Pricing Strategies of Remanufacturing Business with Replacement Purchase

Abstract

A special feature of remanufacturing business is the existence of large proportion of replacement customers. This is due to the fact that many durable product markets are highly saturated and customers who return their end-of-life products need to do replacement purchase. At the same time, pricing strategies have been widely adopted by remanufacturing companies to balance supply and demand. In this study, the joint decision of acquisition, trade-in, and selling price is considered. The objective is to maximize the expected profit. It is shown that a remanufacturing firm should offer higher rebates to replacement customers when this customer segment has high return quality and high price sensitivity. The optimal pricing policies under uncertain return yield rate are studied. The profitability of different pricing schemes is also investigated.

Appendix

Proof of Lemma 1

Define H(x, ρ) = θ( f ) + ω(p) − ρ(η(r) + θ(f)) and $K\left(\mathrm{x}\right)=\frac{\theta (f)+\omega (p)}{\theta (f)+\eta (r)},$ where x = (r, f, p). It is obvious that E[Π(x)] is differentiable for H(x, B) < 0 and H(x, B) > 0. The only thing that needs to be proven is whether E[Π(x)] is differentiable at H(x, B) = 0. Let x₀ = {x | H(x, B) = 0}; it can be shown that the partial derivatives at x₀ exist and are continuous.

Denote Δ as a vector so that H(x₀ + Δ, B) > 0 and H(x₀−Δ, B) < 0. Consider the special case where Δ _i  = te _i  = (0, …, t, …, 0), i ∈ {r, f, p}.

$$ \begin{array}{c}E\left[\prod \left({\mathrm{x}}_0\right)\right]-E\left[\prod \right({\mathrm{x}}_0-{\boldsymbol{\Delta}}_{\mathbf{r}}\left)\right]=\eta \left({r}_0-t\right)\left({r}_0-t+d\right)-\eta \left({r}_0\right)\left({r}_0+d\right)\\ {}+\left({c}_r-c\right)\left({\displaystyle {\int}_A^{K\left({\mathbf{x}}_0\right)}H\left({\mathbf{x}}_{\mathbf{0}},\rho \right)g\left(\rho \right) d\rho}\right)\\ {}-{\displaystyle {\int}_A^{K\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}}\right)}H\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}},\rho \right)g\left(\rho \right) d\rho \Big)}\end{array} $$

Let |Δ _r | → 0:

$$ \begin{array}{l}\kern0.6em {\displaystyle {\int}_A^{K\left({\mathbf{x}}_0\right)}H\left({\mathbf{x}}_{\mathbf{0}},\rho \right)g\left(\rho \right) d\rho}-{\displaystyle {\int}_A^{K\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}}\right)}H\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}},\rho \right)g\left(\rho \right) d\rho}\\ {}={\displaystyle {\int}_A^{K\left({\mathbf{x}}_0\right)}\left(H\left({\mathbf{x}}_{\mathbf{0}},\rho \right)-H\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}},\rho \right)\right)g\left(\rho \right) d\rho}+{\displaystyle {\int}_{K\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}}\right)}^{K\left({\mathbf{x}}_0\right)}H\left({\mathbf{x}}_{\mathbf{0}},\rho \right)g\left(\rho \right) d\rho}\\ {}={\displaystyle {\int}_A^{K\left({\mathbf{x}}_0\right)}t{H}_r^{\prime}\left({\mathbf{x}}_{\mathbf{0}},\rho \right)g\left(\rho \right) d\rho}+o(t)+\left(K\left({\mathbf{x}}_{\mathbf{0}}\right)-K\left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}}\right)\right)H\left({\mathbf{x}}_{\mathbf{0}},\xi \right)\end{array} $$

where K(x ₀  − Δ _r ) < ξ < K(x ₀ ) and ξ → K(x ₀ ) as |Δ _r | → 0.

Then we can obtain

$$ \begin{array}{l}\underset{\left|{\boldsymbol{\Delta}}_{\mathbf{r}}\right|\to 0}{ \lim}\frac{E\left[\prod \left({\mathbf{x}}_{\mathbf{0}}\right)\right]-E\left[\prod \left({\mathbf{x}}_{\mathbf{0}}-{\boldsymbol{\Delta}}_{\mathbf{r}}\right)\right]}{\left|{\varDelta}_r\right|}\\ {}=-\eta \left({r}_0\right)-{\eta}_r^{\prime}\left({r}_0\right)\left({r}_0+d\right)+\left({c}_r-c\right)\left({\displaystyle {\int}_A^{K\left({\mathbf{x}}_{\mathbf{0}}\right)}{H}_r^{\prime}\left({\mathbf{x}}_{\mathbf{0}},\rho \right)g\left(\rho \right) d\rho +{K}_r^{\prime}\left({\mathbf{x}}_{\mathbf{0}}\right)H\left({\mathbf{x}}_{\mathbf{0}},K\left({\mathbf{x}}_{\mathbf{0}}\right)\right)}\right)\\ {}=-\eta \left({r}_0\right)-{\eta}_r^{\prime}\left({r}_0\right)\left({r}_0+d\right)+\left(c-{c}_r\right){\eta}_r^{\prime}\left({r}_0\right)\mu \end{array} $$

The last equality comes from the fact that K(x₀) = B. It is easy to show that

$$ \begin{array}{l}\kern0.48em \underset{\left|{\boldsymbol{\Delta}}_{\mathbf{r}}\right|\to 0}{ \lim}\frac{E\left[\prod \left({\mathbf{x}}_{\mathbf{0}}+{\boldsymbol{\Delta}}_{\mathbf{r}}\right)\right]-E\left[\prod \left({\mathbf{x}}_{\mathbf{0}}\right)\right]}{\left|{\varDelta}_r\right|}\\ {}=-\eta \left({r}_0\right)-{\eta}_r^{\prime}\left({r}_0\right)\left({r}_0+d\right)+\left(c-{c}_r\right){\eta}_r^{\prime}\left({r}_0\right)\mu \end{array} $$

Similarly, it can be proven that the partial derivatives exist and are continuous with respect to f and p. Hence, E[∏(x)] is differentiable at x₀.

Poof of Proposition 2

It is easy to show that E[∏(r, f, p)] is concave when (θ + η)A ≤ θ + ω ≤ (θ + η)B. For (θ + η)A ≤ θ + ω ≤ (θ + η)B, it is necessary to show E[−∏ (r, f, p)] is convex in r, f, and p. Applying Sylvester’s criterion, it is equivalent to prove

• 1.
  $$ {H}_1=\frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial {r}^2}>0; $$
• 2.
  $$ {H}_2=\left|\begin{array}{cc}\hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial {r}^2}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial r\partial f}\hfill \\ {}\hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial r\partial f}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial {f}^2}\hfill \end{array}\right|>0; $$

  and

• 3.
  $$ {H}_3=\left|\begin{array}{ccc}\hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial {r}^2}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial r\partial f}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial r\partial p}\hfill \\ {}\hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial f\partial r}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial {f}^2}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial f\partial p}\hfill \\ {}\hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial p\partial r}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial p\partial f}\hfill & \hfill \frac{\partial^2E\left[-\prod \left(r,f,p\right)\right]}{\partial {p}^2}\hfill \end{array}\right|>0. $$

Define $C={\scriptscriptstyle \frac{c-{c}_r}{\eta +\theta }}g\left({\scriptscriptstyle \frac{\omega +\theta }{\eta +\theta }}\right),x={\scriptscriptstyle \frac{\beta \left(\omega +\theta \right)}{\eta +\theta }},$ and $y={\scriptscriptstyle \frac{\gamma \left(\eta -\omega \right)}{\eta +\theta }}$. Since c > c _r , it is straightforward that C > 0. It can be obtained that H ₁ = 2β + Cx ² > 0, H ₂ = 4βγ + 2Cβy ² + 2Cγx ² > 0, and H ₃ = 4Cby ² β + 4Cbx ² γ + 8bβγ + 4Cb ² βγ > 0. Therefore, the expected profit function is concave on (θ + η)B ≤ θ + ω. Combing Proposition 1, it can be concluded that E[∏(r, f, p)] is concave.