We endogenize the discount rate via wealth and provide evidence that wealth affects the discount rate negatively. We give a necessary and sufficient condition for endogenous discounting to lead to the Twin-Peaks of economic growth and show that improvements in technology help avoid them.endogenous discounting, Twin-Peaks of economic growth, multiple equilibria.

Ingmar Schumacher

English

Research Papers in Economics

 
 
ENDOGENOUS DISCOUNTING VIA WEALTH, 
TWIN-PEAKS AND THE ROLE OF TECHNOLOGY 
 
 
 
Ingmar SCHUMACHER 
 
 
 
 
January 2009 
 
 
Cahier n° 2009-03 
 
 
 
                              ECOLE POLYTECHNIQUE                       
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE 
 
 
DEPARTEMENT D'ECONOMIE 
Route de Saclay 
91128 PALAISEAU CEDEX 
(33) 1 69333033 
http://www.enseignement.polytechnique.fr/economie/  
mailto:chantal.poujouly@polytechnique.edu 
 
 
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9Endogenous Discounting via Wealth,
Twin-Peaks and the Role of Technology
Ingmar Schumacher1
Ecole Polytechnique, Paris
January 23, 2009
Abstract
We endogenize the discount rate via wealth and provide evidence that wealth affects the discount rate neg-
atively. We give a necessary and sufﬁcient condition for endogenous discounting to lead to the Twin-Peaks of
economic growth and show that improvements in technology help avoid them.
JEL classiﬁcation: D90, C61, O41.
Keywords:endogenous discounting, Twin-Peaks of economic growth, multiple equilibria.
1 Introduction
We are becoming more and more aware that convergence between country’s income levels is far from obvious.
Empirical observations have forcefully shown that rich countries are becoming richer, while poor ones are not
able to catch up. Jones [14], Azariadis [1] and Quah [21] ﬁnd that the world income distribution changed from
a somewhat normal distribution shape toward a twin-peak shape. We provide new answers to the following two
questions: One, what can be a potential source of these twin-peaks? Two, why were some of the countries that were
believed to belong to the group of low steady state countries (like Taiwan, South Korea) able to reach a convergence
path which led them to a high steady state?
Theoretical models have attempted to draw some conclusions on the possible sources of these twin-peaks of
economic growth. These models have either built on non-convexities in technologies to arrive at multiple steady
states (Azariadis and Drazen [2]) or on incomplete markets (Durlauf [8]; Quah [21]).
In this article we propose endogenous discounting as another potential source of the twin-peaks of economic
growth. Speciﬁcally, we argue that wealth affects the discount rate by proposing that this is a shortcut for suggesting
that wealthier countries have better health standards (Pritchett [20]) and better insurance markets (Carroll [4]), all
of which generally affects the agent’s discount rate negatively (Frederick et al. [12]).
There exist now an increasing number of endogenous discounting models, which nearly exclusively have con-
sumption as the source of endogenity (e.g. Epstein and Hynes [9]; Obstfeld [17]; Das [5]). In a recent working
paper, Drugeon [7] allows for consumption and capital to inﬂuence the discount rate, but focuses on increasing
returns in the production function and does not provide a thorough analysis of multiple steady states.
Our treatment of wealth here will be based upon a broad view of wealth combining physical capital and human
capital (see Mankiw [15]). Fielding and Torres [11] estimate the relationship between wealth, health and education
for 41 developing countries. They ﬁnd that improvements in physical and human wealth lead to lower mortality
rates. Their results are robust even across countries, pointing at a uniform effect of the variables in question. Similar
1Department of Economics, Ecole Polytechnique, 91128 Palaiseau Cedex, France. email: ingmar.schumacher@polytechnique.edu, tel:
0033169333038.
The author is grateful for helpful comments by Raouf Boucekkine. Financial support from the Chaire Lhoist Berghmans in Environmental
Economics in CORE and from the Chair AXA Assurance et Risques Majeurs at the Ecole Polytechnique is kindly acknowledged. This
article was mostly written during my stay in CORE. Any errors lie with the author.
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9results are due to Grossman [13], Feldman et al. [10], Pappas et al. [18], Preston and Elo [19], Richards and Barry
[22]. Wealth as a source of endogenous discounting has also been proposed by Becker and Mulligan [3] as well as
Deaton and Paxson [6], who show that ﬁnancial assets and human capital inequality grow as cohorts age. These
results are interpreted by Becker and Mulligan [3] as a potential consequence of endogenous time preference.
Wederiveanecessaryandsufﬁcientconditionfortheexistenceofmultiplesteadystatesandshowthatdiscount-
ing endogenized via wealth can provide another explanation for the development of the twin-peaks of economic
growth. We believe this is a particular useful extension as existing contributions to the literature on twin-peaks
exhaustively focus on the twin-peaks as a result of assumptions on technologies or uncertainty. Here, however, the
twin-peaks arise through the preferences of the agents.
We answer our second question by demonstrating how improvements in technology can help avoid the twin-
peaks. This result ought to be particularly interesting for the case of developing countries.
2 The Model
The model is based on an inﬁnitely-lived agent approach where the agent obtains utility u() from consumption
c(t). In addition, his wealth k(t) affects the discount rate (k(t)) negatively. Wealth can be accumulated by
investing but is reduced by consumption and constant depreciation  > 0. The inﬁnitely-lived agent then solves the
following problem.
max
fc(t)g
U(c;k) =
Z 1
t=0
u(c(t))e (t)dt s.t.
8
> > <
> > :
_ k(t) = f(k(t))   c(t)   k(t); 8t
_ (t) = (k(t)); 8t
k(t)  0; c(t)  0; 8t;
withk(0) given.
(1)
We make use of the following assumptions.
(A1) We impose that the production function f : R+ ! R+ follows standard assumptions of concavity, such that
f(k)  0, f(0) = 0, f0(k) > 0, f00(k) < 0, with limk!0 f0(k) = 1, and limk!1 f0(k) = 0. We deﬁne  k as the
level of k which solves f(k) = k, and ~ k as the level of k that solves f0(k) = .
(A2) The utility function u : R+ ! R+ is at least twice continuously differentiable with u0(c) > 0, u00(c) < 0, 8c,
and limc!0 u0(c) = 1.
(A3) We take (k) : R+ ! R+ with 0(k) < 0 and 00(k) > 0, where we impose limk!1 (k) > 0.
This model generalizes the neoclassical Ramsey-Caas-Koopmans model by introducing a discount rate endo-
genized via wealth. It reduces to the classical model if we assume that (k(t)) = r, 8t. Integrating _  = (k(t))
from t 2 [0;] then leads to () = r. Substituting this into (1) gives the classical model.
2.1 Solving the Model
We deﬁne the value function
V (k) = max
fcg
U(c;k) s.t. _ k = f(k)   c   k; k(0) > 0 given; (2)
which leads to the following Bellman equation
0 = max
c
n
u(c)   (k)V (k) + V 0(k)_ k
o
: (3)
The ﬁrst order conditions are
u0(c) = V 0(k): (4)
By equation (4) we can write c(k), substituting this in (3) gives
0 =  0(k)V (k) + V 00(k)_ k + V 0(k)(f0(k)      (k)): (5)
2
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9We know that V 0(k) = , with  being the shadow price of capital. Also, V 00(k)_ k =
@V 0(k)
@k
@k
@t, implying V 00(k)_ k =
_ . V (k) is the prospective optimal value of utility. Rewriting leads to the following system
_ c =  
u0(c)
u00(c)

f0(k)      (k)  
0(k)
(k)

u(c)
u0(c)
+ _ k

; (6)
_ k = f(k)   c   k: (7)
The last term of equation (6) is obtained by making use of the correspondence between ﬁrst-order conditions from
the Hamiltonian method and the Bellman method as well as using Michel’s [16] transversality condition. That this
correspondence indeed holds is because the conditions from Stokey and Lucas [24], are satisﬁed. For 0(k) = 0
we see that equation (6) reduces to the Keynes-Ramsey rule of optimal consumption.
We deﬁne m(k) =
00(k)(k) 0(k)2
(k)2 with m(k) >
< 0, and n(k) =
u0(c)2 u00(c)u(c)
u0(c)2 > 0.
Proposition 1 A necessary and sufﬁcient condition for multiple steady states1 is 9k > 0 such that
f00(k) > (1 + n(c))0(k) + m(k)
u(c)
u0(c)
+
0(k)2
(k)2 n(c)
u(c)
u0(c)
; (8)
with c = f(k)   k.
In a standard Ramsey-Caas-Koopmans model with a concave production function this condition will never be
satisﬁed since the right-hand side of equation (8) will be equal to zero. We denote the steady states as 0 < kl <
km(< ~ k) < kh(<  k). When one linearizes the dynamical system around the non-trivial steady states, fcl;klg,
Figure 1 Figure 2
fcm;kmg and fch;khg, one gets the following differential system:
J =
2
4 (k) +    f0(k)  
u
0(c)
u00(c)

f00(k)   0(k)   m(k)
u(c)
u0(c)   (f0(k)   )

0(k)
(k)

 1 f0(k)   
3
5: (9)
As the trace is Tr(J) = 1 + 2, and the determinant is Det(J) = 12, with 1 and 2 denoting the eigenvalues
of the Jacobian, saddle-path stability occurs if the trace is positive and the determinant is negative. The trace of this
matrix is given by Tr(J) = (k) > 0, while the determinant is negative if
f00(k) < 0(k) + m(k)
u(c)
u0(c)
+
0(k)
(k)
(f0(k)   )n(k): (10)
From the conditions for multiple equilibria we know that this condition does not hold for the middle steady state,
which is thus instable. It holds for the low and high steady states, which implies saddle path stability. Complex
1The case of unique steady state has been treated in Schumacher [23].
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9dynamics arise if Tr(J)2 < 4Det(J). As Det(J) < 0 for the low and high steady state, this excludes the possibility
of complex dynamics for these steady states. However, for the medium steady state complex dynamics occur if
(k)2 <  4
u0(c)
u00(c)

f00(k) 0(k) m(k)
u(c)
u0(c)  n(k)
0(k)
(k) (f0(k) )

. This corresponds to the recent twin-peaks
of economic growth hypothesis raised by researchers like Quah [21], Jones [14] and Azariadis [1] and is depicted
in Figure 1.
In order to understand where the multiplicity of equilibria comes from we re-write equation (6) by solving for
V (k), which gives
_ c(t) =  
u0(c(t))
u00(c(t))

f0(k(t))      (k(t))  
0(k(t))
R 1
t u(c(s))e (s)ds
u0(c(t))

(11)
The multiplicity of steady states occurs since when an agent is very poor then increases in consumption are nec-
essary for survival (u0(0) = 1), and the agent is so impatient that the preferences are nearly exclusively directed
toward today, implying that most wealth will be directly consumed. Similarly, if the future expected utility is
very low, then our agent will prefer to consume current wealth. For k > km however, this implies that overall
wealth has already been built up sufﬁciently in order to incorporate far-sighted goals, such that the discount rate
is relatively low (or the life expectancy high enough). For example, with a low enough mortality rate agents will
need to carefully plan ahead to the distant future, whereas a high mortality rate implies that agents don’t expect to
become old and thus do not plan ahead. If we use Stone-Geary preferences to analyze the impact of changes in
the subsistence level of consumption, then we ﬁnd that increases in the subsistence level decrease the last term in
equation (11). This means that the higher the subsistence level or the closer is consumption to subsistence the less
willing are agents to postpone consumption and the less important becomes the impact of wealth on the discount
rate. In other words, consumption close to subsistence and a low expected future utility, both a likely phenomena
in less developed countries, are likely to trap countries in the low wealth equilibrium.
One can easily show that improvements in total factor productivity lead to shifts in the steady state curves as
depicted in Figure 2. The intuition is that if wealth is more productive, then it is more efﬁcient to increase the
level of wealth in order to raise overall utility by reducing the discount rate. In recent years, some of the countries
that were believed to belong to the group of low steady state countries (South Korea, Taiwan, etc.) were able to
reach a convergence path which led them to a high steady state. It is plausible that this is due to improvements
in technology or health (mortality). This should provide a new and interesting possibility for policy interventions
which focus on international convergence of wealth.
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9

Are Educational Differentials In Mortality Increasing in the United States?

Buffer-stock saving and the life cycle/permanent income hypothesis.

Education and nonmarket outcomes.

Endogenous discounting and the domain of the felicity function.

Fertility, Education and Inequality.

Hynes. The rate of time preference and dynamic economic analysis.

Intertemporal Choice and Consumption Inequality.

Intertemporal dependence, impatience, and dynamics.

National trends in educational differentials in mortality.

Nonergodic Economic Growth.

On the evolution of the world income distribution.

On the transversality condition in inﬁnite horizon optimal problems.

Optimal growth with decreasing marginal impatience.

Recursive methods in economic dynamics.

The economics of poverty traps part one: Complete markets.

The endogenous determination of time preference.

The growth of nations. NBER Reprints

The increasing disparity in mortality between socioeconomic groups in the united states,

Threshold Externalities in Economic Development.

Time discounting and time preference: A critical review.

Twin peaks: Growth and convergence in models of distribution dynamics.

US Life Tables for 1990 by Sex, Race, and Education.

Wealthier is Healthier.

Endogenous discounting via wealth, Twin-Peaks and the role of technology

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Endogenous discounting via wealth, Twin-Peaks and the role of technology

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