Additionality When REDD Contracts Must be Self-Enforcing

Abstract

This paper examines self-enforcing contracts as a financial mechanism for reducing carbon emissions from deforestation and forest degradation when the opportunity cost of the land (i.e., landholder type) is private information and is imperfectly correlated over time (i.e., partially persistent types). Because self-enforcement limits the feasible incentives, the conservation levels are constrained by the surplus created. Regardless of the degree of persistence of such opportunity costs across contracting periods, a first-best self-enforcing contract can deliver “additional” carbon sequestration beyond the business as usual scenario only if the value of forest conservation is sufficiently high. Otherwise, self-enforcing contracts can induce some, suboptimal level of carbon sequestration. The degree of persistence of opportunity costs across periods does not affect the amount of total payments provided in the optimal menu of contracts, but greater persistence of opportunity cost types leads to contracts that feature more of the total payment as a bonus in contracts for landholders with a high opportunity cost for their land and more of the total payment as an upfront fixed payment for landholders with a low opportunity cost.

Appendix

Proof of Proposition 1

Following the arguments for incentive compatibility constraint (ICC) the $h -type$ IRC does not bind while the ICC binds. From the ICCs we get equation: $P_ h \ge g(f_ h ; \theta _ h )+ \Delta _ l$ as $P_{\theta _ l }=0$. A buyer makes the highest payment to the $h -type$ landholder and the lowest payment to the $l -type$ landholder. Self-enforcement dictates that the difference between the highest possible payment and the lowest payment should be lower or equal to the gains from the contracts $\frac{\delta }{1-\delta } (E(S(\theta ^ h _{t-1}))-r) \ge P_{ h }$ and $\frac{\delta }{1-\delta } (E(S(\theta ^ l _{t-1}))-r) \ge P_{ h }$, where $E(S(\theta ^ h _{t-1}))=\frac{\delta }{1-\delta } (\Phi ^ h _{t-1} S_ h + (1-\Phi ^ h _{t-1}) S_ l )$ and $E(S(\theta ^ l _{t-1}))=\frac{\delta }{1-\delta } (\Phi ^ l _{t-1}) S_ h + (1-\Phi ^ l _{t-1}) S_ l )$ are the expected surplus given that the landholder was a $h -type$ or $l -type$ in the previous period respectively. Combining this with ICCs we get the relationships in Proposition 1.

Proof of Proposition 2

To get the payment structure for the $l -type$, we combine the IRC into the DICC as the ICC does not bind: $p_ l =g(f_ l ; \theta _ l )- b_ l$ and as $U_ h =\Delta _ l$ and $U_ l =0$, $b_ l \ge g(f_ l ; \theta _ l )- \frac{\delta \Phi ^ l _{t-1} \Delta _ l }{1-\delta }$. Substituting and arranging we get the minimum bonus and the maximum fixed price that satisfy the constraints. By solving in the buyer’s DICC and using the total payments derived by the landholder’s IRC constraints we obtain the maximum bonus and the minimum fixed price:

$$\begin{aligned}&g(f_ l ^{*}; \theta _ l ) +\frac{\delta (\Phi ^ l _{t-1}\Delta _ l ^*-ES(\theta ^ l _{t-1})-r)}{1-\delta } \le p^*_{ l } \le \frac{\delta \Phi ^ l _{t-1} \Delta _ l ^*}{1-\delta } \text {; and} \end{aligned}$$

(22)

$$\begin{aligned}&\quad \frac{\delta ( ES(\theta ^ l _{t-1})-r-\Phi ^ l _{t-1}\Delta _ l ^*)}{1-\delta }\ge b_ l \ge g(f_ l ^{*}; \theta _ l )- \frac{\delta \Phi ^ l _{t-1} \Delta _ l ^*}{1-\delta } \text {; and} \end{aligned}$$

(23)

$$\begin{aligned}&\quad P_ l = g(f_ l ^{*}; \theta _ l ). \end{aligned}$$

(24)

Similarly, to get the payment structure for the $h -type$, we take the same steps but use the ICC and DICC as the IRC does not bind: $p_ h =\Delta _ l +g(f_ h ; \theta _ h )- b_ h$ and $b_ h \ge g(\theta _ h ; \theta _ h )- \frac{\delta \Phi ^ h _{t-1} \Delta _ l }{1-\delta }$. This results in the minimum bonus and the maximum fixed price that satisfy the constraints. By solving in the buyer’s DICC and using the total payments derived by the landholder’s ICC constraint we obtain the maximum bonus and the minimum fixed price:

$$\begin{aligned}&g(f_ h ^{*}; \theta _ h )+\frac{\Delta _ l ^*(1-\delta +\delta \Phi ^ h _{t-1}) -\delta (ES(\theta ^ h _{t-1})-r)}{1-\delta } \le p^*_{ h } \le \frac{ \Delta _ l ^*(1-\delta +\delta \Phi ^ h _{t-1}) }{1-\delta } \text {; and} \nonumber \\\end{aligned}$$

(25)

$$\begin{aligned}&\quad \frac{\delta (ES(\theta ^ h _{t-1})-r-\Phi ^ h _{t-1} \Delta _ l ^*)}{1-\delta }\ge b_ h \ge g(f_ h ^{*}; \theta _ h )- \frac{\delta \Phi ^ h _{t-1} \Delta _ l ^*}{1-\delta } \text {; and} \end{aligned}$$

(26)

$$\begin{aligned}&\quad P_ h = g(f_ h ^{*}; \theta _ h )+ \Delta _ l ^* \end{aligned}$$

(27)

The payments in Proposition 2 are derived by assuming that the buyer pays the minimum bonus required to meet the landholders’ DICC constraints. Finally, self-enforcement is sustainable if inequalities (17) and (18), are satisfied. The maximization problems that are set in the paper lead to the following Lagrangians. The first Lagrangian is given by:

$$\begin{aligned} L= & {} \frac{ \Phi ^ h _{t-1}(V(f_ h -\theta _ h )-g(f_ h ; \theta _ h ))+ (1- \Phi ^ h _{t-1})(V(f_ l -\theta _ l )-g(f_ l ; \theta _ l ))}{1-\delta }\\&+\mu \Big [\frac{\delta }{1-\delta } (\Phi ^ h _{t-1} (V(f_ h -\theta _ h )-g(f_ h ; \theta _ h )) + (1-\Phi ^ h _{t-1})(V(f_ l -\theta _ l )-g(f_ l ; \theta _ l )) -r)\\&- g(f_ h ; \theta _ h )-\Delta _ l \Big ]+\beta (f_ h -f_ l ) \end{aligned}$$

where $\mu$ and $\beta$ are the Lagrangian multipliers. The following are the FOC

$$\begin{aligned} \frac{d L}{d f_ h }= & {} \frac{ \Phi ^ h _{t-1} }{1-\delta } \left( \frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h }\right) \\&+\mu \left[ \frac{\delta }{1-\delta } \Phi ^ h _{t-1} \left( \frac{d V(f_ h -\theta _ h )}{d f_ h } - \frac{d g(f_ h ; \theta _ h )}{d f_ h }\right) - \frac{d g(f_ h ; \theta _ h )}{d f_ h }\right] \\&+\beta =0 \frac{d L}{d f_ l } = \frac{(1-\Phi ^ h _{t-1})}{1-\delta } \left( \frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l }\right) \\&+\mu \bigg [\frac{\delta }{1-\delta } (1-\Phi ^ h _{t-1}) \left( \frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l }\right) \\&-\left( \frac{d g(f_ l ; \theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _H)}{d f_ l }\right) \bigg ]-\beta =0 \\ \frac{d L}{d \mu }= & {} \frac{\delta }{1-\delta } (\Phi ^ h _{t-1} (V(f_ h -\theta _ h )-g(f_ h ; \theta _ h ))\\&+(1-\Phi ^ h _{t-1})(V(f_ l -\theta _ l )-g(f_ l ; \theta _ l )) -r) - g(f_ h ; \theta _ h )-\Delta _ l =0\\ \frac{d L}{d \beta }= & {} f_ h - f_ l =0 \end{aligned}$$

Case 1 Suppose $\mu =0$.

This means that the self-enforcing constraint is slack at the solution: $\frac{\delta }{1-\delta } ( \Phi ^ h _{t-1} S_ h + (1- \Phi ^ h _{t-1}) S_ l -r) \ge g(f_ h ; \theta _ h )+ \Delta _ l$. We consider maximizing the joint surplus subject to the monotonicity constraint, $f_ h \ge f_L$. If $\beta >0$, $f_ h =f_ l$. This condition implies that $\frac{ \Phi ^ h _{t-1} }{1-\delta }(\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })+ \frac{(1- \Phi ^ h _{t-1} )}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })=0$ which is not possible. As f is increasing in the type, it maximizes the joint surplus at the optimum for both types and satisfies monotonicity. That is, if $\beta =0$ then $f_ h ^{*}>f_ l ^{*}$; and from $\frac{d L}{d f_ h }$, $\frac{d V(f_ h -\theta _ h )}{d f_ h }=\frac{d g(f_ h ; \theta _ h )}{d f_ h }$ and from $\frac{d L}{d f_ l }$, $\frac{d V(f_ l -\theta _ l )}{d f_ l }= \frac{d g(f_ l ; \theta _ l )}{d f_ l }$. Then, $f_ l =f_ l ^{*}$ and $f_h=f_ h ^{*}$ and $f_ h ^{*}>f_ l ^{*}$.

Case 2 Suppose $\mu >0$.

This means that the self-enforcement constraint is binding at the solution: $\frac{\delta }{1-\delta } ( \Phi ^ h _{t-1} S_ h + (1- \Phi ^ h _{t-1}) S_ l -r) \ge g(f_ h ; \theta _ h )+ \Delta _ l$. To have $f_ h >f_ l$, $\beta =0$ and the self-enforcing constraint must allow to cover for information rents for the $h -type$. This implies that $\frac{d L}{d f_ h }=\frac{\Phi ^ h _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })+\mu [\frac{\delta }{1-\delta } \Phi ^ h _{t-1} (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })- \frac{d g(f_ h ; \theta _ h )}{d f_ h }] =0$ and $\frac{d L}{d f_ l } = \frac{(1-\Phi ^ h _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _ l )}{d f_ l })+\mu [\frac{\delta (1-\Phi ^ h _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })-(\frac{d g(f_ l ; \theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _ h )}{d f_ l }]=0$. Solving from each constraint we get $\frac{d V(f_ h -\theta _ h )}{d f_ h }=\frac{d g(f_ h ; \theta _ h )}{d f_ h } [(1+\frac{\mu (1-\delta )}{\Phi ^ h _{t-1} (1+\mu \delta )}]$ which implies that $f_ h <f_ h ^{*}$ and $\frac{d V(f_ l -\theta _ l )}{d f_ l }=\frac{d g(f_ l ; \theta _ l )}{d f_ l } (1+ \frac{\mu (1-\delta )}{(1-\Phi ^ h _{t-1}) (1+\mu \delta )})+ \frac{d g(f_ l ; \theta _ h )}{d f_ l } \frac{\mu (1-\delta )}{(1-\Phi ^ h _{t-1}) (1+\mu \delta )}$ which implies that $f_ l <f_ l ^{*}$. Therefore, $f_ h >f_ l$ if the self-enforcing constraint is not too restrictive and the expected surplus is enough to cover some information rents.

If the self-enforcement constraint is too restrictive, forest conservation decreases up to the level in which incentives are cover by such constraint, resulting in the same level forest conservation delivered by both landholders. That is $\beta >0$, $\frac{d L}{d f_ h }=\frac{\Phi ^ h _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h }) +\mu [\frac{\delta \Phi ^ h _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })- \frac{d g(f_ h ; \theta _ h )}{d f_ h }] +\beta =0$ and $\frac{d L}{d f_ l } = \frac{(1-\Phi ^ h _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l }) +\mu [\frac{\delta (1-\Phi ^ h _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })-(\frac{d g(f_ l ; \theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _ h )}{d f_ l }]-\beta =0$. This implies that $\Phi ^ h _{t-1} (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })+ (1-\Phi ^ h _{t-1}) (\frac{d V(f_L-\theta _L)}{d f_L}- \frac{d g(f_L; \theta _L)}{d f_L})=\frac{\mu (1-\delta )}{(1+\mu \delta )} (\frac{d g(f_ h ; \theta _ h )}{d f_ h }+\frac{d g(f_L; \theta _L)}{d f_L}-\frac{d g(f_L; \theta _ h )}{d f_L}$ and $\frac{\delta }{1-\delta } (E(S)_ l -r) = g(f_i; \theta _ l )$. Then, $f_ h =f_ l$ is possible.

The second Lagrangian is given by:

$$\begin{aligned} L= & {} \frac{\Phi ^ l _{t-1}(V(f_ h -\theta _ h )-g(f_ h ; \theta _ h ))+(1-\Phi ^ l _{t-1})(V(f_ l -\theta _ l )-g(f_ l ; \theta _ l ))}{1-\delta }\\&\quad +\mu \bigg [\frac{\delta }{1-\delta } (\Phi ^ l _{t-1} (V(f_ h -\theta _ h )-g(f_ h ; \theta _ h ))\\&\quad +(1-\Phi ^ l _{t-1})(V(f_ l -\theta _L)-g(f_ l ; \theta _ l )) -r) \\&\quad - g(f_ h ; \theta _ h )-\Delta _ l \bigg ]+\beta (f_ h -f_ l ) \end{aligned}$$

where $\mu$ and $\beta$ are the Lagrangian multipliers. The following are the FOC

$$\begin{aligned} \frac{d L}{d f_ h }= & {} \frac{\Phi ^ l _{t-1}}{1-\delta } \left( \frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h }\right) \\&\quad + \mu \left[ \frac{\delta \Phi ^ l _{t-1}}{1-\delta } \left( \frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h }\right) - \frac{d g(f_ h ; \theta _ h )}{d f_ h }\right] \\&\quad +\beta =0 \frac{d L}{d f_ l } = \frac{(1-\Phi ^ l _{t-1})}{1-\delta } \left( \frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l }\right) \\&\quad +\mu \left[ \frac{\delta (1-\Phi ^ l _{t-1})}{1-\delta } \left( \frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l }\right) -\frac{d g(f_ l ; \theta _ l )}{d f_ l }\right. \\&\quad \left. -\frac{d g(f_ l ; \theta _ h )}{d f_ l }\right] -\phi =0\\ \frac{d L}{d \mu }= & {} \frac{\delta }{1-\delta } (\Phi ^ l _{t-1} (V(f_ h -\theta _ h )-g(f_ h ; \theta _ h ))\\&+(1-\Phi ^ l _{t-1})(V(f_ l -\theta _ l )-g(f_ l ; \theta _ l )) -r) - g(f_ h ; \theta _ h )\\&-\Delta _ l =0 \frac{d L}{d \beta } = f_ h - f_ l =0 \end{aligned}$$

Case 1: Suppose $\mu =0$.

This means that the self-enforcing constraint is slack at the solution: $\frac{\delta }{1-\delta } (E(S(\theta _{t-1}^ l )-r)> g(f_ h ; \theta _ h )+ \Delta _ l$. We consider maximizing the joint surplus subject to the monotonicity constraint, $f_ h \ge f_ l$. If $\beta >0$, $f_ h =f_ l$. This condition implies that $\frac{\Phi ^ l _{t-1}}{1-\delta }(\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })+ \frac{(1-\Phi ^ l _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })=0$ which is not possible. As f is increasing in the type, it maximizes the joint surplus at the optimum for both types and satisfies monotonicity. That is, if $\phi =0$ then $f_ h ^{*}>f_ l ^{*}$; and from $\frac{d L}{d f_ h }$, $\frac{d V(f_ h -\theta _ h )}{d f_ h }=\frac{d g(f_ h ; \theta _ h )}{d f_ h }$ and from $\frac{d L}{d f_ l }$, $\frac{d V(f_ l -\theta _ l )}{d f_ l }= \frac{d g(f_ l ; \theta _ l )}{d f_ l }$. Then, $f_ l =f_ l ^{*}$ and $f_h=f_ h ^{*}$ and $f_ h ^{*}>f_ l ^{*}$.

Case 2: Suppose $\mu >0$.

This means that the self-enforcement constraint is binding at the solution: $\frac{\delta }{1-\delta } (E(S(\theta _{t-1}^ l )-r)> g(f_ h ; \theta _ h )+ \Delta _ l$. To have $f_ h >f_ l$, $\beta =0$ and the self-enforcing constraint must allow to cover for information rents for the $h -type$. This implies that $\frac{d L}{d f_ h }=\frac{\Phi ^ l _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })+\mu [\frac{\delta \Phi ^ l _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })- \frac{d g(f_ h ; \theta _ h )}{d f_ h }] =0$ and $\frac{d L}{d f_ l } = \frac{(1-\Phi ^ l _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _ l )}{d f_ l })+\mu [\frac{\delta (1-\Phi ^ l _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })-(\frac{d g(f_ l ; \theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _ h )}{d f_ l }]=0$. Solving from each constraint we get $\frac{d V(f_ h -\theta _ h )}{d f_ h }=\frac{d g(f_ h ; \theta _ h )}{d f_ h } [(1+\frac{\mu (1-\delta )}{\Phi ^ l _{t-1} (1+\mu \delta )}]$ which implies that $f_ h <f_ h ^{*}$ and $\frac{d V(f_ l -\theta _ l )}{d f_ l }=\frac{d g(f_ l ; \theta _ l )}{d f_ l } (1+ \frac{\mu (1-\delta )}{(1-\Phi ^ l _{t-1}) (1+\mu \delta )})+ \frac{d g(f_ l ; \theta _ h )}{d f_ l } \frac{\mu (1-\delta )}{(1-\Phi ^ l _{t-1}) (1+\mu \delta )}$ which implies that $f_ l <f_ l ^{*}$. Therefore, $f_ h >f_ l$ if the self-enforcing constraint is not too restrictive and the expected surplus is enough to cover some information rents.

If the self-enforcement constraint is too restrictive, the levels of forest conservation must decrease up to the level in which incentives are cover by such constraint. This means that it is possible that both landholders are required to deliver the same level forest conservation. That is $\beta >0$, $\frac{d L}{d f_ h }=\frac{\Phi ^ l _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h }) +\mu [\frac{\delta \Phi ^ l _{t-1}}{1-\delta } (\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })- \frac{d g(f_ h ; \theta _ h )}{d f_ h }] +\beta =0$ and $\frac{d L}{d f_ l } = \frac{(1-\Phi ^ l _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l }) +\mu [\frac{\delta (1-\Phi ^ l _{t-1})}{1-\delta } (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })-(\frac{d g(f_ l ; \theta _ l )}{d f_ l }-\frac{d g(f_ l ; \theta _ h )}{d f_ l }]-\beta =0$. This implies that $\Phi ^ l _{t-1}(\frac{d V(f_ h -\theta _ h )}{d f_ h }- \frac{d g(f_ h ; \theta _ h )}{d f_ h })+ (1-\Phi ^ l _{t-1}) (\frac{d V(f_ l -\theta _ l )}{d f_ l }- \frac{d g(f_ l ; \theta _ l )}{d f_ l })= \frac{\mu (1-\delta )}{(1+\mu \delta )} (\frac{d g(f_ h ; \theta _ h )}{d f_ h }+\frac{d g(f_ l ; \theta _ l )}{d f_ l }-\frac{d g(f_v; \theta _ h )}{d f_ l }$ and $\frac{\delta }{1-\delta } (E(S(\theta ^ l _{t-1})-r) = g(f_i; \theta _ l )$. Then, $f_ h =f_ l$ is possible.