A Note on Ramsey, Harrod-Domar, Solow, and a Closed Form
Saddle Path
Halvor Mehlum∗
Abstract
Following up a 50 year old suggestion due to Solow, I show that by including
a Ramsey consumer in the Harrod-Domar model, the knife-edge problem is solved.
More surprisingly, however, the derived saddle path happens to be a closed form
expression. The existence of a closed form expression greatly simplifies the analysis
of how the parameters of the utility function affect investments and growth.
Keywords: Ramsey growth model
JEL: D91, O41
∗
Department of Economics, University of Oslo P.O. Box 1095, Blindern N-0317 Oslo, Norway. E-mail:
halvor.mehlum@econ.uio.no.
1
A note on Ramsey and a Closed Form Saddle Path
1
2
Introduction
One of the main contributions to growth theory is the Solow model (1956). As a backdrop
for his own modeling Solow expresses uneasiness with the knife-edge feature of the HarrodDomar model and in the introduction to his article Solow writes “he bulk of this paper is
devoted to a model of long-run growth which accepts all the Harrod-Domar assumptions
except that of fixed proportions. Instead I suppose that the single composite commodity
is produced by labor and capital under the standard neoclassical conditions [. . . ], to see if
the Harrod instability appears.”(p. 66) Indeed he finds that the knife-edge feature of the
Harrod-Domar model goes away when using the neoclassical production function. This
was not the first time, however, that Solow expressed his uneasiness with the implications
of the Harrod-Domar model. Already in a note in 1953 Solow points out several problems
in “Mr. Harrod’s system”. Problems that eventually are handled in the 1956 article.
However, that the solution would lay in allowing for substitution in production is not
pointed out in the 1953 note. Here Solow’s attention is rather on the assumption of a
fixed savings rate. In the concluding section of the 1953 note, when pointing out possible
ways of building causal dynamics he writes: “A mechanical first step in this direction
could be made by letting the choice between consumption and investment depend on the
interest rate and price level in some arbitrary but simple way. Another possibility would
be to think of investors as Ramsey-type utility maximisers over time.” (p. 79)
In the present note I take up the tread from Solow’s 50 year old note and replace
Mr. Harrod’s assumption about a fixed savings rate with savings behavior a la Ramsey
(1928). Paraphrasing Solow (1956): The bulk of this note is devoted to a model of
long-run growth which accepts all the Harrod-Domar assumptions except that of fixed
savings rate. Instead I suppose that the savings decision is made by Ramsey-type utility
A note on Ramsey and a Closed Form Saddle Path
3
maximizers.
This note has two main results. (1) Not surprisingly Solow was right: the Harrod
instability disappears when the savings rate is flexible. (2) Most surprisingly: in this
problem, in contrast to most other cases, the Ramsey saddle path has a closed form
solution.
The last result is both interesting in its own right and as a contribution to the literature
on the Ramsey model. An explicit saddle path makes it a lot easier to get to grips with the
Ramsey framework in general - the essential feature of which is the optimizing consumer’s
behavior in a specific technological environment.
2
The model
Production per unit of labor is given as a function of capital per unit of labor, K, by the
fixed coefficient production function
¡
¢
X = a min K, K̄ ,
(1)
where a is the output-capital ratio and K̄ is the maximal capital stock that can be used
productively. K̄ is determined by the productivity of labor relative to capital. This
production function is concave and has the essential properties needed for the Ramsey
problem. It can be shown that (1) is the limit of a constant elasticity of substitution
production function as the elasticity of substitution between labor and capital goes to
zero.
The dynamics of the economy depends on the capital accumulation. All income is
earned by a representative consumer who is the owner of the firms and the owner of
capital. The supply of capital accumulation is thus determined by this consumer’s sav-
A note on Ramsey and a Closed Form Saddle Path
4
ings/investment decision. The consumer maximizes a constant relative risk aversion utility
function
U=
Z
∞
t=0
1− 1
Ct σ − 1 −θt
e dt,
1 − σ1
(2)
where C is consumption, θ the rate of time preferences, and σ the intertemporal elasticity
of substitution. When including depreciation δ and using (1), investments (the time
derivative of capital) is simply
(a − δ) K − C when K < K̄
dK
.
=
dt
aK̄ − δK − C when K > K̄
(3)
By using standard methods of dynamic optimization1 , maximizing (2) with respect to (3)
and using from (1), the optimal consumption path is characterized by
Cσ [a − δ − θ] when K < K̄
dC
= Cσ [∂X/∂K − δ − θ] =
dt
−Cσ [δ + θ] when K > K̄
(4)
it further follows that the two steady state terminal conditions are
K = K̄ and C = (a − δ) K̄ ≡ C̄
Throughout it is assumed that the consumer wants to hold some capital, hence a−δ −θ >
0. Then consumption grows exponentially with rate σ (a − δ − θ) as long as K < K̄.
When K > K̄, however, consumption will decline exponentially with the rate −σ (δ + θ).
Figure 2 shows the complete phase diagram when also incorporating the capital dynamics
1
In order to keep the note short and as I assume that the readers are familiar with the problem I avoid
technicalities. For a careful presentation of the Ramsey problem see for example Barro and Sala-I-Martin
(1995).
5
A note on Ramsey and a Closed Form Saddle Path
Figure 1: The saddle path
C
dC
dt
=0
......
...........
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... ...................................... ..........................
...
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.........
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.......... ....................................... dK = 0
....... ...........
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dt
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...... ........... ...
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K
K̄
from (3). The phase diagram shares all the essential features of the textbook exposition of
the Ramsey growth model. All paths except the indicated saddle path bumps into either
zero capital or zero consumption constraints and are obviously not solutions.
Along the lower branch of the saddle path (K starting below K̄) both the capital
stock and consumption grow over time until the steady state is reached. Along the upper
branch (K starting above K̄) both capital and consumption decline over time until the
steady state is reached. It should in passing be noted that the steady state condition
K = K̄ rules out the knife-edge problem. Savings will adjust exactly so that over time
there is neither idle capital nor labor.
As the consumption growth is exponential, the transition time is finite. At the time
of termination all capital is employed and the return to capital drops down to the level
where the consumer has no incentive for further savings nor for dissavings. Hence from
the point of termination and onwards consumption is stable. The two linear differential
equations (3) and (4), in combination with the initial condition K = K0 and the two
terminal conditions C = C̄ and K = K̄, are sufficient for solving for the two transition
paths and for the transition time.
6
A note on Ramsey and a Closed Form Saddle Path
2.1
The Saddle Path
The saddle path is found by applying standard tools for differential equations. The upper
and the lower branch has to be treated separately. Here I will concentrate on the lower
branch, which, from a growth perspective, is the most interesting. The solution for this
branch is found by transforming the differential equations (3) and (4) to one differential
equation between K and C. This elimination of time from the system is possible since both
K and C are strictly increasing in t. Let K = K (C) indicate the function that describes
the saddle path in the phase diagram.2 It follows from the rules for differentiation that
dK
∂K (C) dC
∂K (C)
dK dC
=
⇐⇒
=
/
dt
∂C dt
∂C
dt dt
By inserting from (3) and (4) the result is a general linear first-order differential equation
∂K (C)
(a − δ) K (C) − C
=
∂C
Cσ (a − δ − θ)
(5)
Hence, the solution to (5) is found by using standard formulas. Following from the two
terminal conditions C = C̄ and K = K̄, the terminal steady state condition for K (C) is
¡ ¢
K C̄ = K̄,which determines the constant of integration. It can be confirmed by taking
the derivative that the solution is simply
K (C) =
2
C̄ 1−β C β − βC
,
(a − δ) (1 − β)
β=
a−δ
σ (a − δ − θ)
(6)
I choose to express K as a function of C as this has a closed form solution while C as a function of
K does not.
A note on Ramsey and a Closed Form Saddle Path
7
This explicit expression for the saddle path is the main finding of this paper.3 It gives a
closed form solution for the Ramsey saddle path. In addition (4) can be solved to give
the transition time T until the steady state is reached, where C = C̄. Let the present
consumption level be C then T is given by the equation C̄ = CeT σ(a−δ−θ) . Hence
µ ¶
C̄
T (C) = ln
σ (a − δ − θ)
C
(7)
The closed form solution provided in (6) and (7) gives anyone working with the Ramsey
problem an accessible way to investigate the consequences of altering the parameters
of the utility function. One can employ standard methods from calculus and need not
use numerical or approximate methods. It is for example easy to find the saddle path by
solving for K for different C. Comparative statics shifts in the saddle path can be analyzed
by taking the derivative of (6) with respect to the parameters. The saddle path (6) and
the transition time (7) give capital and transition time as functions of consumption. In
order to find the effect of parameter changes on C and T for a given K one has to approach
(6) and (7) using implicit differentiation.
Figure 2 gives two examples of shifts in parameter values - easily and accurately
plotted using a spreadsheet: When the intertemporal elasticity of substitution σ is high,
consumer is willing to forego consumption today in order to achieve faster transition to
higher income. Hence C is low relative to production and the growth rate is high. On the
other hand, if the rate of time preferences θ is high, the future does not matter that much
and the consumer want to consume more today. Hence, consumption is high relative to
production and the growth is low.4
£
¡
¢¤
In the case where β = 1 the solution becomes K (C) = C 1 + ln C̄/C / (a − δ). Along the upper
branch of the saddle path both C and K are declining in t and the solution is found by the same method.
4
In the numerical example I use the following parameter values: a −δ = 1/3, δ = 0.05, K̄ = 3, σ = 0.5,
3
8
A note on Ramsey and a Closed Form Saddle Path
Figure 2: The saddle path
C
1
0
....
......... ........................................... dK = 0
...... .......
.
dt
.
.
.
.
.
.
.
..
.
....... . ............
.
.
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.
.
.. ..
...
θ high ............................................................................................
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.
... . ...
..
...... ... ..... ................. .........
.
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. ... .......................... σ high
.
.
...... ..... ......... .........
....... ..... ..... ................ .... ...
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.....
....... .... .........
.............. ........................ ... .... ....
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. . . ..... .
........... .............
........................................ ..... ....
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.
.............. ..... ..
.............. .....
K
1
2.2
2
3
Transition Time
The saddle path gives the relationship between K and C. In order to find the consumption
level for a given starting value of the capital stock one needs to solve the equation K0 =
K (C0 ). Here one needs to revert to numeric methods, though of the less demanding sort
which can be done even by hand. Once C0 is known (4) gives the exponential growth of
C
C (t) = C0 etσ(a−δ−θ)
(8)
Once C (t) is known it can be inserted into (6) to also get K as a function of time. The
economic growth over time is illustrated in Figure 3 . When the intertemporal elasticity
of substitution σ is high, investments are high and the growth is fast. If the rate of time
preferences θ is high, however, investments are low and the growth is slow.
This concludes the main analysis. To summarize: The main result is the existence of
the function (6) that gives the closed form solution for the saddle path. Combined with
(8) and the initial condition on the capital stock it is simple to calculate the time paths
for consumption, capital and production.
θ = 0.05, σ (high) = 1, θ (high) = 0.20, and in addition (for the next figure) K0 = 1.5.
9
A note on Ramsey and a Closed Form Saddle Path
Figure 3: Transition time
X
1
.... ..... ..... ..... .......................................................................................................................................
.... ..... ...................... .
.
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... ..... ..
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....
..... ..... ..
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σ high ..... ..... .....................
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...
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.... ..... ..
..........
. ...
..... .... . θ high
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.... ......
. .... .....
................................. ..... ..... ....
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............. ....
0
t
0
3
5
10
15
Concluding Remarks
Following up a 50 year old suggestion due to Solow, I have shown that by including a
Ramsey consumer in the Harrod-Domar model, the knife-edge problem is solved. More
surprisingly, however, the derived saddle path happens to be a closed form expression.
The literature contains some other examples of explicit solutions to the Ramsey problem, (see e.g. Benhabib and Rustichini 1994). These are all quite restrictive: Complete
depreciation over one or two periods, crosscutting conditions on parameters in production and consumption et c. The assumptions in the analysis above may also be characterized as restrictive - given the fixed coefficient production function. The question is:
restrictive compared to what? One quite general base-case for the Ramsey model, which
satisfies all accepted neoclassical assumptions, is the combination of a general constant
returns to scale CES production function, a general CRRA utility function, and depreciation/population growth/technical progress less than unity. The present model is within
this domain, though at the border, as the fixed coefficient production is the limit of the
CES as the elasticity of substitution goes to zero. Hence, compared to the base-case only
one parameter is fixed - the elasticity of substitution- while all the other parameters are
A note on Ramsey and a Closed Form Saddle Path
10
free. The cost of fixing this single parameter should be weighted against the benefit. An
explicit solution makes it a lot easier to get to grips with the essential feature of the
Ramsey model: The optimizing consumer’s behavior in a specific technological environment. The present model allows the analysis of changes in the parameters of the utility
function: the intertemporal elasticity of substitution and the rate of time preferences. In
addition, important aspects of technology can be analyzed, such as efficiency, a, capacity
K̄ and depreciation δ. Obviously, all relevant questions cannot be addressed satisfactorily
without substitution, e.g. questions related to factor prices.
The explicit function describing the saddle path K (C) in (6) follows from the linear
differential equation (5). The essential condition for this result is that the marginal
productivity of capital is constant during the transition; a property that follows from the
fixed coefficient production function. Given that this condition is satisfied the model’s
assumptions may be altered in a number of ways. It can be shown that the following
alterations can be done without the sacrifice of an explicit expression for the saddle
path. First, the utility function may be modified to include a minimum consumption,
a la Stone-Geary, or be changed to the constant absolute risk aversion type. Second,
the production function need not start out in the origin but at the constant b. That is:
¡
¢
X = b + a min K, K̄ . Readers who are unhappy with the lack of substitution may use
this production function instead of (1), to approximate production functions with more
curvature.
A note on Ramsey and a Closed Form Saddle Path
11
References
Barro, Robert and Xavier Sala-I-Martin (1995) Economic Growth, McGraw Hill, New
York.
Benhabib, Jess and Aldo Rustichini (1994) ”A Note on a New Class of Solutions to
Dynamic Programming Problems Arising in Economic Growth,” Journal of Economic
Dynamics and Control, 18, 808-813.
Ramsey, Frank (1928) “A mathematical theory of saving,” Economic Journal, 38, 543-559.
Solow, Robert (1953) “A Note on the Price Level and Interest Rate in a Growth Model,”
The Review of Economic Studies, 21(1), 74-79.
Solow, Robert (1956) “A Contribution to the theory of economic growth,” Quarterly
Journal of Economics, 70, 65-94.