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The Deterministic Neoclassical-GrowthModel: a Quick Introduction
Christos Koulovatianos
University of Nottingham (CFCM, CPE)
and
Center for Financial Studies (Frankfurt)
e-mail: [email protected]
March 6, 2010*
0 * These notes heve bene�ted from comments and corrections by Sebastian Schmidt and Leonid Silberman.
1
Assumptions and Description of the Model
Assumptions concerning Households
Time is discrete and the horizon is infinite, i.e. t = 0, 1, .... The economy is populated by
a large number of identical infinitely-lived households. All households own the same stock of
physical-capital claims in period 0, k0 > 0 that they can sell in a free capital market. Each
household derives utility from the consumption of a single final consumable good throughout
the infinite horizon. All households have the same preference profiles over potential infinite
sequences of consumption, {ct}∞t=0, represented by the following utility function:
U ({ct}∞t=0) =
∞∑t=0
βtu (ct) , (1)
with β ∈ (0, 1), dudc
≡ u1 > 0, and d2udc2
≡ u11 < 0, for all c > 0. Moreover, we impose the
“Inada conditions” on u (from the famous Japanese economist who first suggested them),
limc→0
u1 (c) = ∞ , (2)
limc→∞
u1 (c) = 0 . (3)
Remark 1: the role of the Inada conditions
The first Inada condition is placed in order to create a consumer-optimization tendency
towards “interior” solutions, by discouraging “corner” consumption choices where ct = 0
for some t ∈ {0, 1, ...}. The second Inada condition guarantees that the lowest bound of
marginal utility is zero. Thus, equalization of marginal utility with any strictly positive price
is guaranteed, helping to avoid the possibility of non-existence of competitive equilibria.
2
Remark 2: additive separability of momentary utility leads to path indepen-
dence of a single-period’s marginal utility of consumption
The functional representation of household preferences that Ramsey, Cass, and Koop-
mans have used, is a discounted sum of additively-separable “momentary” utility functions
u, referring to the one-period’s utility derived by one-period’s consumption. Additive sep-
arability gives a key economic property to preferences: the marginal utility of any period’s
consumption is path-independent, meaning that
∂U
({ct}
∞t=0t�=τ
, cτ
)∂cτ
=
∂U
({ct}
∞t=0t�=τ
, cτ
)∂cτ
= βτu1 (cτ ) , (4)
for all possible sequences {ct}∞t=0t�=τ
, {ct}∞t=0t�=τ
. Path independence of the marginal utility of a
certain period’s consumption stems from the fact that the marginal utility of a single-period’s
consumption is not affected by changes in any other period’s consumption. In other words,
∂2U ({ct}∞t=0)
∂ci∂cj= 0 for all i, j ∈ {0, 1, ...} , with i �= j. (5)
Remark 3: (im)patience of households
The constant discount factor β links geometrically momentary utilities in the household’s
utility function and imposes an exogenous and constant degree of (im)patience over time. In
particular, the assumption that β ∈ (0, 1) means that consumers are impatient: they prefer
to consume a given quantity today, rather than consuming the same quantity in any other
future period.
To see why households are impatient, define the rate of time-preference as: ρd ≡1−β
β⇔
β = 11+ρd
(ρd = rate of time-preference in discrete time). The condition that β ∈ (0, 1) holds
if and only if ρd > 0. When a household considers a consumption plan where the same
3
quantity is consumed today and tomorrow, the rate of time preference ρd can be interpreted
as the percentage of utility loss one period ahead due to that the household has to wait for
one period. Impatience leads to the tendency for having the same good being cheaper next
period.
Assumptions concerning Firms and their technology
There is a very large number of small firms, each having the same size and operating the
same technology for producing the single consumable good. Production of a single firm is
achieved through a neoclassical production function with two inputs, physical capital and
labor, given by,
y = F (k, l) . (6)
A summary of its properties is:
• Both factors of production are necessary in strictly positive amounts in order to achieve
strictly positive production quantities:
F (0, l) = F (k, 0) = 0, for all (k, l) ∈ R2+. (7)
• For all (k, l) ∈ R2+, the marginal returns of each of the two inputs are positive, i.e.
∂F (k, l)
∂k≡ F1 (k, l) > 0 , and
∂F (k, l)
∂l≡ F2 (k, l) > 0 , (8)
and the law of diminishing marginal returns holds, so:
∂2F (k, l)
∂k2≡ F11 (k, l) < 0 , and
∂2F (k, l)
∂l2≡ F22 (k, l) < 0 . (9)
4
• For all (k, l) ∈ R2+, F (·) exhibits constant returns to scale, i.e.
F (ξk, ξl) = ξF (k, l) , for all ξ > 0 . (10)
• The marginal product of each productive input goes to infinity (zero) as the same input
goes to zero (infinity). These are the Inada conditions pertaining to production.
Specifically,
limk→0
F1 (k, l) =liml→0
F2 (k, l) = ∞ (11)
and
limk→∞
F1 (k, l) = liml→∞
F2 (k, l) = 0 . (12)
Household- and firm-population normalization for simpler analysis
In order to simplify our analysis and to economize on notation, we can assume, without
loss of generality, that the total size of the very large number of households is equal to Lf ,
a fixed total labor supply.1 If we denote the aggregate quantities of all variables with bold
upper-case characters, aggregation gives,
Y = yLf ,
C = cLf ,
K = kLf .
Yet, it is better to transform all aggregate variables into aggregates per-unit of labor, i.e. to
divide all variables by Lf . So,
y ≡Y
Lf= y ,
1 Assume, for simplicity, there is no population growth, even though an exogenously-driven constant-population growth process can be easily incorporated in the analysis.
5
c ≡C
Lf= c ,
k ≡K
Lf= k .
Although variables denoted by lower-case bold characters are equal to our initially defined
individual variables, these two categories of variables are not the same. Variables denoted by
bold characters are still aggregate. Dividing by Lf changed the total size of the households
from Lf to 1. This normalization does not mean that the number of households is still not
“very large.” Without loss of generality, the population size of households in this economy
is equal to one. Our assumption of all households being identical means that the household
distribution is degenerate. Our assumption that the population number is “quite large,”
means that the size of any single household tends to zero. In other words, each household
is a “drop in the ocean,” enforcing a perfectly competitive behavior for households on the
demand side of the final-good market. Therefore, households are price-takers.
We can move along the same lines with respect to the supply side of the final good.
Aggregate production is given by,
Y = F(K,Lf
)⇒ y =F
(K
Lf,1
).
Defining the univariate function f as,
f (x) = F (x, 1) ,
it is,
y =f (k) . (13)
Taking the partial derivatives of the aggregate production function expressed as LfF(
K
Lf ,1)
with respect to K and Lf , we obtain,
∂Y
∂K= F1
(K
Lf,1
),
6
∂Y
∂Lf= F
(K
Lf,1
)− F1
(K
Lf,1
)K
Lf,
resulting to,
∂Y
∂K=f1 (k) and
∂Y
∂Lf=f (k)− f1 (k)k . (14)
In this way, we have managed to eliminate variable Lf from both the expression of production
and the expressions of marginal products that are necessary for determining equilibrium
prices. We have also managed to normalize the large number of firms. Without loss of
generality, the total population size of firms in this economy is equal to one. Our assumption
of all firms being the same means that the firm distribution is degenerate. Our assumption
that the number of firms is “quite large,” means that the size of any single firm tends to
zero. In other words, each firm is a “drop in the ocean,” enforcing a perfectly competitive
behavior for firms on the supply side of the final-good market.
Whereas the small size of households is enough to rationalize their price-taking behavior,
the organization of firms in a competitive supply market is not straightforward. Technology
may generate tendencies for other organizational structures: for example, increasing returns
to scale can encourage mergers or monopolistic competition. The key assumption that ratio-
nalizes perfect competition among firms in this model is this of having a constant-returns-
to-scale technology. Constant returns to scale drive all firms to achieving zero maximum
profits and being price-takers. Moreover, production-input markets (these for capital and
labor) are perfectly competitive as well.
To summarize, we have managed to eliminate the labor variable from both the demand-
and the supply side. Also, the fact that the total measure of households and firms is equal
to one for both (degenerate) distributions allows us to use one of all identical households as
a representative household and one of all identical firms as a representative firm. We will
now see that one can also solve alternatively one of two a-priori different problems that give
7
the same result: (i) the general competitive-equilibrium problem; and (ii) the social-planner
problem. Of course, it is the first and second welfare theorems that allow us to do so.
Competitive general equilibrium versus the social planner’s solution
Suppose that we have solved the competitive general-equilibrium problem. Remarks (i)
and (ii) help us see the link between the competitive general equilibrium and the social
planner’s solution.
(i) Preferences are convex. We know this, since the law of diminishing marginal utility
holds for the preferences of households. Moreover, production technology exhibits constant
returns to scale and there are no externalities. These are sufficient conditions for the first
welfare theorem to hold in general equilibrium. This means that any competitive
equilibrium resulting from this model is Pareto efficient.
(ii) Our assumption that all firms and households are identical, provides a symmetry
among all individual maximization problems of firms and households. This symmetry
means that the resulting competitive equilibrium is characterized by a degenerate
distribution of relative personal utilities among households.
Remarks (i) and (ii) lead us to the conclusion that the model’s unique competitive equi-
librium (we will take uniqueness as a granted property for the moment) is Pareto efficient
with a degenerate relative personal-utility distribution.
Now if we consider a social planner with an equi-weighted utilitarian social-welfare func-
tion, the symmetry among all households and firms will induce her to pick a Pareto-efficient
equilibrium with a degenerate relative personal-utility distribution. There is one more re-
mark to be made, even though a formal proof of this claim is beyond the scope of the present
8
analysis: that there is only one Pareto-efficient equilibrium corresponding to each possible
relative distribution of personal utilities. In other words, the correspondence mapping the
domain of relative personal utility distributions to Pareto efficient equilibria is a function.
Since both the social-planner’s equilibrium and the competitive equilibrium are unique
and Pareto efficient both implying a degenerate relative personal-utility distribution, the two
problems are equivalent, i.e., their unique solutions coincide. This result gives us substantial
practical facility with respect to solving the model, especially on the computer: a-priori, we
can pick the social planner’s problem, which is easier, in order to solve for the competitive
equilibrium of the model. Historically, this facility has been a major breakthrough at a time
when computational power was low. For the purposes of this note we will characterize the
social planner’s solution analytically, knowing that we are characterizing the competitive
equilibrium at the same time.
The General Two-period Neoclassical Growth Model
In order to comprehend the key dynamic properties of the neoclassical growth model, it
is better to have a chance to draw some graphs that remind us of the typical microeconomics
we know so far. This purpose is met by studying the two-period model first, i.e. we will
restrict attention to the case of t = 0, 1. One key feature that we can see through the
two-period-model analysis is the fact that in a finite-horizon neoclassical world production
possibilities are given solely by the initial conditions k0 > 0, i.e. the initial physical capital by
which the economy is endowed. Thus, the problem of growth in the neoclassical model is one
of optimal resource management and not of growth-enhancing strategies via technological
investments. Another key result is that the general-equilibrium conditions resulting from
9
the two-period model are the same as these of the infinite-horizon model. So, the two-period
model provides the backbone for a neoclassical general-equilibrium analysis in the long-run.
Commodity space and the production possibility frontier
Once more for simplicity, assume that labor is exogenous. In other words, independently
of how prices may evolve over time, households provide a fixed fraction of their time endow-
ment as labor services, because (by assumption) they do not value leisure. Denote this fixed
level of labor supply with Lf = 1. This takes us to using equations (13) and (14).
By having households not valuing leisure, we have restricted the commodity space into a
two-dimensional one, since there are only two goods: (a) consumption of a single composite
good in period 0, c0; and (b) consumption of the same good in period 1, c1. Time is what
makes the same good be valued differently by consumers.
There are two boundary conditions concerning the capital stock in this economy: (a) in
period 0, the available capital stock is positive, i.e. k0 > 0; and (b) after the end of period 1,
i.e. at the beginning of period 2, the representative household cannot owe physical-capital
claims, i.e. it cannot die in debt, so k2 ≥ 0. With respect to boundary condition (b),
even though the household can borrow resources in order to consume more in period 0,
this condition secures that it has enough time to pay off its debt before it dies. Efficient
utilization of resources implies that households will exhaust their available capital (since
providing it to the firms gives back interest income to them). Therefore, condition k2 ≥ 0
will “bind,” i.e. it will become k2 = 0.
Consumption and capital goods have the same price within the same time period.2 The
2 Notice that this assumption of fixing an exogenous intra-temporal price between consumable goods andcapital dominates all existing growth literature: both the “neoclassical-growth literature,” and “new growththeory.”
10
one type of good can be transformed into the same number of units of the other type of
good at no cost. The economy-resource constraints are given by,
ct + it = yt , t = 0, 1. (15)
The two constraints given by (15) represent the expenditure identity in the national accounts
of a closed economy without a public sector. Now observe that i0 = k1 − (1− δ)k0, where
δ ∈ [0, 1] is the capital depreciation rate, and also i1 = − (1− δ)k1, since k2 = 0, as
explained above. Substituting these last two equations into the two relationships given by
(15) and taking into account that yt = f (kt), we obtain,
c0 + k1 = f (k0) + (1− δ)k0 , (16)
c1 = f (k1) + (1− δ)k1 . (17)
Solving (16) for k1 and substituting into (17), we have,
c1 = f
(f (k0) + (1− δ)k0 − c0
)+ (1− δ)
[f (k0) + (1− δ)k0 − c0
]. (18)
Equation (18) fully describes the production-possibility frontier (PPF). The latter is the set
of all possible bundles (c0, c1) under the constraint that production resources are utilized
efficiently. The PPF is given by the curve drawn in Figure 1. Some simple calculus validates
the shape of the PPF as in Figure 1. In particular,
dc1
dc0= −f1
(f (k0) + (1− δ)k0 − c0
)− (1− δ) < 0 , (19)
d2c1
dc20= f11
(f (k0) + (1− δ)k0 − c0
)< 0 . (20)
An important remark is that the PPF reflects the decentralized-equilibrium production
possibilities under perfect competition. This is due to the fact that production exhibits
11
constant returns to scale. Therefore, competitive pricing of the productive inputs leads,
through Euler’s theorem, to zero profits. Notice that our discussion above about the equiv-
alence between the social planner’s problem and the problem of finding competitive general
equilibrium applies here, too.
0
Production Possibility Frontier (PPF)
C0
C1f(f(K0)+(1−δ) K0)+
+ (1- δ) [f(K0)+(1- δ) K0]
f(K0)+(1- δ) K0
Figure 1 The Production Possibility Frontier in the 2-period Neoclassical growth
model
12
The Social Planner’s Problem
Households have preferences over the consumption of two periods given by additively-
separable “momentary” utility functions as:
U (c0, c1) = u (c0) + βu (c1) ,
with β ∈ (0, 1) and dudc
≡ u1 > 0 and d2udc2
≡ u11 < 0. At time t = 0, households own
physical-capital claims (assets) equal to K0 > 0.
Therefore, the household problem is stated as:
maxc0,c1
u (c0) + βu (c1)
subject to:
c0 + k1 = f (k0) + (1− δ)k0 , (21)
c1 = f (k1) + (1− δ)k1 , (22)
given k0 > 0.
The Lagrangian is given by,
L = u (c0) + βu (c1) + λ0 [f (k0) + (1− δ)k0 − c0 − k1] + λ1 [f (k1) + (1− δ)k1 − c1]
and the first-order conditions are,
∂L
∂c0= 0 : u1 (c0) = λ0 (23)
∂L
∂c1= 0 : βu1 (c1) = λ1 (24)
∂L
∂k1= 0 : λ0 = [1 + fK (k1)− δ]λ1 (25)
∂L
∂λ0= 0 : c0 + k1 = f (k0) + (1− δ)k0 (26)
∂L
∂λ1= 0 : c1 = f (k1) + (1− δ)k1 (27)
13
Using (23)-(25), it is straightforward that the general equilibrium conditions are given by:
u1 (c0)
βu1 (c1)= 1 + f1 (k1)− δ , (28)
and (26) and (27). Obviously, the indifference curve is tangent on the PPF, as it would
be the case in a static model. As it is clear from Figure 2, the general-equilibrium point is
interior, with the (negative value of the) gross-effective interest rate of the second period,
1 +R1 − δ, being equal to the slope of the tangent line passing from both the PPF and the
indifference curve.
PPF
C0
C1
0 C0
C1
K1
Umax
slope=-(1+R1-δ)
Figure 2 General equilibrium in the 2-period Neoclassical growth model
14
The Two-period Model with CES utility and production functions and δ = 1
Consider the following functional forms for the two-period model,
Momentary utility:
U (c0, c1) =c1− 1
σ
0 − 1
1− 1σ
+ βc1− 1
σ
1 − 1
1− 1σ
,
with σ > 0.
Consumable/capital goods production function:
y =[αk1− 1
σ + (1− α)L1− 1
σ
] σσ−1
,
with labor being supplied exogenously with L = 1.
Under the assumption that δ = 1, by direct application of (26)-(28), it is straightforward
to show that,
c0 =1
1 + (αβ)σf (k0) ,
and
k1 =(αβ)σ
1 + (αβ)σf (k0) ,
which comprise the solution to the two-period problem.
15
The General Infinite-horizon Neoclassical Model
First-order conditions using a Lagrangian
The social planner’s problem can be described as:
max{ct,kt+1}
∞t=0
∞∑t=0
βtu (ct)
subject to:
kt+1 + ct ≤ f (kt) + (1− δ)kt
given k0 > 0 ,
limt→∞
kt+1
t∏j=0
(1 +Rj − δ)
= 0 ,
where 1+Rj−δ is the gross-effective interest rate in period j ∈ {0, 1, ...}. The last boundary
constraint means that the present discounted value of aggregate capital as time goes to
infinity must be zero. In other words, eventually the economy should not have debt. This
is rather straightforward in this economy, since homogeneity among households means that
there are no net borrowers and lenders, and, as non-negative amounts of capital are necessary
for production, the economy will never be in debt. Since all variables are normalized so that
the total population- and household-measure is equal to one, and since all firms are identical,
it is k = k and c = c. We can therefore use the above structure of the optimization problem,
after having imposed these aggregation rules, i.e. we can use a representative household
and a representative firm solving the social planner’s problem. We do so, for forming the
Lagrangian.
L =∞∑t=0
βtu (ct)+∞∑t=0
λt [f (kt) + (1− δ) kt − ct − kt+1]
16
The first-order conditions are:
∂L
∂ct= 0 ⇒ βtu1 (ct) = λt , (29)
∂L
∂kt+1= 0 ⇒ λt = [f1 (kt+1) + 1− δ]λt+1 , (30)
∂L
∂λt
= 0 ⇒ ct + kt+1 = f (kt) + (1− δ) kt . (31)
Combining (29) and (30) we can eliminate the Lagrange multipliers and obtain,
u1 (ct) = β [f1 (kt+1) + 1− δ]u1 (ct+1) . (32)
Equations (31) and (32) together with the two terminal conditions k0 > 0 and limt→∞
kt+1
t∏
j=0
(1+Rj−δ)=
0, constitute the necessary conditions for a maximum. We will refer to other arguments for
showing that these conditions are also sufficient for a maximum in a later section. In brief,
equation (32) says that the necessary condition for a maximum sets the marginal rate of
substitution between any two subsequent periods equal to the gross effective interest rate
linking these two periods, f1 (kt+1) + 1− δ. On the other hand, equation (31) says that the
budget constraint is never slack; it is always met with equality.
Steady state(s)
In the steady state the growth rate of all variables is constant. A good conjecture for
the steady-state growth rate of variables in the neoclassical model may come from what we
know from the Solow model: given that the function f (kt) is strictly concave, the Solow
model says that in the steady state the growth rate of all variables equals 0. Is this the case
here? Does such a steady state exist?
As we do with linear difference-equation systems, we check if there is a particular solution,
17
with,
kt+1 = kt and ct+1 = ct for all t ∈ {τ , τ + 1, ...} for some τ ∈ {0, 1, ...} .
We denote the equilibrium part that corresponds to the particular solution as (kss, css), the
steady-state. The two equations in the steady state become,
css = f (kss)− δkss , (33)
f1 (kss) =
1− β
β+ δ . (34)
kt
0
ct
( )c f k k= − δ
kk fss =−
+⎛
⎝⎜
⎞
⎠⎟−
11 1 β
βδ
css
( )k f* = −1
1 δ
Figure 3 Graphical exposition of the argument that css > 0
Equation (34) comes from imposing the condition that csst+1 = csst . With the Inada
conditions f1 (0) = ∞ and f (∞) = 0, there should be only one possible non-zero and
18
bounded value for kss, as f1 (k) is a strictly decreasing function and 0 = f (∞) < 1−β
β+
δ < f (0) = ∞. So, the intermediate-value theorem guarantees existence of a kss with
0 < kss < ∞, and strict monotonicity the uniqueness of this kss. In particular,
kss = f−11
(1− β
β+ δ
). (35)
Equation (33) comes from imposing the condition that ksst+1 = kss
t . To see that css > 0,
observe that the Inada condition f1 (0) = ∞ together with (33) imply that there is a value
k > 0 such that f(k)− δk = 0 and that c > 0 if k ∈
(0, k
). In Figure 1 the function
c = f (k) − δk is drawn over the interval[0, k
]. Apparently the maximum point of this
function k∗ ∈[0, k
]is given by: k∗ = f−1
1 (δ). Since f1 is strictly decreasing and because
f−11 preserves the monotonicity of f1, it is,
1− β
β+ δ > δ ⇒ f−1
1
(1− β
β+ δ
)< f−1
1 (δ) ⇒ kss < k∗ . (36)
This property is known as the “dynamic inefficiency” of the optimal solution to the neoclas-
sical model, as css < f (k∗)−δk∗, which corresponds to Solow’s “golden rule” of steady-state
consumption. As it can be seen from (36), the reason for this inefficiency is the impatience
of households, namely that β < 1. Since kss ∈ (0, k∗) ⊂(0, k
), it follows that 0 < css < ∞.
Figure 3 summarizes this argument graphically. In other words, there is a unique steady
state (kss, css) with 0 < kss, css < ∞.
19
Stability Analysis of the Dynamic System
We can first focus on the 2-equation, 2-unknown first-order non-linear dynamic system
given by equations (31) and (32), of the form:
h (kt+1, ct+1) = φ (kt, ct) ,
where h : R2+ → R
2, φ : R2+ → R
2 with:
h1 (kt+1, ct+1)
h2 (kt+1, ct+1)
≡
≡
kt+1
β [f1 (kt+1) + 1− δ]u1 (ct+1)
=
=
f (kt) + (1− δ) kt − ct
u1 (ct)
≡
≡
φ1 (kt, ct)
φ2 (kt, ct)
Stability properties of the system
Now we will prove that the linearized dynamic system given by equations (31) and (32)
within a neighborhood (open ball) around the point (kss, css) with an arbitrary radius ε > 0
is a saddle, rather than a sink or a source. We can take a first-order Taylor expansion of
functions h and φ around (kss, css). The linearized system will look like:
Jh (kss, css)
⎡⎢⎣ kt+1 − kss
ct+1 − css
⎤⎥⎦ = Jφ (k
ss, css)
⎡⎢⎣ kt − kss
ct − css
⎤⎥⎦ ,
where Jh (kss, css) and Jφ (k
ss, css) are the Jacobian matrices of functions h and φ evaluated
at point (kss, css), with,
Jh (kss, css) =
⎡⎢⎣ 1 0
βf11 (kss) u1 (c
ss) u11 (css)
⎤⎥⎦ ,
and
Jφ (kss, css) =
⎡⎢⎣
1β
−1
0 u11 (css)
⎤⎥⎦ .
20
In order to characterize the system for stability, we should focus on the linear system of
equations:⎡⎢⎣ kt+1 − kss
ct+1 − css
⎤⎥⎦ = J−1
h (kss, css) Jφ (kss, css)
⎡⎢⎣ kt − kss
ct − css
⎤⎥⎦ ⇐⇒
⇐⇒
⎡⎢⎣ kt+1 − kss
ct+1 − css
⎤⎥⎦ =
⎡⎢⎣
1β
−1
−f11(kss)u1(css)u11(css)
βf11(kss)u1(css)u11(css)
+ 1
⎤⎥⎦⎡⎢⎣ kt − kss
ct − css
⎤⎥⎦ .
Following the stability analysis by Azariadis (1993, Ch. 7) for 2× 2 linear dynamic systems,
we should calculate the trace and determinant of matrix J−1h (kss, css) Jφ (k
ss, css). The reason
is that, for 2 × 2 systems, the trace equals the sum of the system’s eigenvalues and the
determinant equals the product of the two eigenvalues.. By comparing them to certain key
values, we can reach a concrete conclusion about whether the eigenvalues are positive or
negative, and greater or less than 1 in absolute value. So, let
T = tr(J−1h (kss, css) Jφ (k
ss, css))= 1 +
1
β+
βf11 (kss)u1 (c
ss)
u11 (css)> 2 ,
and
D =∣∣J−1
h (kss, css) Jφ (kss, css)
∣∣ = 1
β> 1 .
The positive sign of the determinant implies that the two eigenvalues have the same sign,
and since their sum is greater than zero, we know they should be both positive. At the
same time we can see that the two eigenvalues are both real, since the discriminant of the
characteristic polynomial equation
λ2 − Tλ+D = 0 ,
is,
∆ = T 2 − 4D >
(1 +
1
β
)2
−4
β=
(1−
1
β
)2
> 0 .
21
But we can be more specific about the eigenvalues. In order to find out whether both
eigenvalues are greater than 1 or if it is only one of the two, we can set λ = 1 in the
polynomial equation and observe its sign. In other words,
λ2 − Tλ+D∣∣λ=1
= 1− T +D ⇒
⇒ λ2 − Tλ+D∣∣λ=1
= −βf11 (k
ss)u1 (css)
u11 (css)⇒
⇒ λ2 − Tλ+D∣∣λ=1
< 0 .
Given that the sign of the factor multiplying λ2 is positive, the fact that the value of the
polynomial is negative at λ = 1, means that 1 lies between the two real eigenvalues, λ1 and
λ2. In brief,
0 < λ1 < 1 < λ2 .
Therefore we know that the linearized system around the steady state can behave so that
the steady state is a saddle, i.e. it can be either a sink or a source. The conclusion that
the steady state can be either a sink or a source is drawn from equations (31) and (32), i.e.
only from a subset of the necessary conditions: the two “boundary conditions,” the initial
conditions and the transversality condition, have been ignored so far. If we impose the
transversality condition, then, (i) for any initial condition k0 > 0, the steady state is a sink,
and (ii) it can be proved that there can be only one stable path. We will take statements
(i) and (ii) as granted for the moment, since their proofs are advanced for the scope of this
introductory note.
A heuristic analysis through the construction of a phase diagram can help us see the
position of a stable part of the decision correspondence ct = C (kt) in the planar field (kt, ct).
A formal analysis is more involved. So, we will proceed with a characterization of the
22
equilibrium path with the aid of a phase diagram.
Building the phase diagram
A 2 × 2 dynamic system like this given by equations (31) and (32) is often hard to
characterize by naked eye. A useful device for characterizing the period-by-period motion
of the system’s two variables is the phase diagram. Its goal is to define and depict areas
of combinations of the system’s two variables in an arbitrary (current) period t ∈ {0, 1, ...}
with clear-cut dynamic properties: such areas should point out distinct directions of these
two variables in the next period. So, the phase diagram for a system consisting of first-
order difference equations is a picture of key information about the system’s period-by-period
mechanics.
One of the easiest ways to build a phase diagram in a 2-dimensional Euclidean vector
space is to isolate the motion of each variable in the 2-dimensional Cartesian plane. The idea
of motion is easy to comprehend and even easier to depict on the phase diagram. Given that
we work on a 2-dimensional Cartesian plane, we can view it in two ways: (i) as a planar field,
that gives us the position of the two variables in a certain arbitrary period t ∈ {0, 1, ...}; and
(ii) as a vector field, where we can use vectors of motion in the planar field, referring to the
direction of change of each variable in the next period.3 In this way, in a given Cartesian
plane, we can blend two planar fields in one graph: one referring to period t ∈ {0, 1, ...},
and one referring to period t + 1. If we think of the Cartesian plane of variables k and c,
ordered as (k, c), the one-period transition from a point (kt, ct) to the point (kt+1, ct+1), can
3 The planar field of all the Cartesian ordered pairs (k, c) is straightforward. On the other hand, the definitionof the vector field rests upon equations (31) and (32). You can find a definition of a vector field in any realanalysis book, (a introductory text is this of Lang (1983), and the definition is on page 332). It is a mapfrom (k, c) to R
2 in the case we examine here, and the interpretation of the two numbers can be (∆k,∆c),the change in two subsequent periods.
23
be depicted by drawing an arrow with the tip of the arrowhead at point (kt+1, ct+1) and the
end of its tail at point (kt, ct).
Meeting our goal through a phase diagram means to find areas in the planar field of
(kt, ct) with clear-cut period-by-period dynamics for k and c. This means that we should
find areas where the direction of the arrows is qualitatively similar. In order to do so in
a two-dimensional planar field, it is easier to search through the motion of each variable
in isolation. In particular, we should identify areas where the system given by equations
(31) and (32) dictates that ∆c > 0 (perpendicular vectors pointing upwards), or ∆c < 0
(perpendicular vectors pointing downwards), or ∆c = 0 (usually a “phase curve” instead of a
whole area). Then we can identify areas where the system drives k in isolation so that∆k > 0
(flat vectors pointing to the right), or ∆k < 0 (flat vectors pointing to the left), or ∆k = 0
(usually a “phase curve” instead of a whole area). At the end we can put our conclusions
together in one graph, combining qualitatively the directions of the perpendicular and flat
vectors in areas where the combined vector directions are unambiguous. In this way we
can locate areas where choices of the control variable (c), make the system unambiguously
unstable, refining in this way areas for candidate solutions. This is what we will do in the
following subsections.
Isolating the motion of consumption (the control variable)
Consider the Cartesian plane of variables kt and ct, ordered as (kt, ct) at any t ∈ {0, 1, ...}.
A perpendicular vector with upwards direction and with the starting point of its tail po-
sitioned at the point (kt, ct), means that, at this point, ∆ct ≡ ct+1 − ct > 0 (equivalently,
a perpendicular vector starting from this point showing downwards direction means that
24
∆ct < 0).
Because we are interested in the two-period motion of ct, we will focus on equation (32),
which is a first-order difference equation of c. At this point of our analysis there is no use
in examining equation (31), since this is a first-order difference equation of k.
kt
0
ct
( ) ( ) ( )( )= = ⇔ + − − =−
+⎧⎨⎩
⎫⎬⎭
+k c c c f f k k ct t t t t t t, 1 1 11
δβ
βδ
( ) ( ) ( )( )k f f k k,0 11
1 + − =−
+⎧⎨⎩
⎫⎬⎭
δβ
βδ
phase curve 1
area A ( ) ( ) ( )( )= + − − >−
+ ⇒ >⎧⎨⎩
⎫⎬⎭
+k c f f k k c c ct t t t t t t, 1 111
δβ
βδ
area B ( ) ( ) ( )( )= + − − <−
+ ⇒ <⎧⎨⎩
⎫⎬⎭
+k c f f k k c c ct t t t t t t, 1 111
δβ
βδ
( ) ( )01
1,c f c− =−
+⎧⎨⎩
⎫⎬⎭
ββ
δ
∆ct > 0
∆ct < 0
Figure 4 Diagrammatic analysis of the motion of the control variable c in two arbitrary
subsequent periods, t, t+ 1, where t ∈ {0, 1, ...}
In order to identify clear-cut areas where ∆ct > 0 or ∆ct < 0, we should start from
the set of points (kt, ct) for which ∆ct = 0. Therefore, we start by imposing the condition
∆ct = 0 ⇔ ct = ct+1 on equation (32), which gives,
f1 (f (kt) + (1− δ) kt − ct) =1− β
β+ δ .
25
So, the first phase curve (phase curve 1) is defined as,
phase curve 1 =
{(kt, ct)
∣∣∣∣ct = ct+1 ⇔ f1 (f (kt) + (1− δ) kt − ct) =1− β
β+ δ
}, (37)
and it is depicted in Figure 4. We can see that phase curve 1 divides the planar field (kt, ct)
in two areas, one “on the left” (area A) and one “on the right” (area B) as shown in Figure
4. Taking into account that f11 < 0, we can define the two areas as,
area A =
{(kt, ct)
∣∣∣∣f1 (f (kt) + (1− δ) kt − ct) >1− β
β+ δ
}, (38)
and
area B =
{(kt, ct)
∣∣∣∣f1 (f (kt) + (1− δ) kt − ct) <1− β
β+ δ
}. (39)
Now let’s examine the sign of ∆ct in area A.
f1 (f (kt) + (1− δ) kt − ct) >1−β
β+ δ ⇒ β [1 + f1 (kt+1)− δ] > 1
(32) ⇒ β [1 + f1 (kt+1)− δ] = u1(ct)u1(ct+1)
⎫⎪⎬⎪⎭
u1>0=⇒
u1>0=⇒ u1 (ct) > u1 (ct+1)
u11<0=⇒ ct < ct+1 ⇒ ∆ct > 0 .
Therefore, in area A it is∆ct > 0, which justifies the perpendicular arrow pointing upwards in
Figure 4. Similarly, we can show that in area B it is∆ct < 0, which justifies the corresponding
perpendicular arrow pointing downwards in Figure 4.
A last remark about Figure 4 and the motion of variable c pertains to points (kt, ct) that
are on the phase curve 1. We have shown that if (kt, ct) ∈ phase curve 1 at an arbitrary
point in time t ∈ {0, 1, ...}, it is indeed the case that ct = ct+1. However, it is not necessarily
the case that ct+1 = ct+2, unless (kt, ct) = (kss, css). It is important to remember that Figure
4 provides information on the motion of ct only for one period ahead. This happens because
Figure 4 was derived by a first-order difference equation.
26
Isolating the motion of capital (the state variable)
Consider again the Cartesian plane of variables kt and ct, ordered as (kt, ct) at any
t ∈ {0, 1, ...}. A flat vector starting from point (kt, ct) with direction to the right, means
that ∆kt > 0.
Because we are interested in the two-period motion of kt, we will focus on equation (31),
which is a first-order difference equation of k. There is no use in examining equation (32),
since this is a first-order difference equation of c.
In order to identify clear-cut areas where ∆kt > 0 or ∆kt < 0, we should start from
the set of points (kt, ct) for which ∆kt = 0. Therefore, we start by imposing the condition
∆kt = 0 ⇔ kt = kt+1 on equation (31), which gives,
ct = f (kt)− δkt .
So, the second phase curve (phase curve 2) is defined as,
phase curve 2 = {(kt, ct) |kt = kt+1 ⇔ ct = f (kt)− δkt} , (40)
and it is depicted in figure 5. We can see that phase curve 2 divides the planar field (kt, ct)
in two areas, one “above” (area C) and one “below” (area D) as shown in Figure 5.
It is straightforward to define the two areas as,
area C = {(kt, ct) |ct > f (kt)− δkt} , (41)
and
area D = {(kt, ct) |ct < f (kt)− δkt} , (42)
Now let’s examine the sign of ∆kt in area C.
ct > f (kt)− δkt
(31) ⇒ ct = f (kt) + (1− δ) kt − kt+1
⎫⎪⎬⎪⎭ ⇒ kt > kt+1 ⇒ ∆kt < 0 .
27
Therefore, in area C it is ∆kt < 0, which justifies the flat arrow pointing to the left in Figure
5. Similarly, we can show that in area D it is ∆kt > 0, which justifies the corresponding flat
arrow pointing to the right in Figure 5.
kt
0
ct
( ) ( ){ }= = ⇔ = −+k c k k c f k kt t t t t t t, 1 δphase curve 2
area C ( ) ( ){ }= > − ⇒ <+k c c f k k k kt t t t t t t, δ 1
area D ( ) ( ){ }= < − ⇒ >+k c c f k k k kt t t t t t t, δ 1
∆kt < 0
∆kt > 0
Figure 5 Diagrammatic analysis of the motion of the state variable, k, in two arbitrary
subsequent periods, t, t+ 1, where t ∈ {0, 1, ...}
Again, it is important to remember that, since Figure 5 was derived by a first-order
difference equation (equation (31)), Figure 5 provides information on the motion of kt only
for one period ahead. This means that although (kt, ct) ∈ phase curve 2 at an arbitrary
point in time t ∈ {0, 1, ...} implies kt = kt+1, it is not necessarily the case that kt+1 = kt+2,
unless (kt, ct) = (kss, css). Yet, we will see that when we set up the whole phase diagram we
28
can characterize the long-run behavior of the model. This is the task we will deal with in
the following section.
The complete phase diagram
In our analysis above, we showed that in a neighborhood around the steady state, the
system can behave in a way that the steady state can be a sink or a source. We also
mentioned (without proof) that if the transversality condition is imposed, then the steady
state will become a sink, and in particular, with a unique, globally stable equilibrium path,
that we call “stable arm.” Although the phase diagram neither provides a formal proof of
stability, nor it provides a proof of uniqueness of an equilibrium path, the phase diagram
can point out the areas in which the stable arm may lie.
By putting Figures 4 and 5 together in a single diagram, we can see in Figure 6 that the
two phase curves (1 and 2) intersect at one point only. By the definition of the phase curves,
their intersection point is the steady state (kss, css).
The planar field (kt, ct) is divided into four areas (denoted by Latin numbers) with specific
vector-field directions for both variables pertaining to the motion for one-period ahead only.
The vector-field directions of motion are depicted in Figure 6.
If we consider any initial conditions k0 > 0 and start examining alternative initial equi-
librium levels of consumption c0 > 0, then the only areas that could guarantee that, in a
repeated, recursive fashion, an equilibrium path could lead to the steady state, are the areas
II (northeast) and IV (southwest) of the phase diagram. In Figure 6 we can see a candidate
placement of the stable arm.
29
kt
0
ct
css
( ): f k ss1
1=
−+
ββ δk ss
I
IV
II
III
Stable arm
Figure 6 The complete phase diagram and the stable arm
The economic intuition stemming from this phase diagram is direct and substantial. First
of all, the stable arm is a decision rule for consumption, a function of the form ct = C (kt).
Given that the production function yt = f (kt) bonds capital and income tightly together,
the stable arm can be viewed and interpreted as a consumption function as well. Although
all households take into account their long-run income in order to decide how much to
consume in this model (permanent-income considerations), intuitively, when agents are poor,
then they consume less. The study of issues such as how big is the average and marginal
propensity to consume while starting as richer or poorer (higher or lower k0) is not easy to
undertake analytically for such a general setting. Some analytical results are possible for
30
specific parametric functional forms for u and f , but these results are known only for the
continuous-time version of the neoclassical model. For the discrete-time neoclassical model,
Figures 7-9 provide some numerical results for u (c) =(c1−θ − 1
)/ (1− θ) and f (k) = Akα,
(the numerical values used are, α = 0.34, β = 0.96, δ = 0.05, A = 1), conducting also a
sensitivity analysis with respect to varying the values of θ, with θ ∈ {0.5, 1, 2, 3, 5}.
REFERENCES
Azariadis, C. (1993), “Intertemporal Macroeconomics,” Blackwell Publishing.
Lang, S. (1983), “Undergraduate Analysis,” first edition, Springer, UndergraduateTexts in Mathematics.
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