamma _{\rm loss};q(t),p(t)):=\int_{0}^{\infty }e^{-\alpha t} \bigg[(p(t)-c)Q(t)-q^{2}(t)\bigg]\,\textrm{d}t~, $$

where Q is given by Eq. 5 and γ is given by Eq. 6 and define

$$ G\left(\gamma _{\rm gain},\gamma _{\rm loss}\right):=\sup_{q(t),p(t)}g\left(\gamma _{\rm gain},\gamma _{\rm loss};q(t),p(t)\right)~. $$

(20)

Lemma 3

$$ G\left(\gamma _{\rm gain},\gamma _{\rm loss}\right)\leq G\left(\tilde{\gamma},\tilde{\gamma}\right)\text{ \ \ \ \ for all }\gamma _{\rm gain}\leq \tilde{\gamma}\leq \gamma _{\rm loss}.\qquad \label{eq:GG} $$

(21)

Proof

The function g(γ _gain, γ _loss; q(t), p(t)) increases monotonically in γ _gain and decreases monotonically in γ _loss. Thus,

$$ g\left(\gamma _{\rm gain},\gamma _{\rm loss};q(t),p(t)\right)\leq g\left(\tilde{\gamma},\tilde{\gamma };q(t),p(t)\right) \quad \text{for all }\gamma _{\rm gain}\leq \tilde{\gamma}\leq \gamma _{\rm loss}. $$

(22)

Then, by Eq. 22 we have that

$$ \sup_{q(t),p(t)}g\left(\gamma _{\rm gain},\gamma _{\rm loss};q(t),p(t)\right) \! \leq \! \sup_{q(t),p(t)}g\left(\tilde{\gamma},\tilde{\gamma};q(t),p(t)\right) \ \ \ \text{for all }\gamma _{gain} \! \leq \! \tilde{\gamma} \! \leq \! \gamma _{\rm loss}, $$

(23)

• 1. and since the right hand side of Eq. 23 is G(γ _gain, γ _loss) and the left hand side of Eq. 23 is $G( \tilde{\gamma},\tilde{\gamma})$ we get

  $$ G\left(\gamma _{\rm gain},\gamma _{\rm loss}\right)\leq G(\tilde{\gamma},\tilde{\gamma})\text{ \ \ \ \ for all }\gamma _{\rm gain}\leq \tilde{\gamma}\leq \gamma _{\rm loss}.\qquad $$

□

We now turn to proving the proposition in which we have three possible cases with respect to the relationship between the initial reference quality $r_{q}^{0}$ and the steady-state quality level q ^ss:

• 1. We start with the first case and show that if $r_{q}^{0}<q^{\rm ss}(\gamma _{\rm gain})$ then, q _optimal(t), p _optimal(t) given by Eqs. 9 and 12 with γ = γ _gain is the optimal solution for the asymmetric case. As we show by Lemma 2, $\left( q_{\rm optimal}(t)-r_{q}(t)\right)$ does not change its sign and thus, substituting γ = γ _gain in q _optimal(t) we have that q _optimal(t) ≥ r _q (t) for all t ≥ 0. Therefore,

  $$\begin{array}{rll} g\left(\gamma _{\rm gain},\gamma _{\rm loss};q_{\rm optimal}(t),p_{\rm optimal}(t)\right) &=& g\left(\gamma _{\rm gain},\gamma _{\rm gain};q_{\rm optimal}(t),p_{\rm optimal}(t)\right) \\ &=& G\left(\gamma _{\rm gain},\gamma _{\rm gain}\right) \end{array} $$

  (24)

  where the last equality follows because g(γ _gain, γ _gain; q _optimal(t), p _optimal(t)) is symmetric in γ. Thus, q _optimal(t), p _optimal(t) is the optimal strategy that maximizes g. On the other hand, from Eq. 20 and the monotonicity of G(γ _gain, γ _loss) given by Lemma 3 we have that

  $$ g\left(\gamma _{\rm gain},\gamma _{\rm loss};q_{\rm optimal}(t),p_{\rm optimal}(t)\right)\leq G\left(\gamma _{\rm gain},\gamma _{\rm loss}\right)\leq G\left(\gamma _{\rm gain},\gamma _{\rm gain}\right). $$

  (25)

  Combining Eqs. 24 and 25 yields

  $$ g\left(\gamma _{\rm gain},\gamma _{\rm loss};q_{\rm optimal}(t),p_{\rm optimal}(t)\right)=G\left(\gamma _{\rm gain},\gamma _{\rm loss}\right)~, $$

  which proves that the symmetric solution q _optimal(t), p _optimal(t) with γ _gain is the optimal solution for the asymmetric problem.

• 2. The second case is where $r_{q}^{0}>q^{\rm ss}(\gamma _{\rm gain})$. Then, q _optimal(t), p _optimal(t) given by Eqs. 9 and 12 with γ = γ _loss is the optimal solution for the asymmetric case is proved similarly.

• 3. Last, we show that for the case where $q_{\rm optimal}^{\rm ss}(\gamma _{\rm loss})\leq r_{0}\leq q_{\rm optimal}^{\rm ss}(\gamma _{\rm gain})$, then $q(t)\equiv r_{q}(t)\equiv {r_{q}^{0}}$ and $p(t)=p_{\rm optimal}^{\rm ss}(\tilde{\gamma})$ for specific $\tilde{\gamma}$ is the optimal solution for the asymmetric case. Let $\tilde{\gamma}$ be the solution to $q_{\rm optimal}^{\rm ss}(\tilde{\gamma})={ r_{q}^{0}}$. By the monotonicity of $q_{\rm optimal}^{\rm ss}(\gamma )$ given in Lemma 1 there exists $\tilde{\gamma}$, $\gamma _{\rm gain}\leq \tilde{\gamma}\leq \gamma _{\rm loss}$ that solves this equation. Therefore, when we use $q(t)={r_{q}^{0}}$ and $p(t)=p_{\rm optimal}^{\rm ss}(\tilde{ \gamma})$ there is no loss or gain and thus,

  $$ g\left(\gamma _{\rm gain},\gamma _{\rm loss};{r_{q}^{0}},p_{\rm optimal}^{\rm ss}\left(\tilde{\gamma} \right)\right)=g\left(\tilde{\gamma},\tilde{\gamma};{r_{q}^{0}},p_{\rm optimal}^{\rm ss}\left(\tilde{\gamma }\right)\right)=G\left(\tilde{\gamma},\tilde{\gamma}\right)~. $$

  On the other hand, $g(\gamma _{\rm gain},\gamma _{\rm loss};{r_{q}^{0}} ,p_{\rm optimal}^{\rm ss}(\tilde{\gamma}))\leq G(\gamma _{\rm gain},\gamma _{\rm loss})\leq G(\tilde{\gamma},\tilde{\gamma})$. Combining the last two relationships gives g(γ _gain,γ _loss;p _optimal(t)) = G(γ _gain,γ _loss).