2.2.1. Contributions to a public good

Private contributions can be important on at least two grounds. NGOs are important lobbyists for the environment and they depend on private donations but also on public money, at least in Europe – and that to a quite significant extent. A government may use private donations as a gauge for its population's willingness to pay for a particular project and therefore how much to back it up or to motivate the population to contribute (e.g., by offering tax deductions). Donations are collected and given to victims of earthquakes, floods, other natural disasters (at home and abroad) and of wars and conflicts, and recently to refugees, etc.

Wichman (Reference Wichman2016) considers voluntary contributions to a public good $( x)$ that deliver the benefit of a warm glow (Andreoni, Reference Andreoni1990). Ma and Burton (Reference Ma and Burton2016) find evidence of the warm glow for Australian electricity consumers and the response of many to the refugee crisis in 2015 is a confirmation of the importance of warm glow against own economic interest. The principal's benefit is linear in the (total) contributions, and the principal values the donation of one dollar more than the most empathic agent,

(7)\begin{equation} V=vx,\;v>\bar{\theta}=\bar{t}>1. \end{equation}

Therefore, the action $x$ is a good one. Following Remark 1, we use the plural, agents, in the case of voluntary contributions to a public good. Incentives are in the form of subsidies, tax breaks and the promise to double one's contribution. However, Benabou and Tirole (Reference Benabou and Tirole2006) warn that monetary incentives can be counterproductive if they lower the value of the signal of showing one's green or social attitude.

Transforming the setup in Wichman (Reference Wichman2016) into a principal-agent problem requires abstracting from free riding. An agent's benefit from his contribution is to enjoy the warm glow,

\[ b( x) =wx,\;w=1, \]

linear for simplicity and normalized (to avoid unnecessary parameters). In an environmental context, $x$ may reflect a costly but environmentally friendly action, such as buying an electric car, installing photovoltaic panels, and donating to wildlife protection. Such actions do not only deliver a warm glow but could offer other benefits such as fuel saving and signalling an investor's green attitude. That is, $b$ includes all benefits and not only the warm glow. However, this benefit is not free due to the associated loss of hedonistic (logarithmic) utility given his wage income $l$ (identical across types),

\[ c( x) =\beta \left[ \ln ( l) -\ln ( l-x) \right]. \]

Assuming that the cost parameter $t$ is an agent's private information and dropping the constant term $( \beta \ln ( l) )$, an agent has the following objective:

(8)\begin{equation} W( x,t) =x+\frac{\beta \ln ( l-x) }{t}, \end{equation}

in which $( 1/t)$ measures the loss of hedonistic utility so that the principal prefers higher (i.e., less greedy) types. The superscript used in equation (2) is dropped because the argument (here $t$) is given explicitly; this applies also to the case of $\theta$ below.

Alternatively, the private information parameter may be attached to the benefit, i.e., to the warm glow. Then an agent has the objective:

(9)\begin{equation} W( x,\theta ) =\theta x+\beta \ln ( l-x), \end{equation}

and the application of proposition 1 yields the claimed observation equivalence,

(10)\begin{equation} x_{0}( t) =\max \left\{ 0,l-\beta /t\right\} =x_{0}( \theta ) =\max \left\{ 0,l-\beta /\theta \right\}, \end{equation}

after accounting for the non-negativity constraint, $x\geq 0$, because contributions cannot turn negative. It seems a priori unclear which of the two specifications, equations (8) or (9), one should favor.

2.2.2. Rainforest

The protection of tropical forests mitigates climate change (as a sink for negative emissions) and maintains biodiversity. Incentives are only successful if (financial) benefits from conservation measures are larger than costs for relevant decision-makers (Kremen et al., Reference Kremen, Niles, Dalton, Daily, Ehrlich, Fay, Grewal and Guillery2000). Incentives include support to eco-tourism, the compensation of willing landowners for not developing their land and for losses from wildlife depredation.Footnote ¹

Harstad and Mideksa (Reference Harstad and Mideksa2017) investigate how donors can protect natural habitats. A principal is concerned about conservation, say Norway's government about a tropical rainforest of original size $R$ and an agent who enjoys the benefits from logging $( x)$ and from the remaining parts of the rainforest $( R-x)$. Although different conservation projects (such as wetlands or wildlife) fit the framework, the reference will be to a rainforest. Norway's benefit from protecting a particular patch of a rainforest $( V)$ is presumably linear in the protected size given the many options including those in other countries,

(11)\begin{equation} V=v( R-x),\;v>0. \end{equation}

Clearly, the agent's action, logging, is a negative one in the sense of harming the principal's interest $( V^{\prime }<0)$. The agent's benefit from logging is linear if he is a price taker ($p$ denotes the profit per unit),

\[ b( x) =px,\;p>0, \]

but logging is also costly if the agent appreciates the remaining rainforest. Assuming a standard (logarithmic) utility function (Harstad and Mideksa (Reference Harstad and Mideksa2017) assume linear relations and ignore private information), the recipient country's convex cost from logging is

\[ c( x) =\ln ( R) -\ln ( R-x), \]

due to the loss associated with a shrinking rainforest. Combining and dropping the constant term $( \ln ( R) )$ yields the agent's objective in terms of his private information about the costs,

(12)\begin{equation} W( x,t) =px+t\ln ( R-x). \end{equation}

The cost parameter $t$ enters now linearly so that the principal again prefers high types since logging is an action that harms the principal. The superscript $t$ (and $\theta$ below) is again dropped because the argument $t$ is made explicit in $W$.

Alternatively, assume that the principal knows the agent's valuation of the rainforest but not his need for cash from timber exports, which leads to the agent's objective:

(13)\begin{equation} W( x,\theta ) =\frac{1}{\theta }( px) +\ln ( R-x), \end{equation}

and a high value of $\theta$ indicates less need for cash. Proposition 1 implies observation equivalence,

(14)\begin{equation} x_{0}( t) =R-\frac{t}{p},\;x_{0}( \theta ) =R-\frac{ \theta }{p}, \end{equation}

and logging $x_{0}$ remains positive as long as $t<pR$.

The crucial difference from the other example is that it seems easier to choose the private information parameter, namely the developing country's evaluation of its rainforests, i.e., the cost parameter $t$.

3.1.1. Cost is private information

(15)\begin{equation} U( \hat{t},t) :=W( x( \hat{t}),t) +s( \hat{t}) \end{equation}

defines the net payoff of an agent of type $t$ pretending the type $\hat {t}$ using $W=W^{t}$ from equation (2) but dropping the superscript since the argument $t$ is explicitly indicated. The revelation principle allows us to restrict the analysis to incentives that ‘force’ the agent to the tell the truth. This adds the inequality,

(16)\begin{equation} U( \hat{t},t) \leq U( t,t) =:U( t) \;\forall t,\hat{t}\in \left[ \underline{t},\overline{t}\right], \end{equation}

to the principal's optimization problem equation (1).

Assuming that the cost parameter $t$, which scales the agent's consumption benefit, is private information, the principal's optimal offer can be derived by solving the following optimal control problem with the constraints equations (18) and (19):

(17)\begin{align} &\underset{\left\{ x( t) \geq 0\right\} }{\max }\int_{\underline{t}}^{{\overline{t}}} \left[ vx( t) +x( t) +\frac{\beta \ln ( l-x( t) ) }{t} -U( t) \right] dF(t), \end{align}

(18)\begin{align} &\dot{U}(t) =W_{t}={-}\frac{\beta \ln ( l-x( t) ) }{t^{2}}, \end{align}

(19)\begin{align} &U(t)\geq U_{0}( t) :=\max \left\{ \frac{\beta \ln ( l) }{t},l-\frac{\beta }{t}+\frac{\beta \ln \left( \frac{\beta }{ t}\right) }{t}\right\} \quad \forall t\in [ \underline{t},\overline{t}]. \end{align}

The objective equation (17) results from subtracting the subsidies, $s=U-W$ due to the definition in equation (15) using equation (2) for $W$, from the principal's gross payoff $V$ from equation (7). The control constraint, $x( t) \geq 0$, stipulates that only non-negative contributions are possible. The differential equation (18) results from applying the envelope theorem to compute the total derivative of $U( t)$ from equation (16) and is called the incentive compatibility constraint; recall that dots denote total derivatives with respect to the private information parameter. Although equation (18) is only derived from the first order condition that truth telling is optimal, we verify that it is also sufficient. The last constraint equation (39) is called the participation (or individual rationality) constraint and it requires that the principal's offer must exceed the agent's outside option, the reservation price $U_{0}$, which is given by substituting $x_{0}$ from equation (10) into equation (19); the two cases in equation (19) reflect that an agent may contribute, $x_{0}>0$, or not, $x_{0}=0$. While most principal-agent models assume a constant (and normalized to $0$) reservation price, $U_{0}$ here is type dependent even if $x_{0}=0$ because $\dot {U}_{0}=-({\beta \ln ( l-x_{0}) }/{t^{2}})\neq 0$ and possibly declining as well as $U( t)$.

Setting up the Hamiltonian ($H$, $\lambda$ denoting the costate and dropping the argument $t$) in general terms,

\[ H=\left[ V+W-U\right] f+\lambda W_{t}, \]

the first order optimality conditions for interior solutions are

(20)\begin{align} ( &V_{x}+W_{x}) f+\lambda W_{tx} =0, \end{align}

(21)\begin{align} &\dot{\lambda} =f,\;\lambda ( \overline{t}) =0. \end{align}

Integrating equation (21) yields $\lambda =F-1,$ and substituting it into equation (20) yields

(22)\begin{equation} V_{x}+W_{x}=\frac{W_{xt}}{h}. \end{equation}

Equating the left hand side in equation (22) to zero determines the first best,

(23)\begin{equation} x^{1}:=\arg \underset{x}{\max }V+W, \end{equation}

from maximizing the joint objective of principal and agent (and ignoring possible constraints, such as non-negativity). The right hand side is positive (given a positive action, as in the case of private contributions to public good, $W_{xt}=({c}/{t^{2}})>0$) and determines the agency cost that leads to an allocation below the first best except at $t=\bar {t}$, at which point the right hand side in equation (22) vanishes due to $( 1/h( \bar {t}) ) =0$. Thus, $x( \bar {t}) =x^{1}( \bar {t})$, which is known as no distortion at the top. The solution of equation (22), denoted $x^{r}$, is called the relaxed program (see Fudenberg and Tirole, Reference Fudenberg and Tirole1992), thus the use of the superscript $r$). It is optimal only if it satisfies all constraints: of the control (non-negativity), of participation equation (19) and of monotonicity, here $\dot {x}\geq 0$, since the principal wants higher contributions from the agent.

Applying the condition equation (22) to the objectives equations (7) and ( 8),

(24)\begin{equation} v+1-\frac{\beta }{t( l-x) }=\frac{\beta ( \overline{t} -t) }{t^{2}( l-x) }, \end{equation}

allows us to compute the interior (= relaxed program) solution,

(25)\begin{equation} x^{r}( t) =l-\frac{\beta \bar{t}}{t^{2}( 1+v) }. \end{equation}

Therefore, positive contributions, $x^{r}( t) >0$, result only for the types

(26)\begin{equation} t>t^{\min }:=\sqrt{\frac{\beta \bar{t}}{l( 1+v) }}. \end{equation}

And once the relaxed program is positive, it always exceeds the agents’ own contributions, but remains below the first best (except at the top),

\[ x_{0}\leq x^{r}\leq x^{1}=\max \left\{ 0,l-\frac{\beta }{t( 1+v) } \right\}. \]

The type $t^{\min }$ determines the marginal type, i.e., only the types $t\geq t^{\min }$ participate. The reason is that $U$ cuts $U_{0}$ at $t=t^{\min }$ from below,

\[ \dot{U}( t) ={-}\frac{\beta \ln ( l-x^{r}( t) ) }{t^{2}}>\dot{U}_{0}( t) ={-}\frac{\beta \ln ( l) }{t^{2}},\;t>t^{\min }\text{ and close to }t^{\min }, \]

so that the participation constraint equation (19) is met first locally and then for all $t>t^{\min }$ due to $x^{r}>x_{0}$.

Proposition 2 The optimal mechanism consists of the relaxed program $x^{r}( t)$ from equation (25) coupled with the boundary solution ($x_{0}$ and thus no contribution),

(27)\begin{equation} x\left( t\right)= \begin{array}{ccc} 0 && \leq\\ &\text{ if }t&\max \left\{ \underline{t},t^{\min }\right\} \\ l-\frac{\beta \bar{t}}{t^{2}\left(1+v\right)}&&< \end{array}, \end{equation}

and the subsidies assuming $t^{\min }>\underline {t}$,

(28)\begin{equation} s(t) = \begin{array}{ccc} 0 &\leq \\ &\text{ if }t \quad t^{\min }\\ \dfrac{2\sqrt{\beta l(1+v)/t}}{\sqrt{\bar{t}}}+\dfrac{\beta \bar{t}}{ t^{2}(1+v) }-l-\dfrac{2\beta }{t} &> \end{array}. \end{equation}

Integrating the incentive compatibility constraint equation (18) yields the payoff of an agent of type $t$,

\begin{align*} U( t) &=U_{0}( t^{\min }) +\int_{t^{\min }}^{{t}}W_{z}( x^{r}( z),z) dz\\ &=\frac{\beta ( \ln ({(\beta \bar{t}}/{( 1+v) t^{2})}) -2) }{t }+\frac{2\sqrt{\beta ( l( 1+v) ) }}{\sqrt{\bar{t}}}, \end{align*}

and substracting,

\[ W( x^{r}( t),t) =l+\frac{\beta \ln ({(\beta \bar{t}}/{( 1+v) t^{2})}) }{t}-\frac{\beta \bar{t}}{ t^{2}( 1+v) }, \]

determines the subsidies equation (28) if $t^{\min }>\underline {t}$, because $s=U-W$.

Figure 1 shows the contract in a schematic form although it is based on the numerical example used in figure 2 for a comparison with the alternative representation equation (9) of the agents’ payoffs. The panel at the top shows how the contributions depend on the type: $x( t)$ is increasing and concave. Absent incentives, only a minority contributes and only a little $( x_{0})$. This may justify the incentive because it increases participation as well as contributions; for completeness, the first best $( x^{1})$ is included. The panel on the bottom left side shows the reservation price $( U_{0})$ and the agents’ payoffs, if incentivized $( U)$ but only for the participating types, $t\geq t^{\min }$; note that both are declining but $U$ less than $U_{0}$ for $t\geq t^{\min }$ so that the participation constraint is met (as claimed). Replacing in equation (28) by solving equation (24) for

\[ t=\sqrt{\frac{\beta \bar{t}}{( 1+v) ( l-x) }}, \]

yields subsidies depending on the contributions (i.e., $s( x)$, which is suppressed due to its cumbersome expression), increasing and convex.

Figure 1. Optimal incentives in order to increase private contributions to a public good if the cost parameter t is private information.

Figure 2. Comparing the contributions to a public good and the necessary subsidies if either the cost parameter (t, bold, measuring the consumption benefit) or the benefit parameter (θ about the warm glow) is private information, ν = 2, β = 10, l = 10.

3.1.2. Benefit – that is, the warm glow – is private information

Assuming that the agents’ payoffs are given by equation (9) and following the above outlined procedure, the optimal contract can be derived by solving the following optimal control problem:

(29)\begin{align} &\underset{\left\{ x( \theta ) \geq 0\right\} }{\max }\int_{\underline{\theta }}^{{\overline{\theta }}}\left[ vx( \theta ) +\theta x( \theta ) +\beta \ln ( l-x( \theta ) ) -U( \theta ) \right] dF, \end{align}

(30)\begin{align} &\dot{U}( \theta ) =W_{\theta }=x( \theta ) >0, \end{align}

(31)\begin{align} &U( \theta ) \geq U_{0}( \theta ) :=\max \left\{ \beta \ln ( l),\;l\theta -\beta +\beta \ln \left( \frac{\beta }{ \theta }\right) \right\} \text{ }\forall \theta \in \left[ \underline{\theta },\overline{\theta}\,\right]. \end{align}

The maximization of the expected value of all contributions over and above the subsidies, i.e., the objective equation (29), is constrained by incentive compatibility equation (30) and individual rationality equation (31) in which the reservation price $U_{0}( \theta )$ is again type dependent but this time increasing, $\dot {U}_{0}=x_{0}>0$, except at the boundary $x_{0}=0$.

Expressing the contracting problems as optimal control problems reveals some differences in spite of the equivalence out of contract: first, between the integrands, objective equation (29) versus equation (17); second, different ‘dynamic’ constraints resulting from the incentive compatibility constraints, equation (30) versus equation (18); and third, different path constraints because of the different reservation prices, equation (31) versus equation (19).

The derivation is as in section 3.1.1. Therefore, we can start with the computation of the relaxed program (using again the superscript $r$) from (22) using the specification equation (8),

(32)\begin{equation} v+\theta -\frac{\beta }{l-x}=\overline{\theta }-\theta \Longrightarrow x^{r}( \theta ) =l-\frac{\beta }{v-( \overline{\theta } -2\theta ) }\leq x^{1}=l-\frac{\beta }{v+\theta }. \end{equation}

The contributions are of course again below the first best (except at the top, $\theta =\overline {\theta }$) and cannot exceed the wage $( l)$. Hence, $v+2\theta >\overline {\theta }$ for which $v>\overline { \theta }$ is sufficient, i.e., the principal always values the contribution more than the agent with the highest warm glow (actually, $v>\overline { \theta }-\theta$ is sufficient). Given this sufficient condition, the relaxed program exceeds the unincentivized contributions but must in addition be positive,

(33)\begin{equation} x^{r}( \theta ) =l-\frac{\beta }{v-( \overline{\theta } -2\theta ) }\geq 0\Longleftrightarrow \theta \geq \theta ^{\min }:= \frac{\overline{\theta }-v}{2}+\frac{\beta }{2l}, \end{equation}

so that only higher types, $\theta \geq \theta ^{\min }$, contribute at least if incentivized (assuming $\theta ^{\min }>\underline {\theta }$). And indeed, the intersection of the relaxed program with the abscissa at $\theta ^{\min }$ determines the marginal type since $\dot {U}( \theta ) =x^{r}>\dot {U}_{0}( \theta ) =x_{0}$ for all $\theta \geq \theta ^{\min }$ ensures that the participation constraint holds for all $\theta \geq \theta ^{\min }$. Therefore, the subsidy policy enlarges the set of contributing types since $x_{0}( \theta ) >0\Longleftrightarrow \theta >\beta /l>\theta ^{\min }$, compare equation (9).

Proposition 3 Assuming $\theta ^{\min }>\underline {\theta }$, the optimal contract asks the types $\theta \geq \theta ^{\min }$ from equation (33) to contribute $x^{r}( \theta )$, increasing with respect to the type $\theta$, from equation (32) and pays the subsidies,

(34)\begin{equation} s( \theta ) =\left\{ \begin{array}{c} 0 \\ \dfrac{1}{2}\left[ l( v-\overline{\theta }) -\dfrac{\beta }{ v+2\theta -\overline{\theta }}+\beta \ln \dfrac{l( v+2\theta -\overline{ \theta }) }{\beta }\right] \end{array} \text{ for }\theta \begin{array}{c} < \\ \geq \end{array} \right\} \theta ^{\min }, \end{equation}

which are increasing and convex with respect to types and contributions,

(35)\begin{equation} \sigma ( x) =( v-\overline{\theta }) x+\beta \ln \frac{ l}{l-x}\text{ if }x>0. \end{equation}

This mechanism includes more types than its counterpart equation (2), i.e.,

(36)\begin{equation} t^{\min }>\theta ^{\min }\Longleftrightarrow l\geq \frac{\beta ( 1+v-2 \overline{t}) }{( v-\overline{t}) ( 1+v) }\text{ since }\theta =t. \end{equation}

Inequality equation (36) depends on the wage level and the following corollary gives a sufficient condition for $t^{\min }>\theta ^{\min }$ that renders the numerator in equation (36), $( 1-\overline {t}) +( v-\overline {t})$, negative.

Figure 2 compares the agents’ voluntary contributions without incentives $( x_{0})$ and with incentives depending on which parameter, $t$ or $\theta$, is treated as private information and includes the necessary subsidies. The first and crucial observation is that the incentives differ across the two representations of the agents’ objectives equations (8) and ( 9) although both descriptions imply the same actions in the absence of incentives (observation equivalent). Incentives based on private information about the warm glow $( \theta )$ include more participants but the induced contributions over and above the necessary subsidy are very small. Indeed, subsidies can even exceed an agent's contribution (e.g., for all types $\theta >1.62$ in figure 2). It is hard to recommend such incentive schemes in particular if one accounts in addition for deadweight costs of public funds. However, the principal still benefits because she values the contribution above the agent's warm glow. If the consumption benefit is private information $( t)$, participation is more selective. Although it does not yield higher (except for the high types and less average) contributions, it comes much cheaper in terms of subsidies. Hence, the increment raised is substantially larger. Nevertheless, the above conclusion about $\theta$ extends to $t$: it is also a poor instrument and this is without accounting for the crowding out effect a la Frey (see, e.g., the empirical application to an environmental issue in Frey and Oberholzer-Gee (Reference Frey and Oberholzer-Gee1997)) or the depreciation of the signal of a truly voluntary contribution as argued in Benabou and Tirole (Reference Benabou and Tirole2006).

Why is there this difference, given that: (i) both formulations yield the same contributions absent incentives, (ii) the enumeration of types is identical, $\theta =t$, and (iii) moreover, normalized, $E\theta =Et=1$? The left hand sides of the relaxed program conditions are declining (due to concave objectives) and characterize the first best if equated to zero. Since $W( x,\theta ) =tW( x,t)$, the left hand side of equation (25) exceeds its counterpart in equation (32) for $t=\theta <1$ and the converse holds for $t=\theta >1$. The right hand sides of the equations for the relaxed program, equations (32) and (25), determine the distortions due to the agency costs. These agency costs are declining with respect to types and they are lower for the $t$-types (from equation (32)),

\[ \bar{t}-t<\frac{\beta ( \overline{\theta }-\theta ) }{\theta ^{2}( l-x) }\Longleftrightarrow v>\delta. \]

The equivalence follows after using the identity $t=\theta$ and substituting the relaxed program $x^{r}( \theta )$. Therefore, combining a larger first best with lower agency costs implies:

3.2.1. Cost – $t$, that is, the evaluation of the rainforest – is private information

Subtracting from the objective $V$ from equation (11) the subsidies using the definition in equation (15), and $W$ from equation (12), the principal has to solve the following optimal control problem:

(37)\begin{align} &\underset{\left\{ x_{0}( t) \geq x( t) \geq 0\right\} }{\max }\int_{\underline{t}}^{{\overline{t}}} \left[ v( R-x( t) ) +px( t) +t\ln ( R-x( t) ) -U( t) \right] dF, \end{align}

(38)\begin{align} &\dot{U}( t) =W_{t}=\ln ( R-x( t) ), \end{align}

(39)\begin{align} &U( t) \geq U_{0}( t) :=pR+( \ln \frac{t}{p} -1) \text{ }\forall t\in \left[ \underline{t},\overline{t}\right]. \end{align}

The control constraints stipulate that incentives are only offered for a reduction of logging ($x( t) \leq x_{0}( t)$) and that logging must be non-negative (no reforestation). The other constraints account, as in the above cases, for incentive compatibility equation (38) and individual rationality equation (39). The reservation price $U_{0}$ is again type dependent and characterizes the agent's outside option from logging as he likes.

In this example, equation (22) implies the relaxed program,

(40)\begin{equation} x^{r}( t) =R+\frac{( \bar{t}-2t) }{p-v}\geq x^{1}( t) =R-\frac{t}{p-v}, \end{equation}

which now exceeds the first best, because $x$ is a negative action and the principal finds it too costly to reduce logging to the first best. Proper monotonicity, here $\dot {x}\leq 0$, requires $p>v$, and the participation constraint equation (39) binds again in many if not most cases at least for a subset of types.

Proposition 5 Assuming that: (i) $p>v\Longleftrightarrow \dot {x} ^{r}<0$, i.e., the relaxed program $( x^{r})$ satisfies monotonicity, (ii) $R>({\overline {t}}/{p-v})=({1+\delta }/{ p-v})\Longleftrightarrow x^{r}>0\forall t$ ensures positive logging, and (iii) the appreciation of the rainforest is sufficiently high,

(41)\begin{equation} t>t^{\min }:=\frac{p( 1+\delta ) }{p+v}, \end{equation}

then the relaxed program is optimal, $x( t) =x^{r}( t)$. The relaxed program is continuously joined with the boundary solution $x_{0}$ from equation (14) at $t=t^{\min }\geq$$t$ and incentives are restricted to the types $t\geq t^{\min }$. They receive the compensation,

(42)\begin{equation} s( t) =\left\{ \begin{array}{c} 0 \\ \dfrac{2( ( p+v) t-p\bar{t}) +\bar{t}( p-v) \ln ({\bar{t}}/{2t-\bar{t}})({p-v}/{p+v})}{2( p-v) } \end{array} \text{ for }t \begin{array}{c} < \\ \geq \end{array} \right\} t^{\min }, \end{equation}

as a function of the type or conditional on logging (denoted by the function $\sigma$),

(43)\begin{equation} \sigma ( x) =\frac{1}{2}( ( p+v) ( R-x) -\bar{t}( \ln \frac{\bar{t}}{( p+v) ( R-x) }-1) ) \text{ if }x<x_{0}, \end{equation}

in order to reduce their logging from $x_{0}$ to $x^{r}$ equation (40).

The proof is straightforward. All properties along the relaxed program follow from the explicit solution. A necessary condition for the provision of incentives is that the principal values the protection less than the agent values logging, $v<p$, because otherwise the value of logging is negative even leaving the agent's own appreciation of the rainforest aside. The subsidies follow from the procedure outlined in section 3.1.

The restriction of incentives to a subset of types, $t>t^{\min }$, is a consequence of the participation constraint equation (39) and of the type dependency of the reservation price. More precisely and economically intuitive, handing out subsidies must be conditional on reduced logging, i.e., $x<x_{0}=R-t/p$, arithmetically: starting at a type $\tilde {t}$ with $x^{r}( \tilde {t}) >x_{0}( \tilde {t})$ violates the participation constraint because then $\dot {U}( \tilde {t}) =\ln ( R-x^{r}( \tilde {t}) ) <\dot {U}_{0}=\ln ( R-x_{0}( \tilde {t}) )$. The second constraint, $t>\bar {t} /2$, is necessary to ensure that some of the rainforest remains after logging, $x^{r}<R$, and this follows from the construction of $t^{\min }> \bar {t}/2\Leftrightarrow 2p>v+p$ and the assumption $p>v$. A continuous joining of the boundary policy, $x=x_{0}$, with the relaxed program, $x^{r}$, at the point of intersection $t^{\min }$ in equation (41) is optimal (i.e., it satisfies all necessary and sufficient optimality conditions).

However, the relaxed program can turn negative at high valuations or small forest areas $( R)$. Therefore, the inequality (ii) assumed in proposition 5 ensures (positive) logging.

Corollary 2 The share of agents receiving incentives,

\[ \frac{\bar{t}-t^{\min }}{\bar{t}-\underline{t}}=\frac{( 1+\delta ) v}{2\delta }, \]

increases with the principal's appreciation of the rainforest $( v)$ and is reduced by a larger variation of the agent types $( \delta )$.

The optimal incentive scheme is shown in figure 3, which compares the two different incentives resulting from the two observation equivalent characterizations of the agent. Logging is declining (linearly) in the agent's type and this decline is compensated by increasing and convex subsidies to the agent with respect to types and logging and, of course, less logging earns higher subsidies. However, only a fraction ($t^{\min }=8/9$ and thus only $2/3$) of the types receive incentives and they reduce logging only modestly although the principal's valuation is significant (at one-half of the profit that the agent gets).

Figure 3. Logging the rain forest with either the conservation (t and bold) or revenue (θ) as the agent's private information parameter, p = 1, a = 1/2, R = 5.

3.2.2. Benefit (i.e., importance of earnings) is private information

The alternative but observation equivalent assumption – the benefit from logging is the agent's private information – leads again to a different incentive contract. The principal has to solve the following control problem (listing all arguments in order to highlight the difference from equations (37)–(39)),

(44)\begin{align} &\underset{\left\{ x_{0}\geq x\geq 0\right\} }{\max }\int_{\underline{ \theta }}^{{\overline{\theta }}}\left[ v( R-x( \theta ) ) +\frac{p}{\theta }x+\ln ( R-x( \theta ) ) -U( \theta ) \right] dF( \theta ), \end{align}

(45)\begin{align} &\dot{U}( \theta ) =W_{\theta }={-}\frac{px( \theta ) }{ \theta^{2}}, \end{align}

(46)\begin{align} &U( \theta ) \geq U_{0}( \theta ) :=\frac{pR}{\theta } +\ln \frac{\theta }{p}-1\text{ }\forall \theta \in \left[ \underline{\theta },\overline{\theta }\right], \end{align}

and the constraints reflect incentive compatibility equation (45) and individual rationality equation (46) as in the above case.

Proposition 6 Assuming $v<v^{\max }:=\frac {p}{\theta }-\frac {1}{R}$ and $\theta \geq \theta ^{\min }:=\frac {\sqrt {p( p+4\bar {\theta }v) }-p }{2v}>\theta$, the relaxed program,

(47)\begin{equation} x^{r}( \theta ) =R-\frac{\theta ^{2}}{p\bar{\theta}-\theta ^{2}v}, \end{equation}

determines the amount of logging. This program differs from its counterpart (proposition 5 and the agent's valuation of the resource is private information) in terms of logging and participation: a contract in terms of $\theta$ is more inclusive, i.e., $t^{\min }\geq \theta ^{\min }$, iff

(48)\begin{equation} ( \bar{t}-1) \geq \frac{v}{p}. \end{equation}

In terms of volume, $x^{r}( \bar {t}) >x^{r}( \bar { \theta }) \Longleftrightarrow p>\bar {\theta }v$ and by continuity for types close to the top. Hence, a contract in terms of $\theta$ is (locally at least) more conservationist if the agent's revenues exceed the donor's valuation times the agent's highest valuation of the rainforest (= least valuation of the revenues).

The proof is along the lines in 3.1.1 and thus brief. Starting now directly with equation (22),

\[ -vx+\frac{p}{\theta }-\frac{1}{R-x}=\frac{p( \theta -\bar{\theta} ) }{\theta ^{2}}, \]

the solution $x^{r}( \theta )$ in equation (47) is verified. Comparing the marginal types and applying a few elementary but tedious operations yields equation (48).

However, condition equation (48) is not met in the example in figure 3 since $\delta =1/3<v/p=1/2$. The panel at the top shows logging within, $x( t)$and $x( \theta )$, outside of the contracts $( x_{0})$, and the relaxed programs outside of their range of applicability ($x^{r}$ and dotted lines). Hence, the contract in terms of $t$ includes slightly more types while the contract in terms of $\theta$ is significantly more conservationist for efficient types (as claimed) but not for intermediate types due to its concave shape. The bottom panels compare the subsidies depending on the type (left hand side) and on the amount of logging (right hand side). The subsidies cover a much larger range in terms of types and logging if the agent's private information is about his need for revenues (i.e., the benefit parameter $\theta$).