Lecture notes on modern growth theory
Part 2
Mario Tirelli
Very preliminary material
Not to be circulated without the permission of the author
November 9, 2016
Contents
1.
2.
3.
Introduction
Optimal economic growth: the Ramsey-Cass-Koopmans model
Qualitative predictions
1
1
5
1. Introduction
We are now ready to analyze the balanced growth equilibrium of a Ramsey-Cass-Koopmans
(RCK) economy and to compare it with Solow-Swan’s. The main difference in the two contributions is that, unlike in the Solow’s economy, in RCK consumption and saving decisions are
determined endogenously. More precisely, we find the optimal allocation as a solution of the
Ramsey problem (P); then, by the II Welfare Theorem, we conclude that this allocation can be
decentralized as a competitive equilibrium of a Walrasian economy (i.e. one with a complete
set of future commodity markets, or with spot commodity markets and a complete set of asset
markets allowing for sequential trade). Finally, we discuss how accounting for consumption and
saving decisions might sharpen Solow’s qualitative predictions, and why some early criticisms
remain valid.
2. Optimal economic growth: the Ramsey-Cass-Koopmans model
For expositional simplicity, consider the RCK economy with inelastic labor supply. As for the
Solow model, balanced growth –with variables growing at an equal, positive rate– is achieved introducing some (exogenous) growth in the fundamentals, population and technological progress
(or factors productivity). Otherwise, we shall argue, the only possible balance growth is the
steady state (i.e. zero growth of output, capital and consumption, with constant per period investment used to replace depreciated capital). To help the comparison with the Solow’s model,
we maintain our previous assumptions and consider the case of a Harrod-neutral, Cobb-Douglas
technology. Thus, with a per-period, population growth n and an exogenous technological
progress µ, we now express all the variables in efficiency units (keep using lower case letters):
yt = f (kt ) :=
Yt
=
N̄t
Kt
At Nt
α
= ktα
Accordingly, the economy resource constraint (in efficiency units) is,
ct = ktα + (1 − δ)kt − kt+1 (1 + η)
(1)
where, as before, η := n + µ.
Since ct := Ct /N̄t , with N̄t = (1 + η)t N0 , (P) can be rewritten as follows,
max
X
β t u ct N̄t
t∈T
such that, for all t in T ,
ct = f (kt ) + (1 − δ)kt − kt+1 (1 + η), k0 > 0 given
1
Forming the Lagrangian, at all t in T , we derive the following necessary conditions for an
interior solution of (P), with (λt )t∈T being a sequence of Lagrange multipliers,
∂L0
= β t u0 (ct N̄t )N̄t − λt = 0
∂ct
∂L0
= λt+1 f 0 (kt+1 ) + (1 − δ) − λt = 0
∂kt+1
∂L0
= ct + kt+1 (1 + η) − f (kt ) − kt (1 − δ) = 0
∂λt
From the first condition one obtains,
u0 ct+1 N̄t+1 N̄t+1
λt+1
=β
λt
u0 (ct N̄t )
N̄t
u0 ct+1 N̄t+1
(1 + η)
=β
u0 ct N̄t
From the second condition one obtains,
−1
λt+1
= (1 + η) f 0 (kt+1 ) + 1 − δ
λt
Putting the two together yields the Euler equation,
βu0 ct+1 N̄t+1
0
(E)
f
(k
)
+
1
−
δ
=1
t+1
u0 ct N̄t
where, we can define an implicit interest rate, rt := f 0 (kt ) − δ.
As before, we can approximate marginal utility and write explicitly the consumption dynamics. First-order approximation of marginal utility around Ct , for a sufficiently small length
period to make ∆Ct+1 small, yields: u0 (Ct+1 ) = u0 (Ct ) + u00 (Ct )∆Ct+1 . Dividing through by
u0 (Ct ),
u0 (Ct+1 )
u00 (Ct ) ∆Ct+1
=
1
+
Ct
u0 (Ct )
u0 (Ct )
Ct
Using the elasticity of substitution of the per-period utility, σ(Ct ) = − (u00 (Ct )/u0 (Ct )) Ct ,
u0 (Ct+1 )
∆Ct+1
β 0
= β 1 − σ(Ct )
u (Ct )
Ct
Using this to rewrite the Euler equation gives,
∆Ct+1
1
1
(E’)
=
1−
Ct
σ(Ct )
β(1 + rt+1 )
where, by definition, Ct+1 /Ct = (1 + η)ct+1 /ct and the following can be easily derived,
∆Ct+1
∆ct+1
∆ct+1
−η =
+ gtc η ≈
Ct
ct
ct
This provides an important insight on the existence of a balanced growth. Indeed, by (E 0 ),
Ct grows at a constant rate gC := η only if σ(Ct ) is also constant over time. Equivalently, a
balanced growth exists only if preferences are iso-elastic, with σ(Ct ) = σ > 0. Thus, we the
2
aim of studying efficient dynamics entailing balance growth, we let the per-period utility be,
u(C) = C 1−σ /(1 − σ), σ > 0, with associated intertemporal marginal rate of substitution,
u0 (Ct+1 )
β 0
=β
u (Ct )
Ct+1
Ct
−σ
= β(1 + η)
−σ
ct+1
ct
−σ
Substituting into (E), we obtain a new version of the Euler equation,
(E(σ))
ct+1
βb
ct
−σ
(1 + rt+1 ) = 1
where βb := β(1 + η)−σ . Letting consumption growth, in efficiency units, be measured by
gtc := ct+1 /ct − 1, we can rewrite (E(σ)) as,
b + rt+1 )1/σ
1 + gtc = β(1
the optimal consumption growth is increasing in the return to investing in physical capital.
The resource constraint (1) can be rewritten as follows, simply adding and subtracting ηkt
from both sides, dividing through by kt and using η∆kt+1 /kt ≈ 0,
(∗∗)
∆kt+1
yt
ct
=
− (δ + η) −
kt
kt
kt
We can use the law of motion of capital to find the relationship between savings and growth.
Since,
ct
st := 1 −
yt
∆kt+1 = yt − kt (δ + η) − ct
= yt − kt (δ + η) − yt (1 − st )
= st − kt (δ + η)
or
∆kt+1
1
= st − (δ + η)
kt
kt
This implies that the saving rate is positively correlated with the growth rate of capital. At
balance growth (i.e. in the ‘long-run’), when ∆kt+1 = 0, income per capita is negatively
correlated with savings, as in the Solow’s model: (∗∗) yields, y∗ = (δ + η)k∗ /s∗ .
Steady-state. For the exact same reasons as in the Solow’s economy, the Ramsey’s in efficiency
units has no balanced growth too, only a steady state might exists. Thus, assuming that a
steady state exists, we now compute it, letting (yt , ct , kt , rt ) = (y∗ , c∗ , k∗ , r∗ ) at all t in T .
From the Euler equation (E(σ)),
βb αk∗α−1 + 1 − δ = 1
3
αk∗α−1 = δ − 1 +
=δ−1+
1
βb
1
β(1 + η)−σ
= δ − 1 + (1 + θ)(1 + η)σ
≈ δ − 1 + (1 + θ)(1 + ση)
≈ δ + θ + ση
where the third condition uses the definition of the individual discount rare θ (= β −1 − 1), the
penultimate one exploits a first-order approximation and the last condition assumes σθη ≈ 0.1
Solving for k∗ ,
1
1−α
α
k∗ =
δ + θ + ση
α
1−α
α
α
y∗ = k∗ =
δ + θ + ση
Implying,
k∗
α
=
y∗
δ + θ + ση
Next, we solve for c∗ , given y∗ = k∗α , and finally we compute the steady-state saving rate s.
From the feasibility constraint,
c∗ = y∗ + (1 − δ)k∗ − (1 + η)k∗ = y∗ − (δ + η)k∗
From the law of motion of capital, at steady state, the saving rate is,
s∗ :=
α(δ + η)
k∗
(δ + η) =
y∗
δ + θ + ση
The saving rate is higher the more patient is the household (θ close to zero). If, after a permanent shock to one of the parameters, the saving rate increases, capital increases only temporarily, adjusting the capital-output ratio. In fact, in the long-run the steady state equilibrium
predicts no capital accumulation.
Balanced growth of the original economy. Finally, going back to the original variables,
we can derive a balanced growth equilibrium. For capital, as the condition found studying the
Solow economy applies here too; hence, at steady state,
∆Kt+1
∆kt+1
g :=
≈
+η =η
Kt
kt
ss
that is, the rate of growth of the original economy g is (exogenously given) by η = n + µ.
The ratio of capital stock to output is,
Kt
α
kt
=
= k∗1−α =
Yt
yt ss
δ + θ + σ(n + µ)
1To clarify, suppose we approximate around a point in which there is neither population nor productivity
growth, η = 0. Then, at the first order,
(1 + η)σ ≈ (1 + η)σ |η=0 +σ(1 + η) |η=0 (η − 0)
yielding, (1 + η)σ ≈ 1 + ση.
4
Hence, the ratio of capital to output is constant; this is so because they grow at the same rate
(the balanced growth g). Indeed, at a balanced growth,
α
1−α
α
t
α
Yt = y∗ N̄t = k∗ (1 + n + µ) N0 =
(1 + n + µ)t N0
δ + θ + σ(n + µ)
1
1−α
α
Kt = k∗ N̄t =
(1 + n + µ)t N0
δ + θ + σ(n + µ)
Ct = c∗ N̄t = [y∗ − (δ + n + µ)k∗ ] (1 + n + µ)t N0
(Ct , Yt , Kt ) have a growth rate that equals η, the growth rate of N̄t . Aggregate savings do also
grow at η, simply because St = s∗ Yt = It .
3. Qualitative predictions
We can now use the results from our previous analysis to summarize the qualitative predictions obtained with the RCK model and compare them with Solow’s.
The RCK model predicts that,
• an economy experiences a higher balanced growth the higher is technological progress
and population growth, g = η := n+µ; implying that without (exogenous) technological
progress the per-capita variables display zero growth, in the long-run.
• capital-output ratio tends to be constant in the long-run, and it is lower the higher is
g, σ, θ, δ (and s∗ ) and the lower is α.
• A permanent shock to any of the parameters (δ, θ, σ, α) that makes the saving rate
increase, will also determine a temporary adjustment of the great ratios; namely, it
will increase the capital-output ratio and decrease the consumption-output ratio; this
adjustment is such that the balanced growth remains constant at g (remember g = n+µ
is independent of the saving rate). This is again as in the Solow model.
• The steady-state value of capital is exactly as in the Solow model (just use s∗ to substitute into k∗ and find),
1
1−α
s∗
k∗ =
δ+n+µ
with the caveat that k∗ is always below the golden rule capital stock k G , for β < 1 (or
θ > 0).
Therefore, the Solow model and the Ramsey-Cass-Koopmans’ have very similar qualitative
predictions. However there are two major differences:
(1) In RCK the saving rate is endogenously determined, increasing in the share of capitalincome f 0 (k∗ )k∗ /f (k∗ ) = α, decreasing in θ (impatience) and in n + µ.
(2) In the RCK model the golden rule is never an equilibrium (for β < 1 or θ > 0), because
it would violate optimality; conversely, it could be an equilibrium in the Solow model,
for a sufficiently high (exogenous) saving rate.
Moreover, there is a clear difference of the two models outside of the steady state, as in the
optimal growth model the saving rate adjusts in response to economic shocks, and so do the
5
great ratios. This is important as it raises the speed of adjustment of the economy when
a permanent shock to fundamentals leads to a new steady state. In particular, it changes
considerably the response of consumption.
You should try some more comparative statics using both the graphical and the algebraic
analysis.
6