On the Impact of Input Prices on an Entrant’s Profit Under Multi-Product Competition

Abstract

We study the impact of input prices on an entrant’s profit when firms are engaged in multi-product competition. We consider a setting with both horizontal and vertical differentiation, in which a vertically integrated firm controls the input that is required for the supply of the high-quality product. We establish the conditions under which the entrant is better off from an increase in the wholesale price of this critical input. This possibility contrasts with the existing literature that, under linear demands, finds a non-positive relationship between the input price and the single-product entrant’s profit.

Appendices

Appendix 1

Throughout the paper we assume that the consumer who is indifferent between choosing products $\left( {ij} \right)$ and $\left( {ml} \right)$, with either $i \ne m$ and $j = l$ or $i = m$ and $j \ne l$, where $i,m = I,E$ and $j,l = L,H$ is located in [0,1], i.e., $0 \le \hat{x}_{ij/ml} \le 1$. In particular:

$$0 \le \hat{x}_{IH/EH} \le 1 \Rightarrow R1 \equiv - \frac{(4 - \rho )}{(1 - \rho )} \le w \le \frac{(4 - \rho )}{(1 - \rho )} \equiv R2$$

(18)

$$0 \le \hat{x}_{IL/EL} \le 1 \Rightarrow R1 \le w \le R2$$

(19)

$$\begin{aligned} 0 &\le \hat{x}_{IH/IL} \le 1 \Rightarrow \hfill \\ R3 &\equiv \frac{{(4 - \rho )\left[ {\delta (2 - \rho ) - (4 - 3\rho )} \right]}}{3\rho (2 - \rho )} \le w \le \frac{{(4 - \rho )\left[ {\delta (2 - \rho ) + (4 - 3\rho )} \right]}}{3\rho (2 - \rho )} \equiv R4 \hfill \\ \end{aligned}$$

(20)

$$\begin{aligned} 0 & \le \hat{x}_{EH/EL} \le 1 \Rightarrow \\ R5 & \equiv \frac{{(4 - \rho )\left[ {\delta (2 - \rho ) - (4 - 3\rho )} \right]}}{8 + (3\rho - 8)\rho } \le w \le \frac{{(4 - \rho )\left[ {\delta (2 - \rho ) + (4 - 3\rho )} \right]}}{8 + (3\rho - 8)\rho } \equiv R6. \\ \end{aligned}$$

(21)

Therefore, the value of $w$ should be such that all inequalities (18)–(21) hold, or $max\{ R1,R3,R5,0\} \le w \le min\{ R2,R4,R6\}$. It is straightforward to show that R5 > R1 and R6 < R4. Moreover, R5 > R3 if and only if $\delta < \underline{\delta } = (4 - 3\rho )/(2 - \rho )$ and R6 < R2 if and only if $\delta < \overline{\delta } = (4 - \rho )/\left[ {(2 - \rho )(1 - \rho )} \right]$, with $\overline{\delta } > \underline{\delta }$ for all $\rho \in (0,1]$.

Hence, the condition that ensures that (18)–(21) hold is $max\{ R3,R5\} \le w \le min\{ R2,R6\}$ and can be subdivided into three cases:

• 1. When $\delta > \overline{\delta }$, the necessary and sufficient condition is $R3 \le w \le R2$.

• 2. When $\underline{\delta } < \delta < \overline{\delta }$, the necessary and sufficient condition is $R3 \le w \le R6$.

• 3. When $\delta < \underline{\delta }$, the necessary and sufficient condition is $R5 \le w \le R6$.

As $R2 > R3 \Leftrightarrow \delta < \overline{\delta }$, $R6 > R3 \Leftrightarrow \delta < \overline{\delta }$ and $R6 > R5$ for all parameter values, this implies that: (1) The admissible values of $\delta$ are in the interval [$0,\overline{\delta }$]; and (2) The equilibrium retail prices presented in Lemma 1, as well as the derived demand function of each product hold when $w \in [\underline{w} ,\bar{w}]$, where $\underline{w} = max\{ R3,R5\}$ and $\bar{w} = R6$.

We now show that the condition for the entrant’s profit to increase in the fiber access price is not inconsistent with both the upper and lower limits in Assumption 1(ii). Indeed, with respect to the upper limit, it is shown that $\hat{w} < \bar{w}$ since

$$\bar{w} - \hat{w} = \frac{(4 - \rho )(4 - 3\rho )(4 + \delta \rho )}{{4\left[ {8 + (3\rho - 8)\rho } \right]}} > 0.$$

(22)

As for the lower limit, when $\delta < \underline{\delta }$, $\hat{w}$ is higher than R5 if and only if $\delta < 4/\rho$, which always holds when $\delta < \underline{\delta }$. When $\delta > \underline{\delta }$, $\hat{w}$ is higher than R3 if and only if $\delta < \widetilde{\delta } = 4/(2 - \rho )$. Note that $\underline{\delta } < \widetilde{\delta } < \overline{\delta }$ for $\rho \ne 0$. Therefore, if $\delta \in [0,\widetilde{\delta }]$, then $\underline{w} \le \hat{w} < \bar{w}$. If $\delta \in [\widetilde{\delta },\overline{\delta } ]$, then $\hat{w} < \underline{w}$ which means that the entrant’s profit is positively correlated with the fiber access price, for all admissible $w$. These results prove Proposition 1.

Appendix 2

The functions $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$, and $f_{5}$ are as given below:

$$\begin{aligned} f_{1} & = \frac{{(1 - \rho )(2 - \rho )(9\rho^{2} - 32\rho + 32)}}{{(4 - \rho )^{2} (4 - 3\rho )^{2} }},\quad f_{2} = \frac{{(1 - \rho )(7\rho^{2} - 20\rho + 16)}}{{(4 - \rho )(4 - 3\rho )^{2} }} \\ f_{3} & = \frac{{(2 - \rho )(1 - \rho )^{2} }}{{(4 - 3\rho )^{2} }},\quad f_{4} = \frac{{16(1 - \rho )^{2} (2 - \rho )}}{{(4 - \rho )^{2} (4 - 3\rho )^{2} }},\quad f_{5} = \frac{{8(1 - \rho )^{2} (2 - \rho )}}{{(4 - \rho )(4 - 3\rho )^{2} }}. \\ \end{aligned}$$

Appendix 3

In this Appendix, we explain why the total demand for a given firm is constant in equilibrium. As the market is assumed to be covered, it is sufficient to show that the total number of consumers served by the two firms is equal in equilibrium.

When setting the price for a given product, each firm faces the usual value versus volume trade-off: Increasing the price of one product increases the profit from each unit of that product that the firm sells, but it also affects all products’ sales volume. In equilibrium, these two effects compensate each other. Hence, from the first-order conditions, the quantity that is sold of a given product is equal to the aggregate net loss in all products’ sales due to a marginal increase in the price of that product.

For the incumbent, this net loss in sales is due to the decrease in the sales of the product with a higher price plus the increase in (retail or wholesale) sales of other products. For the entrant, the net loss in sales results from the decrease in the product’s sales, but is compensated by a reduction in wholesale costs. The total output that is sold by each multi-product firm then corresponds to adding the net losses in the value of sales due to a unit increase in the prices of all its products.

If there were no input sales involved, firms would be symmetric, and each firm would have the same aggregate quantity. In fact, firm asymmetry in our model results solely from the fact that one firm sells an input to its competitor: The incumbent has wholesale revenues, whereas the entrant incurs wholesale costs. However, as is explained below, the effects of increasing the prices of their products on wholesale revenue or cost are the same.

When the incumbent marginally raises its prices, there is an increase in the wholesale revenue of the high-quality product that is given by $w\left( {\frac{{\partial q_{EH} }}{{\partial p_{IH} }} + \frac{{\partial q_{EH} }}{{\partial p_{IL} }}} \right)$. As for the entrant, when it marginally raises retail prices, it faces a similar reduction in its input costs (instead of an increase in wholesale revenue). The increase in the high-quality product’s retail price reduces the input costs by $\left( {w\frac{{\partial q_{EH} }}{{\partial p_{EH} }}} \right)$, and the increase in the low-quality product’s retail price increases the input costs by $\left( {w\frac{{\partial q_{EH} }}{{\partial p_{EL} }}} \right)$ since some “firm-oriented” consumers switch from high-quality to low-quality products and vice versa.

Thus, the impact of an increase in the wholesale revenue due to an increase in the incumbent’s retail prices is given by $w\left( {\frac{{\partial q_{EH} }}{{\partial p_{IH} }} + \frac{{\partial q_{EH} }}{{\partial p_{IL} }}} \right)$, which reflects the additional wholesale revenue from those consumers who switch from the incumbent to the entrant. This effect can be compared to the reduction in the entrant’s input costs due to an increase in its retail prices: $w\left( {\frac{{\partial q_{EH} }}{{\partial p_{EH} }} - \frac{{\partial q_{EH} }}{{\partial p_{EL} }}} \right)$.

With $\frac{{\partial q_{EH} }}{{\partial p_{EH} }} = \frac{{\partial q_{EH} }}{{\partial p_{IH} }} + \frac{{\partial q_{EH} }}{{\partial p_{EL} }} + \frac{{\partial q_{EH} }}{{\partial p_{IL} }}$, which in the general linear demand function $D_{ij} = A_{j} - bp_{ij} + cp_{mj} + dp_{il} + ep_{ml}$ amounts to $b = c + d + e$, the impact on the entrant’s equilibrium aggregate output of the wholesale activity is equal to the impact on the incumbent’s output. As the single source of asymmetry between the entrant and the incumbent impacts the two firms equally, they both sell the same quantity in equilibrium. This follows from: (1) demand symmetry; and (2) the fact that a similar increase in all prices leaves the quantity demanded unchanged (i.e., the market is fully covered).