DBJ Research Center on Global Warming
Discussion Paper Series No. 53 (10/2015)

The Revelation of the Time Preference Rate andIntertemporal Negative Externality
Morio Kuninori
Masayuki Otaki

This paper is a preliminary material in the draft form to stimulate discussion
and comments from academics and policy makers.Views expressed in thispaper are those of the author(s) and do not necessarily reflect the official
views of the Development Bank of Japan.

The Revelation of the Time Preference Rate andIntertemporal Negative Externality
Morio Kuninori
Hosei University
kuninori@hosei.ac.jp
Masayuki Otaki
The University of Tokyo
ohtaki@iss.u-tokyo.ac.jp

Abstract
Different from the overlapping-generations model, it is allowable to discount the future
utility in a dynasty model without intergenerational conflicts. While the social utility
discount rate should be equal to the time preference rate, such a rate is unobservableand hence must be estimated based on observables and measurables based on a theory.
Much precedent research uses the Ramsey type optimal growth theory for thisestimation. However one must note almost all estimations neglect the existence ofintertemporal negative externalities. This problem is vital when one analyzes the global
warming problem mainly caused by the excess concentration of carbon dioxide (CO2).
This is because there emerges another effect of capital accumulation besides the
improvement of product capacity that is expressed by the rate of interest (orequivalently, marginal productivity of capital). There exists the negative externality to
the future productivity that is originated from the excess emissions of CO2. Accordingly,
the rate of time preference is always lower than the rate of interest even in asustainable growth path where there is no growth in consumption.

Keywords: Time Preference; Interest Rate; Negative Externality by the Emissions of
CO2
JEL Classifications: Q54;Q58;D61;D62

11. Introduction
It requires some elaborations to determine the rate of time preference in a dynastic
social planning problem because such a rate is unobservable. Determining its value
equivalently means to find a substitutable form for the rate of time preference byobservable and/or measurable variables such as the rate of interest. Many precedingstudies 1 are based on the Ramsey type optimal growth theory for the estimation.
Whenever the rate of preference exceeds the rate of interest, the optimal consumption
proportionately decreases because current consumption is more advantageous than the
future. Accordingly the following well-known formula is obtained:  r cc

(1)where  is the rate of time preference, and r is the (real) rate of interest. and1cdenotec

In discounting the far-distant future events, two approaches are noted (Arrow, et al.

1996). One is the descriptive approach, which typically focus on the opportunity cost ofcapital whose level is observable as the rate of return on alternative investments in themarket (Nordhaus, 2013). The advocates of this approach employ the opportunity cost ofcapital as the discount rate since they claim that investments in reducing climatechange must compete with alternative investments in the market. See also Nordhaus(2007) and Weitzman (2007). The other is the prescriptive approach, where theadvocates maintain the view that the market interest rates fail to indicate the trade-offs
of consumptions across generations and that the discount rate should be derived fromethical point of view. Those advocates include Cline (1992) and Stern (2007).
Proponents in both approaches use the same Ramsey rule of Equation (1) ininterpreting their discount rates. As we argue later in the article, main problem of thisexpression is the fact that there is no negative externality postulated in the rule. (SeeAppendix I to find the limitation of the standard Ramsey rule. See also Appendix II for
its scope of validity in relation with the technological change.)
In addition, this formulation does not provide the solutions for the time preference
rates  or elasticity of marginal consumption  endogenously, leaving the eachadvocate in both approaches to assign appropriate values as parameters according tohis/her observations or value judgement.
Examples of other approaches where social discount rate is endogenously derived arefound in Otaki (2013 and 2015).
2

the elasticity of marginal (instantaneous) utility2 and the increase rate of consumption,
respectively. According to Equation (1), except for a stationary state, the time preference
rate is lower than the rate of interest as far as an economy grows. Those who infer thatthe time preference in a social planning might be lower than the market interest ratefocus on refining the estimation of the second term of Equation (1)3.
However, when one analyzes the efficient allocation of public bads such as theemissions of CO2, he or she finds that Equation (1) alienates from the effect caused bysuch intertemporal negative externalities. This asks us for an extension of the theory.
Suppose that a unit current capital investment not merely strengthens the production
capacity of an individual firm but also, via the aggravation of the global warming, if notoffsetting the former positive effect, lowers the productivity as a whole. Let this rate bedenoted . In this case, the social rate of return is less than the rate of interest that isequal to the marginal productivity of capital of an individual firm.
This study obtains the following modified formula: Kt 
ct ct
 Kt 

  r  * (2)where  * is the optimal carbon tax rate that is solved in the next section. denotes theelasticity of CO2 stock to capital accumulation. g n is the autonomous emissions
associated with the incumbent capital equipment. Thus, one finds that the time
preference  is lower than the rate of interest r even in a stationary state where

c
0.
c

This article is organized as follows. Section 2 derives the formula in Equation (2) and2

Three kinds of interpretation of the elasticity of marginal utility are known (Stern,2008). They are intratemporal distribution, intertemporal distribution and attitudes torisk, respectively. Directly related to our argument is the intertemporal distribution andin this analysis we postulate that we can set the appropriate values for  . Since, under
the stationary state wherec 0 , the role of  plays the secondary role, we leave howcto treat  to a future study.
3

Introduction of uncertainty into discounting leads us to a theory of declining discountrates (Arrow, et al., 2013). For example, uncertainty of the growth rates of future
consumption ( See Gollier, 2012) and uncertainty of future discount rates such as therates of return on investment (See Weitzman, 1998 and Gollier-Weitzman, 2010) will
typically cause the discount rates to decline as time goes by.
3

discusses the implications. Section 3 concludes with brief remarks.
2. The Model
2.1 The Derivation of the New Formula
Since Equations (1) and (2) are derived from the local maximization on the optimal
path, it is sufficient to consider the optimality condition between two sequential periods.
Let the aggregate production function over the world, f ,be denoted4

yt  f  Kt , Et  , Et    Kt  ,

f
f

 0,
 0,
 0.
K
E
K

(3)

yt denotes the volume of current output, Kt , Et are the capital stock and theaccumulated emissions of CO2 that prescribes the environmental condition, respectively.

 is the function that represents how much emissions are accumulated by production
activities by using capital.5
Consequently, the social planning problem which one must solve is

 





1
max It u f Kt* ,   Kt*   I t t 
u f  Kt t ,   Kt t    I t*t t 
1  t









(4)

subject to

Kt t  Kt*  It  t ,(5)

where u is a strictly concave utility function. K t* is the optimal capital stock at the
beginning of period t that has been determined beforehand by the past decisions. I t*t is
the given optimal investment during period t  t .  denotes the instantaneous time
preference rate.
Since the productivity slowdown brought about the CO2 emissions is external as foreach firm, the profit-maximization condition requiresf 1  r  t 
K

(6)

where r is the instantaneous interest rate. In addition, if the proportional carbon tax tothe increase rate of CO2 stock, the rate of which is , is levied to firms to internalize the

We assume that each economy in the world has the same production and utility
function, and endowments. Accordingly, such functions can be aggregated.
5 We do not exclude the possibility that an economy-wide carbon neutral technology
might be available, which effectively absorbs emitted CO2, and/or the usage of
non-carbon emitting energy resources fully in order to avoid the increase of the CO2
concentration.
44

negative externalities originated from CO2 emissions, this lowers the return from the
new capital investment by 

Et . The optimal tax rate should be equalized to theE tmarginal negative productivity of capital brought by the emissions. Thus, one obtains

f   K 
* E




  K t E t K  K 
f  K  K Et   *t   *  t  K K t t K K t 

(7)Thus, the optimal tax rate * is equal to how many outputs are curtailed by one percent
increase in the accumulate stock of CO2. That is, it is evident from Equation (7) that

*  

f

(8)holds.
Furthermore, one must note that the following relationships:

 K c o  K   o  t   o  t  , o  c   o  t   o  t  .
 t t 

(9)

Differentiating Equation (4) with respect to I t , and taking the relationships inEquations (6), (7) and (8) into consideration, we obtain

 

*

 u ' ct t* E
1  t  1  r t tt  u '  ct*   K  K  o  K  o  t    c 1   r  * t  o  c   t  1  tt   
ct

 
 K K t 

(10)Dividing both sides of Equation (9) by t , one can ascertain K  K 
c o  t 


.

ct
t
 K K t 

  r  

(11)

Taking the limit t  0 of Equation (11), finally we obtain the following representationform: Kt 
ct ct
 Kt 

  r  * 

where is the elasticity of the accumulated CO2 to capital stock6.

6

For example, if the form of the production function is5(12)

2.2 The Implications of the Derived FormulaFirst, we shall consider the economic meaning of the formula in Equation (12). The mostprominent feature of the formula is that the social rate of return from unit capital
accumulation is lower than the real interest rate r . This is because an additional capital
reduces productivity via the accumulation of CO2 emissions. This negative effectappears in the second term of the right-hand side of Equation (12). The inside of thebracket comprises the total increase rate of emitted CO2. Since  * is determined
optimally in order that it completely internalizes such a negative diseconomy, thesecond term, as a whole, corresponds to the exact carbon price that has to been paid.

 Kt 
 in comparison with the market
 Kt 

Thus, the social rate of return is lowered by  *interest rate. Accordingly, the time preference rate  is strictly smaller than the rate ofinterest, r , even in the stationary state wherect 0 holds. Such a constantct

consumption path is dubbed sustainable by Dasgupta and-Heal (1974) and Solow(1986) 7 . This definition of sustainability seems to be characterized by theintergenerational fair opportunity for consumption, although the analysis is formulated
as the optimization problem of representative individual and firm, both of which live
infinitely.8
From Equation (12), the capital growth rate, g k* , which sustains the zero consumption
growth is expressed asg k* 

r

 *

(13)

.

Equation (13) indicates that the sustainable growth path is affected by the following
two factors, given that the difference r   is positive.
First, when the optimal carbon tax rate  * is high, which implies that theyt    f  kt   ,   Kt , t      Kt , t   kt   , 0      1,

  Kt , t    Kt  , Kt  10 kt    d ,,and the utility function belongs to the CRRA family, the formula in Equation (12)becomes an exact solution.
7 Note that Solow (1992) is skeptical about the arbitrariness of the concept ofsustainability.
8 Find Appendix III to see if there exists a feasible path with a constant consumption
growth.
6accumulation of CO2 exacerbates the productivity more seriously, the sustainablegrowth rate becomes lower. This is because a higher tax rate curtails the social rate of
return from capital. Second, if the elasticity of CO2 emissions to capital investment ishigh, the growth rate also becomes lower. This result comes from the fact that a higher
elasticity substantially raises the carbon price.
2.3 The Limit of Economic Growth: The Case of Linear Homogenous Individual
Production Function with Negative Externality
We heretofore assume that the rate of interest, r , is constant over time. However, as
Equation (6) shows, this varies along with capital accumulation. Since we assume that

2 f
 2 f dE 0,
K 2 K E dK(14)

the relationship between capital stock, K , and the rate of interest, r , becomes
downward sloping curve KK as illustrated in Figure 1. It is clear from Equation (13)that if the world interest rate is located above Line LL , capital accumulation advances.
Otherwise, disinvestment should be fastened. Ultimately, the world economy converges
to Point E . This asserts that there is a limit of the world economy’s growth when aintertemporal negative externality such as the global warming exists.
What one must note is that even though the production function of each economy is alinear function on capital stock, which means

2 f
=0,
K 2

(15)

Equation (14) holds, and thus the property of Figure 1 is preserved. This means that ouratmosphere is not renewable resources as human being as a whole, and therefore, eventhough the artificial technological system in the short run contains no non-renewable
resources, the scarcity of the quality of atmosphere becomes the bottleneck of the
production/consumption enjoyment in the long run.

3．Concluding Remarks
The representation theory concerning the time preference rate under an intertemporal
negative externality (e.g., caused by the excess emissions of CO2) has been consideredwith the following results. First, even under a sustainable equilibrium where theaggregate consumption level is kept constant, the time preference rate is lower than theinterest rate. This stems from the fact that the negative externality associated with thecapital accumulation lowers the social rate of return from capital. It is notable that such7

a negative effect is not taken into consideration in the preceding articles analyzing the
global warming problem despite of its importance.
Second, the optimal rate of the proportional carbon tax is also derived explicitly. The
rate is equal to the marginal decrease in product per one percentage increase in CO2 in
atmosphere. That is, if the emissions exacerbate productivity more seriously, the tax
rate should be heightened.
Finally, besides the optimal carbon tax rate, the following economic factor affects thesocial rate of return from capital that is defined at the sustainable stationary state in
the sense of Dasgputa-Heal and Solow: the elasticity of the density of CO2 to the capital
accumulation . Whenever  increases, which means that more CO2 is emitted by aunit capital accumulation (i.e., lower efficiency in emissions reduction process), lowers
the social rate of return from capital because this incurs higher costs for capital
accumulation.
References
Arrow, K.J., W.R. Cline, K-G. Maler, M. Munasinghe, R. Squitieri, and J.E. Stiglitz(1996)“Intertemporal

Equity,Discounting,andEconomic

Efficiency,

Intergovernmental Panel on Climate Change, Working Group III, pp. 127-144.
Arrow, K., M. Cooper, C. Gollier, B. Green, G. Heal, R. Newell, W. Nordhaus, R. Pindyck,
W. Pizer, P. Portney, T. Steiner, R.S.J. Tol and M. Weitzman (2013) “Determing
Benefits and Costs for Future Generations,” Science, Vol. 341, pp. 349-350.
Cline, W.R. (1992) The Economics of Global Warming, Washington DC: Institute forInternational Economics.
Dasgupta, P. and G. Heal (1974) “The Optimal Depletion of Exhaustible Resources,”

Review of Economic Studies, Symposium: pp. 3-28.
Gollier, C. (2012) Pricing the Planet’s Future: The Economics of Discounting in an

Uncertain World, Princeton, NJ: Princeton University Press.
Gollier, C. and M. Weitzman (2010) “How Should the Discount Future be Discounted
when Disount Rates are Uncertain?,” Economics Letters, Vol. 107, pp.350-353.
Nordhaus, W. (2007) “A Review of the Stern Review on the Economics of Climate
Change,” Journal of Economic Literature, Vol. 45, pp. 686-702.
Nordhaus, W. (2013) The Climate Casino: Risk, Uncertainty, and Economics for a

Warming World, New Haven, CT: Yale University Press.
Otaki, M. (2013) “Endogenous Social Discount Rate, Proportional Carbon Tax, andSustainability: Do We Have the Right to Discount Future Generations’ Utility?,”8

Environmental Systems Research, 2:1, pp. 1-8.
Otaki, M. (2015) Local Altruism as an Environmental Ethic in CO2 Emissions Control,

Atmospheric and Climate Sciences, 5:4, pp. 433-440.
Ramsey, F.P. (1928) A Mathematical Theory of Saving,” Economic Journal, Vol. 38, pp.
543-559.
Solow, R.M. (1986) “On the Intergenerational Allocation of Natural Resources,”

Scandinavian Journal of Economics, Vol.88, pp.141-149.
Solow, R.M. (1992). “Sustainability: an Economist's Perspective, National Geographic

Research and Exploration, Vol. 8: pp. 10-21.
Stern, N. (2007) The Economics of Climate Change: The Stern Review, Cambridge, UK:
Cambridge University Press.
Stern, N. (2008) “The Economics of Climate Change,” American Economic Review:

Papers & Proceedings, Vol. 98, pp. 1-37.
Weitzman, M.L. (1998) “Why the Far-Distant Future Should be Discounted at Its
Lowest Possible Rate,” Journal of Environmental Economics and Management, Vol.
36, pp. 201-208.
Weitzman, M.L. (2007) “Review of the Stern Review on the Economics of Climate
Change,” Journal of Economic Literature, Vol. 45, pp. 703-724.

910Appendix I: The Formula  r c
Cannot Describe the Global Warming ProblemcIntact
The Ramsey formula   r  
the case that    0 and*c
is a special case of our formula (2) which corresponds tocF r  const. One must note that since the production
K

function of each industry is linear on capital, accordingly, the aggregation is feasible.
Thus, we obtain the macroeconomic production function F asyt  rKt(A.1)

Equation (A.1) implies that there is no other scarce production resource than capital.
This fact asserts that there is a serious drawback in the standard Ramsey model toanalyze the global warming problem.
This is because the scarcity, which stems from the quality of our atmosphere(measured by the mass of CO2 contained in the atmosphere), is the vital issue in thisproblem. As such, unless we assume some optimistic exogenous emissions-absorbing
technological progress, it is a plausible theoretical formulation that the aggregatedproduction function is subject to the diminishing return to scale owing to the
Marshallian negative externality from CO2 emissions. That is, theoretically, thecongestion in availing the atmosphere is the very core of the global warming problem.

Appendix II: The Formula  r cand Technological ProgresscPreserving the relevance of the formula   r  cin the context of the global warmingcproblem, we need to introduce some exogenous technological progress that reduces theconcentration of CO2 at a constant rate  . In such a case, for example, a macroeconomic
production becomesYt  Kt  Kt e  t   .



(A.2)

The inside of the bracket indicates the external effect associated with theconcentration/reduction of CO2, which corresponds to the exogenous total factor
productivity (TFP) for each firm. Accordingly, as far as the TFP is kept constant, theworld economy can achieve a steady growth rate because the macroeconomic production
function becomes linear.
By (A.2) such a rate is represented as
11

dKt e  t
K0 t   ,
dt
Kt

(A.3)

F '  K *   r *  const. .

(A.4)

Applying Equations (A.3) and (A.4) to the formula (2), one obtainscc  r*  .

(A.5)

One here must note that Equation (A.5) presumes an interior solution in the sense thatthe increase rate of consumption,ct, can take an arbitrary value. However, from thectbudget constraint,

Kt   Kt  r * Kt  ct
 Kt 

ct
K c
  t  t
r 
Kt ct

(A.6)

*

holds. Thus, (A5) necessarily becomes at the steady state equilibrium such as  r  .

(A.7)

In other words, Equation (A.7) implies that the feasibility of the future increase in
consumption is entirely relies on the autonomous technological progress in the
reduction of the stock of CO2.
The stability of the stationary state can be proved as follows. Again, from the budget
constraint,

Kt  gt Kt  rt Kt  ct
 Kt ct
gt
c (A.8)
 gt 
 trt  gt
 rt  gt  ct

holds.
On the other hand, sincert   Kt e  t  ,(A.9)

logarithmically differentiating Equation (A.9) with respect to time, one obtainsrt     gt  .
rt(A.10)

Thus, our economy can be described by the two differential equations on  gt , rt  as inEquations (A.8) and (A.10). The phase diagram of this system is illustrated in Figure
12A.1. Accordingly, the economy has saddle-point property, and the planning economy canachieve these stable paths, which converges to the stationary equilibrium, E . Thus,
equilibrium growth rate of consumption in the steady state is equal to  , and theformula (A.7) is upheld.
Appendix III: There Is No Feasible Path with a Positive Constant Consumption GrowthRate
In this Appendix III, we show that any positive constant consumption growth rate pathis unfeasible in our model, and hence the only feasible path is the sustainable path (i.e.

c  0 ).
Assume that the growth rate is denoted g c . Then Equation (12) implies that there
must exist K * in the stationary state such as gc  r  K *     0(A.11)c  F ' K  K  K  0(A.12)

However,holds, if there is a constant K * which satisfies Equation (A.11). This is a contradiction.

13

14