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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.

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Notes

  1. 1.

    For literature discussing the goal of firms employing corporate hedging see, among others, Géczy et al. (1997), Kajüter (2012).

  2. 2.

    Even though the Cournot model has been criticized for its theoretical foundations, the model is simple and has pleasing comparative statics and has proved useful in the literature (see, e.g. Mas-Colell et al., 1995; Martin, 1994; Maggi, 1996; Larue & Yapo, 2000).

  3. 3.

    The forward market is called unbiased if the forward price equals to the expected spot price.

  4. 4.

    Since in reality forward contracts also have limited maturities, we do not include contracts with a maturity of two periods or longer in our model.

  5. 5.

    Note that this is only relevant if decision makers are risk-neutral or even risk-seeking.

  6. 6.

    Besides demanders for the homogeneous good, speculators may have incentives to engage in forward trading and take offsetting positions of the firms.

  7. 7.

    Cyert and DeGroot (1970) make a similar argument, that the counterpart’s choices cannot be observed in a simultaneous-move game and, hence, subjective expectations have to be considered.

  8. 8.

    We provide the first-period forward price that take the equilibrium choices into account in Eq. (5) below.

  9. 9.

    Note, that at this point we do not allow firms to enter production in time \(t=1\). Hence, firms cannot produce additional goods for the second period. Possible justifications for this limitation include the possibility to produce large lot sizes, high setup costs, or seasonal limitations in production.

  10. 10.

    For more detailed justification of the (\(\mu\), \(\sigma\))-approach see, e.g., Robison and Barry (1987).

  11. 11.

    The preference function describes risk-averse behavior as long as utility increases in expected profits and decreases in risk and marginal utility in expected profits does not increase (\(\partial \Phi /\partial \mu > 0\), \(\partial \Phi /\partial \sigma < 0\), and \(\partial ^2 \Phi /\partial \mu ^2 \le 0\)).

  12. 12.

    Note that Eq. (10) is equivalent to Eq. (4) from the Single Production setting in Sect. 4.1.

  13. 13.

    This study is registered in the AEA RCT Registry (unique identifying number: AEARCTR-0003940), and full experimental instructions are available at http://www.socialscienceregistry.org/trials/3940. The experimental data supporting this study are available from the corresponding author upon request.

  14. 14.

    Due to a computer crash, one session had only 19 rounds.

  15. 15.

    As will be discussed in our regression results in the next section, we find no significant correlation between current decisions and lagged feedback on the decisions of previous rival players.

  16. 16.

    This approach has been used to aid subject understanding in many other experimental studies of complicated strategic environments (e.g. Durham et al. 2004; Healy 2006; McIntosh et al. 2007; Gächter and Thöni 2010; Van Essen 2012; Van Essen et al. 2012).

  17. 17.

    To account for correlation between observations within a session, we use block-bootstrap t-tests, re-sampling at the session level (Fréchette 2012; Moffatt 2016).

  18. 18.

    We use multilevel Tobit models to account for the bounds on the decision variables (between 0 and 50) with random effects at the individual and session levels to control for correlation within individuals and within sessions (Fréchette 2012; Moffatt 2016). To examine whether current decisions are correlated with previously observed feedback, we also ran models that are similar to Models (1) and (2) in Tables 45, and 6 including controls for the lagged decisions of the other player in the previous round. We found no significant correlations, and chose to omit these results for brevity. We also tried alternative specifications using the log of round or round fixed effects, and we found very similar results to those presented in Tables 45, and 6.

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Acknowledgements

This work was supported by the DUBS Seedcorn Fund from Durham University. We are grateful for comments from editor Lawrence J. White and two anonymous referees, as well as Udo Broll, Jack Wahl, Bernhard Ganglmair, and participants of the 2014 Cambridge Business & Economics Conference, the Experimental Finance North American Finance Meeting 2016, and the 2017 AEA Annual Meeting in Chicago. A previous version of this paper was circulated under the title “Strategic corporate hedging”. We thank Minh-Lý Liêu and Sonja Warkulat for outstanding research support. Matthias Pelster gratefully acknowledges the warm hospitality of the Durham University Business School during his visits. Any errors, misrepresentations, and omissions are our own.

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Appendix: Expected Equilibrium Values: Numerical Example

Appendix: Expected Equilibrium Values: Numerical Example

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

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.

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.

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Cox, C., Karam, A. & Pelster, M. Two-Period Duopolies with Forward Markets. Rev Ind Organ (2021). https://doi.org/10.1007/s11151-021-09839-6

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Keywords

  • Corporate hedging
  • Duopoly
  • Dynamic setting
  • Game theory
  • Laboratory experiments
  • Strategic application

JEL Classification

  • D21
  • D22
  • D43
  • D53