Complementarity of willingness to pay and cost heterogeneity under vertical product differentiation: A welfare analysis

Abstract

Due to the fact that a consumer’s willingness to pay differs between segments, many unregulated industries are price constrained, although the specific costs of market segments also differ. If the product quality is endogenously chosen, we find that third-degree price discrimination increases welfare if a sufficiently pronounced complementarity between the willingness to pay and variable cost heterogeneity is given. This is due to the fact that the monopolist’s incentive for employing a pronounced price dispersion strategy is directly influenced by the consumers’ willingness to pay for the quality of a product. With endogenous product quality, the paper shows that the standard welfare result of third-degree price discrimination compared to uniform monopoly pricing (e.g. that total welfare and consumer surplus both fall if total output does not rise) can be only reversed given the complementarity is sufficiently pronounced.

Appendices

Appendix 1

Table 2 presents the fixed quality game’s results.

Table 2 Fixed quality game’s results summary

Appendix 2

Proof Proof of Proposition 1

Based on Table 2, we are able to verify the following results:

$$\begin{array}{@{}rcl@{}} {y_{1}^{D}}>{y_{1}^{U}}&\Leftrightarrow&\bar{q}<\frac{2 (\alpha -1)}{v_{2}-v_{1}}\:\text{given}\:\ v_{1}>v_{2} \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} {y_{1}^{D}}>{y_{1}^{U}}&\Leftrightarrow& v_{2}\geq v_{1} \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} {y_{2}^{D}}<{y_{2}^{U}}&\Leftrightarrow&\bar{q}<\frac{2 (\alpha -1)}{v_{2}-v_{1}}\:\text{given}\:\ v_{1}>v_{2} \end{array} $$

(14)

$$\begin{array}{@{}rcl@{}} {y_{2}^{D}}<{y_{2}^{U}}&\Leftrightarrow& v_{2}\geq v_{1} \end{array} $$

(15)

$$\begin{array}{@{}rcl@{}} y^{D}=y^{U}&& \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} p_{}^{U}>{p_{1}^{D}}&\Leftrightarrow&\bar{q}<\frac{2 (\alpha -1)}{v_{2}-v_{1}}\:\text{given}\:\ v_{1}>v_{2} \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} p_{}^{U}>{p_{1}^{D}}&\Leftrightarrow& v_{2}\geq v_{1} \end{array} $$

(18)

$$\begin{array}{@{}rcl@{}} {p_{2}^{D}}>p_{}^{U}&\Leftrightarrow&\bar{q}<\frac{2 (\alpha -1)}{v_{2}-v_{1}}\:\text{given}\:\ v_{1}>v_{2} \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} {p_{2}^{D}}>p_{}^{U}&\Leftrightarrow& v_{2}\geq v_{1} \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \pi^{D}>\pi^{U} \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} C{S_{1}^{D}}\leq(>)C{S_{1}^{U}}&\Leftrightarrow& \bar{q}\in\left[\frac{2 (\alpha -1)}{v_{2}-v_{1}},\frac{2 \alpha (3 \alpha +1)}{(\alpha +2) v_{1}+\alpha v_{2}}\right]\:\text{given}\:\ v_{1}> v_{2} \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} C{S_{1}^{D}}\geq(<)C{S_{1}^{U}}&\Leftrightarrow& \bar{q}\leq(>) \frac{2 \alpha (3 \alpha +1)}{(\alpha +2) v_{1}+\alpha v_{2}}\:\text{given}\:\ v_{2}\geq v_{1} \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} C{S_{2}^{D}}\geq (<)C{S_{2}^{U}}&\Leftrightarrow& \bar{q}\in\left[\frac{2 (\alpha -1)}{v_{2}-v_{1}},\frac{2 (\alpha +3)}{(2 \alpha +1) v_{2}+v_{1}}\right]\:\text{given}\:\ v_{1}> v_{2} \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} C{S_{2}^{D}}\geq (<)C{S_{2}^{U}}&\Leftrightarrow& \bar{q}\geq(<) \frac{2 (\alpha +3)}{(2 \alpha +1) v_{2}+v_{1}}\:\text{given}\:\ v_{2}\geq v_{1} \end{array} $$

(25)

$$\begin{array}{@{}rcl@{}} CS_{}^{D}\geq CS_{}^{U}&\Leftrightarrow& \bar{q}\in\left[\frac{2 (\alpha -1)}{v_{2}-v_{1}},\infty^{+}\right]\:\text{given}\:\ v_{1}> v_{2} \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} CS_{}^{D}\geq CS_{}^{U}&\Leftrightarrow& \bar{q}\in\left[\frac{6 (\alpha -1)}{v_{1}-v_{2}},\infty^{+}\right]\:\text{given}\:\ v_{2}> v_{1} \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} CS_{}^{D}<CS_{}^{U}&& \forall\: \bar{q}\:\text{given}\:\ v_{1}= v_{2} \end{array} $$

(28)

If we now use the set of expressions (12)-(28) together with Eqs. 2, 3and 4, we can derivein a straightforward way the relationships depicted with the expressions (5) and (7). □

Appendix 3

The equilibrium values for the endogenous quality game are recorded in Tables 3 and 4.

Table 3 Endogenous product quality game’s results summary. Note that $\Delta \equiv \sqrt {\alpha ^{2} \left (\left (4 \alpha ^{2}-3\right ) {v_{1}^{2}}+\left (4-3 \alpha ^{2}\right ) {v_{2}^{2}}+2 \alpha v_{2} v_{1}\right )}$

Table 4 Endogenous product quality game’s results summary. Note that ${\Gamma }\equiv \sqrt {\alpha \left ((\alpha -3) {v_{1}^{2}}+8 \alpha v_{2} v_{1}+(1-3 \alpha ) \alpha {v_{2}^{2}}\right )}$