Two-Period Duopolies with Forward Markets

Abstract

We experimentally consider a dynamic multi-period Cournot duopoly with a simultaneous option to manage financial risk and a real option to delay supply. The first option allows players to manage risk before uncertainty is realized, while the second allows managing risk after realization. In our setting, firms face a strategic dilemma: They must weigh the advantages of dealing with risk exposure against the disadvantages of higher competition. In theory, firms make strategic use of the hedging component, enhancing competition. Our experimental results support this theory, suggesting that hedging increases competition and negates duopoly profits even in a simultaneous setting.

Appendix: Expected Equilibrium Values: Numerical Example

This appendix derives the equilibrium values that are shown in Table 3.

1.1 Single Production Setting

We begin with the Single Production setting (see Sect. 4.1). First, we derive second-period choices. We present mathematical representations for firm A. As the duopoly is symmetric, similar representations apply for firm B. At this point in the model, all decisions from the first period, the realization of the demand uncertainty of the first period, and the forward rate for the second period are common knowledge. Thus, firms maximize

$$\begin{aligned} \Phi ^{A}_L & {} = \underbrace{(50 - (s_1^A+s_1^B)) s_1^A}_{\text {revenues from sales in } t=1} - \underbrace{q_1^A}_{\text {fixed production costs}} + \underbrace{{\check{h}}^A_1( f(0,1)- (50 - (s_1^A + s_1^B))}_{\text {profits from t=0 hedging position}}-\underbrace{0.25({\check{q}}_1^A-s_1^A)}_{\text {storage costs}} \nonumber \\&+\, \{\underbrace{(50 - ({\check{q}}_1^A+{\check{q}}_1^B-s_1^A-s_1^B))({\check{q}}_1^A-s_1^A)}_{\text {revenues from sales in } t=2} - \frac{\alpha ^A}{2} \ 200 \ ({\check{q}}_1^A-s_1^A-h^A_2)^2\} \end{aligned}$$

(13)

by choosing $s_1^A$ and $h^A_2$. The first-order conditions are

$$\begin{aligned} 50 - (2 s_1^A+s_1^B)+ {\check{h}}^A_1 + 0.25 +\{-50 + (2 {\check{q}}_1^A+{\check{q}}_1^B - 2 s_1^A-s_1^B) +\alpha ^A \ 200 \ ({\check{q}}_1^A-s_1^A-h^A_2)\} = 0 \end{aligned}$$

(14)

and

$$\begin{aligned} \alpha ^A \ 200 \ ({\check{q}}_1^A-s_1^A-h^A_2) =0. \end{aligned}$$

(15)

Subtracting these yields Eq. (4). The optimal supply decision in $t=1$ fulfills ($\text{ E }[\varepsilon _1] = 50$)

$$\begin{aligned} 50 - (2 s_1^A+s_1^B) + {\check{h}}^A_1 = 50 -(2 {\check{q}}_1^A- 2 s_1^A +{\check{q}}_1^B -s_1^B) - 0.25. \end{aligned}$$

(4 revisited)

For the second-period hedging decision, the full hedge immediately follows from Eq. (15):

$$\begin{aligned} h^A_2 = {\check{q}}_1^A - s_1^A. \end{aligned}$$

(16)

Equation (4) yields a system of two equations (firm A and firm B) with six unknowns. Solving for spot-sale decisions, this leads to

$$\begin{aligned} s_1^A = \frac{1}{24} + \frac{1}{2} {\check{q}}_1^A + \frac{1}{3} h^A_1 - \frac{1}{6}h^B_1. \end{aligned}$$

(17)

Taking these future decisions into account, decision makers initially maximize

$$\begin{aligned} \Phi ^{A}_L & {} = \underbrace{(50- {\hat{s}}_1^A-{\hat{s}}_1^B) \cdot {\hat{s}}_1^A}_{\text {revenues from sales in } t=1} - \underbrace{q_1^A}_{\text {production costs}} - \underbrace{0.25 (q_1^A - {\hat{s}}_1^A)}_{\text {storage costs}} - \frac{\alpha ^A}{2} 100 ({\hat{s}}_1^A-h^A_1)^2\\&+\, \{\underbrace{(50 - (q_1^A+q_1^B- {\hat{s}}_1^A- {\hat{s}}_1^B))(q_1^A- {\hat{s}}_1^A)}_{\text {revenues from sales in } t=2} - \frac{\alpha ^A}{2}(200 (\underbrace{q_1^A- {\hat{s}}_1^A- {\hat{h}}^A_2}_{=0 \text {, full hedge}})^2 + 100 ({\hat{h}}^A_2)^2)\}. \end{aligned}$$

based on the expectations in time $t=0$.

Simplified first-order conditions that take second-period choices (that are made in $t=1$) into account are

$$\begin{aligned} 48.875 - q_1^A - \frac{q_1^B}{2} - 100 \cdot \alpha ^A \cdot \left( \frac{3}{4}q_1^A - \frac{1}{48} - \frac{2}{3} h^A_1 + \frac{h^B_1}{12}\right) = 0 \end{aligned}$$

(18)

and

$$\begin{aligned} \frac{2\cdot 100}{3} \cdot \alpha ^A \cdot \left( \frac{1}{24} + \frac{q_1^A}{2} - \frac{2}{3} h^A_1 - \frac{h^B_1}{6}\right) + \frac{1}{72} - \frac{2}{9} h^A_1 - \frac{1}{18} h^B_1 =0. \end{aligned}$$

(19)

This system of four equations (similar equations apply for firm B) with four unknowns (considering that risk aversion is a fixed parameter) yields expected equilibrium choices in $t=0$.

For $\alpha ^A = 0.15$, we obtain

$$\begin{aligned} q_1^A = q_1^B = 6.47 \end{aligned}$$

and

$$\begin{aligned} h^A_1 = h^B_1 = 3.80. \end{aligned}$$

Equation (17) yields resulting spot sales1 in $t=1$, $s_1^A = s_1^B = 3.91$. Based on the spot-sale decisions, the equilibrium forward price is

$$\begin{aligned} f(0,1) = 50 - \frac{1}{2} (q_1^A + q_1^B) - \frac{1}{6} (h^A_1 + h^B_1) - \frac{1}{3} \cdot 0.25 = 42.18. \end{aligned}$$

(5 revisited)

Expected prices are given by the inverse demand function,

$$\begin{aligned} p_1= 50 - (s_1^A + s_1^B) = 42.18 \end{aligned}$$

and

$$\begin{aligned} p_2= 50 - (q_1^A + q_1^B - s_1^A - s_1^B) = 44.88 \end{aligned}$$

Finally, expected profits read

$$\begin{aligned} \text{ E }[{{\tilde{\pi }}}_1^A]&= \text{ E }[{\tilde{p}}_1] s_1^A - q_1^A + h^A_1(f(0,1)-\text{ E }[{\tilde{p}}_1])-0.25(q_1^A-s_1^A) \\&= 42.18 \cdot 3.91 - 6.47 + 0 - 0.25 \cdot (6.47-3.91) \text { and} \\ \text{ E }[{{\tilde{\pi }}}_2^A]&= \text{ E }[p_2] (q_1^A-s_1^A) +h^A_2({\tilde{f}}(1,2) -\text{ E }[{\tilde{p}}_2]) \\&= 44.88 \cdot (6.47-3.91) + 0 \end{aligned}$$

and consequently

$$\begin{aligned} \text{ E }[{{\tilde{\pi }}}^A] = \text{ E }[{{\tilde{\pi }}}^A_1] + \text{ E }[{{\tilde{\pi }}}^A_2] = 272.71. \end{aligned}$$

For risk-neutral decision makers ($\alpha ^A = 0$), the system of equations (18) and (19) simplifies to

$$\begin{aligned} q_1^A = 48.875 - \frac{q_1^B}{2} \end{aligned}$$

and

$$\begin{aligned} h^A_1 = \frac{1}{16} - \frac{1}{4} h^B_1, \end{aligned}$$

which yields

$$\begin{aligned} q_1^A = \frac{2}{3} \cdot 48.875 \end{aligned}$$

and

$$\begin{aligned} h^A_1 = \frac{3}{60}. \end{aligned}$$

Second-period decisions, prices, and profits are obtained in the same manner as above.

1.2 Double Production Setting

Next, we turn to the equilibrium values in the Double Production setting (see Sect. 4.2). We again start with the second-period decision problem. Firms maximize

$$\begin{aligned} \Phi ^A & {} = \underbrace{p(Q_1) s_1^A}_{\text {revenues from sales in } t=1}- \underbrace{{\check{q}}_1^A}_{\text {first-period production costs}}\\&+\underbrace{{\check{h}}_1^A(f(0,1)- (50 - (s_1^A + s_1^B)))}_{\text {profits from t=0 hedging decision}}-\underbrace{0.25({\check{q}}_1^A-s_1^A)}_{\text {storage costs}} \\&+\, \{\underbrace{{\bar{p}}(Q_2)(q_2^A+{\check{q}}_1^A-s_1^A)}_{\text {revenues from sales in } t=2}- \underbrace{q_2^A}_{\text {second-period production costs}} - \frac{\alpha ^A}{2} \ 200 \ (q_2^A+{\check{q}}_1^A-s_1^A-h_2^A)^2\}, \end{aligned}$$

where $Q_1$ and $Q_2$ denote the industry supply in $t=1$ and $t=2$, respectively. To obtain the interior solution, we derive the first-order conditions (${\bar{\varepsilon }}_1 = 50$):

$$\begin{aligned}&50 - (2s_1^A+s_1^B) + {\check{h}}^A_1 +0.25 - (50 - (2q_2^A+q_2^B+2 {\check{q}}_1^A+{\check{q}}_1^B-2s_1^A-s_1^B) \nonumber \\&\quad -\alpha ^A \ 200 \ (q_2^A+ {\check{q}}_1^A-s_1^A-h_2^A)) = 0, \end{aligned}$$

(20)

$$\begin{aligned}&50-(2q_2^A+q_2^B+2 {\check{q}}_1^A+ {\check{q}}_1^B-2s_1^A-s_1^B) - 1 -\alpha ^A \ 200 \ (q_2^A+ {\check{q}}_1^A-s_1^A-h_2^A) = 0 \end{aligned}$$

(21)

and

$$\begin{aligned} \alpha ^A \ 200 \ (q_2^A+ {\check{q}}_1^A-s_1^A-h_2^A) = 0. \end{aligned}$$

(22)

The second-period hedging decision immediately follows from Eq. (22). The addition of Eqs. (21) and (22) yields Eq. (9): marginal costs = marginal revenues. Thus, firms sell 49/3 units (= standard Cournot quantity) in the second period and adjust their second-period production accordingly. Also, we obtain Eq. (10):

$$\begin{aligned} 50 - (2 s_1^A+s_1^B) + {\check{h}}^A_1 = 1 - 0.25 = 0.75. \end{aligned}$$

(10 revisited)

Solving for the equilibrium, we obtain

$$\begin{aligned} s_1^A = \frac{50 - 0.75}{3} + \frac{2}{3} {\check{h}}^A_1 - \frac{1}{3} {\check{h}}^B_1 \le q^A_1. \end{aligned}$$

(23)

Then, decision makers initially maximize

$$\begin{aligned} \Phi ^A & {} = \ \underbrace{(50 - ({\hat{s}}_1^A+{\hat{s}}_1^B)){\hat{s}}_1^A}_{\text {revenues from sales in } t=1}- \underbrace{q_1^A}_{\text {production costs}} - \underbrace{0.25(q_1^A-{\hat{s}}_1^A)}_{\text {storage costs}} -\frac{\alpha ^A}{2} \ 100 \ ({\hat{s}}_1^A-h_1^A)^2 \\&+\, \underbrace{\text {prof}}_{\text {net revenues from sales in } t=2} + \underbrace{(q_1^A-{\hat{s}}_1^A)}_{\text {non-incurred second-period production costs}} -\frac{\alpha ^A}{2} \ 100 \ (q_1^A-{\hat{s}}_1^A)^2. \end{aligned}$$

Simplified first-order conditions to determine the interior solution, considering the second-period choices, are

$$\begin{aligned} q_1^A - \frac{49.25}{3} - \frac{2}{3} h^A_1 + \frac{1}{3} h^B_1 + \frac{0.25}{100 \ \alpha ^A} = 0 \end{aligned}$$

and

$$\begin{aligned} \alpha ^A \ 100 \ \left( \frac{2}{3} q_1^A - \frac{49.25}{9} - \frac{5}{9} h^A_1 + \frac{1}{9} h^B_1\right) + \frac{49.25}{9} - \frac{4}{9} h^A_1 - \frac{1}{9} h^B_1 = 0. \end{aligned}$$

This system of four equations with four unknowns (again considering that risk aversion is a fixed parameter) yields expected equilibrium choices in $t=0$. For $\alpha ^A = 0.15$, we obtain $q^A_1 = q^B_1 = 23.89$ and $h^A_1 = h^B_1 = 22.42$.

As a result, spot sales1 amount to $s_1^A=23.89$ [see Eq. (23)]. With nothing in storage, the firms produce $q^A_2 = q^B_2 = 16.33$ and hedge the entire production on the second-period forward market $h^A_2 = h^B_2 = 16.33$. Prices and firm profits follow immediately from the inverse demand function and Eqs. 7 and 8.

For risk-neutral decision makers ($\alpha ^A = 0$), the decision problem collapses to a repeated single-shot duopoly under certainty with a corner solution, which is consistent with the notion of Broll et al. (2011) that strategic considerations are absent in a setting where firms decide on their production and hedging decision at the same time.