Abstract
In this paper, we consider a non-cooperative differential game, in which each of the two competing firms (privately) holds and manages a renewable natural resource in order to produce a homogeneous good. We suppose that each firm’s resource stock grows at a different rate, depending on environmental factors or on firms’ technical experience and skills. We find an (asymmetric) linear feedback Nash equilibrium, in which each player’s strategy depends only on its available resource stock. We then carry out both short-run and steady-state comparative static analyses, from a social welfare point of view as well.
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Notes
Tending is the term applied to pre-harvest silvicultural treatment of forest crop trees at any stage after initial planting or seeding.
Regeneration is basic to the continuation of forests. It can take place through self-sown seed (natural regeneration), by artificially sown seed, or by planted seedlings.
Therefore, by definition, FNE is subgame perfect Nash equilibria. For more formal definitions of differential games, related information structures, strategies and equilibrium concepts, see Basar and Olsder (1999).
More precisely, to explicitly account for the non-negativity constraint on the output levels would yield the piecewise linear FNE \([\phi _i(s_i)]^+=\max \{\phi _i(s_i),0\}\) [see, e.g., Colombo and Labrecciosa (2013a)], meaning that firms do not produce at all whenever their resource stock is very low. For simplicity of exposition, following again (Colombo and Labrecciosa 2013b), we assume throughout that initial resource stocks are big enough, such that along the equilibrium path \(\phi _i(s_i)>0\) for \(i\in \{1,2\}\). Thus, the following analysis rests on the assumption that resource stocks and output levels of both firms are strictly positive quantities over time: in other words, we are abstracting from issues concerning resource exhaustion or monopoly position of one firm.
In the short-run, as expected, if \(s_2\) is sufficiently large, the highest production levels occur in the case of symmetric efficient firms. As we will show, the same holds in steady state.
Indeed, to this end, it is sufficient to impose \(r<\delta _2<\delta _1\).
Differently from what we observed in the previous footnote, for F2 the assumption \(r<\delta _2<\delta _1\) does not guarantee positive values for the resource stock in steady state. To this end, condition (11) must necessarily be satisfied.
The computations are shown in details in Appendix C.
In Appendix D we prove that, for \(r\rightarrow 0\): \(\frac{\partial Q^*}{\partial \alpha }>0\).
See Appendix E.
Subscript sym stands for “symmetric”.
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Acknowledgements
The authors are grateful to the Associate Editor and two anonymous referees for their useful comments and suggestions. This research has been financially supported by the Italian National Research Project PRIN, year 2012, Titled: Long Life, High Sustainability.
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Appendices
Appendix A: proof of Proposition 1
From Eqs. (7) and (8), replacing them in (6) we obtain:
Similarly, for F2:
Replacing Eqs. (25) and (26) in the HJB equation of player 1 and after some algebra we obtain the following system:
In a similar way, replacing Eqs. (25) and (26) in the HJB equation of F2, we obtain the system for \(v_0,v_1,v_2,v_{1,1},v_{1,2},v_{2,2}\).
Thus, we have a Riccati algebraic system of 12 equations in 12 unknowns. It is well known [see, e.g., Basar and Olsder (1999)] that such system admits only one solution that stabilizes the state for all initial conditions. In what follows, we show that the stable linear FNE is such that the strategy for each firm depends only on its own resource stock. To look for such equilibrium, from Eqs. (25) and (26), amounts to impose the following conditions: \(v_{1,1}=2k_{1,2}\) and \(k_{1,1}=2v_{1,2}\). Under these conditions, Eq. (31) yields:
whose solutions are \(k_{1,1}=0\) or \(k_{1,1}=2(r-2\delta _1)\). It is trivial to show that the first solution leads to the Cournot equilibrium: \(\phi _1=\phi _2=\displaystyle \frac{1}{3}\), which is unstable. Considering the second solution, it holds:
and for \(v_{1,1}\ne 0\) it holds:
From Eq. (30) we have:
Using such results, from Eq. (27) and from the corresponding equation for F2, we obtain the following linear system in \(k_1\) and \(v_1\):
whose solutions are:
Replacing such values in (25), we obtain the linear FNE for F1. Proceeding in a similar way, we obtain the FNE strategy for F2. The global asymptotic stability of the steady state can be easily checked by deriving the expressions for the equilibrium trajectories of the resource stocks (see Eq. (14) for F1).
Appendix B: proof of Corollary 1
We compute:
where \(\tilde{s}\) is given by (12). Moreover,
It holds:
Such a value is not dependent on \(s_1\), and is greater than zero for \(r<\displaystyle \frac{2\delta _1(2\alpha ^2+4\alpha -3)}{4\alpha -1}\), which is satisfied under Assumption (11).
We have:
where \(\tilde{s}\) is given by (13).
It is easy to show:
which is satisfied \(\forall \alpha \in ]\displaystyle \frac{2}{3},1[\) under Assumption (11).
For the short-run aggregate output, given by: \(\Phi (s_1,s_2)=\phi _1(s_1)+\phi _2(s_2)\), it holds:
which for \(r\rightarrow 0\) gives:
Furthermore we compute:
It is trivial to show that such a quantity is positive \(\forall \alpha \in ]\frac{2}{3},1[\) in the case \(r\rightarrow 0\).
Appendix C: proof of Corollary 2
We compute:
Under our parametric restrictions, we have: \(\displaystyle \frac{\partial s^*_1}{\partial \delta _1}>0\iff \alpha \delta _1^2(2\alpha -3)+2r\delta _1-r^2>0\). It holds: \(\alpha \delta _1^2(2\alpha -3)+2r\delta _1-r^2\le -(\delta _1-r)^2\le 0\).
Then, we compute:
We have: \(\displaystyle \frac{\partial s^*_1}{\partial \alpha }<0\iff 2\delta _1(2\alpha ^2+4\alpha -3)-r(4\alpha -1)>0\iff r<\frac{2\delta _1(2\alpha ^2+4\alpha -3)}{4\alpha -1}\), which is always the case under our Assumption (11).
As for the less efficient firm, let us compute:
Such quantity is positive for \(\delta _1\alpha (3\delta _1 - 2r) - 2\delta _1^2 + r^2 < 0\), which is satisfied if \(r>\delta _1(2\alpha - \sqrt{2(2\alpha ^2 - 3\alpha + 2)})\)—which is compatible with Assumption (11).
Finally, we compute:
Such quantity is positive if \(2 r^2 (\alpha ^2 - 1) - \delta _1(4\alpha ^3 + 5\alpha ^2 - 8\alpha - 4)r + \delta _1^2(12\alpha ^3 - 15\alpha ^2 + 4\alpha - 4) < 0\), which is satisfied for \(r<\delta _1\frac{\sqrt{16\alpha ^6 - 56\alpha ^5 + 81\alpha ^4 - 48\alpha ^3 - 64\alpha ^2 + 96\alpha - 16} + 4\alpha ^3 + 5\alpha ^2 - 8\alpha - 4}{4(\alpha ^2 - 1)}\), which is always the case under Assumption (11).
Equation (19) follows, for \(r\rightarrow 0\), from:
For \(r\rightarrow 0\), we have:
and
where \(4\alpha ^4+20\alpha ^3-21\alpha ^2+4\alpha -4<0\) for \(\displaystyle \frac{2}{3}<\alpha <\tilde{\alpha }\), whereas the opposite is true for \(\tilde{\alpha }<\alpha <1\), with \(\tilde{\alpha }\) given by (20).
Appendix D: proof of Corollary 3
It is trivial to show: \(\displaystyle \frac{\partial q^*_1}{\partial \alpha }\propto \displaystyle \frac{\partial s^*_1}{\partial \alpha }<0\). Moreover:
As to the less efficient firm, we have:
Moreover \(\frac{\partial q^*_2}{\partial \alpha }>0\), since \(q^*_2=\alpha \delta _1 s^*_2\) and \(\frac{\partial s^*_2}{\partial \alpha }>0\).
As to the aggregate output in steady state, we can compute:
Such quantity is positive for \(r<\delta _1\frac{3 \alpha - \sqrt{\alpha (-2 + (5 - 2 \alpha ) \alpha )}}{1 + \alpha }\), which is always the case under Assumption (11).
Moreover, for \(r\rightarrow 0\), we have:
Here, we also compute the steady-state price and profits, which turn out to be useful to get the results of Corollary 4.
The steady-state price is:
which, for \(r\rightarrow 0\) gives:
which is decreasing in \(\alpha \).
The steady-state profits are:
For \(r\rightarrow 0\), we have:
and we compute:
Finally, the steady-state aggregate profit is given by:
where:
For \(r\rightarrow 0\), we have:
and we can compute:
Thus, aggregate steady-state profit decreases as the efficiency gap decreases: the increase in F2’s profits does not counterbalance the reduction in F1’s profit.
Appendix E: proof of Corollary 4
In Colombo and Labrecciosa (2013b), the steady-state resource stock for each firm is given byFootnote 12: \(s^*_{i,sym}=\displaystyle \frac{1}{3\delta }\) (\(i\in \{1,2\}\)). In our model we have:
Indeed, it is easy to check that the first inequality is satisfied for every \(r>0\), whereas the third one holds true for every \(r<\frac{\delta _1(-2\alpha ^2 - \alpha + 6)}{2(1 - \alpha )}\), which is always the case under Assumption (11).
In Colombo and Labrecciosa (2013b), the global steady state for the resource is \(S^*_{sym}=\displaystyle \frac{2}{3\delta }\). If both firms in the symmetric model are “inefficient” that is: \(S^*_{sym}=\displaystyle \frac{2}{3\delta _2}=\displaystyle \frac{2}{3\alpha \delta _1}\), it holds: \(S^*<S^*_{sym}\) for all \((4-\alpha )r^2 + 2\delta _1(5\alpha ^2 - 6\alpha - 2)r - \alpha \delta _1^2(6\alpha ^2 + \alpha - 10) > 0\), which is satisfied for all \(\alpha \in ]\frac{2}{3},1[\).
As for the steady-state output levels, in the symmetric model, each firm produces the static Cournot quantity \(q^*_{i,sym}=\displaystyle \frac{1}{3}\), whereas in our asymmetric model:
Specifically, the first inequality is satisfied for all \(r<\frac{2\alpha \delta _1(4 - \alpha )}{\alpha + 2}\), whereas the second one holds true for \(r<\frac{2\delta _1(4\alpha - 1)}{2\alpha + 1}\), and both the requirements are satisfied under Assumption (2).
However, as compared with the Cournot levels, the lower production of the inefficient firm outweighs the higher output of the efficient firm. In fact, the aggregate steady-state output level [given by (23)] is smaller than the aggregate Cournot quantity (\(\frac{2}{3}\)) for every \(r<\delta _1( 2\alpha + 2-\sqrt{2(2\alpha ^2 - \alpha + 2)})\), which is always the case under Assumption (11). It is worth noting that symmetric firms, regardless of their efficiency, produce more than asymmetric firms, since \(Q^*_{sym}\) does not depend on the resource growth rate.
To get analytical results concerning aggregate profits and social welfare, we consider the limit case in which \(r\rightarrow 0\).
In this case, \(\Pi ^*\) is a decreasing function w.r.t. \(\alpha \) and \(\lim _{\alpha \rightarrow 1}\Pi ^*=\frac{2}{9}\), as a result: \(\Pi ^*>\Pi ^*_{sym}=\frac{2}{9}\).
Compared to the symmetric model, in our asymmetric game the aggregate profit tends to increase since the efficient firm’s higher profit is not counterbalanced by the inefficient firm’s lower profit. In particular:
Finally, let us consider the social welfare in steady state of the two considered models:
In our asymmetric model we have:
It holds: \(\forall \alpha \in ]\displaystyle \frac{2}{3},1[: \frac{d SW}{d \alpha }>0\), therefore the social welfare is maximum for \(\alpha \rightarrow 1\). In particular, for \(\alpha \rightarrow 1\): \(SW\rightarrow \displaystyle \frac{4}{9}\), that is the same value for the symmetric model, regardless of firms’ efficiency.
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Grilli, L., Bisceglia, M. A dynamic private property resource game with asymmetric firms. Decisions Econ Finan 43, 109–127 (2020). https://doi.org/10.1007/s10203-019-00266-7
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DOI: https://doi.org/10.1007/s10203-019-00266-7