In this note I develop two approaches to rule out the overconsumption paths in the Ramsey model with irreversible capital. The ørst focuses on the multiplier of the irreversible constraint and is applied to the situation where preferences are CES and the production function is Cobb-Douglas. The second, relies on a revealed preference argument and is used to rule out overconsumption paths when the preferences are strictly concave and the initial level of per effective capital is below its steady state level.RAMSEY GROWTH MODEL; IRREVERSIBLE CAPITAL; OVERCONSUMPTION PATHS

Comin, D.

English

Research Papers in Economics

ECONOMIC RESEARCH REPORTS
C.V. Starr Center for Applied Economics
Department of Economics
Faculty of Arts and Science
New York University
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rd Floor
New York, New York 10003-6687
Two Ways to Rule Out
the Overconsumption Paths in the
Ramsey Model with Irreversible
Investment
by
Diego Comin
RR#:  2000-17                                            October 2000Two Ways to Rule Out the Overconsumption
Paths in the Ramsey Model with Irreversible
Investment.
Diego Com¶ in¤
First Version: January, 1998
This Version: October, 2000.
Abstract
In this note I develop two approaches to rule out the overconsumption
paths in the Ramsey model with irreversible capital. The ¯rst focuses on
the multiplier of the irreversible constraint and is applied to the situation
where preferences are CES and the production function is Cobb-Douglas.
The second, relies on a revealed preference argument and is used to rule
out overconsumption paths when the preferences are strictly concave and
the initial level of per e®ective capital is below its steady state level.
JEL: O4, C6.
Keywords: Ramsey growth model, Irreversible capital, Overconsump-
tion paths.
1. Introduction
In the in¯nite horizon Ramsey model with reversible investment all overconsump-
tion paths of the Ramsey model lead to the consumption of all the capital, at
some ¯nite time. This cannot be optimal since the marginal product of capital is
¤I thank Robert Barro for suggesting me this exercise and him and John Leahy for useful
discussions. I also gratefully acknowledge the ¯nancial support of the Bank of Spain. Comments
are very welcome at dcomin@kuznets.fas.harvard.edu. The usual disclaimer applies.in¯nite at k = 0 and the neoclassical agent would havebeen better o® by investing
the last unit of capital instead of consuming it. In short, overconsumption paths
can be ruled out because they violate the Euler equation.1 This argument does
not extend to the situation where capital is irreversible (i.e. c · f(k)). Then, the
locus c = f(k) is reached at some ¯nite time and from then on capital follows the
di®erential equation
_ k = ¡(± + n +x)k
where ± is the depreciation rate, k is capital per e®ective unit of labor (i.e.
K
LA), and x and n are the exogenous rates of technological and population growth.
For future reference, c is going to denote consumption per e®ective unit of labor
(i.e.
C
LA). Note that with this law of motion the k = 0 locus is reached only
asymptotically and the usual argument cannot be made.
In this note I present two arguments that, under di®erent circumstances, rule
out the overconsumption paths in the Ramsey model with irreversible capital.
The ¯rst approach is applied to the case where the utility function is CES and
the production technology is Cobb Douglas and attacks the problem by showing
that the multiplier of the irreversible constraint eventually becomes negative at
any overconsumption path. Remember that the multiplier of any constraint can
be interpreted as the marginal value of relaxing the constraint. Hence the neg-
ative multiplier in the irreversible constraint implies that the marginal value of
consumption is smaller than the marginal value of saving at some point along the
c = f(k) locus, i.e. the agent would be better o® by saving more. As a result,
overconsumption cannot be optimal. The second technique relies on a revealed
preference argument. In particular, if the constraint on the irreversibility of in-
vestment is not binding at any instant along the optimal path for the unrestricted
problem, and this optimal path is unique, it must also solve the problem once
the constraint is introduced. As a corollary of this proposition, when the irre-
versibility constraint on investment is not binding along the solution to Ramsey
problem with reversible investment, overconsumption paths are not optimal in
the restricted problem. In order to compare the applicability of the two meth-
ods I show that when the initial level of e±ciency units of capital is below the
steady state level the irreversible constraint on investment does not bind along
the optimal path in the traditional Ramsey problem.
1See Blanchard and Fisher (1989) or Barro and Sala-i-Mart¶ in (1995) among others.
2The exercise conducted in this paper is related to two other papers. In his
seminal analysis of the Ramsey model, Cass [1965] imposes the constraint on irre-
versible investment but he does not rule out overconsumption paths. Arrow and
Kurz [1970] are interested in understanding when the economies with irreversible
capital that converge to the steady statefrom above consume all the output. Their
main conclusion is that, in general, it is very hard to characterize the number of
intervals in which the nonnegativity constraint on investment is binding. I am
concerned about the optimality of overconsumption paths. My analysis focuses
more on economies that converge from below, and for the CES utility function
and Cobb Douglas production function case it turns out that, even if the economy
converges from above, the constraint on nonnegative investment is never binding.
2. The Ramsey problem with irreversible investment
The social planner enforces the optimal intertemporal allocation of resources given
the initial capital (per e®ective unit) (k0), the law of motion for capital accumula-
tion (equation 2), the no-Ponzi game restriction on borrowing (inequality 3), and
the irreversibility constraints on investment (inequality 4).2
(1) max V =
R 1
0 e¡(½¡n¡x(1¡µ))t
h
c1¡µ
t ¡e¡xt
1¡µ
i
dt
fct;ktg1
t=0
s.t.
(2) _ kt = f(kt)¡ ct ¡ (± + n +x)kt
(3) limT!1kTe¡
R T
0 (f0(kv)¡±¡n¡x)dv ¸ 0
(4) f(kt) ¸ ct , 8t
given k0 > 0
where f(kt) is a neoclassical production function.3
3. The multiplier method.
If condition (4) is not imposed, the ¯rst order conditions of this problem are given
by:4
2Note that for simplicity I write the program with a CES utility function. As will be seen
latter, this is only relevant for the ¯rst approach.
3See Barro and Sala-i-Mart¶ in (1995), page 16, for a precise description of the properties that
must satisfy a neoclassical production function.
4See Barro and Sala-i-Mart¶ in (1995), Chapter 2.
3(5) Hc = 0
(6) Hk = ¡_ ¹t
(7) limT!1 ¹TkT = 0
(8) ¹t ¸ 0
where the Hamiltonian (H) is
H = e
¡(½¡n¡x(1¡µ)) t
·
c1¡µ
t ¡ e¡xt
1¡ µ
¸
+¹t[f(kt)¡ ct ¡ (± + n + x)kt]
Hz is thepartial derivativeof the Hamiltonianwithrespect to z, ¹t is theLagrange
multiplier associated with the law of motion of capital (equation 2), equation (7)
is the transversality condition, and, for future reference, the mutiplier associated
with the irreversibility constraint on investment (equation 4), will be denoted by
¸t:5
When the constraint on the irreversibility of investment (equation (4)) is bind-
ing the ¯rst order necessary conditions of the problem are going to be altered in
three ways. The marginal utility of consumption is going to exceed (by ¸t) the
marginal utility of investment. The other side of this e®ect is that by increasing
thelevel of capital pere®ectiveworker, the representativeagent reduces the extent
to which her consumptiondecisions are constrained, by equation (4), inthe future.
Moreover, the Kuhn-Tucker theorem states that the multiplier of the irreversibil-
ity constraint on investment must be non-negative and will be strictly positive
only when the constraint is binding. Lemma 3.1 formalizes these intuitions which
are proved in the appendix.
Lemma 3.1. The F.O.N.C. of the Ramsey model with irreversible investment
and CES utility function are given by
(5') Hc = ¸t
(6') Hk = ¡_ ¹t ¡ ¸tf0(kt)
(7') limT!1 ¹TkT = 0;
5The solution to this problem is given by the system
_ ct
ct = 1
µ [f 0(kt) ¡ ± ¡ ½¡ µx]
_ kt = f(kt) ¡ ct ¡ (± + n + x)kt
with boundary conditions
kt = k0 > 0; at t = 0:
limT !1 kTe¡
R T
0 (f0(kv)¡±¡n¡x)dv = 0
See Barro and sala-i-Mart¶ in (1995), page 71.
4(8') ¹t ¸ 0
(9.a') f(kt) ¸ ct; (9.b') ¸t ¸ 0; (9.c') ¸t [f(kt) ¡ ct] = 0:
Since the conditions stated in Lemma 1 are necessary for optimality, in order
to rule out the overconsumption paths it is su±cient to show that condition (9.b')
is eventually violated. By following this strategy, we are able to proof that the
marginal utility of consumption at the overconsumption paths is eventually lower
than the marginal utility of saving and therefore the agent would be better o®
by increasing her saving rate along these paths. This result is stated and proved
in Proposition 3.2 for the particular case of a Cobb-Douglas production function
and CES utility function. The argument used in the proposition does not depend,
a priori, on these speci¯cations; however, it involves several ¯rst order di®erential
equations and the assumed functional forms are very convenient.
Proposition 3.2. (Ruling out overconsumption paths) Suppose that the prefer-
ences are CES, and the production function is Cobb-Douglas. Then, the over-
consumption paths in the Ramsey model with irreversible investment are not
optimal.
Proof. Figure 3.1 illustrates two possible overconsumption paths drawn in the
(k;c) space. Overconsumption paths in an in¯nite horizon world are characterized
by reaching in a ¯nite time the ct = f(kt) locus. Let's denote by t¤ this instant.
Inspecting the phase diagram we can see that the ct = f(kt) locus is above the
_ k = 0locus and therefore aneconomy on an overconsumption path will eventually
reach the Northern quadrants. Once the economy is in this region the dynamics
implied by the F.O.N.C. tend to reduce the capital stock per e®ective unit of
labor. From t¤ to in¯nity, the representative agent will consume ct = f(kt): We
can use the law of motion for capital together with this restriction to pin down
the path of capital starting at t¤: In particular, ct = f(kt) for t ¸ t¤ together with
equation (2) imply that
_ k
k
= ¡(± + x+ n); 8t ¸ t¤
and therefore
(10) kt = kt¤e¡(±+n+x)t; for all t ¸ t¤:
A Cobb-Douglas production function takes the form f(kt) = Ak®
t : Hence for
t ¸ t¤; ct = Ak®
t¤e¡(±+n+x)® t:
5                  c=f(k) 
        . 
        c=0 
  c 
 
 
 
 
 
   
 
 
                  . 
                  k=0 
 
 
 
 
 
 
 
 
 
 
                       
                           k 
                        
Figure 3.1: Phase Diagram
Now we can rewrite conditions (5') to (9') using the de¯nition of the Hamil-
tonian and the expressions of ct and kt : In particular, for t ¸ t¤
(11) (Ak®
t¤)¡µe¡(½¡n(1+®µ)¡x(1¡µ+®µ)¡®µ±)t = ¹t + ¸t
(12) (¹t + ¸t)®Ak
®¡1
t¤ ¡ ¹t(± + n +x) = ¡_ ¹t
(13) limT!1 ¹Tkt¤e¡(±+n+x)T = 0;
(8') ¹t ¸ 0
(14.a) ct = Ak®
t¤e¡(±+n+x)® t; (14.b) ¸t ¸ 0:
Only three stepsremainto concludethe proof. FirstI derivethe evolution of¹t
and thenI use the TVC to restrict the parameters. Finally, I reach acontradiction
by showing that under this condition inequality (14.b) cannot hold.
Plugging equation (11) into (12) we obtain the di®erential equation
((Ak®
t¤)¡µe¡(½¡n(1+®µ)¡x(1¡µ+®µ)¡®µ±)t)®Ak®¡1
t¤ ¡ ¹t(± +n + x) = ¡_ ¹t; 8t ¸ t¤
The solution to this di®erential equation takes the form
¹t = ¡®A
1¡µk
®(1¡µ)¡1
t¤ e
(±+n+x)t
·
e»t
»
+b
¸
; 8t ¸ t
¤ (15)
6where b is an integration constant which must be pin down by some boundary
condition, and » ´ ¡½ + n+ x(1¡ µ) ¡ ®(1¡ µ)(± + n+ x):
Using this expression for ¹; and the expression for the level of physical capital;
the transversality condition (13) can be written as
lim
T!1
¡®A1¡µk
®(1¡µ)
t¤
·
e»T
»
+ b
¸
= 0 (16)
Since both b and » are ¯nite constants, the only possibility for the TVC to
hold is that » < 0; and b = 0:
Finally, using these restrictions and equations (11) and (15) I can obtain an
expression for the multiplier of the irreversibility constraint.
¸t = e»t
0
B B
@(Ak®
t¤)¡µ
| {z }
> 0
+
®A1¡µk
®(1¡µ)¡1
t¤
»
e(±+n+x)t
| {z }
< 0
1
C C
A; 8t ¸ t¤ (17)
Notethatsince(±+n+x) > 0thesecondterm intheparenthesis will eventually
dominate the ¯rst and therefore 9^ T s.t. ¸t < 0; 8t > ^ T:¤
The intuition behind this result comes from equation (11). When ¸t is nega-
tive, the marginal utility of investment (¹t) exceeds the marginal utility of con-
sumption ((Ak®
t¤)¡µe¡(½¡n(1+®µ)¡x(1¡µ+®µ)¡®µ±)t) and therefore the agent would be
better of by reducing her consumption (i.e. not following an overconsumption
path).
4. The revealed preference argument
A second approach consists in contrasting the solution to the traditional Ramsey
model with the Ramsey model with irreversible investment. If the constraint does
not bind along the optimal path, and this is unique,6 the unrestricted solution
coincides with the solution to the problem once constraint (4) is incorporated.
Next I state precisely and prove this result.
Theorem 4.1. Whenever the initial level of capital per e®ective unit of labor
does not exceed the steady state level (i.e. k0 · ks:s) and the instantaneous utility
6A su±cient condition for uniqueness of the optimal consumption path is that the instanta-
neous utility function is strictly concave.
7function is strictly concave, the solution to the Ramsey model with irreversible
investment is exactly the same as the solution to the traditional Ramsey model.
Proof. Let c(t) be the optimal path of consumption in the problem de¯ned
by (1) to (3) with a given k0; and let cR(t) be the optimal path for the con-
sumption level per unit of e®ective worker of the representative agent when the
irreversibility constraint on investment (4) is added to the problem. Since the set
of possible paths for consumption for the unrestricted problem (C) is a superset
of the set of possible paths for the restricted problem (CR); 7the utility attained
by cR(t) cannot exceed the utility of an agent that consumes according to c(t),
i.e. V(c(t)) ¸ V (cR(t)):
Next I use the law of motion for capital , equation (2), to show that the
irreversibility constraint on investment (4) does not bind along the optimal path
for the unrestricted problem. We know from the analysis of the reversible Ramsey
model that if k0 · kss; then _ kt ¸ 0;8t: But, from equation (2), _ kt ¸ 0 implies that
f(kt) ¸ ct;and therefore there is no need to consume the existing capital along
the optimal path of the unconstrained problem.
Hence c(t) 2 CR;and since V(c(t)) ¸ V (cR(t)); and the optimal path is unique
(by assumption) then it must be the case that c(t) = cR(t):¤
But since overconsumption is suboptimal in the Ramsey problem with re-
versible capital so is when capital is irreversible.
Corollary 4.2. The overconsumptionpathsinthein¯nite horizon Ramsey model
with irresversible investment are not optimal when k0 < kss.
5. Conclusions
In this note, I have presented two approaches to rule out overconsumption paths
in the Ramsey model with irreversible investment. The ¯rst is based in showing
that along any overconsumption path the multiplier of the irreversible constraint
is eventually negative and therefore the marginal utility of saving exceeds the
marginal utility of consumption. The second, relies on a revealed preference argu-
ment. In particular, if the solutionto theunconstrainedRamseyproblem is unique
and along the optimal path there is no need to consume the existing capital, the
optimal path of the constrained and unconstrained problem must coincide.
7i.e. C ¾ CR:
8Both proofs are complementary. I use the ¯rst to rule out overconsumption
paths when preferences are CES and the production technology is Cobb-Douglas;
however, there is nothing speci¯c to these functional forms, apart from being
convenient to conduct the prove with pencil and paper, that is necessary to use
the multiplier argument. The second method just requires that the initial level of
capital is below its steady state level. In this sense the multiplier method might
be more general than the revealed preference argument.
References
[1] Arrow, K. and M. Kurz (1970), \Optimal Growth with Irreversible Investment
in a Ramsey Model" Econometrica, Volume 38,2 pp. 331-344.
[2] Barro, R. and X. Sala-i-Mart¶ in Economic Growth, Advanced Series in Eco-
nomics. New York; London and Montreal: McGraw-Hill, 1995.
[3] Blanchard, O. and S. Fisher Lectures on Macroeconomics, Cambridge, Mass.
and London: MIT Press, 1989.
[4] Cass, D. (1965), Optimum Growth in an Aggregative Model of Capital Accu-
mulation" Review of Economic Studies XXXII (3), No. 91 (July), pp. 233-40.
9Appendix
In this appendix I prove Lemma 3.1.
Lemma 3.1 The F.O.N.C. of the Ramsey model with irreversible investment
and CES utility function are given by
(5') Hc = ¸t
(6') Hk = ¡_ ¹t ¡ ¸tf0(kt)
(7') limT!1 ¹TkT = 0;
(8') ¹t ¸ 0
(9.a') f(kt) ¸ ct; (9.b') ¸t ¸ 0; (9.c') ¸t [f(kt) ¡ ct] = 0:
Proof. In the proof I follow the perturbation method used by Barro and Sala-i-
Mart¶ in (1995) (Appendix 1.3).
In particular, I start with the ¯nite horizon problem
(A.1) max V =
R T
0 e¡(½¡n¡x(1¡µ))t
h
c
1¡µ
t ¡e¡xt
1¡µ
i
dt
fct;ktgT
t=0
s.t.
(A.2) _ kt = f(kt)¡ ct ¡ (± + n +x)kt
(A.3) kTe¡
R T
0 (f0(kv)¡±¡n¡x)dv ¸ 0
(A.4) f(kt) ¸ ct , t 2 [0;T]
given k0 > 0
Next I de¯ne the Lagrangian (L)
L =
Z T
0
e¡(½¡n¡x(1¡µ))t
·
c
1¡µ
t ¡ e¡xt
1¡ µ
¸
dt+ (A.5)
+
Z T
0
h
¹t(f(kt)¡ ct ¡ (± + n + x)kt ¡ _ kt)
i
dt+
+vTkTe
R T
0 (¡f0(k¿)+±+n+x)d¿ +
Z T
0
¸t(f(kt) ¡ ct)dt
Integration by parts of the term
RT
0 ¹t_ ktdt gives
Z T
0
¹t_ ktdt = [¹tkt]
T
0 ¡
Z T
0
_ ¹tktdt (A.6)
10and by plugging it into (A.5) the Lagrangian can be written as
L =
Z T
0
(H(ct;kt;t) + _ ¹tkt)dt+ ¹0k0 ¡ ¹TkT (A.7)
+vTkTe
R T
0 (¡f0(k¿)+±+n+x)d¿ +
Z T
0
¸t(f(kt) ¡ ct)dt
where
H(ct;kt;t) = e¡(½¡n¡x(1¡µ))t
·
c
1¡µ
t ¡ e¡xt
1¡ µ
¸
+¹t(f(kt) ¡ct ¡ (± +n + x)kt)
Let c¤
t and k¤
t be the optimal time paths for consumption and capital. If we
perturb the optimal path c¤
t by an arbitrary perturbation function, p1t; then we
can generate a neighboring path for consumption,
ct = c¤
t + " p1t
When ct is perturbed, there must be a corresponding perturbation of kt and
kT so as to satisfy the budget constraint:
kt = k
¤
t + " p2t
kT = k
¤
T + " dkT
If the initial paths are optimal then @L
@" = 0 when L is evaluated at the optimal
paths. Let's rewrite the Lagrangian as a function of " by substituting ct; kt and
kT into (A.7).
L =
Z T
0
(H(c¤
t + " p1t;k¤
t + " p2t;t) + _ ¹t(k¤
t + " p2t))dt +¹0k0
+(k
¤
T + " dkT)
³
vTe
R T
0 (¡f0(k¿)+±+n+x)d¿ ¡ ¹T
´
+
Z T
0
¸t (f(k
¤
t + " p2t) ¡ (c
¤
t + " p1t))dt
@L
@"(0) = 0 implies that
Z T
0
[(Hc ¡ ¸t)p1t + (Hk + _ ¹t +¸tf
0(kt)p2t)]dt+dkT(vTe
R T
0 (¡f0(k¿)+±+n+x)d¿¡¹T) = 0
(A.8)
11Equation (A.8) can hold for all perturbation paths described by p1t; p2t and
dkT if each of the components in the equation vanishes, i.e.
Hc = ¸t (A.9)
Hk = ¡_ ¹t ¡ ¸tf
0(kt) (A.10)
vTe
R T
0 (¡f0(k¿)+±+n+x)d¿ = ¹T (A.11)
Using (A.11), (A.3) and the complementary slackness condition associated
with (A.3) we can derive that
¹TkT = 0
If T ! 1 this condition takes the form (7'). Conditions (8') and (9') are
necessary conditions from the Kuhn Tucker Theorem.¤
12

Lectures on Macroeconomics,

Optimal Growth with Irreversible Investment in a Ramsey Model&quot;

Optimum Growth in an Aggregative Model of Capital Accumulation&quot;

Two Ways to Rule Out the Overconsumption Paths in the Ramsey Model with Irreversible Investment

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Two Ways to Rule Out the Overconsumption Paths in the Ramsey Model with Irreversible Investment

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