Capacity Accumulation Games with Technology Constraints

Abstract

This chapter examines the conduct and performance of large mutually dependent firms. Its objective is to study contractual relationships in a dynamic bilateral monopoly, where producers’ investment choices must obey a technology constraint. This is in contrast to previous studies of accumulation games, in which technological interdependence was not explicitly allowed for. The analysis focuses on investment incentives and payoff allocation under two regimes: (1) contracting based on input quantities, and (2) contracting based on final revenues. The technologically feasible equilibrium strategies and the terms of trade that support them are characterized with intuitive necessary conditions which reflect the players’ intertemporal trade-offs. To assess the factors that influence efficiency and market power, the chapter presents a linear-quadratic example. Our simulations indicate that contracts based on input quantities generate higher joint payoffs and tend to benefit the input producer.

Appendices

Appendix A: Markov Allocation Equilibrium Conditions of the Bilateral Monopoly Game

This appendix derives the necessary conditions that characterize the Markov equilibrium of the bilateral monopoly game.

1.1 A.1 Euler Equation of the Final Good Producer

First consider the problem of firm A. Differentiating Bellman equation (9) yields the first-order condition:

$$ V_{1}^{A\prime}=\frac{C_{1}^{A}}{\delta}. $$

(38)

By assumption the equilibrium strategies of firm A and firm B are respectively f ^A(x,y) and f ^B(x,y). Therefore, these strategy functions satisfy the recursive equation

$$\begin{aligned} V^{A}(x_{t},y_{t}) =&R(x_{t})-S^{B} \bigl(z(x_{t},y_{t}),g(x_{t},y_{t}) \bigr) -C^{A}\bigl(f^{A}(x_{t},y_{t}) \bigr) \\ &{}+ \delta V^{A} \bigl(\mu^{A}x_{t}+f^{A}(x_{t},y_{t}), \mu ^{B}y_{t}+f^{B}(x_{t},y_{t}) \bigr). \end{aligned}$$

(39)

Differentiating with respect to x _t gives us

$$\begin{aligned} V_{1}^{A} =&R_{1}-S_{1}^{B}z_{1}-S_{2}^{B}g_{1}-C_{1}^{A}f_{1}^{A} \\ &{}+ \delta\bigl(\mu^{A}+f_{1}^{A} \bigr)V_{1}^{A\prime}+\delta f_{1}^{B}V_{2}^{A\prime}. \end{aligned}$$

(40)

Substituting $V_{1}^{A}(x,y)$ from the first-order condition into (40) forwarded one period yields an equation for $V_{2}^{A}(x,y)$:

$$ V_{2}^{A\prime\prime}=-\frac{1}{\delta f_{1}^{B\prime}} \biggl\{ R_{1}^{\prime}-S_{1}^{B\prime}z_{1}^{\prime}-S_{2}^{B\prime }g_{1}^{\prime}- \frac{C_{1}^{A}}{\delta}+\mu^{A}C_{1}^{A\prime } \biggr\} . $$

(41)

Furthermore, differentiating (39) with respect y _t−1 delivers

$$\begin{aligned} V_{2}^{A} =&R_{1}-S_{1}^{B}z_{2}-S_{2}^{B}g_{2}-C_{1}^{A}f_{2}^{A} \\ &{}+ \delta f_{2}^{A}V_{1}^{A\prime}+\delta \bigl(\mu ^{B}+f_{2}^{B}\bigr)V_{2}^{A\prime}. \end{aligned}$$

(42)

Substituting $V_{1}^{A}(x,y)$ from (38) and $V_{2}^{A}(x,y)$ from (41) into (42) yields (14).

1.2 A.2 Euler Equation of the Intermediate Good Producer

Now consider the decision problem of firm B. Bellman equation (10) implies that the optimal strategy solves the first-order condition

$$ V_{2}^{B\prime}=\frac{C_{1}^{B}}{\delta}. $$

(43)

Furthermore, by assumption the optimal strategies of firm A and firm B are respectively f ^A(x,y) and f ^B(x,y). Therefore, these strategy functions satisfy the recursive equation

$$\begin{aligned} V^{B}(x_{t},y_{t}) =&S^{B} \bigl(z(x_{t},y_{t}),g(x_{t},y_{t}) \bigr) -C^{B}\bigl(f^{B}(x_{t},y_{t}) \bigr) \\ &{}+ \delta V^{B} \bigl(\mu^{A}x_{t}+f^{A}(x_{t},y_{t}), \mu ^{B}y_{t}+f^{B}(x_{t},y_{t}) \bigr). \end{aligned}$$

(44)

Differentiating (44) with respect to y _t yields

$$ V_{2}^{B}=S_{1}^{B}z_{2}+S_{2}^{B}g_{2}-f_{2}^{B}C_{1}^{B}+ \delta f_{2}^{A}V_{1}^{B\prime}+\delta\bigl( \mu^{B}+f_{2}^{B}\bigr)V_{2}^{B\prime }. $$

(45)

Substituting $V_{2}^{B}$ from the first-order condition and solving for $V_{1}^{B}$ we get

$$ V_{1}^{B\prime\prime}=-\frac{1}{\delta f_{2}^{A\prime}}\biggl\{ S_{1}^{B\prime }z_{2}^{\prime}+S_{2}^{B\prime}g_{2}^{\prime}- \frac {C_{1}^{B}}{\delta} +\mu^{B}C_{1}^{B\prime}\biggr\} . $$

(46)

Similarly, differentiating (44) with respect to x _t gives us

$$ V_{1}^{B}=S_{1}^{B}z_{1}+S_{2}^{B}g_{1}-f_{1}^{B}C_{1}^{B}+ \delta\bigl(\mu ^{A}+f_{1}^{A}\bigr)V_{1}^{B\prime}+ \delta f_{1}^{B}V_{2}^{B\prime}. $$

(47)

After substitution of (43) and (46) into (47) we obtain (15).

Appendix B: Dynamically Efficient Investment

This appendix derives the necessary condition for joint surplus maximization.

Bellman equation (20) yields the first-order condition

$$ -F_{1}C_{1}^{A}-C_{1}^{B}+ \delta F_{1}W_{1}^{\prime}+W_{2}^{\prime }=0. $$

(48)

Differentiation with respect to x gives us the envelope condition

$$ W_{1}=R_{1}-\bigl(F_{2}-\mu^{A} \bigr)C_{1}^{A}+\delta F_{2}W_{1}^{\prime}. $$

(49)

Furthermore, differentiation with respect to y gives us the envelope condition

$$ W_{2}=-\bigl(\mu^{B}F_{1}+F_{3} \bigr)C_{1}^{A}+\delta\bigl(\mu ^{B}F_{1}+F_{3} \bigr)W_{1}^{\prime}+\delta\mu^{B}W_{2}^{\prime}. $$

(50)

Multiplying (49) by F ₁ and adding it to (50) yields

$$\begin{aligned} F_{1}W_{1}^{\prime}+W_{2}^{\prime} =&F_{1}C_{1}^{A}-C_{1}^{B} \\ =&\delta F_{1}R_{1}^{\prime}-\delta F_{1}\bigl(F_{2}-\mu^{A}\bigr)C_{1}^{A\prime }- \bigl(\mu^{B}F_{1}^{\prime}+F_{3}^{\prime} \bigr)C_{1}^{A\prime} \\ &{}+ \delta^{2}\mu^{B}\bigl(F_{1}^{\prime}W_{1}^{\prime\prime}+W_{2}^{\prime \prime} \bigr)+\delta^{2}\bigl(F_{1}F_{2}^{\prime}+F_{3}^{\prime} \bigr)W_{1}^{\prime \prime}. \end{aligned}$$

(51)

Substituting $F_{1}^{\prime}W_{1}^{\prime\prime}+W_{2}^{\prime \prime}$ from the first-order condition into (51) gives us an equation for W ₁:

$$\begin{aligned} W_{1}^{\prime\prime} =&\frac{1}{\delta^{2}(F_{1}F_{2}^{\prime }+F_{3}^{\prime})}\bigl\{ F_{1}C_{1}^{A}-C_{1}^{B}- \delta F_{1}R_{1}^{\prime } \\ &{}+\delta F_{1}\bigl(F_{2}^{\prime}- \mu^{A}\bigr)C_{1}^{\prime A}+F_{3}^{\prime }C_{1}^{A\prime}- \delta\mu^{B}C_{1}^{B\prime}\bigr\} . \end{aligned}$$

(52)

Finally, substituting (52) into (49) delivers Euler equation (21).