Price Discrimination in Input Markets and Quality Differentiation

Abstract

This paper examines the welfare implications of input price discrimination in a vertically-related market, which is composed of a monopolistic upstream market and a duopolistic downstream market. The downstream duopolists produce quality-differentiated products at different marginal costs. We show that the equilibrium input prices are closely related to the downstream quality gap and cost difference. When the monopolist simply charges a unit wholesale price for its input product, discriminatory pricing could be socially desirable even though the aggregate output remains unchanged. Nevertheless, if a two-part tariff is feasible, then banning price discrimination could increase the aggregate output and social welfare.

Appendices

Appendix 1: Proof of Lemma 2

When solving the unit wholesale price under uniform pricing, we need to check the monopolist’s incentive for serving both downstream firms and the participation constraint for each downstream firm. If the monopolist serves only firm 1 or firm 2, then its profits are respectively:

$$\Omega_{1} = \frac{{(s - c)^{2} }}{4s},\quad \Omega_{2} = \frac{1}{4}.$$

(29)

Given a unit wholesale price, downstream firms’ profits gross of fixed fees are:

$$\pi_{1} (w) = \frac{{(2s - 2c - w - 1)^{2} }}{{(4s - 1)^{2} }},\quad \pi_{2} (w) = \frac{{(s + c + w - 2sw)^{2} }}{{(4s - 1)^{2} }}.$$

(30)

Note that Eqs. (29) and (30) are used to derive the relevant parameter ranges for the equilibrium uniform wholesale prices.

We shall derive the optimal uniform wholesale price in three regimes: First, if firm 2’s participation constraint is binding, then the monopolist solves:

$$\mathop {\text{Max}}\limits_{w} \, \Omega \text{ }(w) = \text{ }w\left( {q_{1} (w) + q_{2} (w)} \right) + 2\pi_{2} (w ).$$

(31)

From the first-order condition for profit maximization, the optimal wholesale price is:

$$w = \frac{{4s^{2} + 5c + 1 - 12sc - 3s}}{4(3s - 1)}$$

(32)

The second-order condition is satisfied as: $\partial^{2} \Omega /\partial w^{2} = - 4(3s - 1)/(4s - 1)^{2} < 0.$ The optimal fixed fee is $F = \pi_{2} (w)$.

By substituting (32) into (31) and rearranging, the monopolist profit is specified as follows:

$$\, \Omega \text{ } = \text{ }\frac{{(s - 3c)^{2} + 2s + 10c + 1}}{8(3s - 1)}.$$

(33)

To ensure that the monopolist will serve both downstream firms—i.e., the profit in (33) is larger than that in (29)—the cost disadvantage must be larger than the threshold level:

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} = \frac{{ - 3s(s + 1) + (2s + 1)\left[ {2s(3s - 1)} \right]^{1/2} }}{3s + 2}.$$

(34)

Substituting (32) into (30) and rearranging yield:

$$\pi_{1} = \frac{{s(5s - 3c - 3)^{2} }}{{16(3s - 1)^{2} }},{\kern 1pt} \quad \pi_{2} = \frac{{(2s^{2} + c + 1 - 5s - 6sc)^{2} }}{{16(3s - 1)^{2} }}.$$

(35)

From (35), to ensure that firm 1’s participation constraint is satisfied (i.e., $\pi_{1} - \pi_{2} > 0$), the cost difference must be smaller than the threshold level:

$$c^{\prime} = \frac{{12s^{3} - 47s^{2} + 20s - 1 + 8(3s - 1)s^{3/2} }}{{36s^{2} - 21s + 1}} .$$

(36)

A comparison of (34) and (36) shows that $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} < c^{\prime}$. Hence, if $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} < c < c^{\prime}$, then (32) is the optimal unit wholesale price under uniform pricing. Note that in this case the uniform wholesale price may turn to be negative under some parameter combinations. The equilibrium wholesale price is positive only if: $c < \tilde{c} = {{(4s^{2} - 3s + 1)} \mathord{\left/ {\vphantom {{(4s^{2} - 3s + 1)} {(12s - 5)}}} \right. \kern-0pt} {(12s - 5)}}$. However, if $s > \tilde{s} \approx 1.97$, then $\tilde{c} < c^{\prime}$ and (32) is negative for $\tilde{c} < c < c^{\prime}$. Hence, to ensure that the equilibrium wholesale price is always positive, we further assume that $s < 1.97$.

Second, if firm 1’s participation constraint is binding, then the monopolist’s objective function is: $\Omega \text{ }(w) = \text{ }w\left( {q_{1} (w) + q_{2} (w)} \right) + 2\pi_{ 1} (w ).$ Proceeding as previously, we can respectively solve for the unit wholesale price, the monopolist’s profit, and the downstream firms’ profits as follows:

$$w = \frac{{4s^{2} + 4sc - 3s + c + 1}}{8s(2s - 1)} > 0 ,$$

(37)

$$\Omega \text{ } = \frac{{(16s + 1)c^{2} - (32s^{2} - 18s - 2)c + 16s^{3} - 15s^{2} + 2s + 1}}{16s(2s - 1)}\text{ } ,$$

(38)

$$\pi_{1} = \frac{{(8s^{2} - 8sc - 7s + c + 1)^{2} }}{{64s(2s - 1)^{2} }},\quad {\text{and}}\quad \pi_{2} = \frac{{(s + c + 1)^{2} }}{{64s^{2} }}.$$

(39)

The fixed fee is $\pi_{ 1} (w )$. The second-order condition satisfies: ${{\partial^{2} \Omega } \mathord{\left/ {\vphantom {{\partial^{2} \Omega } {\partial w^{2} }}} \right. \kern-0pt} {\partial w^{2} }} = {{ - 8s(2s - 1)} \mathord{\left/ {\vphantom {{ - 8s(2s - 1)} {(4s - 1)^{2} }}} \right. \kern-0pt} {(4s - 1)^{2} }} < 0.$

From (29) and (38), the monopolist serves both downstream firms only if:

$$c < \bar{c} = \frac{{16s^{2} - 9s - 1 - 2\left[ {s(8s + 1)(2s - 1)} \right]^{1/2} }}{16s + 1}.$$

On the other hand, from (39), firm 2’s participation constraint is satisfied (i.e., $\pi_{2} - \pi_{1} > 0$) only if:

$$c > c^{\prime\prime} = \frac{{64s^{4} - 60s^{3} + 15s^{2} - 4s + 1 - 16(2s - 1)s^{5/2} }}{{64s^{3} - 20s^{2} + 5s - 1}}.$$

As $c^{\prime\prime} < \bar{c}$, we thus conclude that given a quality level, if $c^{\prime\prime} < c < \bar{c}$, then (37) is the optimal unit wholesale price.

Note that if $c^{\prime} < c < c^{\prime\prime}$, then neither (32) nor (37) meet all of the downstream participation constraints. Under such circumstances, the equilibrium unit wholesale price is determined by the profit-equalization condition: $\pi_{1} (w ) { = }\pi_{2} (w )$. By substituting the profits in (30) into the condition and solving for w, we obtain:

$$w = \frac{{c - (s - c - 1)s^{1/2} }}{s - 1} .$$

(40)

The optimal fixed fee is $F = \pi_{1} (w ) { = }\pi_{2} (w )$. The monopolist’s profit is: $\, \Omega \text{ } = {{\left[ {(s - c - 1)(\varPsi - \varUpsilon )} \right]} \mathord{\left/ {\vphantom {{\left[ {(s - c - 1)(\varPsi - \varUpsilon )} \right]} {\left[ {(s - 1)(4s - 1)} \right]}}} \right. \kern-0pt} {\left[ {(s - 1)(4s - 1)} \right]}}^{2}$, where $\varPsi = - 4s^{3} + (6 + 16c)s^{2} - (2 + 9c)s + c$, and $\varUpsilon = \left[ {4s^{3} - (7 + 12c)s^{2} + (4 + 5c)s - c - 1} \right]s^{1/2}$. Given $s < 1.97$, in the cost range the uniform wholesale price is positive, and the monopolist’s profit $\Omega \text{ }$ is larger than $\, \Omega_{1}$ and $\, \Omega_{2}$ in (29).

By summarizing the results in (32), (37), and (40), we obtain Lemma 2.

Appendix 2: Proof of Proposition 5

By substituting the corresponding marginal consumers from (23) and (28) into the social welfare function in (14), we can calculate the equilibrium welfare under either pricing policy as follows:

$$SW^{d} = \text{ }\frac{{3\left[ {(s - c)(s - c - 1) + c} \right]}}{8(s - 1)},\quad SW^{u} = \text{ }\frac{{\alpha c^{2} + \beta c + \gamma }}{{32(3s - 1)^{2} }},$$

where $\alpha = - 36s^{2} + 111s - 31$, $\beta = 24s^{3} - 154s^{2} + 144s - 30$, and $\gamma = - 4s^{4} + 87s^{3} - 37s^{2} + s + 1$.

The welfare difference ($\Delta SW = \text{ }SW^{d} - SW^{u}$) is then specified as follows:

$$\Delta SW = \frac{{\tilde{\alpha }c^{2} + \tilde{\beta }c + \tilde{\gamma }}}{{32(s - 1)(3s - 1)^{2} }}.$$

where $\tilde{\alpha } = 36s^{3} - 39s^{2} + 70s - 19$, $\tilde{\beta } = - 2(s - 1)(3s + 1)(4s^{2} + 9s - 3)$, and $\tilde{\gamma } = (s - 1)^{2} (4s^{3} + 25s^{2} - 10s + 1)$.

Letting $\Delta SW$ equal zero and solving for the threshold levels of cost difference, we obtain:

$$\tilde{c}^{ + , - } = \frac{{(3s + 1)(4s^{2} + 9s - 3) \pm 2(s - 1)(3s - 1)(56s^{2} - 23s + 7)^{1/2} }}{{\tilde{\alpha }}}.$$

This shows that the welfare difference is negative for $\tilde{c}^{ - } < c < \tilde{c}^{ + }$; otherwise, it is positive. If $s < 1.97$, then the cost ranking $\tilde{c}^{ - } < \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} < c^{\prime} < \tilde{c}^{ + }$ always holds true. Hence, if $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} < c < c^{\prime}$, then $\Delta SW < 0$, which completes the proof of Proposition 5.