Coordinating Marketing and Production with Asymmetric Costs: Theory and Estimation

Abstract

This paper proposes a theoretical framework for decision making when the firm needs to make marketing and production/order decisions simultaneously before demand uncertainty is resolved. We discuss the theoretical properties of the framework for single and multi-product firms; in addition, we show that the framework can be extended to allow for competitive reaction in a duopoly setting. We propose an empirical method to operationalize the model and compare the results to those from extant methods. The empirical results for both single and multi-product firms show that the proposed method outperforms decision making using standard econometric methods. In particular, depending on customer lifetime value (CLV) and other error costs and price elasticities, the loss in potential profits by using the standard regression-based methodology or quantile regression can be considerable.

Appendices

Appendix 1. Proofs of Propositions

1.1 Proof of Proposition 3 (Multi-Product Model)

First, we need the following Lemma:

Suppose Z = f(x₁, x₂) is a twice differentiable continuous function. Let $\frac{d{x}_2}{d{x}_1}{\vert}_{f_1=0}$ and $\frac{d{x}_2}{d{x}_1}{\vert}_{f_1=0}$, respectively, denote the loci of all (x₁, x₂) pairs satisfying the first-order conditions. Then, $\frac{d{x}_2}{d{x}_1}{\vert}_{f_1=0}-\frac{d{x}_2}{d{x}_1}{\vert}_{f_2=0}>0$ if f₁₂ > 0 and $\frac{d{x}_2}{d{x}_1}{\vert}_{f_1=0}-\frac{d{x}_2}{d{x}_1}{\vert}_{f_2=0}<0$ if f₁₂ < 0.

Proof:

The first-order conditions are $\frac{\partial Z}{\partial {x}_1}={f}_1\left({x}_1,{x}_2\right)=0$ and $\frac{\partial Z}{\partial {x}_2}={f}_2\left({x}_1,{x}_2\right)=0$. By taking total differentials, f₁₁dx₁ + f₁₂dx₂ ≡ 0. Hence, $\frac{d{x}_2}{d{x}_1}{\vert}_{f_1=0}=\frac{-{f}_{11}}{f_{12}}$ >0 if f₁₂ > 0 and < 0 if f₁₂ < 0.

Similarly, f₂₁dx₁ + f₂₂dx₂ ≡ 0. Hence, $\frac{d{x}_2}{d{x}_1}{\vert}_{f_2=0}=\frac{-{f}_{21}}{f_{22}}$ >0 if f₁₂ > 0 and < 0 if f₁₂ < 0.

Subtracting, we get, $\frac{d{x}_2}{d{x}_1}{\vert}_{f_1=0}-\frac{d{x}_2}{d{x}_1}{\vert}_{f_2=0}=\frac{-{f}_{11}}{f_{12}}+\frac{f_{21}}{f_{22}}=\frac{-\left({f}_{11}{f}_{22}-{f}_{12}^2\right)}{f_{12}{f}_{22}}$ > 0 if f₁₂ > 0 and < 0 if f₁₂ < 0. Q.E.D.

1.1.1 Comparison of Accounting and Economic Profit Models

We distinguish two cases based on the signs of $\frac{\partial^2A}{\partial {p}_1\partial {p}_2}$ and $\frac{\partial^2E\left(\pi \right)}{\partial {p}_1\partial {p}_2}$ where A denotes accounting profit and π denotes economic profit. In our model, $\frac{\partial^2A}{\partial {p}_1\partial {p}_2}=\frac{\partial^2E\left(\pi \right)}{\partial {p}_1\partial {p}_2}=\frac{\partial {f}_1}{\partial {p}_2}+\frac{\partial^2{f}_1}{\partial {p}_1\partial {p}_2}\left({p}_1-{c}_1\right)+\frac{\partial^2{f}_2}{\partial {p}_1\partial {p}_2}\left({p}_2-{c}_2\right)+\frac{\partial {f}_2}{\partial {p}_1}$.

Suppose f₁and f₂ are additively separable in p₁and p₂. Then, $\frac{\partial^2A}{\partial {p}_1\partial {p}_2}>0$ and $\frac{\partial^2E\left(\pi \right)}{\partial {p}_1\partial {p}_2}>0$ if the products are substitutes and $\frac{\partial^2A}{\partial {p}_1\partial {p}_2}<0$ if the products are complements.

1.1.2 Case 1: Products Are Substitutes (ϕ₁ > 0, ϕ₂ > 0)

In Fig. 1, let AB and CD respectively denote the loci of all (p₁, p₂) pairs satisfying $\frac{\partial A}{\partial {p}_1}=0$ and $\frac{\partial A}{\partial {p}_2}=0$. Then, both AB and CD are upward sloping and AB is steeper than CD (see Lemma and Proof).

Fig. 1

Multi-product model (substitutes)

Similarly, let A′B′ and C′D ′, respectively, denote the loci of all (p₁, p₂) pairs satisfying $\frac{\partial E\left(\pi \right)}{\partial {p}_1}=0$ and $\frac{\partial E\left(\pi \right)}{\partial {p}_2}=0$. Then, both A′B′ and C′D ′ are upward sloping and A′B′ is steeper than C′D ′ (see Lemma and Proof).

Let $\widehat{p_1}$ and $\widehat{p_2}$ denote any pair that satisfies $\frac{\partial A}{\partial {p}_1}=0$. At this point, $\frac{\partial E\left(\pi \right)}{\partial {p}_1}=\frac{\partial A}{\partial {p}_1}-E\left({u}_1>0\right)+E\left({u}_1<0\right)$. Hence, by concavity, holding p₂ constant, p₁ must be reduced so that $\frac{\partial E\left(\pi \right)}{\partial {p}_1}=0$. This implies that A′B′ is always to the left of AB.

Similarly, we can show that C′D ′ is always below CD.

Hence, the economic profit model always leads to lower prices for both products than the accounting profit model if the demand functions are additively separable and the products are substitutes (Proposition 3).

Case 2: Products are Complements (ϕ₁ < 0, ϕ₂ < 0)

Proceeding as in Case 1, we can show that both AB and CD are downward sloping and AB is steeper than CD (see Fig. 2). Similarly, both A′B′ and C′D ′ are downward sloping and A′B′ is steeper than C′D ′. Hence, we can only exclude the case that both products are priced higher in the economic profit model than in the accounting profit model.

Fig. 2

Multi-product model (complements)

1.2 Proof of Proposition 4 (Duopoly Model)

Let ${\overline{\uppi}}^{(1)}$:

: expected profit for Firm 1

${\overline{\uppi}}^{(2)}$ :

: expected profit for Firm 2

${\overline{\pi}}_1^{(1)}$ :

: first derivative for Firm 1 w.r.t own price p₁

${\overline{\uppi}}_2^{(2)}$ :

: first derivative for Firm 2 w.r.t own price p₂

${\overline{\uppi}}_{12}^{(1)}$ :

: cross partial for Firm 1 w.r.t prices p₁ and p₂

${\overline{\uppi}}_{21}^{(2)}$ :

: cross partial for Firm 2 w.r.t prices p₁ and p₂

${\overline{\uppi}}_{11}^{(1)}$ :

: second derivative for Firm 1 w.r.t price p₁

${\overline{\pi}}_{22}^{(2)}$ :

: second derivative for Firm 2 w.r.t price p₂

$$ {\displaystyle \begin{array}{c}{\overline{\pi}}^{(1)}=\left({p}_1-{c}_1\right){f}_1\left({p}_1\right)+{c}_1{\delta}_1E\left({u}_1<0\right)+{p}_1E\left({u}_1<0\right)-{\theta}_1\left({p}_1-{c}_1\right)E\left({u}_1>0\right)\\ {}{\overline{\pi}}^{(2)}=\left({p}_2-{c}_2\right){f}_2\left({p}_2\right)+{c}_2{\delta}_2E\left({u}_2<0\right)+{p}_2E\left({u}_2<0\right)-{\theta}_2\left({p}_2-{c}_2\right)E\left({u}_2>0\right)\end{array}} $$

Suppose the demand function is additively separable in its arguments. Thus,

$$ {\displaystyle \begin{array}{c}{f}_1\left({p}_1\right)={\alpha}_1-{\upbeta}_1{p}_1+{\phi}_1{p}_2\\ {}{f}_2\left({p}_2\right)={\alpha}_2-{\upbeta}_2{p}_2+{\phi}_2{p}_1\end{array}} $$

Then, ${\overline{\uppi}}_{12}^{(1)}={\phi}_1>0$ and ${\overline{\uppi}}_{21}^{(2)}={\phi}_2>0$, where ϕ₁ ≠ ϕ₂ in general.

At ${\overline{\uppi}}_1^{(1)}=0$ the total differential is ${\overline{\uppi}}_{11}^{(1)}d{p}_1+{\overline{\uppi}}_{12}^{(1)}{p}_2\equiv 0$.

Thus, ${\left.\frac{d{p}_2}{d{p}_1}\right|}_{{\overline{\uppi}}_1^{(1)}}=\frac{-{\overline{\uppi}}_{11}^{(1)}}{{\overline{\uppi}}_{12}^{(1)}}>0$ and similarly, ${\left.\frac{d{p}_2}{d{p}_1}\right|}_{{\overline{\uppi}}_2^{(2)}}=\frac{-{\overline{\uppi}}_{21}^{(2)}}{{\overline{\uppi}}_{22}^{(2)}}>0$.

Subtracting, ${\left.\frac{d{p}_2}{d{p}_1}\right|}_{{\overline{\uppi}}_1^{(1)}}-{\left.\frac{d{p}_2}{d{p}_1}\right|}_{{\overline{\uppi}}_2^{(2)}}=\frac{-{\overline{\uppi}}_{11}^{(1)}}{{\overline{\uppi}}_{12}^{(1)}}+\frac{{\overline{\uppi}}_{21}^{(2)}}{{\overline{\uppi}}_{22}^{(2)}}=-\frac{{\overline{\uppi}}_{11}^{(1)}{\overline{\uppi}}_{22}^{(2)}-{\overline{\uppi}}_{12}^{(1)}{\overline{\uppi}}_{21}^{(2)}}{{\overline{\uppi}}_{12}^{(1)}{\overline{\uppi}}_{22}^{(2)}}>0$ by the second-order conditions.

Proposition 4 follows immediately following the same method as in the multi-product case for substitutes. Hence, the economic profit model leads to lower prices than the accounting profit model if demand is additively separable.

Appendix 2. Optimal Price Derivation

1.1 Optimal Price for Proposed Model

Suppose based on our proposed method, the quantity estimate for a price point p is $\widehat{\mu}=\alpha +\upbeta p$, and the difference between realized demand y and $\widehat{\mu}$ is $u=y-\widehat{\mu}$. Furthermore, assume that the demand function is linear: y = α₀ + β₀p + ε, and ε~N(0, σ²). Then, the demand function can be estimated by ordinary least squares. At each point p, y follows a normal distribution with density function f(y) and cumulative distribution function F(y).

At price point p, the expected profit is as follows:

$$ {\displaystyle \begin{array}{ll}E\left(\pi \right)={\int}_{u<0}\pi du+{\int}_{u\ge 0}\pi du& ={\int}_{u<0}\left[\left(p-c\right)\left(\alpha +\beta p+u\right)+ c\delta u\right] du+{\int}_{u\ge 0}\left[\left(p-c\right)\left(\alpha +\beta p\right)-\varphi \left(p-c\right)u\right] du\\ {}& =\left(p-c\right)\left(\alpha +\beta p\right)+\left(p+ c\delta \right){E}_{u<0}(u)-\varphi \left(p-c\right){E}_{u\ge 0}(u)\\ {}& =\left(p-c\right)\left(\alpha +\beta p\right)+p{E}_{u<0}(u)-c{E}_{u<0}(u)+ c\delta {E}_{u<0}(u)-\varphi p{E}_{u\ge 0}(u)+\varphi c{E}_{u\ge 0}(u)\end{array}} $$

Taking the derivative with respect to p, we have

$$ {\displaystyle \begin{array}{l}\frac{\partial \left(p-c\right)\left(\alpha +\upbeta p\right)}{\partial p}=\alpha +2\upbeta p-c\upbeta \\ {}\frac{\partial p{E}_{u<0}(u)}{\partial p}=\frac{\partial p{\int}_{-\infty}^{\widehat{\mu}}\left(y-\alpha -\upbeta p\right)f(y) dy}{\partial p}\\ {}\frac{\partial p{\int}_{-\infty}^{\widehat{\mu}}\left(\alpha \right)f(y) dy}{\partial p}=\left(\alpha \right)F\left(\widehat{\mu}\right)+p\left(\left(\alpha \right)f\left(\widehat{\mu}\right)\upbeta -\frac{\upbeta_0\left(\alpha \right)}{\sqrt{2{\uppi \upsigma}^2}}\exp \left(\frac{-{\left(\widehat{\mu}-{\alpha}_0-{\upbeta}_0p\right)}^2}{2{\upsigma}^2}\right)\right)\\ {}\frac{\partial p{\int}_{-\infty}^{\widehat{\mu}}\left(\left(\upbeta \right)p\right)f(y) dy}{\partial p}=\left(\upbeta \right) pF\left(\widehat{\mu}\right)+p\left( pF\left(\widehat{\mu}\right){\upbeta}^2+\left(\upbeta \right)F\left(\widehat{\mu}\right)-\frac{\upbeta_0\upbeta p}{\sqrt{2{\uppi \upsigma}^2}}\exp \left(\frac{-{\left(\widehat{\mu}-{\alpha}_0-{\upbeta}_0p\right)}^2}{2{\upsigma}^2}\right)\right)\\ {}\frac{\partial p{\int}_{-\infty}^{\widehat{\mu}}(y)f(y) dy}{\partial p}=\left({\alpha}_0+{\upbeta}_0p\right)F\left(\widehat{\mu}\right)-\frac{\sigma }{\sqrt{2\uppi}}\exp \left(\frac{-{\left(\widehat{\mu}-{\alpha}_0-{\upbeta}_0p\right)}^2}{2{\upsigma}^2}\right)+p\left(\left(\widehat{\mu}\right)f\left(\widehat{\mu}\right)\upbeta -\left(\widehat{\mu}\right)f\left(\widehat{\mu}\right){\upbeta}_0+F\left(\widehat{\mu}\right){\upbeta}_0\right)\\ {}\frac{\partial p{E}_{u\ge 0}(u)}{\partial p}=\frac{\partial p{\int}_{\widehat{\mu}}^{+\infty}\left(y-\alpha -\upbeta p\right)f(y) dy}{\partial p}\\ {}\frac{\partial p{\int}_{\widehat{\mu}}^{+\infty}\left(\alpha \right)f(y) dy}{\partial p}=\left(\alpha \right)\left(1-F\left(\widehat{\mu}\right)\right)+p\left(-\left(\alpha \right)f\left(\widehat{\mu}\right)\upbeta +\frac{\upbeta_0\left(\alpha \right)}{\sqrt{2\uppi {\sigma}^2}}\exp \left(\frac{-{\left(\widehat{\mu}-{\alpha}_0-{\upbeta}_0p\right)}^2}{2{\upsigma}^2}\right)\right)\\ {}\frac{\partial p{\int}_{\widehat{\mu}}^{+\infty}\left(\left(\upbeta \right)p\right)f(y) dy}{\partial p}=\left(\upbeta \right)p\left(1-F\left(\widehat{\mu}\right)\right)+p\left(p\left(1-F\left(\widehat{\mu}\right)\right){\upbeta}^2+\left(\upbeta \right)\left(1-F\left(\widehat{\mu}\right)\right)+\frac{\upbeta_0\upbeta p}{\sqrt{2{\uppi \upsigma}^2}}\exp \left(\frac{-{\left(\widehat{\mu}-{\alpha}_0-{\upbeta}_0p\right)}^2}{2{\sigma}^2}\right)\right)\\ {}\frac{\partial p{\int}_{\widehat{\mu}}^{+\infty }(y)f(y) dy}{\partial p}=\left({\alpha}_0+{\upbeta}_0p\right)\left(1-F\left(\widehat{\mu}\right)\right)+\frac{\upsigma}{\sqrt{2\uppi}}\exp \left(\frac{-{\left(\widehat{\mu}-{\alpha}_0-{\upbeta}_0p\right)}^2}{2{\upsigma}^2}\right)+p\left(-\left(\widehat{\mu}\right)f\left(\widehat{\mu}\right)\upbeta +\left(\widehat{\mu}\right)f\left(\widehat{\mu}\right){\upbeta}_0+\left(1-F\left(\widehat{\mu}\right)\right){\upbeta}_0\right)\end{array}} $$

The optimal price p* is then the value that makes the derivative equal to 0.

1.2 Optimal Price for OLS and Quantile Regression

For both OLS and quantile regression, E_u < 0(u) and E_u ≥ 0(u) are, by definition, independent of p. Therefore, the derivation of the optimal price is much simpler than for the proposed method.

At the optimal price point p, the expected profit is as follows:

$$ E\left(\uppi \right)={\int}_{u<0}\uppi du+{\int}_{u\ge 0}\uppi du={\int}_{u<0}\left[\left(p-c\right)\left(\alpha -\upbeta p+u\right)+c\updelta u\right] du+{\int}_{u\ge 0}\left[\left(p-c\right)\left(\alpha -\upbeta p\right)-\upvarphi \left(p-c\right)u\right] du=\left(p-c\right)\left(\alpha -\beta p\right)+\left(p+c\updelta \right){E}_{u<0}(u)-\upvarphi \left(p-c\right){E}_{u\ge 0}(u) $$

In order to maximize expected profit, we take the derivative of the expected profit function and set it to 0:

$$ \frac{\partial E\left(\uppi \right)}{\partial p}=\alpha -2\upbeta p+c\upbeta +{E}_{u<0}(u)-\upvarphi {E}_{u\ge 0}(u)=0 $$

Then, the optimal price${p}^{\ast }=\frac{\alpha }{2\upbeta}+\frac{c}{2}+\frac{1}{2\upbeta}{E}_{u<0}(u)-\frac{\upvarphi}{2\upbeta}{E}_{u\ge 0}(u)$

For both OLS and quantile regression:

$$ {E}_{u<0}={\int}_{-\infty}^{\upmu}\left(y-\upmu \right)f(y) dy=\frac{-\upsigma}{\sqrt{2\uppi}}\exp \left(\frac{-{\upmu}^2}{2{\upsigma}^2}\right)-\upmu F\left(\upmu \right) $$

Appendix 3. Algorithm for Estimation

We used a recently proposed Normalized Adaptive Gradient Descent (NAG) algorithm for model training[16]. The algorithm scales gradients automatically in each step in order to reach the minimum fast while keeping the estimates stable. The detailed model training steps are as follows:

First, set values for the learning rate α. We set α = 2 for our model training, as suggested by the authors of NAG. Set the initial parameter value β_i = 0. We also need to set the initial values for the tuning parameters: s_i = 0, G_i = 0, N = 0.

For each step t, we randomly pick a sample (x, y)

$$ {\displaystyle \begin{array}{l}\mathrm{for}\ \mathrm{each}\ \mathrm{independent}\ \mathrm{variable}\ i\ \left(\mathrm{including}\ \mathrm{the}\ \mathrm{intercept}\right),\mathrm{if}\mid {x}_i\mid >{s}_i\\ {}1.\ {\upbeta}_i\leftarrow \frac{\upbeta_i{s}_i}{\mid {x}_i\mid}\\ {}2.\ {s}_i\leftarrow \mid {x}_i\mid \\ {}\mathrm{end}\\ {}\widehat{y}=\sum \limits_i{\upbeta}_i{x}_i\\ {}N\leftarrow N+\sum \limits_i\frac{x_i^2}{s_i^2}\\ {}\mathrm{for}\ \mathrm{each}\ i,\\ {}1.\ {G}_i\leftarrow {G}_i+{\left(\frac{\partial L\left(\widehat{y},y\right)}{\partial {\upbeta}_i}\right)}^2\\ {}2.\ {\upbeta}_i\leftarrow {\upbeta}_i-\alpha \sqrt{\frac{t}{N}}\frac{1}{s_i\sqrt{G_i}}\frac{\partial L\left(\widehat{y},y\right)}{\partial {\upbeta}_i}\\ {}\mathrm{end}\end{array}} $$

Appendix 4. Additional Simulation Results

1.1 One-Product Case

Table 4 shows the prices, quantities, and expected economic profits for different estimators for different cost-price ratios (i.e., 0.3, 0.5, and 0.7) and CLV parameters (i.e., 1 and 2). These results support the conclusion in the text that, in general, the proposed model leads to higher prices and quantities than OLS and quantile regression. Furthermore, regardless of parameter values, the proposed model leads to higher expected economic profits than OLS or quantile regression.

Table 4 Optimal prices, quantities, and expected economic profits