Welfare reducing vertical licensing in the presence of complementary inputs

Abstract

This research explores the welfare implications of vertical licensing when the final goods are produced by multiple complementary inputs. We spotlight the importance of two-part tariff input terms when there is a buyer–seller relationship after vertical licensing, which has different welfare ramifications depending on the product differentiation. When the products are less differentiated, our result shows welfare improving licensing, but when the products are more differentiated, the wholesale price is set above the supplier’s marginal cost through licensing, leading to the problem of double marginalization and reducing welfare. This study offers various policy implications, which go up against conventional wisdom that welfare improving licensing may not be attainable by considering multiple complementary inputs.

Appendices

Appendix 1: Outside option for firm 1

In the absence of licensing, firm 1 produces the core input in-house. Each firm chooses its output in order to maximize its profits as $\pi_{i}^{N} = \left( {1 - q_{i}^{N} - rq_{j}^{N} - c - w_{B}^{N} } \right)q_{i}^{N} , i = 1, 2, i \ne j$, where superscript “N” denotes the equilibrium with no licensing. Solving the resulting system of first-order conditions, we obtain the Cournot-Nash equilibrium quantities, $q_{1}^{N} = q_{2}^{N} = \frac{{\left( {2 - r} \right) + r\left( {c + w_{B}^{N} } \right) - 2\left( {c + w_{B}^{N} } \right)}}{{4 - r^{2} }}$. In the following stage, supplier U_B maximizes the profit, which is $\pi_{B}^{N} \equiv \pi_{B}^{N} \left( {q_{1}^{N} \left( {w_{B}^{N} } \right),q_{2}^{N} \left( {w_{B}^{N} } \right),w_{B}^{N} } \right) = w_{B}^{N} \left( {q_{1}^{N} + q_{2}^{N} } \right)$, to determine the price of input B. The first-order condition of $\pi_{B}^{N}$ leads to:

$$\frac{{d\pi_{B}^{N} }}{{dw_{B}^{N} }} = \left( {\frac{{\partial \pi_{B}^{N} }}{{\partial q_{1}^{N} }}\frac{{\partial q_{1}^{N} }}{{\partial w_{B}^{N} }}} \right) + \left( {\frac{{\partial \pi_{B}^{N} }}{{\partial q_{2}^{N} }}\frac{{\partial q_{2}^{N} }}{{\partial w_{B}^{N} }}} \right) + \frac{{\partial \pi_{B}^{N} }}{{\partial w_{B}^{N} }} = 0,$$

(11)

where $\left( {\frac{{\partial \pi_{B}^{N} }}{{\partial q_{1}^{N} }}\frac{{\partial q_{1}^{N} }}{{\partial w_{B}^{N} }}} \right) = \left( {\frac{{\partial \pi_{B}^{N} }}{{\partial q_{2}^{N} }}\frac{{\partial q_{2}^{N} }}{{\partial w_{B}^{N} }}} \right) = w_{B}^{N} \left( {\frac{r - 2}{{4 - r^{2} }}} \right) < 0$, and $\frac{{\partial \pi_{B}^{N} }}{{\partial w_{B}^{N} }} = \left( {q_{1}^{N} + q_{2}^{N} } \right) > 0$. Therefore, we have the equilibrium input price of B and the respective equilibrium profits as:

$$w_{B}^{N} = \frac{{\left( {1 - c} \right)}}{2}\quad {\text{and}}\quad \pi_{1}^{N*} = \pi_{2}^{N*} = \frac{{\left( {1 - c} \right)^{2} }}{{4\left( {2 + r} \right)^{2} }}.$$

(12)

Appendix 2: Two-part tariff input pricing

In the second stage, firm S and firm 1 negotiate over $\left( {w_{s} ,{\text{ T}}} \right)$ to determine T and $w_{s}$. Substituting T into (3), it follows that $w_{s}$ is chosen to maximize the following profits as ${\text{ U}} = \pi_{1}^{L} \left( {w_{s} } \right) + \pi_{s}^{L} \left( {w_{s} } \right) - \pi_{1}^{N}$. The first-order condition of (3) is:

$$\begin{aligned} \frac{dU}{{dw_{s} }} & = \left\{ {\frac{{\partial \pi_{1}^{L} }}{{\partial q_{1}^{L} }}\left( {\frac{{\partial q_{1}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }} + \frac{{\partial q_{1}^{L} }}{{\partial w_{s} }}} \right) + \frac{{\partial \pi_{1}^{L} }}{{\partial q_{2}^{L} }}\left( {\frac{{\partial q_{2}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }} + \frac{{\partial q_{2}^{L} }}{{\partial w_{s} }}} \right) + \frac{{\partial \pi_{1}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }} + \frac{{\partial \pi_{1}^{L} }}{{\partial w_{s} }}} \right\} \\ & \quad + \left\{ {\frac{{\partial \pi_{s}^{L} }}{{\partial q_{1}^{L} }}\left( {\frac{{\partial q_{1}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }} + \frac{{\partial q_{1}^{L} }}{{\partial w_{s} }}} \right) + \frac{{\partial \pi_{s}^{L} }}{{\partial w_{s} }}} \right\} = 0. \\ \end{aligned}$$

(13)

Using the envelop theorem, we derive $\frac{{\partial \pi_{1}^{L} }}{{\partial q_{1}^{L} }}\left( {\frac{{\partial q_{1}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }} + \frac{{\partial q_{1}^{L} }}{{\partial w_{s} }}} \right) = 0$, $\frac{{\partial \pi_{1}^{L} }}{{\partial w_{s} }} = - q_{1}^{L} < 0$, and $\frac{{\partial \pi_{s}^{L} }}{{\partial w_{s} }} = q_{1}^{L} > 0$. Therefore, we obtain the first-order condition as (4). Solving (4), we derive the equilibrium wholesale price as (5) and rewrite the wholesale price and equilibrium fixed fee as:

$$\left( {w_{s}^{*} - c} \right) = \frac{{2\left( {1 - c} \right)\left( {2 - r} \right)\left( {2 - 2r^{2} - r} \right)}}{{A_{1} }}.$$

(14)

We can easily check that $w_{s}^{*} > \left( \le \right)c$ if $0 \le r < \overline{r} \equiv 0.780776$ ($\overline{r} \le r \le 1$).

$$T^{*} = \frac{{\left( {1 - c} \right)^{2} \left( { - 2r^{2} - r + 2} \right)\left[ { - 8\left( {r^{2} - 4} \right)^{2} + \beta A_{4} } \right]}}{{4\left( {6 + r} \right)\left( {2 + r} \right)^{2} \left( { - 4r^{2} - r + 10} \right)^{2} }} = \frac{{\left( {1 - c} \right)\left( {w_{s}^{*} - c} \right)\left[ { - 8\left( {r^{2} - 4} \right)^{2} + \beta A_{4} } \right]}}{{8\left( {2 - r} \right)\left( {2 + r} \right)^{2} \left( { - 4r^{2} - r + 10} \right)}},$$

(15)

where $A_{4} = \left( {8r^{4} + 6r^{3} - 27r^{2} - 12r + 20} \right) > \left( \le \right)0$ if $0 \le r < \overline{r}$ ($\overline{r} \le r \le 1$). Therefore, we have $T^{*} > 0$ due to $\left( {w_{s}^{*} - c} \right) < 0$ and $\left[ { - 8\left( {r^{2} - 4} \right)^{2} + \beta A_{4} } \right] < 0$ as $\overline{r} \le r \le 1$. When $0 \le r < \overline{r}$, we find $\left[ { - 8\left( {r^{2} - 4} \right)^{2} + \beta A_{4} } \right] > 0$ if $\beta > \overline{\beta } \equiv \frac{{8\left( {r^{2} - 4} \right)^{2} }}{{A_{4} }}$, where $\overline{\beta } > 1$ due to $0 \le r \le 1$. Therefore, we have $\left[ { - 8\left( {r^{2} - 4} \right)^{2} + \beta A_{4} } \right] < 0$ due to $0 \le {\upbeta } \le 1$. Hence, we have $T^{*} < 0$ due to $\left( {w_{s}^{*} - c} \right) > 0$ and $\left[ { - 8\left( {r^{2} - 4} \right)^{2} + \beta A_{4} } \right] < 0$. Moreover, the net equilibrium profit of firm S is:

$$\pi_{s}^{L} + T^{*} = \frac{{\beta \left( {1 - c} \right)^{2} \left( { - 2r^{2} - r + 2} \right)^{2} }}{{4\left( {2 + r} \right)^{2} A_{1} }} = \frac{{\beta A_{1} \left( {w_{s}^{*} - c} \right)^{2} }}{{16\left( {2 + r} \right)^{2} \left( {2 - r} \right)^{2} }} > 0.$$

(16)

Appendix 3: Welfare implications

The industry profits in the downstream market and in the upstream market for the in-house case are $(\pi_{1}^{N} + \pi_{2}^{N} ) = \frac{{\left( {2 - r} \right)^{2} \left( {1 - c} \right)^{2} }}{{2\left( {4 - r^{2} } \right)^{2} }}$ and $\pi_{B}^{N} = \frac{{\left( {2 - r} \right)\left( {1 - c} \right)^{2} }}{{2\left( {4 - r^{2} } \right)}}$, respectively. The total industry profits for the in-house case are ${\Pi }^{N} = (\pi_{1}^{N} + \pi_{2}^{N} ) + \pi_{B}^{N} = \frac{{\left( {1 - c} \right)^{2} \left( {2 - r} \right)^{2} \left( {3 + r} \right)}}{{2\left( {4 - r^{2} } \right)^{2} }}$. The industry profits in the downstream market in the licensing case are ${\Pi }_{D}^{L} = (\pi_{1}^{L} - T^{*} + F) + \pi_{2}^{L} = \pi_{1}^{L} + \pi_{2}^{L} + \pi_{s}^{L} = \frac{{2\left( {1 - c} \right)^{2} \left( {2 - r} \right)\left( { - 3r^{2} - 5r + 14} \right)}}{{\left( {6 + r} \right)A_{1} }}$, and the profit of complementary input supplier B in the licensing case is $\pi_{B}^{L} = \frac{{2\left( { - 3r^{2} - 5r + 14} \right)^{2} \left( {1 - c} \right)^{2} \left( {2 + r} \right)}}{{A_{1}^{2} }}$, in which the total industry profits in the licensing case are ${\Pi }^{L} \equiv {\Pi }_{D}^{L} + \pi_{B}^{L}$.

Comparing the difference in the profits of the downstream market between the licensing case and the in-house case, we have:

$${\Pi }_{D}^{L} - \left( {\pi_{1}^{N} + \pi_{2}^{N} } \right) = \frac{{\left( {1 - c} \right)^{2} A_{5} \left( { - 2r^{2} - r + 2} \right)}}{{2\left( {2 + r} \right)^{2} \left( {r + 6} \right)A_{1} }} = \frac{{\left( {1 - c} \right)A_{5} \left( {w_{s}^{*} - c} \right)}}{{4\left( {2 + r} \right)^{2} \left( {2 - r} \right)\left( {r + 6} \right)}},$$

(17)

where $A_{5} = \left( { - 6r^{3} - 21r^{2} + 12r + 44} \right) > 0$ due to $0 \le r \le 1$. It shows that ${\Pi }_{D}^{L} > \left( \le \right)\left( {\pi_{1}^{N} + \pi_{2}^{N} } \right)$ as $0 \le r < \overline{r}$ ($\overline{r} \le r \le 1$).

Comparing the difference in the profits of complementary input supplier B between the licensing case and the in-house case, we have:

$$\pi_{B}^{L} - \pi_{B}^{N} = \frac{{ - A_{6} \left( {1 - c} \right)^{2} \left( { - 2r^{2} - r + 2} \right)}}{{2\left( {2 + r} \right)A_{1}^{2} }} = \frac{{ - A_{6} \left( {1 - c} \right)\left( {w_{s}^{*} - c} \right)}}{{4\left( {4 - r^{2} } \right)A_{1} }},$$

(18)

where $A_{6} = \left( {10r^{4} + 27r^{3} - 106r^{2} - 92r + 232} \right) > 0$ due to $0 \le r \le 1$. From (18), it shows that $\pi_{B}^{L} < \left( \ge \right) \pi_{B}^{N}$ if $0 \le r < \overline{r}$ ($\overline{r} \le r \le 1$).

We already showed that $CS^{L} < \left( \ge \right) CS^{N}$ and ${\Pi }^{L} < \left( \ge \right) {\Pi }^{N}$ if $0 \le r < \overline{r}$ ($\overline{r} \le r \le 1$) in Sect. 4. Therefore, for the difference in social welfare between the licensing case and the in-house case, we find that $SW^{L} < \left( \ge \right) SW^{N}$ if $0 \le r < \overline{r}$ ($\overline{r} \le r \le 1$).

Appendix 4: Multiple varieties of complementary inputs

The profit functions of two firms with n varieties of complementary inputs when firm 1 produces the core input in-house are:

$$\pi_{i}^{N} = \left( {1 - q_{i}^{N} - rq_{j}^{N} - c - \mathop \sum \limits_{i = 1}^{n} w_{Bi}^{N} } \right)q_{i}^{N} , i = 1, 2, i \ne j.$$

(19)

Using the first-order conditions from (19), we obtain the equilibrium outputs at the final stage. The profit function of complementary input supplier Bv, v = 1, 2, …, n, in the case of in-house is $\pi_{Bv}^{N} = w_{Bv}^{N} \left( {q_{1}^{N} + q_{2}^{N} } \right)$. The equilibrium wholesale price of input supplier Bv is $w_{Bv}^{N} = \frac{1 - c}{{\left( {1 + n} \right)}}$. The resulting equilibrium outputs and profits of downstream firms in the in-house case (that is, no licensing) are:

$$q_{i}^{N} = \frac{1 - c}{{\left( {1 + n} \right)\left( {2 + r} \right)}}, \pi_{i}^{N} = \frac{{\left( {1 - c} \right)^{2} }}{{\left( {1 + n} \right)^{2} \left( {2 + r} \right)^{2} }},{ }i = 1,2.$$

(20)

When the licensing agreement has been signed, the resulting equilibrium outputs are:

$$q_{1}^{L} = \frac{{\left( {2 - r} \right) - 2w_{s} - 2nw_{B}^{L} + rc + rnw_{B}^{L} }}{{4 - r^{2} }}, q_{2}^{L} = \frac{{\left( {2 - r} \right) + rw_{s} + rnw_{B}^{L} - 2c - 2nw_{B}^{L} }}{{4 - r^{2} }},$$

(21)

where $w_{B1}^{L} = w_{B2}^{L} = \cdots = w_{Bn}^{L} = w_{B}^{L}$ due to symmetry.

Firm 1 and firm S maximize their joint profit. The first-order condition can thus be rewritten as:

$$\frac{dU}{{dw_{s} }} = \left\{ {\frac{{\partial \pi_{1}^{L} }}{{\partial q_{2}^{L} }}\left( {\frac{{\partial q_{2}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }} + \frac{{\partial q_{2}^{L} }}{{\partial w_{s} }}} \right) + \left( {\frac{{\partial \pi_{1}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }}} \right)} \right\} + \left\{ {\frac{{\partial \pi_{s}^{L} }}{{\partial q_{1}^{L} }}\left( {\frac{{\partial q_{1}^{L} }}{{\partial w_{B}^{L} }}\frac{{\partial w_{B}^{L} }}{{\partial w_{s} }}} \right) + \frac{{\partial \pi_{s}^{L} }}{{\partial q_{1}^{L} }}\frac{{\partial q_{1}^{L} }}{{\partial w_{s} }}} \right\} = 0.$$

(22)

From (22), we get the equilibrium wholesale price as (8) and rewrite it as:

$$\left( {w_{s}^{*} - c} \right) = \frac{{2\left( {1 - c} \right)\left( {2 - r} \right)\left( { - r^{2} - nr^{2} - rn + 2n} \right)}}{{\left[ {2\left( {2 - r^{2} } \right) + n\left( {6 - 2r^{2} - r} \right)} \right]\left( {rn + 2n + 4} \right)}}.$$

(23)

It shows that $w_{s}^{*} > \left( \le \right)c$ if $0 \le r < \tilde{r} \equiv \frac{{ - n + \sqrt {n\left( {9n + 8} \right)} }}{{2\left( {1 + n} \right)}}$ ($\tilde{r} \le r \le 1$). Moreover, we have: $\frac{{\partial \tilde{r}}}{\partial n} = \frac{{\left( {5n + 4} \right) - \sqrt {n\left( {9n + 8} \right)} }}{{2\left( {1 + n} \right)^{2} \sqrt {n\left( {9n + 8} \right)} }} > 0$.

The equilibrium wholesale price is $w_{B}^{L} = \frac{{\left( {2 + r} \right)\left( {1 - c} \right)\left[ {n\left( {6 - 2r^{2} - r} \right) + \left( {8 - r^{2} - 4r} \right)} \right]}}{{\left[ {2\left( {2 - r^{2} } \right) + n\left( {6 - 2r^{2} - r} \right)} \right]\left( {rn + 4 + 2n} \right)}}$. The equilibrium outputs are $q_{1}^{L} = \frac{{\left( {2 - r} \right)\left( {1 - c} \right)}}{{\left[ {2\left( {2 - r^{2} } \right) + n\left( {6 - 2r^{2} - r} \right)} \right]}}$ and $q_{2}^{L} = \frac{{\left( {1 - c} \right)\left[ {n\left( {8 - 3r^{2} - 2r} \right) + 2\left( {4 - r^{2} - 2r} \right)} \right]}}{{\left[ {2\left( {2 - r^{2} } \right) + n\left( {6 - 2r^{2} - r} \right)} \right]\left( {rn + 4 + 2n} \right)}}$. We now have the equilibrium profits of firm 1 and firm S in the case of licensing.

The net equilibrium profit of firm 1 is:

$$\pi_{1}^{LN} = \pi_{1}^{L} + \pi_{s}^{L} = \frac{{\left( {2 - r} \right)^{2} \left( {1 - c} \right)^{2} }}{{\left[ {2\left( {2 - r^{2} } \right) + n\left( {6 - 2r^{2} - r} \right)} \right]\left( {rn + 2n + 4} \right)}}.$$

(24)

Using (12), we can derive $\pi_{1}^{LN} - \pi_{1}^{N} > 0$.

Appendix 5: A wholesale price contract

In the second stage, firm S and firm 1 negotiate over $w_{s}$ instead of over $\left( {w_{s} ,{\text{ T}}} \right)$. Under this case, they solve the following generalized Nash bargaining problem:

$${\text{max }}\left[ {\pi_{s}^{L} \left( {w_{s} } \right)} \right]^{\beta } \left[ {\pi_{1}^{L} \left( {w_{s} } \right) - \pi_{1}^{N} } \right]^{1 - \beta } ,$$

(25)

where $\pi_{s}^{L} \left( {w_{s} } \right) = \pi_{s}^{L} \left( {q_{1}^{L} \left( {w_{B}^{L} \left( {w_{s} } \right),w_{s} } \right),w_{s} } \right) = \left( {w_{s} - c} \right)q_{1}^{L} \left( {w_{s} } \right)$ is firm S’s profit, and $\pi_{1}^{L} \left( {w_{s} } \right) = \pi_{1}^{L} \left( {q_{1}^{L} \left( {w_{B}^{L} \left( {w_{s} } \right),w_{s} } \right),q_{2}^{L} \left( {w_{B}^{L} \left( {w_{s} } \right),w_{s} } \right),w_{B}^{L} \left( {w_{s} } \right),w_{s} } \right)$.

Maximizing (25) with respect to $w_{s}$ and solving the first-order condition, we can derive $\hat{w}_{s} \left( {\upbeta } \right)$. It shows that $\hat{w}_{s} \left( {\upbeta } \right) \ge {\text{c}}$ and $\frac{{\partial \hat{w}_{s} }}{\partial \beta } > 0$. If we assume $\beta = 1$, then we can derive the wholesale price as:

$$\hat{w}_{s} = \frac{2cr + 4c - r + 2}{{r + 6}}.$$

(26)

We then substitute (26) into the profit function of firm S: $\pi_{s}^{L} = \left( {w_{s} - c} \right)q_{1}$. We can derive the optimal fixed-fee licensing revenue in the first stage as:

$$\hat{F} = \frac{{\left( {1 - c} \right)^{2} \left( {2 - r} \right)}}{{4\left( {r + 6} \right)\left( {r + 2} \right)}}.$$

Comparing firm 1’s equilibrium profits in the licensing case $\pi_{1}^{LE}$ with its profits $\pi_{1}^{N}$ in the no licensing case, we have:

$\pi_{1}^{LE} - \pi_{1}^{N} = \pi_{1}^{L} \left( {\hat{w}_{s} } \right) + F - \pi_{1}^{N} = \frac{{ - \left( {4r^{2} + 3r + 2} \right)\left( {1 - c} \right)^{2} }}{{16\left( {r + 6} \right)\left( {r + 2} \right)^{2} }} < 0$.

As a result, firm 1 has no incentive to license its core input production technology to firm S with a wholesale price input contract in the extreme case of $\beta = 1$.

Appendix 6: A two-part tariff or pure royalty licensing contract

In the first stage, firm 1 decides whether to license its input technology to external firm S. We consider the case in which the licensing contract is composed of two-part tariffs,${ }\left( {k,{ }F} \right)$, instead of a fixed-fee licensing, where k is the royalty rate and F is the fixed fee. Firm S and firm 1 then negotiate over a vertical contract—that is, $\left( {w_{s} ,{\text{ T}}} \right)$—to determine T and $w_{s}$ in the second stage. Following this setting and routine calculation, we substitute T into (3) in the second stage, and it follows that $w_{s}$ is chosen to maximize the following profits as ${\text{ U}} = \pi_{1}^{L} \left( {w_{s} ,k} \right) + \pi_{s}^{L} \left( {w_{s} ,k} \right) - \pi_{1}^{N}$. The first-order condition is the same as (4). By solving the first-order condition, we derive the equilibrium $w_{s}^{*}$ and $T^{*}$ as:

$$\begin{aligned} & w_{s}^{*} \left( k \right) = c + \frac{{2\left( {1 - c} \right)\left( {2 - r} \right)\left( {2 - 2r^{2} - r} \right)}}{{A_{1} }} + k, \\ & T^{*} = \frac{{\left( {2 - 2r^{2} - r} \right)\left( {1 - c} \right)^{2} \left( {8\beta r^{4} + 6\beta r^{3} - 8r^{4} - 27\beta r^{2} - 12\beta r + 64r^{2} + 20\beta - 128} \right)}}{{4\left( {r + 2} \right)^{2} \left( {4r^{2} + r - 10} \right)^{2} \left( {r + 6} \right)}}, \\ \end{aligned}$$

(27)

where $c + \frac{{2\left( {1 - c} \right)\left( {2 - r} \right)\left( {2 - 2r^{2} - r} \right)}}{{A_{1} }}$ is the same as (14), which is the optimal wholesale price under fixed-fee licensing. Here, (27) shows that $w_{s}^{*}$ is a function of the royalty rate, k, with $\frac{{\partial w_{s} }}{\partial k} = 1$, and fixed fee $T^{*}$ is not related to k. Therefore, the royalty rate can be any value even if it is set to zero. By substituting $k = 0$ into $w_{s}^{*} \left( k \right)$, we can derive the same $w_{s}^{*}$ in (5). The results in the first stage and the incentive for licensing are the same as (7). Therefore, it is found that vertical licensing occurs when the licensing contract is a two-part tariff.

In the case of pure royalty licensing in the first stage, which is $F = 0$, the wholesale price in the second stage is the same as (27). Therefore, the result is similar to the above. We can derive the same $w_{s}^{*}$ in (5) as $k = 0$ in (27), and the results in the first stage are the same as (7). Therefore, vertical licensing still occurs when the licensing contract is a pure royalty. However, a royalty licensing cannot extract all the extra profit from firm S. Therefore, pure royalty licensing is inferior to fixed-fee or two-part tariff licensing.