Lemma 1 Let κ < к be given and assume that condition (10) is satisfied. Then it holds that
β > γq and R — nβ < ρ.
Proof: Using (9) it is easy to see that the (10) is equivalent to β > γq. To prove the second
statement of the lemma, we distinguish two cases. If ρ<R, then it follows from the assumptions
κ < K and ρ > r that (ρ — r)γκ/(κ — κ) > 0 > (ρ — R)/n. On the other hand, if ρ ≥ R, then
we obtain from (10) that (ρ — r)γκ/(κ — κ) > ρ — R +(n — 1)γq ≥ ρ — R ≥ (ρ — R)/n. In both
cases it therefore follows that (ρ — r)γκ/ (κ — κ) > (ρ — R)/n. Using (9) again, it is easy to see
that the latter condition is equivalent to R — nβ < ρ. <ι
Proof of Theorem 1: Suppose that all players j = i use the equilibrium strategies φj. In
this situation, player i faces the optimal control problem P defined as follows:
maximize |
+∞e-ρt[U(ci(t),yi(t)+Yz(t)) — KXi(t)]dt |
subject to |
z( t ) = R z ( t ) — xi ( t ) yi ( t ) = ryi ( t ) + Xi ( t ) — ci ( t ) yj(t) = ryj(t)+ βz(t)— qyj(t) + Yz(t)] for j = i |
where R = R — (n — 1)β. The Hamiltonian function of this problem is
H = U(ci, yi + γz) — KXi + λ(Rz — Xi) + μ(ryi + xi — ci) + V μj [ryj + βz — ¢(Vj + Yz)],
j=i
where λ and μk, k = 1, 2,... ,n, are the costate variables corresponding to z and yk, respectively.
Note that the Hamiltonian is jointly concave in (z, y, ci, Xi). The theorem is therefore proved
if there exist costate trajectories λ : [0, + ∞) → [0, + ∞) and μk : [0, + ∞) → [0, + ∞), k =
1, 2,.. .,n, such that the feasibility conditions of problem P as well as the following optimality
conditions are satisfied by ci(t) = q[yi(t) + γz(t)] and χi(t) = βz(t):
HCi = u ( ci( t ) /[ yi( t ) + Yz( t )]) — μi( t ) = 0,
(11)
(12)
Hxi = μi ( t ) — λ ( t ) — κ = 0,
λ( t ) = ρλ ( t ) — Hz = ( ρ — R) λ ( t ) — γw ( ci ( t ) / [ yi ( t ) + γz ( t )]) — ( β — γq)∑ μj ( t ), (13)
j=i
μ i( t ) = ρμi( t ) — Hyi = ( ρ — r ) μi( t )
— w(ci(t)/[yi(t) + Yz(t)]),
(14)
(15)
μ j( t ) = ρμj( t ) — Hyj = ( ρ — r + q) μj( t )
lim e ρt λ (t) z (t) + ∑ μk (t) yk (t) ≤ 0.
—⅛∙-l-rv^ι ' ∙
t→+∞
(16)
k=1
We define the costate trajectories by λ(t) = K — κ, μi(t) = к, and μj (t) = 0 for all j = i.
Note that all costate trajectories are constant and non-negative. Condition (11) holds because
of u'(q) = к and condition (12) follows immediately from the definitions of λ(t) and μi(t).
Substituting ci(t)/[yi(t) + γz(t)] = q and the definitions of the costate trajectories into (13) we
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