When are Debt for Nature Swaps
Welfare—Improving?
A Note on Chang and Pillarisetti (1997)

Devon A. Garvie
from International Review of Economics and Business, 2002, 49(2), 165-173.

Chang and Pillarisetti (1997), in a recent article published in this journal,
provide suﬃcient conditions for debt for nature swaps to be welfare—improving
for a debtor developing country. They compare a debtor country’s utility levelwith and without a debt for nature swap under two distinct no swap scenarios:(i) the debtor country does not undertake environmental spending in the absenceof a swap and (ii) the debtor country undertakes a positive level of environmental
spending in the absence of a swap. They compare utility levels under these twoscenarios and derive suﬃcient conditions for debt for nature swaps to be welfare—improving for the debtor country.
This paper corrects an error made by Chang and Pillarisetti (1997). Specifically, we prove that debt for nature swaps are unambiguously welfare—improving
whenever a debtor country undertakes a positive level of environmental spending
in the absence of a swap. In this case, debt for nature swaps always result ina first—order welfare gain for the debtor country, namely, the reduction in debt
repayment realized from the swap. The proof involves a straightforward application of the Envelope Theorem. In addition, a simpler and more intuitive suﬃcientcondition is derived to show when debt for nature swaps are welfare—improving if
a debtor country does not undertake environmental spending in the absence of aswap.

1. The Model
Chang and Pillarisetti (1997) construct a model in which a non—profit organization purchases a portion of a developing country’s external debt at a discountfrom a commercial bank holding the country’s debt. The organization and thedebtor country, in turn, exchange the purchased discounted debt for an agreed
upon amount of financial resources (in local currency) to be allocated by thedebtor country to environmental protection activities.
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Following Chang and Pillarisetti (1997), the indebted developing country’s twoperiod utility can be written as a function of N:()

δN (1 + r)
U (N) = G − N + ρ+ B (N)
1−t(1.1)

where N is domestic environmental spending, G is intertemporal utility independent of N, ρ is the country’s subjective discount factor for future consumption,
0 < δ < 1 is the propensity of the organization to purchase the swap, 0 < t < 1is the market discount rate on the debt, 0 < r < 1 is the rate of interest on debt,and B (N) is a strictly concave environmental benefit function.1
Define the following indicator function α: 1α=

 0if debt for nature swap
(1.2)
otherwise

Using the indicator function, write the debtor country’s utility maximization problem as:
1G = E +D −I +ρ {θF (I) − D (1 + r) − H (I)}, where E is the country’s initial endowment,
D is the amount of inherited debt, I is investment, θF (I) is total revenue from production (θ isa commodity price and Y = F (I) is a strictly concave production function), D (1 + r) is debt
repaid in period two, and H (I) is a strictly convex damage function, measuring environmental
damages incurred as a result of first period investment.
Since Chang and Pillarisetti (1997) find that the optimal choices of I and N are independent,
we can, without loss of generality, restrict analysis to the optimal choice of N . The reader isreferred to Chang and Pillarisetti (1997) for analysis pertaining to the optimal choice of I.

3()δN (1 + r)
+ B (N)
max U (N; α) = G − N + ρ α
N1−t(1.3)

The first—order condition for N is given by()δ (1 + r)∂U= −1 + ρ α+ B 0 (N) ≤ 0∂N1−t(1.4)

The choice of environmental spending, N, clearly depends upon the debt for
nature swap parameters. The first term on the right—hand side of (1.4) is themarginal cost of environmental spending, that is, the marginal reduction in resources available for consumption in period one. The second term on the right—
hand side of (1.4) is the discounted marginal benefit of environmental spending:the first term in the bracketed expression is the marginal value of the reduction
in debt repayment realized in period two if a debt for nature swap occurs while
the second term is the direct marginal benefit of environmental spending. Notefrom (1.4) that a debt for nature swap increases a debtor country’s incentive toengage in environmental spending by shifting the marginal benefit curve upwardby ρ δ(1+r), the marginal value of the debt reduction.
1−t¯∂U ¯¯α∈{0,1} < 0,
There are three possible equilibrium outcomes for N. First, if ∂Nthen N = 0, that is, it is never in the interest of the debtor country to undertake4

environmental spending and/or to engage in debt for nature swaps. Second, if∂U∂U< 0 and ∂N||α=1 = 0, then a debt for nature swap provides an additional∂N α=0incentive to a debtor country to undertake environmental spending when it oth∂U|α=0 = 0, then a debtor country will undertake aerwise would not. Finally, if ∂Npositive level of environmental spending whether or not a debt for nature swap isundertaken. Define the possible equilibrium levels of N as follows: N0 = 0¯∂U ¯¯α∈{0,1} < 0if ∂NN =  N e > 0 if ∂U |α=0 = 0∂N

2. Welfare Eﬀects




 N∗ > 0(1.5)∂U|α=1 = 0if ∂N

2.1. No Environmental Funding in Absence of a Debt for Nature Swap
If a debtor country does not undertake environmental spending in the absenceof a debt for nature swap, then it is indeed ambiguous whether or not a debt fornature swap is welfare—improving. The additional resource cost of environmental
spending may be too high to justify the value of writing down an amount of debt.
Nest the debtor country’s two period maximal utility as5"()#δ (1 + r) ∗N + B (N ∗ )U (α) = G (I ) + α −N + ρ1−t∗∗(2.1)∂U= 0 and N ∗ is defined by (1.4) when ∂N|α=1 = 0.
where I ∗ is defined by ∂G∂I

Debt for nature swaps are unambiguously welfare—improving if and only ifdU(α)> 0. By the Envelope Theorem, dU(α)= ∂U∂α(α) . Taking the partial derivative
dα
dαof (2.1) with respect to α yields:()δ (1 + r) ∗∂U (α)= −N ∗ + ρN + B (N ∗ )∂α1−t(2.2)

A suﬃcient condition for ∂U∂α(α) > 0 is(δ (1 + r) B (N ∗ )ρ+1−tN∗)>1(2.3)A simpler and more intuitive suﬃcient condition can be obtained by utilizing
the assumption that B(N) is strictly concave. Since B(N) is strictly concave,∗)
B 0 (0) > B(N, and, hence, a suﬃcient condition for ∂U∂α(α) > 0 isN∗()

δ (1 + r)
ρ+ B 0 (0) > 11−t(2.4)

where the left—hand side of (2.4) is the vertical intercept of the marginal benefit

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curve and the right—hand side of (2.4) is the vertical intercept of the marginal
cost curve. Thus, a simpler suﬃcient condition for a debt for nature swap to bewelfare—improving in this scenario is simply that the marginal benefit curve witha swap cuts the marginal cost curve from above.
The two possible equilibrium outcomes for this case are illustrated in Figure
1. As Figure 1 shows, whether a debt for nature swap is welfare—improving depends upon the value, at the margin, of environmental spending. A swap does
not provide a suﬃciently strong additional incentive to undertake environmental
spending in Figure 1(a), that is, (2.4) fails to hold. In this case, the debtor country
will not engage in environmental spending even when debt for nature swaps arepossible. In Figure 1(b), the suﬃcient condition (2.4) holds and hence the country
will engage in environmental spending according to the terms of the swap. The
welfare gain from a swap is the shaded area depicted in Figure 1(b).
[Insert Figure 1 about here]
Integrating (2.2) yields the utility comparison for this case2 :2Rearranging (2.5) yields Chang and Pillarisetti’s (1997) suﬃcient condition for a debt fornature swap to be welfare—improving when a debtor country does not undertake environmental
spending in the absence of a swap:
ρ > δ(1+r)
1−tN∗N ∗ + B (N ∗ )

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Z 1

∂U (α)dα∂α0()
δ (1 + r) ∗∗∗N + B (N )= −N + ρ1−t∆∗0 = U ∗ − U 0 =(2.5)

Note that ∆∗0 > 0 whenever (2.4) holds.
2.2. Environmental Funding in Absence of a Debt for Nature Swap
Contrary to Chang and Pillarisetti (1997), if a debtor country undertakes
environmental spending in the absence of a debt for nature swap, then a debt for
nature swap is unambiguously welfare—improving.3 At the margin, the developing
country will always benefit from forgiveness of a discrete quantity of debt (afirst—order benefit) in return for a marginal addition to environmental spending(a second—order cost). An unambiguous welfare result can be obtained by astraightforward application of the Envelope Theorem.

3Chang and Pillarisetti (1997) state that a debt for nature swap is welfare—improving in thiscase if and only if the following condition is satisfied:ρ > δN ∗ (1+r)
1−tN∗ − Ne+ B(N ∗ ) − B(N e )8The optimal choice of N is defined by()δ (1 + r)−1 + ρ α+ B 0 (N) = 01−t(2.6)

The first—order condition (2.6) implicitly defines N(α). Apply the Implicit Function Theorem to (2.6) to determine how environmental spending changes when adebt for nature swap is undertaken, that is, dN:dαδ (1 + r)
dN=−>0dα(1 − t) B 00 (N)(2.7)> 0 since B 00 (N) < 0. Hence, environmental spending will always be
Clearly, dNdαhigher when a debt for nature swap is undertaken, that is, N ∗ > N e .
Nest the debtor country’s two period maximal utility as()δ (1 + r)
U (α) = G (I ) − N (α) + ρ αN(α) + B (N (α))
1−t
∗

(2.8)

where N (α = 0) = N e , N (α = 1) = N ∗ , and N e and N ∗ are defined by (2.6).
Debt for nature swaps are unambiguously welfare—improving if and only if dUdα(α) >0. By the Envelope Theorem, dUdα(α) = ∂U(α). Taking the partial derivative of (2.8)∂α
with respect to α yields:

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"()#

∂U (α)
δ (1 + r)= −1 + ρ α+ B 0 (N (β))
∂α1−t

∂N
δ (1 + r)+ρN (α)∂α1−t(2.9)

The first term on the right—hand side of (2.9) vanishes since (2.6) must hold forall values of α. Hence, (2.9) reduces to∂U (α)
δ (1 + r)=ρ
N (α) > 0∂α1−t(2.10)

which is strictly positive for all parameter values. Thus, a debt for nature swap
yields a first—order welfare gain to the debtor developing country. The change inwelfare equals the discounted value of the debt reduction obtainable in period two
as a result of the swap.
Integrating (2.10) yields the utility change for this case:∂U (α)
δ (1 + r) Z 1 ∂Ndα = ρdα∂α
1 − t 0 ∂α0Ã!2ρδ (1 + r)>0= − 00
B (N)1−t∆∗e = U ∗ − U e =Z 1(2.11)

Note again that ∆∗e > 0 for all parameter values. However, the size of the10welfare gain from a debt for nature swap is strictly increasing in ρ, δ, r and t anddecreasing in B 00 (N). The welfare gain from a swap is the shaded area depicted
in Figure 2.
[Insert Figure 2 about here]11References
Chang Y. and Pillarisetti J., “Debt for Nature Swaps: A Simple Economic—
Ecological Approach”, RISEC: International Review of Economics and Business,
1997, 44, pp. 793—806.

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Abstract
This paper proves that debt for nature swaps are unambiguously welfare—improving whenever a debtor country undertakes a positive level of environmental
spending in the absence of a swap – debt for nature swaps always result in a first—
order welfare gain for the debtor country, namely, the reduction in debt repayment
realized from the swap. The proof involves a straightforward application of theEnvelope Theorem. A simple suﬃcient condition is also derived to show when debt
for nature swaps are welfare—improving if a debtor country does not undertakeenvironmental spending in the absence of a swap. (Keywords: debt for natureswaps; JEL: F34, O19)

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