Optimal Forestry Contracts With Private Information
DIDIER TATOUTCHOUP
Université de Montréal, Département de sciences économiques
MAY 2008
Abstract
This article analyzes optimal royalty contracts in the forestry when the firm has private
information on the harvesting cost. We provide the characterization of the optimal royalty
and the optimal rotation period. Perfect correlation and full commitment imply that the
rotation period is longer in asymmetric information case than in symmetric case. We find
that the government can endogenize the highest cost. By doing this, the firm with high
cost would not produce. Then the government can increase its expected profits and make it
attractive for the firm to report truthfully its cost. We provide a comparison of the royalty
in the symmetric and asymmetric information cases. We end by analyzing optimal policy
under a handy uniform contract and characterize the loss in expected welfare surplus when
government chooses a uniform contract instead of a sophisticated mechanism.
1
Introduction
In recent years, many studies have focused on how to grant the rights of ownership in
natural resources. In forestry the owner will lease logging rights to companies specialized
in planting and harvesting, in return for preestablished royalty payments. For example,
Canadian provincial and federal government own 94 percents1 of Canada’s forest land and
delegate the exploitation to firms. The owner will often enter into a contract with forestry
companies which will carry out the exploitation of the forest. The question then arises:
What is the optimal royalty contract from the point of view of the forestry owner? Such
a contract should induce the best time to harvest a tree or stand of trees, determining
hence the optimal rotation period. If the growth function is known and if price, planting
and harvesting costs are constant and known by both the forestry owner and the firm,
the answer is straightforward. The Fausmann rule (Faustmann, 1849) implies that if the
forestry owner wishes to maximize the value of the forest land (land owner’s benefit) from
1
Source: Natural Resources Canada
1
planting and harvesting, the royalty schedule must induce the firm to harvest when the
increase in the net value of the standing forest over a unit time interval (rotation period)
equals the interest on the value of the stand plus the interest on the value of the forest
land. To set a royalty, government want to know the costs of the firm, which are planting
and harvesting costs.
In practice planting and harvesting costs are private information of the forest manager.
This asymmetry of information creates a situation where adverse selection may occur and
the optimal royalty must then take into account some informational constraints. What
will be the optimal rotation period and optimal royalty under adverse selection, compared
to that which would prevail under full information? How will it vary over time? The
problem constitutes an application of the well known principal-agent problem with adverse
selection.
Principal-agent theory, has been applied to natural resource problems by a few authors.
Gaudet, Lasserre, and Long (1995) study optimal nonrenewable resource royalty contracts
when the extracting agent has private information on the costs. Vedel, Lund, Jacobson,
and Helles (2006) apply the existing principal–agent models with a discrete set of types to
the Danish case of subsidized advisory services for private forest owners. Bowers (2003)
puts into a principal–agent framework the problem a government faces when choosing
policy instruments for sustainability in the privately operated forest industry. Adverse
selection issues are now well known, and appear in many papers and in modern textbooks.2
To the best of our knowledge, adverse selection has not yet been applied to the analysis
of the three cutting problem when forest manager has private information on the planting
and the harvesting cost. We will assume throughout the paper that the government is
the owner of the forest. We will analyze the case where asymmetric information is about
harvesting cost and the case where harvesting costs are perfectly correlated over time and,
hence, assume that the government commits itself to current and future royalty rules.
The paper is organized as follows. Section 2 presents the model. We first solve for
the reference scenario of symmetric information. The optimal rotation period satisfies in
that case the Faustmann rule. We end section 2 by characterizing the optimal incentive
mechanism and solve for the case where the firm has private information on harvesting
cost. Section 3 discusses the modified Faustmann rule under asymmetric information.
In section 4 we analyze the case where the highest cost firm is endogenous. We state
some necessary and sufficient conditions for the government to endogenize the highest
cost. Then we interpreted the optimal highest cost and end by analyzing its impact on
welfare. In section 5 we implement and compare the optimal royalty in both symmetric
2
Baron and Besanko (winter 1984) analyze a model of a regulated firm that is better informed about its cost
function than is the regulator. For another studies on adverse selection, see for example Baron and Myerson
(1982), Laffont and Tirole (1988). Useful surveys of various aspects of mechanism design with incomplete
information are contained in Baron (1989), Besanko and Sappington (1987) and Caillaud, Guesnerie and
Torole (1988).
2
and asymmetric information case. Section 6 provides a discussion in the absence of full
commitment. We end with some concluding remarks in section 7.
2
Characterization of the optimal Royalty
In this section we model the problem of determining the optimal royalty when the firm
harvesting cost structure is not known by the government. The firm’s cost at period k is
θk . Let X(T ) be the timber growth function, where T represents the age of a tree, and
assume that it has the following properties:
X 0 (T ) > 0
0 < T < a,
X ∈ C2
(2.1a)
X 0 (a) = 0
(2.1b)
00
(2.1c)
X (T ) < 0.
Suppose that when it is cut down, a stand of trees of age T yields a net profit in present
value given by:
(p − θ)X(T )e−rT − D
(2.2)
where p is the given market price of wood, the parameter D represents the total cost of
planting a unit of land, r is the discount rate and θ is the unit cost of harvesting the trees.
2.1
The model
The government can observe the time when the stand of trees is harvested but it cannot
verify the cost incurred by the firm at the beginning of the first rotation in the infinite
cycle. Therefore it cannot base its royalty schedule on the true harvesting cost of the firm.
We must expect that the firm would, if asked for a report, lie about its true cost function
whenever it is advantageous to do so.
The firm knows θk . The government doesn’t know θk , but the cumulative distribution
of θk , F (θk ), defined on [θL , θH ] is common knowledge. To this distribution function is
associated the density function f (θk ) > 0, assumed differentiable on [θL , θH ]. Knowledge
of this probability distribution is shared by both the government and the firm. We will
assume the monotone hazard rate, that means
h(θk ) =
F (θk )
, is increasing in θk .
f (θk )
(2.3)
Consider a sequence of times T1 < T2 < T3 ... such that at each Tk the tree is cut and
a new tree is planted, over an infinite horizon. Let T0 be the initial date of planting.
The government’s problem is to set a royalty schedule Rk = R(Tk ), that maximizes
expected social welfare. Social welfare is taken to be weighted sum of consumer’s surplus,
the government’s revenue and producer’s surplus.
We assume in the remainder of our analysis that royalties are levied at harvesting time.
3
By using the Faustmann framework [See, Faustmann (1849)], we may write social
welfare as
+∞
X
Rk e−r(Tk −T0 ) + αV
(2.4)
{[(p − θk )X(Tk − Tk−1 ) − Rk ]e−r(Tk −Tk−1 ) − D}e−r(Tk−1 −T0 )
(2.5)
W =
k=1
where
V =
+∞
X
k=1
is the firm’s surplus. The exogenously given price p, the discount rate r and the cost of
planting D are assumed to be known by both the government and the firm.
We adopt the standard assumption that 0 < α < 1: a dollar in government revenue is
valued more highly than a dollar that remains as profits in the hands of the firm.
Let assume that the firm’s costs θk s are perfectly correlated over time (θk = θ ∀ k).
Hence we shall assume that the government commits itself in current and future period.
Indeed after the first rotation, because costs are perfectly correlated, government can
observe the costs of future harvests and then decide to renegotiate the contract.
We model the problem as a direct revelation game. Hence the government chooses
an incentive mechanism, in the form of a pair (Rk (θ̃), Tk (θ̃)), that is optimal given the
optimal response θ̃, of the firm, where θ̃ denotes the value of its cost parameter as reported
by the firm. Given that mechanism, the firm then chooses its optimal response, in the
form of a θ̃; the value of which will depend on θ, the true value of its parameter. According
to the revelation principle, we can restrict attention to mechanisms in response to which
the firm will find optimal to reveal the true value of its cost parameter. Mechanism such
that θ̃ = θ, and knowing Tk (θ), and Rk (θ), we can obtain Rk (Tk ), by inverting θ = θ(Tk ).
In order to be feasible, an incentive scheme must also leave the firm with sufficient
surplus to cover its opportunity costs. Otherwise it would rationally choose not to participate.
Because costs remain constant over time, the rotation will be the same, and then
Rk = R ∀k. Then:
V =
e−rT (p − θ)X(T ) − e−rT R(θ) − D
,
1 − e−rT
(2.6)
and
R(θ)e−rT
e−rT (p − θ)X(T ) − e−rT R(θ) − D
+
α
.
(2.7)
1 − e−rT
1 − e−rT
The social welfare can also be view as a present value of the government for all future
W =
rotation.
2.2
The symmetric information case
Before going on to solve for the optimal royalty scheme under asymmetry of information,
we first derive the properties of the royalty schedule which would maximize social welfare
in the case where the government shares the firm’s information about its cost structure.
4
This symmetric information scenario is a useful benchmark, since it yields a first-best
solution.
The government then wishes to maximize
W =
Re−rT
+ αV
1 − e−rT
(2.8)
subject to V (θ) ≥ 0.
Where V (θ) is given by (2.6). Clearly, since 0 ≤ α < 1, the solution requires that
we set V = 0. In symmetric information, government will extract the entire producer’s
surplus. Hence the solution consists of choosing T to maximize
e−rT (p − θ)X(T ) − D
1 − e−rT
and then set R(θ) so as to collect that maximized value as royalties.
This yields the Faustman formula
(p − θ)X 0 (Ts ) = r(p − θ)X(Ts ) + rWs∗ (Ts ).
(2.9)
Where the subscript s refer to the solution under symmetric information case. A forest
stand shall be harvested when the rate of change of its value with respect to time is equal
to the interest on the value of the stand plus interest on the value of the forest land.
Determining Ts in (2.9), a royalty rule can be specified as follows:
R(θ) = (p − θ)X(Ts ) − DerTs .
2.3
(2.10)
The asymmetric information case
Consider now the situation where the true value of θ is known only to the firm.
We begin by characterizing the class of incentive compatible mechanisms; that is mechanisms in response to which the firm will choose to reveal its true cost.
Let φ(θ̃, θ) be the surplus of the firm if it reports θ̃ when the true cost takes the value θ.
The government asks the firm that reports θ̃ to harvest at T (θ̃) and to pay the government
the total royalty R(θ̃). Hence
φ(θ̃, θ) =
=
e−rT (θ̃) (p − θ)X(T (θ̃)) − e−rT (θ̃) R(θ̃) − D
1 − e−rT (θ̃)
(p − θ)X(T (θ̃)) − R(θ̃) − DerT (θ̃)
erT (θ̃) − 1
.
(2.11)
For the firm to respond truthfully, it is necessary that, for all θ ∈ [θL , θH ],
φ1 (θ̃, θ) = 0 for θ̃ = θ
(2.12)
φ11 (θ̃, θ) ≤ 0 for θ̃ = θ.
(2.13)
and
5
We will drop the argument and use the notation T = T (θ̃); R = R(θ̃), where there is no
risk of confusion. Condition (2.12) implies that the incentive scheme must satisfy
{(p − θ)X 0 (T )
−rerT
dT
dR rT
dT
− rDerT
−
}(e − 1)
dθ̃
dθ̃
dθ̃
dT
{(p − θ)X(T ) − R − erT D} = 0.
dθ̃
(2.14)
In addition, since by condition (2.12) φ1 (θ, θ) = 0 for all θ ∈ [θL , θH ], we have
φ11 (θ, θ) + φ12 (θ, θ) = 0 ∀θ ∈ [θL , θH ].
It follows that condition (2.13) is equivalent to
φ12 (θ, θ)
=
=
dT
{−X 0 (T )(erT − 1) + rerT X(T )}
dθ
rX(T )
dT
{−X 0 (T ) +
} ≥ 0.
dθ
1 − e−rT
(2.15)
Conditions (2.14) and (2.15) are local conditions. However, given the linearity of the cost
function in θ, they are sufficient for global incentive compatibility to hold (See for example
Baron (1989) for the method of proof.).
Note next that if we let V (θ) ≡ φ(θ, θ), then by the envelope theorem, we must have
dV
X(T )
= φ2 (θ, θ) = − rT
≤ 0.
dθ
e −1
(2.16)
Therefore V (θ) is a nonincreasing function of θ: an incentive compatible mechanism must
not leave low cost firms with a smaller surplus than high cost firms. Another way of seeing
this is to integrate (2.16), which gives:
H
Z
θH
V (θ) = V (θ ) +
θ
X(T (τ ))
dτ
erT (τ ) − 1
(2.17)
As it can be seen, V (θ) > V (θH ) for θ < θH .
Conditions (2.16) and (2.17) are local conditions. It must hold in a neighborhood of
θ̃ = θ.
Finally, it is assumed that the firm can decide to opt out in response to the announced
incentive scheme. This means that the combined royalty and time harvesting must satisfy
a participation constraint, we must have:
V (θ) ≥ 0 ∀θ ∈ [θL , θH ].
(2.18)
This set of constraints simply requires that the incentive scheme guarantees each firm a
nonnegative surplus. Since V (θ) is a nonincreasing function of θ by (2.16), constraints
(2.18) may be replaced by the single constraint
V (θH ) ≥ 0.
6
(2.19)
The government’s problem can now be stated as choosing {(R(θ), T (θ)) | θ ∈ [θL , θH ]} to
maximize expected welfare, given by:
Z
θH
EW =
[
θL
R(θ)e−rT (θ)
+ αV (θ)]f (θ)dθ
1 − e−rT (θ)
(2.20)
subject to (2.14), (2.15), (2.16), (2.18) or (2.19) and
R(θ) = (p − θ)X(T (θ)) − DerT (θ) − V (θ)(erT (θ) − 1).
(2.21)
This can be treated as an optimal control problem, with T (θ) the control variable and
V (θ) the state variable.
By substituting (2.21) into (2.20), we obtain:
Z
θH
{
EW =
θL
(p − θ)X(T (θ)) − DerT (θ)
− (1 − α)V (θ)}f (θ)dθ.
erT (θ) − 1
This gives the government’s objective as the expected value of the sum of the royalty
receipts and the producers’ surplus (the first term), minus that part of producers’ surplus
which carries no weight in its objective (the second term). (1 − α)V (θ) is the portion of
the firm’s profits that is not counted in the regulatory objective.
To solve the problem we will do abstract to the incentive compatibility constraint
(2.15), and then verify that the optimal solution satisfies it.
The Hamiltonian for the optimal control problem is
H(V, T, µ, θ) = {
(p − θ)X(T (θ)) − DerT (θ)
X(T (θ))
− (1 − α)V (θ)}f (θ) − µ(θ) rT (θ)
, (2.22)
erT (θ) − 1
e
−1
where µ(θ) is the costate variable corresponding to equation (2.16). Necessary conditions for optimality are given by:
dµ
dθ
dV
dθ
∂H
∂T
∂H
∂V
∂H
∂µ
= −
(2.23)
=
(2.24)
=
0 for interior solution.
(2.25)
Terminal conditions require that:
µ(θL ) = 0
(2.26)
V (θH )µ(θH ) = 0.
(2.27)
Equation (2.23) yields
dµ(θ)
= (1 − α)f (θ).
(2.28)
dθ
Hence the costate variable µ(θ) satisfies µ(θ) = (1 − α)F (θ) + cte. The terminal condition
(2.26), implies that cte = 0 and then
µ(θ) = (1 − α)F (θ).
7
(2.29)
Substituting µ(θ) into (2.22) and Differentiating the Hamiltonian with respect to the
control variable, solution of equation (2.25) must satisfy:
[p − θ − (1 − α)h(θ)]X 0 (Ta )
= r[p − θ − (1 − α)h(θ)]X(Ta )
(2.30)
−rTa
[p − θ − (1 − α)h(θ)]X(Ta )e
−D
.
+r
1 − e−rTa
The subscript a refers to the solution under asymmetry of information.
The terminal condition (2.27) requires that V (θH )µ(θH ) = 0. Since µ(θH ) = 1−α > 0,
this implies that V (θH ) = 0.
The term (1 − α)h(θ) may be interpreted as the marginal cost of dealing with the
informational asymmetry resulting from the firm’s private information (the marginal information rents) and p − θ − (1 − α)h(θ) represents the net price of a cubic meter of wood
in asymmetric information. Then we will assume that p − θ − (1 − α)h(θ) > 0 for all θ.
We can verify that (2.30) satisfies incentive compatibility constraint (2.15). Rewrite
(2.30) as
rD
X(Ta )
]=
.
(2.31)
1 − e−rTa
1 − e−rTa
The right hand side of (2.31) is positive and p − θ − (1 − α)h(θ) > 0, therefore :
[p − θ − (1 − α)h(θ)][−X 0 (Ta ) + r
−X 0 (Ta ) + r
To determine the sign of
dTa
dθ ,
X(Ta )
> 0.
1 − e−rTa
(2.32)
rewrite the first-order condition (2.30) defining the
optimal rotation interval as:
[X 0 (Ta ) − rX(Ta )] = r
[p − θ − (1 − α)h(θ)]X(Ta )e−rTa − D
.
[p − θ − (1 − α)h(θ)](1 − e−rTa )
(2.33)
Totally differentiating this expression yields:
[X 00 (Ta ) − rX 0 (Ta )]dTa = −r
D(1 + (1 − α)h0 (θ))
dθ.
[p − θ − (1 − α)h(θ)]2 (1 − e−rTa )
(2.34)
Because X is a concave and nondecreasing function of T , the left hand side of equation
(2.34) is negative.
Since h0 (θ) ≥ 0, then
dTa
dθ
≥ 0. Hence Ta satisfies incentive compatibility constraint
(2.15).
3
The Modified Faustmann Rule
We can rewrite (2.30) as:
[p − θ − (1 − α)h(θ)]X 0 (Ta )
= r[p − θ − (1 − α)h(θ)]X(Ta )
H ∗ (θ)
+r(
+ (1 − α)V (θ))
f (θ)
where H ∗ (θ) = H(V (θ), Ta (θ), µ(θ), θ).
8
(3.1)
H ∗ (θ)
+ (1 − α)V (θ) is the value of the forest land under asymmetric information.
f (θ)
Equation (3.1) defines the optimal rotation period under asymmetric information.
This is the usual Faustmann Rule, modified in order to take into account for informational constraints. Hence a forest stand shall be harvested when the rate of change of its
value with respect to time is equal to the interest on the value of the stand plus interest on
the value of the forest land for all future rotation. Notice that the net price and the value
of the forest land are properly corrected for the cost of the informational constraints.
Notice that since h(θL ) = 0, the usual Faustmann rule is unmodified for the lowest-cost
firm.
The value of the expected welfare at the optimum is given by
EW ∗ =
Z
θH
{
θL
H ∗ (θ)
+ (1 − α)V (θ)}f (θ)dθ.
f (θ)
(3.2)
Let us analyze the impact of asymmetric information on the optimal rotation period.
Proposition 1 Ta (θ) > Ts (θ) for all θ ∈ (θL , θH ] and Ta (θL ) = Ts (θL ).
Proof. Let
g(T ) = (1 − e−rT )[−X 0 (T ) + r
X(T )
] = −X 0 (T )(1 − e−rT ) + rX(T ) > 0.
1 − e−rT
Then
g 0 (T ) = (1 − e−rT )[−X 00 (T ) + rX 0 (T )] > 0.
Hence g(T ) is strictly increasing in T. We can rewrite (2.31) and (2.9) as:
(p − θ)g(Ts ) = rD
(p − θ − (1 − α)h(θ))g(Ta ) = rD.
This implies that
(p − θ)g(Ta ) = (p − θ)g(Ts ) + (1 − α)h(θ)g(Ta ).
Since h(θL ) = 0, Ta (θL ) = Ts (θL ).
For θ ∈ (θL , θH ], h(θ) > 0 and g(Ta ) > 0. Hence (1 − α)h(θ)g(Ta ) > 0.
Then:
(p − θ)g(Ta ) > (p − θ)g(Ts ) ⇒ g(Ta ) > g(Ts ).
Since g(T ) is strictly increasing in T, then Ta > Ts .
This situation is illustrated in Figure 1.
9
(3.3)
Rotation
O
Ta
.. .. ... .......... .. .. .. . . . . . . .
..
....
......
....
...
.... Ts
.....
.
.
.
.
.
....
......
. .....
/ Cost
θL
θ
H
Figure 1
4
The Commercial forest
The harvesting cost θ can also be viewed as the quality of forest. In asymmetric information case, the total cost of logging an hectare of forest equals the harvesting cost plus the
cost of information constraints which is increasing with θ.
Then a high value of θ may make the exploitation of forest of quality θ unprofitable.
It is the case when the value of forest land is negative. That means there is no alternative
use of land.
We focus here on commercial forest, that is the forest where the value of forest land is
non-negative. Hence we examine the nature of the optimal solution when the government
allows only the exploitation of commercial forest. The government wants to find an optimal
value θ̄ ∈ (θL , θH ] that maximize the expected social welfare. We can then replace θH by
θ̄ in the optimal control problem (2.20) and then determine its optimal value by solving
the following problem:
Z
θ̄
{
max EW =
T (θ),V (θ)
θL
(p − θ)X(T (θ)) − DerT (θ)
− (1 − α)V (θ)}f (θ)dθ
erT (θ) − 1
10
(4.1)
subject to (2.14), (2.15), (2.16), and
≥
V (θ̄)
θ̄
V (θL )
0
≤
θ
free
.
(4.2)
H
(4.3)
(4.4)
A similar argument to that above establishes that the optimal value of T (θ) is the
same as in the preceding problem and it is given by (2.30) .
The terminal value V (θ̄) satisfies V (θ̄) = 0. Optimal value of θ̄ is determining by the
transversality conditions
(θ̄ − θH )H(V (θ̄), T (θ̄), µ(θ̄), θ̄)
θ̄
=
0
≤ θH .
(4.5a)
(4.5b)
H(V (θ̄), T (θ̄), µ(θ̄), θ̄) = 0 gives
(p − θ̄ − (1 − α)h(θ̄))X(T (θ̄)) − DerT (θ̄) = 0.
(4.6)
Then we can state the following proposition which gives necessary and sufficient condition for existence and uniqueness of optimal interior value of θ̄ denote by θa∗ .
Proposition 2 There exist a unique θa∗ ∈ (θL , θH ) such that H ∗ (θa∗ ) = 0 if and only if
H ∗ (θH ) < 0.
Proof. Suppose that H ∗ (θH ) < 0. Let
Z(θ)
=
=
H ∗ (θ)
+ (1 − α)V (θ)
f (θ)
[p − θ − (1 − α)h(θ)]X(Ta ) − DerTa
erTa − 1
(4.7)
By envelope theorem:
dZ(θ)
(1 + (1 − α)h0 (θ))X(Ta )
=−
.
dθ
erTa − 1
Hence Z(θ) is strictly decreasing in θ. In addition Z(θL ) > 0 and Z(θH ) = H ∗ (θH ) <
0. Using intermediate value theorem there exist θa∗ ∈ (θL , θH ) such that Z(θa∗ ) = 0.
Given that V (θa∗ ) = 0, then H ∗ (θa∗ ) = 0.
Because Z(θ) is strictly decreasing in θ, θa∗ is unique.
Let assume that There exist a unique θa∗ ∈ (θL , θH ) such that H ∗ (θa∗ ) = 0. It follows
that Z(θa∗ ) = 0 and H ∗ (θa∗ ) = 0. Since Z(θ) is strictly decreasing in θ, Z(θH ) = H ∗ (θH ) <
0.
Figure 2 summarizes the problem which the government is facing.
11
Z(θ)f (θ)
O
A1
/
θL
θa∗
θ
θH
A2
Figure 2
The welfare is given by EW ∗ = A1 − A2 . The government can increase this welfare
by excluding types belong to an interval (θa∗ , θH ] and the welfare becomes EW ∗ = A1 .
The gain in welfare is A2 .
θa∗ can be interpreted as the limited quality of commercial forest in asymmetric information.
Let Ta∗ = Ta (θa∗ ), substituting (p−θa∗ −(1−α)h(θ̄∗ ))X(Ta∗ ) into (2.30) then the optimal
rotation period Ta∗ satisfies
−X 0 (Ta∗ ) + rX(Ta∗ ) = 0.
(4.8)
The optimal rotation period Ta∗ for the firm with cost parameter θa∗ is independent of
θa∗ and satisfies Wicksell rule for single rotation, that means; Ta∗ = Tw where Tw is the
Wicksell rotation. Having determined Ta∗ and substituting into (4.6) we can implicitly
find θa∗ by solving the equation:
(p − θa∗ − (1 − α)h(θa∗ ))X(Tw ) − DerTw = 0.
(4.9)
Let determine the corresponding value of θa∗ denote by θs∗ in symmetric information case.
θs∗ is such that Ws∗ (θs∗ ) = 0. The resulting rotation period would then satisfies Wicksell
12
rule. Hence we have:
θs∗ = p −
DerTw
.
X(Tw )
(4.10)
It follows that θs∗ − θa∗ = (1 − α)h(θa∗ ) > 0. The difference between θs∗ and θa∗ equals
the marginal cost of information at θa∗ . In the remaining of our analysis, let assume that
θs∗ =θH .
4.1
Interpreting the limited quality of commercial forest
We will focus here in the interesting case where the optimal cost θa∗ < θH .
θa∗ < θH , indicates that the interval [θL , θH ] can be divided into two regions. A region
[θL , θa∗ ] in which the firm accept to produce (commercial forest) an a region (θa∗ , θH ]
in which a firm would not participated and then would earned zero profit. That means
V (θ) = 0 for θ > θa∗ .
V (θa∗ ) = 0, since V (θ) is strictly decreasing in θ, then we have: V (θ) > 0 for θL < θ <
θa∗ .
By determining the limited quality of commercial forest the government also discourages a firm which report a high cost parameter. Indeed, if a firm with θ < θa∗ was to report
θ̃ > θa∗ , it would not participate and then would have zero profit compare to V (θ) > 0.
By doing this, the government raises its expected profits and makes it attractive for
the firm to report truthfully its cost parameter.
By using (4.6), comparative statics results on the limited quality of commercial forest
θa∗ ,
are:
∂θa∗
∂D
∂θa∗
∂α
∂θa∗
∂r
< 0
(4.11a)
> 0
(4.11b)
< 0.
(4.11c)
Expression (4.11a) indicates that as the cost of planting decreases, the commercial forest
increases.
Expression (4.11b) indicates that the greater is the weight the government places on
the firm’s profits, the greater is the commercial forest area. In the limit when α equals
one, government will find optimal to let all the firms participate, because by doing this
the firms will use the available information to maximize their profits and this will yield
the solution in symmetric information case.
Finally, expression (4.11c) indicates that when the interest rate increases, the commercial forest area decreases.
4.2
Impact of the limited quality of commercial forest
To determine the effect of the limited quality of commercial forest on the profit of the
firm, let V ∗ (θ) denotes the profit of the firm when θa∗ with θa∗ < θH occurs.
13
Proposition 3 If θa∗ < θH , the optimal profit of the firm is lower compare to the situation
where the highest cost is exogenous, and is given by
Z
∗
θH
V (θ) = V (θ) −
∗
θa
X(Ta (τ ))
dτ
erTa (τ ) − 1
(4.12)
The difference V ∗ (θ) − V (θ) in the profit of the firm is a constant. Since the profit of
the firm equals the rents it earns on its private information, by supposing that the highest
cost is endogenous is seen to reduce those rents.
The consequence of this proposition is that the optimal royalty is higher when the
terminal cost is endogenous. Hence:
R∗ (θ) = R(θ) + (V ∗ (θ) − V (θ))(erTa − 1).
The difference R∗ (θ) − R(θ) in the royalty of the government is strictly increasing in
θ.
5
Implementing The Optimal Royalty
The determination of the royalty rule is given by the expression
rTa (θ)
R(θ) = (p − θ)X(Ta (θ)) − De
rTa (θ)
− (e
Z
− 1)
θ
∗
θa
X(Ta (τ ))
dτ
erTa (τ ) − 1
(5.1)
We can also express the royalty as function of the optimal rotation period by inverting
θ = θ(T ) in (2.30) and substituting into (5.1). We expect that the optimal royalty
under asymmetric information is nondecreasing in optimal rotation period compare to the
royalty in symmetric information which we expect to be nonincreasing in optimal rotation
period. The intuition is that, since the information rent V (θ) depends on the royalty, the
government can reduce this rent by increasing the optimal rotation period. This is to give
no incentive to the firm to exaggerate its cost.
Now let us compare royalties in symmetric information with those provided in asymmetric information.
Proposition 4 There exist θ+ ∈ (θL , θa∗ ) such that Rs (θ+ ) = Ra (θ+ ) and Rs (θ) > Ra (θ)
for any θL ≤ θ < θ+ .
Proof. See Appendix.
This proposition is illustrated in Figure 3.
14
Royalty
O
..
....................................... . . . . . . . . . .
......
θL
....
...
....
....
..
..
...
.. Ra
...
Rs
θ
θa∗
+
/ Cost
Figure 3
It is more practical to express royalties as function of the commercial stand volume.
Let x = X(T (θ)) be the volume of timber when the forest is cut at time T (θ).
Ra (x) is defined for x ∈ [xL , xa ] x = X(Ta (θ)), θ ∈ [θL , θa∗ ].
Rs (x) is defined for x ∈ [xL , xs ] x = X(Ts (θ)), θ ∈ [θL , θa∗ ].
xL = X(Ts (θL )) = X(Ta (θL )), xs = X(Ts (θa∗ )), xa = X(Ta (θa∗ )).
Since Ta (θa∗ ) > Ts (θa∗ ) and X(.) is strictly increasing then xa > xs .
If x is known, we can find the cost by inverting X(T ). Then we have:
θs = Ts−1 (X −1 (x)) and θa = Ta−1 (X −1 (x)). This implies that Ts (θs ) = Ta (θa ). Thus,
we have the following proposition.
Proposition 5 There exist x̄ ∈ (xL , xs ) such that Rs (x̄) = Ra (x̄) and Rs (x) > Ra (x) for
any xL ≤ x < x̄.
Proof. For simplicity, denote: θa = θa (x) and θs = θs (x).
Ra (x)
=
(p − θa )x − DerTa (θa ) − (erTa (θa ) − 1)V (θa )
Rs (x)
=
(p − θs )x − DerTs (θs ) .
Since Ts (θs ) = Ta (θa ), we can write:
Rs (x) − Ra (x) = (θa − θs )x + (erTa (θa ) − 1)V (θa ).
For x = xL , θa (xL ) = θs (xL ) = θL , from (5.2) we have:
Rs (xL ) − Ra (xL ) = (erTs (θ
15
L
)
− 1)V (θL ) > 0.
(5.2)
For x = xs , θs (xs ) = θa∗ and θa (xs ) = θ̄a , where θ̄a < θa∗ . It follows from (5.2) that:
Rs (xs ) − Ra (xs ) = (θ̄a − θa∗ )xs + (erTa (θ̄a ) − 1)V (θ̄a ),
where
Z
∗
θa
V (θ̄a ) =
θ̄a
By (2.16),
dV
dθa
=
a)
− eX(T
rTa −1
X(Ta (τ ))
dτ.
erTa (τ ) − 1
≤ 0. Then :
d2 V
X 0 (Ta )(erTa − 1) − rerTa X(Ta ) dTa
=
−
> 0.
dθ2
(erTa − 1)2
dθa
Hence
X(Ta )
erTa −1
is a strictly decreasing function of θ. This would then imply that
V (θ̄a ) < (θa∗ − θ̄a )
X(Ta (θ̄a ))
xs
= (θa∗ − θ̄a ) rT (θ̄ )
.
rT
(
θ̄
)
a
a
a
a − 1
e
−1
e
Therefore Rs (xs ) − Ra (xs ) < 0. Using intermediate value theorem we conclude that there
exist x̄ ∈ (xL , x̄∗s ) such that Rs (x̄) = Ra (x̄) and Rs (x) > Ra (x) for any xL ≤ x < x̄.
Royalty
O
Rs
.....
..............
....................................... . . . . . . . . . .
.
...
....
....
..
..
...
.
. . . Ra
/
xL
V olume
xs
xa
x̄
Figure 4 : Royalties as function of volume harvested
6
Uniform contract
The mechanism described above is little practical use because the derived optimal policy
is often too complicated. Hence in this section we analyze optimal policy with incomplete
information when government chooses a uniform contract and characterize the loss in
expected welfare surplus by using uniform contract instead of nonlinear contract above.
A uniform contract offers a payment proportional to the volume of timber. In a uniform
16
policy, the government can only specify a royalty b per unit of the volume of timber
harvested. The firm chooses an optimal rotation period T and pays the government the
total royalty R(T ) = bX(T ).
6.1
Optimal policy under a uniform contract
We can rewrite the firm’s profit and the welfare as :
V (T, b, θ) =
e−rT (p − θ − b)X(T ) − D
,
1 − e−rT
(6.1)
e−rT bX(T )
+ αV.
1 − e−rT
(6.2)
W (T, b, θ) =
We will also assume that the terminal cost is endogenous, let denote it by θ̄ . The
firm chooses a profit-maximizing optimal rotation period which would satisfy Faustmann
rule3 . The government knows that optimal rotation period at θ̄ would yield zero profits
and determines a royalty b and optimal value of θ̄ that maximize expected social welfare
given these constraints. Let assume that b is the same regardless of the types. Then the
problem can be specified as:
Z
θ̄
max
b, θ̄
[
θL
bX(T (b, θ))e−rT (b,θ)
+ αV (T (b, θ), b, θ)]f (θ)dθ
1 − e−rT (b,θ)
(6.3)
subject to
(p − θ − b)X 0 (T )
= r(p − θ − b)X(T ) + rV (T (b, θ), b, θ)
V (T (b, θ̄), b, θ̄) ≥ 0 with equality if θ̄ < θH
θ̄
(6.4)
(6.5)
≤ θH .
(6.6)
Let the subscript u refers to the solution under a uniform contract.
Proposition 6 The solution of optimal policy under a uniform contract can be summarized by:
Case 1: θu∗ = θH
if Vu (θH ) = 0 then, b∗ = p − θH −
DeTw
X(Tw )
if Vu (θH ) > 0 then b∗ solves
Z
θH
θL
H
∂T e−rTu g(Tu ) ∗
X(TuH )e−rTu
[
](b
−
(1
−
α)h(θ))f
(θ)dθ
−
(1
−
α)
= 0.
H
∂b (1 − e−rTu )2
1 − e−rTu
(6.7)
Tu solves
(p − b∗ − θ)g(Tu ) = rD
3
Faustmann rule is given by (p − θ − b)X 0 (T ) = r(p − θ − b)X(T ) + rV (T (b, θ), b, θ)
17
(6.8)
where g(T ) is defined in expression (3.3) and Tw is the Wicksell rotation.
Case 2: θu∗ < θH
θu∗ solves
(p − θu∗ − (1 − α)h(θu∗ ))X(Tw )e−rTw − D
f (θu∗ ) +
1 − e−rTw
Z θu∗
∂T e−rTu g(Tu ) ∗
[
](b − (1 − α)h(θ))f (θ)dθ = 0
−rTu )2
θ L ∂b (1 − e
(6.9)
b∗ satisfies
b∗ = p − θu∗ −
DeTw
X(Tw )
(6.10)
Proof. See Appendix.
6.2
Loss in expected welfare surplus
To analyze the loss in expected welfare surplus induced by a uniform policy, we need
first to compare the optimal free end point (threshold) for the two policies and secondly
compare the optimal rotation period for the both contracts.
According to equation (4.9) and (6.10) we have that:
θa∗ − θu∗ = b∗ − (1 − α)h(θa∗ ) for θu∗ , θa∗ < θH .
(6.11)
Lemma 7 θa∗ = θu∗ = θH or θa∗ < θu∗ < θH
Proof. Let assume that θa∗ > θu∗ .
Let λ(θ) = (p − θ − (1 − α)h(θ))X(Tw )e−rTw − D. λ(.) is strictly decreasing in θ and
λ(θa∗ ) = 0, then λ(θu∗ ) > 0. Given that h(.) is strictly increasing in θ, equation (6.11)
implies that b∗ − (1 − α)h(θ) > 0 for any θ ≤ θu∗ . Using the fact that
g(Tu )
(1−e−rTu )2
∂T
∂b
> 0 and
> 0, (6.9) becomes
λ(θu∗ )
f (θu∗ ) +
1 − e−rTw
Z
∗
θu
[
θL
∂T
g(Tu )
](b∗ − (1 − α)h(θ))f (θ)dθ > 0
∂b (1 − e−rTu )2
(6.12)
contradiction.
The optimal rotation period for the both contracts, are given by:
(p − θ − (1 − α)h(θ))g(Ta ) = rD
(p − θ − b∗ )g(Tu ) = rD.
This implies that (p − θ − (1 − α)h(θ))g(Ta ) = (p − θ − b∗ )g(Tu ). Since g(T ) is strictly
increasing in T , the result depends on the sign of β(θ) = b∗ − (1 − α)h(θ). We can note
that β(θ) is a decreasing function of θ and β(θL ) = b∗ > 0. If β(θH ) > 0 then Tu > Ta . If
β(θH ) < 0 then there exists a unique θ̄ such that β(θ̄) = 0, hence



 Tu > Ta if θ < θ̄
Tu = Ta


 T <T
u
a
18
if
θ = θ̄
if
θ > θ̄
Let turn to the expected welfare surplus.
Let denote
[p − θ − (1 − α)h(θ)]X(T ) − DerT
(6.13)
erT − 1
The optimal social welfare under nonlinear policy and uniform policy are respectively
Z(T (θ), θ) =
given by:
(
W (Ta (θ), θ) =
Z(Ta (θ), θ)
0
if
(
θ ≤ θa∗
u
W (Tu (θ), θ) =
else
Z(Tu (θ), θ)
0
if
θ ≤ θu∗
else
Expected social welfare are E[W ] for nonlinear contract and E[W u ] for a uniform contract.
It is straightforward to verify that the nonlinear contract yields a higher expected social
welfare surplus than that obtained with a uniform contract (E[W ] ≥ E[W u ])4 . The
difference in expected social welfare under the two policies can be approximated for a
value of Tu close to Ta and it is given by the following proposition
Proposition 8 Case 1: for θa∗ = θu∗ = θH
E[W ] − E[W u ] w E[(Tu − Ta )
e−rTu g(Tu ) ∗
(b − (1 − α)h(θ))]
(1 − e−rTu )2
Case 2: for θa∗ < θu∗ < θH
E[W ] − E[W u ] w
Z
∗
θa
[(Tu − Ta )
θL
e−rTu g(Tu ) ∗
(b − (1 − α)h(θ))]f (θ)dθ −
(1 − e−rTu )2
Z θu∗
Z(Tu (θ), θ)f (θ)dθ.
∗
θa
Proof. See Appendix.
7
Conclusion
We have conducted an analysis of optimal royalty contracts in the forestry when the firm
has private information on its harvesting cost, by studying the case where the harvesting
cost are perfectly correlated over time. By assuming full commitment of the government,
we have established that all rotations must be of equal length and will not generally
satisfy Faustmann rule in asymmetric information, which would hold under symmetric
information. We have proved that the rotation period is longer in asymmetric information
case than in symmetric case. The cost of the firm can be seen as the quality of the forest.
Then the high value of a cost may make the exploitation of the forest of such quality
unprofitable. Hence, we show that the government can endogenize the highest cost of
the firm and then determines the limited area of the commercial forest. By making this
decision, the firm with cost parameter above this particular level would not produce.
Then the government can raise its expected profits and makes it attractive for the firm to
4
For proof see appendix.
19
report truthfully its cost parameter. We have proved that there exists a critical value of
the cost and a critical value of the commercial stand volume such that above those values
royalties under asymmetric information case are higher than those provided in symmetric
information. The nonlinear mechanism is too complicated and is little practical. We have
then derived optimal policy under a more useful uniform contract. We show that the
threshold is higher in uniform contract compare to nonlinear contract and the result in
term of optimal rotation interval is indeterminate. We have ended by characterizing the
loss in expected welfare surplus when a uniform contract is used instead of a nonlinear
contract.
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21