Discounting as a double-edged sword: the values of both future goods and present economic growth decrease with the discount rate

ABSTRACT How to compromise between economic growth and sustainable development is a major question. Particularly, climate policy affects capital and production levels (GDP), but it also affects the long-term economic growth, or the development of Total Factor Productivity (TFP) (e.g. technological developments), which enables more effective future production using a given capital. A slowdown in TFP development rate may result from either restrictions on production or climate damages. Such a slowdown results in a long-lasting decrease in GDP that persists long after the restrictions are no longer implied and/or long after the environment recovers from the damages. Therefore, effective climate policy entails analysis that incorporates present and future changes in both capital and TFP. Here, we derive an analytic formula for the economic values of TFP development. The values derived from the formula are consistent with those derived in detailed Integrated Assessment Models. The advantage of the formula is that it reveals the role of some key parameters in determining these values. Specifically, we show that higher discount rates imply lower values to TFP development. Therefore, considering higher discount rates is a double-edged sword, implying that future environmental damages are less costly but also that present economic growth is less valuable.

Endogenous Economic Growth S1.1 General considerations In this section, we prove Theorem 2 (main text). Specifically, we show that, under certain assumptions, choosing x(t) that maximizes social welfare, U T (Eq. 1), subject to a given environmental dynamics (Eq. 2), is equivalent to choosing x T that maximizes the NPV plus a shadow value that can also be expressed in terms of present values equivalence. We assume, like Ramsey, that B(t) c(t) for all t. Substituting Eq. 5b into Eq. 2 implies where u(c) is the instantaneous utility, u (c) is its derivative with respect to c, and c * is the quasiequilibrium consumption level given by Eq. 5a with initial conditions c * (0) = c 0 . It follows that where is the base utility that is obtained when B(t) = H(t) = 0. Namely, c − follows Eq. 5a with H → 0: and initial conditions c − (0) = c 0 .
is the contribution to U T due to change in consumption due to B, and is the deviation of the total utility from the NPV. Note that H affects both U T s and U T c .
In what follows, we express U T c and U T s in terms of the present values of B(t) and H(t), respectively. In other words, we find how much present change in B (or H) at time t would be equivalent S2 to a 'present' change in B (or H) at time zero, and thereby calculating the net present value (or shadow value). The net present shadow value plus the net present actual value gives the Generalized Net Present Value (GNPV) that a social planner is aiming at maximizing. In section S1.2, we derive the expression for U T c in terms of present values. This is similar to the derivation by (Ramsey 1928, Cass 1965, Koopmans 1965, with the difference that we consider H in addition to the regular growth g. Next, in section S1.3, we derive the expression for U T s in terms of present values. Finally, in Section 1.4, we integrate the results to complete the derivation of the formula for the GNPV, and we consider the limit T → ∞ and complete the proof of Theorem 2. S1.2 Regular discount on short-term changes in consumption: contribution of B(t) to the net present value In this subsection, we derive an expression for U T c (Eq. (S5)), the value due to direct change in consumption due to x and P , B(x, P ), in terms of net present equivalent. Namely, we find how much extra consumption does society need at present to achieve an addition of U T c to the utility. Note that we can write Eq. (S5) in the form where It follows that is elasticity of changes in utility with respect to consumption. Eq. (S9) implies where Eq. (S8) implies
Finally, note that Eq. (S13) gives the change in utility due to the perturbation B(t). But since we are interested in the NPV, we need to calculate how much extra consumption at present would result in the same change in utility. Note that a small addition to consumption at time t = 0 adds u(c 0 + ) − u(c 0 ) = u (c 0 ) to utility, and therefore, (S15) S1.3 Contribution of H(t) to the utility via the shadow value S1.3.1 Constant economic growth g

Intuitive considerations
Considering a constant g already yields several insights that we examine here, before moving to the more detailed derivation with a variable g. We begin with an informal discussion to highlight the intuition behind the derivation of the expression for U T s in terms of the present equivalent amount of H. Assume that a small amount is subtracted from H at time t 0 , and the same amount is added to H at a prior time t 0 − ∆t where ∆t 1. Then, there is some gain associated with having a greater consumption during the period between t 0 − ∆t and t 0 . Specifically, the consumption is greater by c(t 0 ) during a period of length ∆t and the gain to utility is given by To find the (t 0 − ∆t)-value of the amount at time t 0 , we need to find by how much we need to discount to compensate for the gain: Instead of adding to H at time t 0 − ∆t, we need to add only where the loss due to δ s has to compensate for the gain. A positive discount, δ s > 0, implies that, for any t > t 0 , consumption is reduced by δ s c(t) from what it could be without a discount. Therefore,

S4
Alternatively, one may look at the loss as the time it takes until consumption recovers. Since consumption decreases by δ s c, and since recovery rate is gc, the time to recovery is (δ s /g)∆t.
U loss comprises three terms: (1) (minus) the value during the time it takes consumption to recover, (ii) the value during the last period before T that is never approached due to the delay, and (iii) if ρ = 0, the decrease in value after recovery due to the delay: Indeed, for a constant g, Eqs. (S17) and (S18) are equivalent as Eq. (S18) can be derived directly from Eq. (S17) via integration by parts (du/dt = (du/dc) · (dc/dt) where dc/dt = gc, and thus, du/dc = (du/dt)/gc). From setting U gain = U loss using Eqs. (S16) and (S17), it follows that (S19) In the following, we derive the expression for δ T s formally and we complete the derivation of the expression for the shadow value. The equivalent expression following from Eq. (S18) is derived formally in Section S3.

Formal derivation
To derive the results above and get the expression for U T s , we assume that T 0 H(t)dt 1. (this is a special case of the assumption in Theorem 2 where g is constant). It follows that It follows that where This implies Thus, for all T , where As in section 1.2, we need to divide by u (c 0 ) to obtain the equivalent value in terms of consumption. Therefore, When g = g(c * ), a change in c * at time t changes the entire trajectory g(c * (t)) for all times t > t .
We would like to find how the function H(t) affects consumption, c * (t), assuming that H(t) is sufficiently small (and to see what 'sufficiently small' means here). We express c * as c − plus the sum of all contributions by H(t ) for all t < t. Specifically, we go backward in time from t = t to t = 0, and, for each t , we examine how g(c * (t )) → g(c * (t )) + H(t ) affects consumption at time t, given that H(t ) = 0 for all t > t . In other words, we consider the equality where, for t ≥ t ,ĉ t (t) is given by with initial conditionsĉ Note that Eq. (S29) is exact and applies to any H(t) as long as the variational term dĉ t (t)/dĉ t (t ) exists for all 0 ≤ t ≤ t.

S6
To calculate the variational term, we derive both sides of Eq. (S30). For simplicity, we omit the subscript t and denoteĉ(t) =ĉ t (t) and g(t) = g(ĉ(t)). For all t > t , d dt Next, note that the three conditions g(t) = 0, |H(t)| |g(t)| and c 0 = 0 together imply that and therefore, d dt which implies that .
Substituting Eq. (S36) into Eq. (S29) and noting that c t (t ) = c − (t ) implies where g(c − (t )) appears in the last denominator becauseĉ t (t ) = c − (t ) (Eq. (S31)). Similarly, for all t < t,ĉ Next, we use the assumption that H is sufficiently small on [0, T ]. Denote namely,ĉ Note that, since for all 0 ≤ t ≤ t ≤ T . From the substitution of Eq. (S40) in Eq. (S37), it follows that And since In summary, it follows from Eq. (S43) that Now that we calculated c * (t) in first-order of 0 (t), we can find the first-order change in welfare.
Note that This implies where

S8
This implies Alternatively, since it follows that

S1.4 Net Present Effective Value and the Price of H
Note that H affects the term U T c in two ways: (1) It alters the discount rate, and (2) it adds another term. Since U T − is independent of B and H (and thus independent of the control x), the objective is to maximize the total change in utility due to the control, where U T c /u (c 0 ) is the actual change in present value and U T s /u (c 0 ) is the shadow value. Substitution of U T c /u (c 0 ) (Eq. (S15)) and U T s /u (c 0 ) (Eq. (S56)) in Eq. (S58) yields

S9
where with δ(t) given by Eq. (S14) and δ T s (t) given by Eq. (S57). Equivalently, we can write where This implies that Eq. 7 holds, where and

S1.5 The limit T → ∞
To complete the proof of Theorem 1, consider the limit T → ∞. Define v(t) = lim T →∞ v T (t).
If v(0) = ∞, then lim T →∞ σ 0 = ∞ and lim T →∞ δ T s = 0. This implies that the NPV becomes negligible compared to the shadow value at T → ∞ and the objective is to maximize On the other hand, if v(0) is finite, the objective becomes to maximize the GNPV with some positive δ and finite δ w and σ 0 . This completes the proof of Theorem 2. S10 S2 Proof of Theorem 1: Expression for the Shadow Value for General Perturbations, H(t), Constant Elasticity, η, and

Constant Exogenous Growth, g
In this section, we prove Theorem 1 (main text). The proof relies on some development from the proof of Theorem 2. Specifically, we do not need to derive again the expression for the NPV (Section 1.2) because that derivation does not rely on our assumption that H is small. We only need to derive an expression for the shadow value in terms of present equivalent changes to economic growth (Eqs. 7-11 in the main text). In Section 2.1, we relax the assumption that H(t) is small, and we derive a general expression for the shadow value allowing any form of H(t). Next, in Section 2.2, we focus on the case where η and g are constant and derive a closed-form expression for the discount function, δ T s (t), and the coefficient σ T . Finally, in Section 2.3, we consider the limit where T → ∞ and complete the proof.
S2.1 Deriving a general expression for the shadow value for general H, u and g First, we express the term as a base term without any H, plus the sum of all contributions due to H(t) starting at t = 0 and going up to t = T : The meaning of the condition H(t ) = 0 for any t > t is that, instead of c * , we consider a consumption function that follows a per-capita growth at a rate g + H if t < t and g if t > t.
Here we sum the contribution due to H(t) in ascending order, from t = 0 to t = T , whereas in Section 1.3.2 we considered the contribution of H(t) in descending order, from t = t to t = 0.]

It follows that
where To further develop the expression for U T s , we introduce the assumption that g and η are constants.
S2.2 Deriving closed-form expressions for δ s and σ 0 assuming constant η and g

Constant g
Assuming that g is a constant (or just assuming that g depends purely on time t but not on c * ) implies that c t (t ) is given by To calculate the variational term dc t (t )/dc t (t), We derive both sides of Eq. (S71). Assuming that g is constant, this implies, for all t > t, d dt [Note that if we do not assume that g is constant but assume instead that g(t) = 0 for all t, then S12 (see section 1.3.2).] On the other hand, for all t < t, a change inc t (t) has no effect onc t (t ). It follows that Substituting this equation into Eq. (S73) implies

This implies
and where a 0 and a 1 are constants (note that, since utility is invariant to multiplication and addition of constants, a 0 and a 1 have no effect on the results). Substitution of Eqs. (S75) and (S82) into Eq.
(S80) implies where It follows that and where v T (t) is given by Eq. (S53) and by Eq. 15 in the main text. Therefore (see also Section 1.3.2), and Next, to calculate v T (t) in the special case where g and η are constants, note that substitution of This term is becoming simpler as we consider the limit where T → ∞. = (η − 1)H(t) + max{(η − 1)g − ρ, 0} and the objective is to maximize the NPV plus the shadow value where δ s (t) = (η − 1)(g + H) + ρ.
Second, where η ≤ 1 − ρ/g, σ T is becoming infinitely large as T → ∞ and the objective becomes to maximize the shadow value, which is proportional to Therefore, maximizing utility is equivalent to This completes the proof of Theorem 1. S15