Input Price Discrimination and Allocation Efficiency

Abstract

This research examines the effects of input price discrimination on allocation efficiency and social welfare. Instead of assuming constant marginal costs, we allow downstream firms to produce under increasing marginal costs. When downstream firms operate in separate markets, even though total output remains unchanged, consumer surplus and social welfare could be greater under discriminatory pricing than under uniform pricing. Moreover, the social desirability of input price discrimination can still hold true when downstream firms compete either in Cournot or Bertrand fashion.

Appendix

1.1 Proof of Lemma 1

From (7) and (8), comparing firms’ outputs under either pricing regime yields:

$$q_{1}^{d} - q_{2}^{d} = \frac{{(1 - \alpha_{1} )(\beta_{2} - \beta_{1} ) - (2 + \beta_{1} )\Delta \alpha }}{{2(2 + \beta_{1} )(2 + \beta_{2} )}},$$

$$q_{1}^{u} - q_{2}^{u} = \frac{{(1 - \alpha_{1} )(4 + \beta_{1} + \beta_{2} )(\beta_{2} - \beta_{1} ) - (2 + \beta_{1} )(8 + \beta_{1} + 3\beta_{2} )\Delta \alpha }}{{2(2 + \beta_{1} )(2 + \beta_{2} )(4 + \beta_{1} + \beta_{2} )}},$$

where $\Delta \alpha = \alpha_{1} - \alpha_{2} > 0$. If $\beta_{1} \ge \beta_{2}$, then $q_{1}^{d} - q_{2}^{d} < 0$ and $q_{1}^{u} - q_{2}^{u} < 0$. If $\beta_{1} < \beta_{2}$, then the two output differences could be positive if $\Delta \alpha$ is sufficiently small. □

1.2 Proof of Proposition 1

By substituting equilibrium outputs (7) and (8) into (10) and (12) under either pricing regime, we specify the comparisons of consumer surplus and social welfare between discriminatory pricing and uniform pricing as:

$$\Delta CS = \frac{\Delta q}{2}\frac{{(1 - \alpha_{1} )(4 + \beta_{1} + \beta_{2} )( \, \beta_{2} - \beta_{1} ) - (2 + \beta_{1} )(6 + \beta_{1} + 2\beta_{2} )\Delta \alpha }}{{(2 + \beta_{1} )(2 + \beta_{2} )(4 + \beta_{1} + \beta_{2} )}},$$

$$\Delta SW = \frac{\Delta q}{2}\frac{{2(1 - \alpha_{1} )(4 + \beta_{1} + \beta_{2} )( \, \beta_{2} - \beta_{1} ) - (2 + \beta_{1} )(20 + 4\beta_{1} + 10\beta_{2} + \beta_{1} \beta_{2} + \beta_{2}^{2} )\Delta \alpha }}{{2(2 + \beta_{1} )(2 + \beta_{2} )(4 + \beta_{1} + \beta_{2} )}}.$$

If $\beta_{1} \ge \beta_{2}$, then the signs of $\Delta CS$ and $\Delta SW$ are negative. If $\beta_{1} < \beta_{2}$, and $\Delta \alpha$ is sufficiently small, then the two signs are positive.□

1.3 Proof of Proposition 2

The two terms $\Delta \alpha + \beta_{1} q_{1}^{d} - \beta_{2} q_{2}^{d}$ and $\Delta \alpha + \beta_{1} q_{1}^{u} - \beta_{2} q_{2}^{u}$ can be rearranged, respectively, as:

$$\frac{{(2 + \beta_{1} )(3 + \beta_{2} )\Delta \alpha + (1 - \alpha_{1} )(\beta_{1} - \beta_{2} )}}{{2(3 + 2\beta_{1} + 2\beta_{2} + \beta_{1} \beta_{2} )}},$$

$$\frac{{(12 + \beta_{1}^{2} + 9\beta_{1} + 7\beta_{2} + 3\beta_{1} \beta_{2} )\Delta \alpha + (2 + \beta_{1} + \beta_{2} )(1 - \alpha_{1} )(\beta_{1} - \beta_{2} )}}{{2(2 + \beta_{1} + \beta_{2} )(3 + 2\beta_{1} + 2\beta_{2} + \beta_{1} \beta_{2} )}},$$

which are positive if $\beta_{1} \ge \beta_{2}$; they are negative if $\beta_{1} < \beta_{2}$ and $\Delta \alpha$ is sufficiently small.

By substituting the above two terms into (18) and rearranging it, we derive the cost difference as:

$$\Delta TC = \frac{\Delta q}{2}\frac{\phi \Delta \alpha + \theta }{\psi },$$

where $\psi = 2(2 + \beta_{1} + \beta_{2} )(3 + 2\beta_{1} + 2\beta_{2} + \beta_{1} \beta_{2} ) > 0$, $\phi = 24 + 21\beta_{1} + 17\beta_{2} + 4\beta_{1}^{2} + 2\beta_{2}^{2} + 10\beta_{1} \beta_{2} + \beta_{1} \beta_{2}^{2} + \beta_{1}^{2} \beta_{2} > 0$, and $\theta = 2(2 + \beta_{1} + \beta_{1} )(1 - \alpha_{1} )(\beta_{1} - \beta_{2} )$. Note that $\theta < 0$ if $\beta_{1} < \beta_{2}$ and otherwise $\theta \ge 0$. Hence, if $\beta_{1} < \beta_{2}$, then the sign of $\Delta TC$ is negative when $\Delta \alpha$ is sufficiently small. Under such circumstances, discriminatory pricing is cost-reducing and thus more socially desirable than uniform pricing.□

1.4 Results under Bertrand competition

From the first-order conditions for profit maximization, the reaction functions under Bertrand competition are $p_{i} (p_{j} ) = \frac{{(1 - \gamma )[1 - \gamma^{2} + \beta_{i} + (1 + \gamma )(\alpha_{i} + w_{i} )]}}{{2 - 2\gamma^{2} + \beta_{i} }} + \frac{{\gamma (1 - \gamma^{2} + \beta_{i} )}}{{2 - 2\gamma^{2} + \beta_{i} }}p_{j} ,i \ne j,i,j = 1,2$, whereby the equilibrium prices are derivable as follows:

$$p_{i} = 1 - \frac{1}{A}\left[ {(2 - 2\gamma^{2} + \beta_{j} )(1 - \alpha_{i} - w_{i} ) + \gamma (1 - \gamma^{2} + \beta_{i} )(1 - \alpha_{j} - w_{j} )} \right],\,\,\,\,i \ne j,\,\,\,i,j = 1,2,$$

where $A = (4 - \gamma^{2} )(1 - \gamma^{2} ) + (2 - \gamma^{2} )(\beta_{1} + \beta_{2} ) + \beta_{1} \beta_{2} > 0$.

The derived demands on input are then given as:

$$q_{i} = \frac{1}{A}\left[ {(2 - \gamma^{2} + \beta_{j} )(1 - \alpha_{i} - w_{i} ) - \gamma (1 - \alpha_{j} - w_{j} )} \right],\,\,\,i \ne j,\,\,\,i,j = 1,2.$$

From these demands, the input monopolist chooses input prices for profit maximization. The equilibrium input prices under corresponding pricing regimes are presented in (19), whereby the equilibrium outputs under either pricing regime are then derivable as:

$$q_{i}^{d} = \frac{1}{2A}[(2 - \gamma^{2} + \beta_{j} )(1 - \alpha_{i} ) - \gamma (1 - \alpha_{j} )],\,\,\,q_{i}^{u} = \frac{{\theta^{\prime}(1 - \alpha_{i} ) - \theta^{\prime\prime}(1 - \alpha_{j} )}}{{2A[2(2 + \gamma )(1 - \gamma ) + \beta_{1} + \beta_{2} ]}},$$

where $\theta^{\prime} = (2 - \gamma^{2} + \beta_{j} )(6 - 3\gamma^{2} - 2\gamma + 2\beta_{i} + \beta_{j} ) - \gamma^{2} > 0$, and $\theta^{\prime\prime} = (2 + \gamma - \gamma^{2} + \beta_{i} )(2 + \gamma - \gamma^{2} + \beta_{j} ) - 4\gamma^{2} > 0$.

To simplify notations, we assume hereafter that $\alpha_{2} = 0$ and $\beta_{2} = 1$. Substituting the above equilibrium outputs into the cost difference defined in (18) and rearranging yields:

$$\Delta TC = \frac{{[\Psi \alpha_{1} - \Gamma (1 - \beta_{1} )]\alpha_{1} }}{{8(6 - 6\gamma^{2} + \gamma^{4} + 3\beta_{1} - \beta_{1} \gamma^{2} )(5 - 2\gamma - 2\gamma^{2} + \beta_{1} )^{2} }},$$

$$\Psi = [(3 - \gamma^{2} )\beta_{1}^{2} + (45 - 14\gamma - 39\gamma^{2} + 4\gamma^{3} + 7\gamma^{4} )\beta_{1} + 144 - 58\gamma - 158\gamma^{2} + 52\gamma^{3} + 67\gamma^{4} - 8\gamma^{5} - 8\gamma^{6} ] > 0,$$

$$\Gamma = 2(2 - \gamma - \gamma^{2} )(5 - 2\gamma^{2} - 2\gamma + \beta_{1} ) > 0.$$

The sign of $\Delta TC$ depends on the sign of the term $[\Psi \alpha_{1} - \Gamma (1 - \beta_{1} )]$, which is always positive if $\beta_{1} \ge 1$. Nevertheless, it is ambiguous if $\beta_{1} < 1$. In this case, when $\alpha_{1}$ approximates to 0, the term $[\Psi \alpha_{1} - \Gamma (1 - \beta_{1} )]$ becomes negative. Moreover, the value of the term is increasing with $\alpha_{1}$. As a result, given $\beta_{1} < 1$, if $\alpha_{1}$ is sufficiently small, then the term is negative, and thus discriminatory pricing leads to a lower production cost relative to uniform pricing.

With differentiated products, social welfare is defined as $W \equiv U - TC_{1} - TC_{2}$, whereby the welfare difference, $\Delta W \equiv W^{d} - W^{u}$, is then derivable as:

$$\Delta W = \frac{{[ - \Psi^{\prime}\alpha_{1} + \Gamma^{\prime}(1 - \beta_{1} )]\alpha_{1} }}{{8(6 - 6\gamma^{2} + \gamma^{4} + 3\beta_{1} - \beta_{1} \gamma^{2} )(5 - 2\gamma - 2\gamma^{2} + \beta_{1} )^{2} }},$$

$$\Psi^{\prime} = [(3 - \gamma^{2} )\beta_{1}^{2} + (33 - 4\gamma - 33\gamma^{2} + 7\gamma^{4} )\beta_{1} + 72 - 14\gamma - 122\gamma^{2} + 14\gamma^{3} + 61\gamma^{4} - 2\gamma^{5} - 8\gamma^{6} ] > 0,$$

$$\Gamma^{\prime} = 2(1 - \gamma^{2} )(5 - 2\gamma - 2\gamma^{2} + \beta_{1} ) > 0.$$

The sign of $\Delta W$ depends on the sign of the term $[ - \Psi^{\prime}\alpha_{1} + \Gamma (1 - \beta_{1} )]$, which is negative if $\beta_{1} \ge 1$, but is ambiguous if $\beta_{1} < 1$. If $\beta_{1} < 1$, then when $\alpha_{1}$ is close to zero, the sign of $[ - \Psi^{\prime}\alpha_{1} + \Gamma^{\prime}(1 - \beta_{1} )]$ becomes positive. Moreover, the value of $[ - \Psi^{\prime}\alpha_{1} + \Gamma^{\prime}(1 - \beta_{1} )]$ is decreasing with $\alpha_{1}$. As a result, if $\beta_{1} < 1$ and $\alpha_{1}$ is not too large, then the term could be positive, whereby discriminatory pricing is more socially desirable than is uniform pricing under Bertrand competition.