Recently, Jones (2002} and Barro and Sala-Ã­-Martin (2004) pointed out that the neoclassical growth model with a Cobb-Douglas technology has a closed-form solution. This note makes a similar remark for the Malthusian model: I develop and characterize a closed-form solution. Moreover, I emphasize structural similarities between the Malthusian and the neoclassical model if the dynamic behavior is governed by a Bernoulli differential equation.

Andreas Irmen

English

Research Papers in Economics

Malthus and Solow − a note on closed−form solutions 
Andreas Irmen
University of Mannheim
Abstract
Recently, Jones (2002} and Barro and Sala−í−Martin (2004) pointed out that the neoclassical
growth model with a Cobb−Douglas technology has a closed−form solution. This note makes
a similar remark for the Malthusian model: I develop and characterize a closed−form
solution. Moreover, I emphasize structural similarities between the Malthusian and the
neoclassical model if the dynamic behavior is governed by a Bernoulli differential equation.
I like to thank Hendrik Hakenes and Tarek Hassan for helpful comments.
Citation: Irmen, Andreas, (2004) "Malthus and Solow − a note on closed−form solutions." Economics Bulletin, Vol. 10, No. 6
pp. 1−6
Submitted: August 27, 2004.  Accepted: September 2, 2004.
URL: http://www.economicsbulletin.com/2004/volume10/EB−04J10002A.pdfMalthus and Solow - A Note on Closed-Form Solutions 2
1 Introduction
It has recently been pointed out that the neoclassical growth model of Solow (1956)
and Swan (1956) has a closed-form solution if the aggregate production function is
Cobb-Douglas. Jones (2002) and Barro and Sala- -Martin (2004) added this result
to the second edition of their books on economic growth.
This note makes a similar remark for the Malthusian model that depicts the inter-
play between population growth and per-capita income. I show that an appropri-
ately chosen Malthusian population equation in conjunction with a Cobb-Douglas
production technology allows for a closed-form solution. Moreover, I highlight sev-
eral striking similarities to the neoclassical growth model. First, I emphasize the
similar role of diminishing returns in both models. They aect the death rate of
the population in the Malthusian setting and the \birth" rate of the capital in the
neoclassical framework. Second, for both models a closed-form solution exists if the
key dierential equation is a Bernoulli dierential equation. Third, for the present
examples, the asymptotic speed of convergence features the constant population
birth rate in the Malthusian setting and the constant \death" rate of the capital
intensity in the neoclassical framework.
2 Malthus - A Closed-Form Solution
2.1 Model and Steady State
At t the economy is endowed with an aggregate production function
Y (t) = AT
 N(t)
1 ; 0 <  < 1; (1)
where Y (t) denotes output, A > 0 the level of the technology, T > 0 the available
amount of arable land, and N(t) the population. Accordingly, per-capita income is
y(t) 
Y (t)
N(t)
= A

T
N(t)

: (2)
It increases with the level of the technology and the land intensity. Let n(t) 
_ N(t)=N(t) denote the population growth rate and assume that n is monotonicallyMalthus and Solow - A Note on Closed-Form Solutions 3
increasing in y according to
n(t) = cb  
cd
y(t)
; with cb; cd > 0: (3)
I henceforth associate cb with the birth rate and cd=y(t) with the death rate. Upon
combining (2) and (3) one obtains
_ N(t)
N(t)
= cb  
cd
A

T
N(t)
 
: (4)
It captures the Malthusian feedback between population growth and the means of
subsistence, which, in turn, depend on current population (see, Malthus (1798)). It
is useful to characterize the steady state dened as any pair (  N;  y) that satises
n(t) = 0.
Lemma 1 There is a unique and globally stable steady state with
 y =
cd
cb
and  N = T

A
 y
 1

: (5)
Proof Equation (5) is immediate from (4) for n(t) = 0. Global stability follows as
(4) implies a negative relationship between n(t) and N(t). Thus, for N(t) ?  N(t)
we have n(t) 7 0. 
At the steady state per-capita income is at the level of Malthusian subsistence,  y,
and population is constant. From (2) and (4) we deduce that decreasing returns
to population are the driving force behind Lemma 1: a larger population than  N
implies a lower land intensity, lower per-capita income than  y, and, in turn, a higher
death rate. For a constant birth rate, population growth is negative and population
declines.
It is interesting to compare this mechanism to the neoclassical growth model. Using
the notation of Barro and Sala- -Martin (2004), p.43, we have
_ k(t)
k(t)
= sAk(t)
 (1 )   (n + d); (6)
which corresponds to (4). Here, k(t)  K(t)=N(t) is the capital intensity at t,
s > 0 the savings rate, A > 0 a technology parameter, n the exogenous populationMalthus and Solow - A Note on Closed-Form Solutions 4
growth rate, d > 0 the depreciation rate, and  the output elasticity of capital in
the underlying Cobb-Douglas production function Y (t) = AK(t) N(t)1 .
The growth rate of the capital intensity can be interpreted as the dierence between a
birth rate and a death rate. The birth rate is gross investment, i.e. aggregate savings
per unit of current capital. Decreasing returns to capital force it to decline as k(t)
increases. The death rate is the sum of two constants. The capital intensity literally
dies because of depreciation. As a per-capita magnitude, it also \dies" because new
workers are born. Thus, contrary to the Malthusian model, here decreasing returns
aect the birth rate rather than the death rate and generate a unique and globally
stable steady state.
2.2 Paths of Population and Per-Capita Income
Equation (4) is a Bernoulli dierential equation, which allows us to express the
evolution of population and per-capita income in closed-form.1 To see this, write
(4) as
_ N(t)N(t)
 (1+) = cb N(t)
   
cd
AT 1  (7)
and dene v(t)  N(t) . Then, (7) can be expressed as
 
_ v(t)

= cb v(t)  
cd
AT  (8)
or, with (5),
_ v(t) =  cb v(t) + cb  N
 
This is a rst-order ordinary dierential equation with constant coecients. The
solution to the initial value problem, where v(0) = v0 = N
 
0 , is2
v(t)  N(t)
  = (N
 
0    N
 )e
 cb t +  N
 : (9)
1Bernoulli equations are non-linear dierential equations of the form _ z(t)+a(t)z(t) = b(t)z(t)r,
where r is any real number dierent from 0 and +1 (see, e.g., Gandolfo (1997)). In view of (4),
we have z(t) := N(t), a(t) :=  cb, b(t) :=  cd=(AT ), and r := 1 + .
2To highlight the structural similarity to the neoclassical model let v(t)  k(t)1 . Then, (6)
can be written as
_ v(t)
1   
=  (n + d)v(t) + sA:
It follows that
v(t) = (k
1 
0    k1 )e (1 )(n+d)t +  k1 ;
where  k is the steady-state level of k(t). These expressions correspond to (8) and (9), respectively.Malthus and Solow - A Note on Closed-Form Solutions 5
Thus, the gap between N(t)  and its steady state value vanishes at the constant
rate cb. Following some straightforward manipulations based on (9), we obtain
Proposition 1 The paths of N(t) and y(t) are
N(t) =
0
@
 N
1  

1  

 N
N0

e cb t
1
A
1

and
y(t) =  y + (y0    y) e
 cb t:
Thus, if N0 ?  N and therefore y0 7  y, population declines (increases) and per-capita
income increases (declines) over time. The steady-state values of (5) appear as the
limits for t ! 1.
The rate cb is the approximate speed of convergence in the neighborhood of the
steady state. It measures by how much n(t) declines as population size increases in
a proportional sense. Denote  the speed of convergence, then
(t)   
@n(t)
@ lnN(t)
:
To determine (t), lnN(t) has to appear in (4), i.e. n(t) = cb (cd=(AT )) e lnN(t).
We infer that
(t) = 
cd
A

N(t)
T

= cb

N(t)
 N

;
where the second step uses (5). As N(t) !  N for t ! 1 it follows that
   lim
t!1 = cb: (10)
Similarly, the asymptotic speed of convergence associated with y is3
lim
t!1y(t)  lim
t!1 
@(_ y(t)=y(t))
@ lny(t)
=  : (11)
Hence, asymptotically, the convergence coecient for n is the same as that for y.
From (10) it is equal to the elasticity of per-capita income with respect to the
3To see this, use (2) and (3) to nd that _ y(t)=y(t) =  n(t) =    + cd e lny(t). Thus,
y(t) = cd=y(t) =    y=y(t). Taking the limit for t ! 1 gives (11).Malthus and Solow - A Note on Closed-Form Solutions 6
land intensity, , times the constant birth rate, cb. The analogous result for the
neoclassical model of (6) is that the convergence coecients for k coincides with the
one for y. Both are equal to the elasticity of the output-capital ratio with respect to
the capital intensity, 1 , times the constant \death" rate of the capital intensity,
n + d.Malthus and Solow - A Note on Closed-Form Solutions 7
References
Barro, R. J., and X. Sala- -Martin (2004): Economic Growth. MIT Press,
Cambridge, MA, 2nd edn.
Gandolfo, G. (1997): Economic Dynamics - Study Edition. Springer Verlag,
Berlin - Heidelberg.
Jones, C. I. (2002): Introduction to Economic Growth. W.W. Norton, New York,
London, 2nd edn.
Malthus, T. R. (1798): An Essay on the Principle of Population. Reprint: Oxford
University Press, Oxford.
Solow, R. M. (1956): \A Contribution to the Theory of Economic Growth,"
Quarterly Journal of Economics, 70, 65{94.
Swan, T. W. (1956): \Economic Growth and Capital Accumulation," Economic
Record, 32, 334{361.

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