Optimal Private and Public Harvesting under Spatial and Temporal Interdependence

Appendix 2. Comparative statics of the Hartman model

The first-order condition for max W can be written as
{T}

A1.1 WT = pf '(T) - rpf (T) - rV + F(T, τ) - rE = 0,

where V = (1 -e^-rT)^-1V^J, V^J = pf(T)e^-rT -c . Assuming that the second-order
condition WTT = pf''(T) - rpf '(T) + FT (T,τ) < 0 holds, A1.1. defines implicitly the
privately optimal rotation age as a function of exogenous parameters, i.e.
T^H = T^H (p,r,c,τ) . Substituting this for T in W_T = 0 gives an identity and its partial
differentiation gives T_α^H = (-W_TT)^-1W_Tα , where α = p,r,c,τ . One gets for c and p

A1.2  sign T_c^H = sign r(1- e^-rT)^-1 > 0 ,

A1.3  sign TpH = sign A = f '( T ) - rf ( T ) - rf ( T ) e^{- r}T (1 - e^{- r}T ) ^-.

On the basis of Lemma 1 presented in the text we have

1. If F_T = 0, then F(T,τ) -rE = 0, ⇒ A = 0.

2. If F_T > 0, then F(T,τ) -rE > 0, ⇒ A < 0.

3. If F_T < 0, then F(T,τ) -rE < 0, ⇒ the sign of A depends whether

rc(1-e^-rT)^-1 +F(T,τ)-rE≥(<) 0 i.e.

<                              >

> 0 as rc(1 - e ^-rT )^-1 + F(T,τ) - rE<

>                              <

As for the effects of the real interest rate r, one has

A1.5 sign T_r^H = sign B,

where B = B₀ + B₁, with B₀ = V + r—V describing the “Faustmann part” and
B₁ = E + r-dE the “Hartman part” of the problem, respectively. The Faustmann part
is B₀ =-pf(T)-V-r(1-e^-rT)Tpf(T)e^-rT -T(pf(T)e^-rT-c)e^-rT and it can be

As for the Hartman part B₁ , note first that

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