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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)-1VJ, VJ = 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.
TH = TH (p,r,c,τ) . Substituting this for T in WT = 0 gives an identity and its partial
differentiation gives TαH = (-WTT)-1WTα , where α = p,r,c,τ . One gets for c and p
A1.2 sign TcH = sign r(1- e-rT)-1 > 0 ,
A1.3 sign TpH = sign A = f '( T ) - rf ( T ) - rf ( T ) e- rT (1 - e- rT ) -.
On the basis of Lemma 1 presented in the text we have
1. If FT = 0, then F(T,τ) -rE = 0, ⇒ A = 0.
2. If FT > 0, then F(T,τ) -rE > 0, ⇒ A < 0.
3. If FT < 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.
< >
A1.4 TpH <
> 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 TrH = 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
rewritten as B0 = -(pf(T) + V)(1
rT
-rT
1) . Applying the L’Hopital’s rule one can
prove that (1
rT
-rT
1)> 0, i.e., B0 < 0 .
As for the Hartman part B1 , note first that
—E = - Te - rT (1
dr
-rT)-1E-(1
-rT
)∫sF(s,τ)e-rsds . Integrating the last term in