In this appendix, we verify the conditions for existence of an equilibrium in affine strategies, given the utility function (17).
Assume the strategies
φ j (
x(
t)) of the
n − 1 players
j ≠
i to be given by
φ j (
x(
t)) =
γ j +
δ j x(
t). Then the equation of motion facing player
i in choosing his own decision rule
c i =
φ i (
x(
t)) becomes:
$$\dot{x} = g(x)-c_{i}-x\sum_{j\neq i}\delta _{\!j} - \sum_{j\neq i}\gamma_j,$$
(43)
and the current value Hamiltonian associated to his problem becomes:
$$H(x,c_{i},\lambda_{i}) = u(c_{i})+\lambda_{i}\left[ g(x)-c_{i}-x\sum_{j\neq i}\delta _{\!j} + \sum_{j\neq i}\gamma_j\right].$$
(44)
Assume
φ i (
x(
t)) =
γ i +
δ i x(
t) to be a solution. Then:
$$\frac{\dot{c}_i}{c_i}=\frac{\delta_i\dot{x}}{\delta_i x + \gamma_i},$$
or, substituting from Eq.
43:
$$\frac{\dot{c}_i}{c_i}=\frac{\delta_i}{\gamma_i + \delta_i x}\left[g(x)-c_{i}-x\sum_{j\neq i}\delta _{\!j} - \sum_{j\neq i}\gamma_j\right].$$
(45)
Furthermore, from the necessary conditions (
7) and (
8), along an interior solution,
$$\frac{\dot{c}_{i}}{c_{i}}=\frac{1}{\eta(c_{i})}\left[ g^{\prime }(x)-\sum\limits_{j\neq i}\delta _{\!j}-r_{i}\right].$$
(46)
Equalizing Eqs.
45 and
46, we find that for any utility function
u(
c i ), in order for
c i =
γ i +
δ i x to be a best response, the following differential equation must be satisfied:
$$[\gamma_i + \delta_i x]g^{\prime }(x)- \delta_i{\eta(\gamma_i + \delta_{i}x)}g(x)=[\gamma_i + \delta_i x]\left( \sum\limits_{j\neq i}\delta_{\!j}+ r_{i}\right) - \delta_i{\eta(\gamma_i + \delta_{i}x)}\left(x\sum_{k=1}^{n}\delta_{k}+ \sum_{k=1}^{n}\gamma_{k}\right),$$
where
δ i ,
γ i , the
δ j ’s and the
γ j ’s,
j ≠
i remain to be determined.
With
η(
γ i +
δ i x) =
θ ≠ 1, this becomes:
$$\left(\gamma _{i} + \delta _{i}x\right) g^{\prime }(x)-\theta \delta _{i}g(x)=Ax+B$$
(47)
where
$$A=\delta _{i}\left[ r_{i}-\theta \delta _{i}+\left( 1-\theta \right) \sum\limits_{j\neq i}\delta _{\!j}\right]$$
and
$$B=\gamma _{i}\left[ r_{i}+\sum\limits_{j\neq i}\delta _{\!j}\right] -\theta \delta _{i}\sum_{k=1}^{n}\gamma _{k},$$
which has as solution:
$$g(x)=\frac{1}{\delta _{i}^{2}}\left[ \frac{A\gamma _{i}-B\delta _{i}}{\theta}-\frac{A\gamma_i}{\left( \theta -1\right) }\right]-\, \frac{A\delta _{i}}{\theta - 1}x + k\left[\gamma _{i} + \delta _{i}x\right] ^{\theta },$$
being the constant of integration.
Therefore, in order for
φ i (
x(
t)) =
γ i +
δ i x(
t) to be a best response to affine strategies on the part of player
i’s rivals, the growth function must be of the form:
$$g(x) = \alpha x + \beta + k[\nu + \sigma x]^{\theta}.$$
Differentiating and substituting into Eq.
47 we get:
$$(1-\theta)\delta_i \beta x + \theta k \left[\sigma\left(\frac{\gamma_i + \delta_i x}{\nu + \sigma x} \right) - \delta_i \right]\times(\nu + \sigma x)^{\theta} + \gamma_i \beta - \theta \delta_i \alpha = Ax + B.$$
Choosing
k = 0, constant equilibrium values of
δ i and
γ i ,
i = 1, ...,
n are obtained by simultaneously solving:
$$\theta \delta_i + (\theta -1)\sum_{j\neq i}\delta_{\!j} - r_i - (\theta -1)\alpha = 0$$
(48)
and
$$\left[\alpha - r_i - \sum_{j\neq i}\delta_i\right]\gamma_i + \theta \delta_i \sum_{k=1}^{n}\gamma_k - \theta \delta_i \beta = 0.$$
(49)
It is easily verified from Eq.
49 that
$$\gamma_i = \frac{\beta \delta_i}{\alpha}$$
is the solution for
γ i , with
δ i obtained from Eq.
48.
The existence of an equilibrium in affine strategies therefore requires that g(x) be an affine function of the stock (β ≠ 0). To be implementable, β should be negative (δ i being positive), as otherwise it would imply that a positive quantity can be harvested from a zero stock. As argued in the text (footnote 5), β < 0 is also necessary for g(x) to be a sensible representation of the growth of a renewable resource stock. Hence, the decision rule will be c i = max {0, γ i + δ i x} so as to satisfy the necessary conditions (Eq. 7). Note that the solution for δ i is the same as in Eq. 21.
Similar calculations can be carried out to solve for an equilibrium in affine strategies in the case of θ = 1.