The Role of Technology in an Endogenous Timing Game with Corporate Social Responsibility

Domenico Buccella; Luciano Fanti; Luca Gori

doi:10.1515/bejte-2024-0122

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The Role of Technology in an Endogenous Timing Game with Corporate Social Responsibility

Domenico Buccella , Luciano Fanti und Luca Gori

Veröffentlicht/Copyright: 3. Juni 2025

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Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 25 Heft 1

Abstract

This research studies the endogenous choice of simultaneous (Cournot) or sequential (Stackelberg) moves in a quantity-setting duopoly in which firms have social concerns and convex technologies. A parsimonious non-cooperative endogenous timing game (ETG) is developed to determine the sub-game perfect Nash equilibrium (SPNE). The article shows that the market structure strictly depends on the (endogenous) adoption of the available technology. In detail, when costs are convex, the sequential move equilibrium can emerge as the SPNE of the ETG. Results contrast the SPNE emerging in a duopoly – with or without CSR – in which firms produce with constant returns to scale technologies (linear costs), where only simultaneous competition occurs. The article also discusses the welfare outcomes corresponding to the SPNE. The main findings also offer empirical and policy implications. The paper finally considers a mixed duopoly and shows that the results of Amir and De Feo (2014. “Endogenous Timing in a Mixed Duopoly.” International Journal of Game Theory 43: 629–58) – sequential move equilibrium – are confirmed. Still, it pinpoints new results about the private firm’s degree of social concern in determining the market leadership when goods are strategic substitutes.

Keywords: endogenous timing; quantity competition; corporate social responsibility

JEL Classification: L1; D4

Corresponding author: Luca Gori, Department of Law, University of Pisa, Via Collegio Ricci, 10, I – 56126 Pisa, PI, Italy, E-mail: luca.gori@unipi.it

Acknowledgments

The authors gratefully acknowledge an anonymous journal reviewer for her/his valuable comments on an earlier manuscript draft. The authors are also indebted to Maria Alipranti, Michael Kopel and participants of the conference entitled “Oligo Workshop 2024,” held at the University of Crete (June 2024), Greece, and seminar participants at the University of Pisa, Italy.

Disclosure of potential conflict of interest: The authors declare that they have no conflict of interest.
Research funding: The authors declare that this study was funded by no institutions.
Declarations of interest: None declared.

Appendix A: The ETG with CSR-Oriented Firms and Linear Costs

This appendix presents the symmetric version of the model developed in the main text by assuming that firm i and firm j adopt constant-return-to-scale technologies. To do this, we assume that w _i = 0 and w _j = 0, and study the cases of symmetric (w _0,i = w _0,j = w ₀) and asymmetric (w _0,i ≠ w _0,j) linear technologies.

The analysis developed in this appendix represents a simplified version of the model of Hamilton and Slutsky (1990) and serves as a basis for comparison with the model in which at least one firm adopts a convex technology.

Assuming symmetric constant marginal costs, and following the notation introduced at the beginning of Section 3 in the main text, the cost functions of firm i and firm j become C _i = w ₀ q _i and C _j = w ₀ q _j, where 0 ≤ w ₀ < 1 is the marginal cost.

Tables A.1 and A.2 summarise the equilibrium output and profits emerging in the simultaneous and sequential sub-games.

The SPNE outcomes of the ETG with CSR-oriented firms and linear technologies are reported in Proposition A.1. Results confirm the main finding pinpointed by the endogenous timing model of Hamilton and Slutsky (1990) under profit-maximising (instead of CSR-oriented) firms, linear demand and equal constant average and marginal costs: the simultaneous-move equilibrium is the only competition mode that can be observed endogenously.

Table A.1:

Output.

Firm j →	L	F
Firm i ↓	L	F
L	q i * C / C , q j * C / C	q i * L / F , q j * L / F
F	q i * F / L , q j * F / L	q i * C / C , q j * C / C

Table A.2:

Profits (payoff matrix) and the ETG.

Firm j →	L	F
Firm i ↓	L	F
L	Π i * C / C , Π j * C / C	Π i * L / F , Π j * L / F
F	Π i * F / L , Π j * F / L	Π i * C / C , Π j * C / C

The entries of the tables are reported below:

(A.1) q i * C / C = q j * C / C = 1 − w 0 3 − 2 b ,

(A.2) q i * L / F = q j * F / L = 1 − w 0 1 + 1 − b 2 4 − 3 b ,

(A.3) q j * L / F = q i * F / L = 1 − w 0 1 + b 1 − b 4 − 3 b ,

and

(A.4) Π i * C / C = Π j * C / C = 1 − w 0 2 1 − 2 b 3 − 2 b 2 ,

(A.5) Π i * L / F = Π j * F / L = 1 − w 0 2 1 − 2 b 1 + 1 − b 2 4 − 3 b 2 ,

(A.6) Π j * L / F = Π i * F / L = 1 − w 0 2 1 − 2 b 1 + b 1 − b 4 − 3 b 2 .

The technical condition that must be satisfied to have well-defined equilibria in pure strategies for every strategic profile (one for each player) in this kind of ETG is b < 1 2 . Then, to derive all possible SPNE of the ETG, one must study the sign of the following two profit differentials for i = 1,2 , i ≠ j , that is:

(A.7) Δ Π A : = Π i * L / F − Π i * F / F = 2 1 − w 0 2 1 − b 1 − 2 b 1 − 8 b + 8 b 2 − 2 b 3 3 − 2 b 2 4 − 3 b 2 ,

and

(A.8) Δ Π B : = Π i * F / L − Π i * L / L = 1 − w 0 2 1 − b 1 − 2 b − 7 + 14 b − 12 b 2 + 4 b 3 3 − 2 b 2 4 − 3 b 2 ,

where Π i * F / F = Π i * L / L = Π i * C / C . The meaning of the thresholds in (A.7) and (A.8) is the same as those defined in the main text.

The analysis of the expression in (A.7) reveals that the sign of ΔΠ_A changes once (irrespective of w ₀) at b = 0.1453, representing the threshold values of the firm social concern such that ΔΠ_A = 0. Then, ΔΠ_A > 0 if b < 0.1453 and ΔΠ_A < 0 if b > 0.1453.

The analysis of the expression in (A.8) reveals that the sign of ΔΠ_B never changes and ΔΠ_B < 0 for any 0 ≤ b < 1 2 irrespective of w ₀.

We can now state the proposition (Proposition A.1) clarifying the endogenous market configuration of the ETG with CSR-oriented firms and linear costs (see also Figure A.1 for a geometrical portrait of the analytical results).

Proposition A.1.

The endogenous market structure of the ETG with CSR-oriented firms and (symmetric) linear costs is the following for any 0 ≤ w ₀ < 1.

If 0 ≤ b < 0.1453 the SPNE of the ETG is the simultaneous-move outcome (Cournot competition), in which both firms prefer to be the leader, L , L .
If 0.1453 < b < 0.5 the SPNE of the ETG is the simultaneous-move outcome (Cournot competition), in which both firms prefer to be the leader or the follower, L , L and F , F .

Proof.

Let 0 ≤ w ₀ < 1 hold. Then, ΔΠ_A > 0 and ΔΠ_B < 0 for any 0 ≤ b < 0.1453; ΔΠ_A < 0 and ΔΠ_B < 0 for any 0.1453 < b < 0.5. Q.E.D.□

Area A: SPNE L , L , Cournot competition. The ETG with symmetric marginal costs is a simultaneous-move game in which both firms prefer to be the leader (first-mover advantage, Π i * L / F > Π i * C / C > Π i * F / L and Π j * F / L > Π j * C / C > Π J * L / F ).
Area B: SPNE L , L and F , F , Cournot competition. The ETG with symmetric marginal costs is a simultaneous-move game in which both firms prefer to be the leader or the follower (the first-mover advantage and the second-mover advantage coexist, Π i * C / C > Π i * L / F > Π i * F / L and Π j * C / C > Π j * F / L > Π J * L / F ).

$Figure A.1: The ETG with CSR-oriented firms and (symmetric) linear costs: SPNE in the space b , w 0 $\left(b,{w}_{0}\right)$ . The figure depicts only the feasible region bounded at b = 0.5. The ETG is a simultaneous-move game irrespective of the values of b and w 0.$

Figure A.1:

The ETG with CSR-oriented firms and (symmetric) linear costs: SPNE in the space b , w 0 . The figure depicts only the feasible region bounded at b = 0.5. The ETG is a simultaneous-move game irrespective of the values of b and w ₀.

We pinpoint that the results studied and discussed in this appendix (Cournot competition) under the assumption of symmetric linear technologies hold also under the assumption of asymmetric linear technologies of the two CSR-oriented firms. In this case, the cost functions are C _i = w _0,i q _i and C _j = w _0,j q _j, where 0 ≤ w _0,i < 1 and 0 ≤ w _0,j < 1 are the marginal costs of firm i and firm j. We do not report the equilibrium results to avoid lengthening the paper further. In addition, assuming different degrees of social concerns of the two firms under linear technologies does not alter the SPNE results of the model, which continue to boil down to the simultaneous-move competition outcome.

Appendix B: The ETG with CSR-Oriented Firms and Mixed Technologies

This appendix considers a version of the ETG in which firms adopt mixed technologies. This implies that one firm (firm i) uses a decreasing-return-to-scale production function implying quadratic costs (increasing marginal costs), C i = w i 2 q i 2 (w _i ≥ 0), and the rival (firm j) uses a constant-return-to-scale production function implying linear costs (constant marginal costs), C _j = w _0,j q _j (0 ≤ w _0,j < 1).

Results of this appendix (i.e. existence of simultaneous-move equilibrium and sequential-move equilibrium in which the leader is the firm with quadratic costs) resemble those in which both firms have convex technologies and are in line with Amir and Grilo (1999), who show that the firm with positive marginal costs becomes the leader if the rival has no cost.

The profit function of firms i and j are respectively Π i = p q i − w i 2 q i 2 and Π i = p − w 0 , j q i .

The Nash equilibrium outcomes of the ETG with mixed technologies are summarised in Figure B.1, depicting the SPNE configurations in the space b , w i for two different values of the marginal cost of firm j, w _0,j = 0.2 and w _0,j = 0.5.

The case w _0,j = 0 exactly replicates the results of Figure 5 (Panel A) according to which Areas A, B, C and D prevail depending on the sign the profit differentials. The general outcome emerging under mixed technologies for any value of w _0,j ≠ 0 resembles this one, but with minor differences (about the shape and position of the loci of points such that ΔΠ_i,A = 0 and ΔΠ_j,A = 0). These differences are outlined in Figure B.1 (Panels A and B).

Results resemble those outlined in the main body of the text with one exception: the Stackelberg equilibrium emerges only with a configuration in which the leader is the firm with quadratic costs (firm i in our example). More specifically, this outcome (resembling one of the results of Amir and Grilo 1999) holds when the extent of increase in marginal costs of the firm adopting the convex technology is sufficiently high. Indeed, for a given value of b, an increase in w _i causes a reduction in the production of firm i and an increase in the production of the rival, firm j. The market configuration implies that firm i becomes the leader because L is a dominant strategy, whereas firm j is the follower because F is a dominant strategy. However, given the reaction of the price, there is a reduction in profits such that the profit of the leader is smaller than that of the follower.

Area A: SPNE L , L , Cournot competition. The ETG with mixed technologies is a simultaneous-move game in which both firms prefer to be the leader (first-mover advantage, Π i * L / F > Π i * C / C > Π i * F / L and Π j * F / L > Π j * C / C > Π J * L / F ). The sign of the profit differentials is the following: ΔΠ_i,A > 0, ΔΠ_i,B < 0, ΔΠ_j,A > 0 and ΔΠ_j,B < 0.
Area E: SPNE L , L , Cournot competition. The ETG with mixed technologies is a simultaneous-move game in which both firms prefer to be the leader (first-mover advantage, Π i * L / F > Π i * C / C > Π i * F / L and Π j * C / C > Π j * F / L > Π J * L / F ). The sign of the profit differentials is the following: ΔΠ_i,A > 0, ΔΠ_i,B < 0, ΔΠ_j,A < 0 and ΔΠ_j,B < 0.
Area C: SPNE L , L and F , F , Cournot competition. The ETG with mixed technologies is a simultaneous-move game in which both firms prefer to be the leader or the follower (the first-mover advantage and the second-mover advantage coexist, Π i * C / C > Π i * L / F > Π i * F / L and Π j * C / C > Π j * F / L > Π J * L / F ). The sign of the profit differentials is the following: ΔΠ_i,A < 0, ΔΠ_i,B < 0, ΔΠ_j,A < 0 and ΔΠ_j,B < 0.
Area D: SPNE L , F , Stackelberg competition. The ETG with mixed technologies is a sequential-move game in which the leader is the firm with quadratic costs (firm i). The firm with linear costs (firm j) has the second-mover advantage ( Π i * L / F > Π i * C / C > Π i * F / L and Π j * L / F > Π j * C / C > Π J * F / L ). The sign of the profit differentials is the following: ΔΠ_i,A > 0, ΔΠ_i,B < 0, ΔΠ_j,A < 0 and ΔΠ_j,B > 0.

$Figure B.1: The ETG with CSR-oriented firms and mixed technologies: SPNE in the space b , w i $\left(b,{w}_{i}\right)$ for different values of w 0,j . Panel A: w 0,j = 0.2. Panel B: w 0,j = 0.5. The white (sand-coloured) region represents the feasible (unfeasible) parameter space.$

Figure B.1:

The ETG with CSR-oriented firms and mixed technologies: SPNE in the space b , w i for different values of w _0,j. Panel A: w _0,j = 0.2. Panel B: w _0,j = 0.5. The white (sand-coloured) region represents the feasible (unfeasible) parameter space.

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Received: 2024-11-05

Accepted: 2025-03-11

Published Online: 2025-06-03

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Artikel in diesem Heft

https://doi.org/10.1515/bejte-2024-0122

Schlagwörter für diesen Artikel

endogenous timing; quantity competition; corporate social responsibility