Slides to Lecture 4 of Introductory Dynamic
Macroeconomics.
Solow and RBC models
Ragnar Nymoen
University of Oslo, Department of Economics
September 5, 2008
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Dynamic systems (cont’d)
The slides to lecture 3 covered a dynamic Keynesian income-expenditure system, as a first example of a dynamic macroeconomic system. Two other important examples of such systems are the Solow growth model and the real
business cycle (RBC) model.
The Solow growth model
The first equation is the macro production function of the Cobb-Douglas type
γ
Yt = Kt Nt1−γ , 0 < γ < 1
(1)
where Yt is GDP in period t, Kt denotes the capital stock, and Nt is population
in period t.
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N is assumed to grow with a constant rate n:
Nt
− 1 = n, n ≥ 0
Nt−1
(2)
Kt denotes the capital stock available at the start of period t. We next assume
that all saving, St, in period t is invested so:
Kt = (1 − δ)Kt−1 + St−1, 0 < δ ≤ 1
(3)
where δ is the rate of depreciation of capital. An essential assumption is that
saving is proportional to income:
St = sYt,
0≤s<1
(4)
where s is the saving rate.
To find the final equation for the dynamic system, we start with equation (3)
and divide on both sides by Nt−1:
Kt
Kt−1
St−1
= (1 − δ)
+
.
Nt−1
Nt−1 Nt−1
3
On the left-and side, multiply by Nt/Nt, and use (2) and (4) to obtain:
kt(1 + n) = (1 − δ)kt−1 + s yt−1
(5)
where kt = Kt/Nt and yt = Yt/Nt. From the production function (1):
γ
yt = kt
so that the dynamic equation for the capital intensity variable kt becomes:
n
o
1
γ
(1 − δ)kt−1 + s kt−1 .
kt =
(1 + n)
(6)
which is the final equation for kt. It is non-linear, but to see the conditions for
stability in (6) subtract kt−1 on both sides to obtain
γ
(1 + n)∆kt = −(δ + n)kt−1 + s kt−1
(7)
and note that
γ
∆kt > 0 ⇐⇒ s kt−1 > (δ + n)kt−1,
γ
∆kt < 0 ⇐⇒ s kt−1 < (δ + n)kt−1,
γ
∆kt = 0 ⇐⇒ s kt−1 = (δ + n)kt−1.
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(8)
The first line in (8) states that the capital intensity is growing in all periods where saving per capita is larger than the amount of saving required to
compensate for capital depreciation and population growth.
Line two states that the capital intensity is falling if saving per capita is less
than that required amount.
γ
Finally, ∆kt = 0 so that kt = kt−1 = k̄ when s kt−1 = (δ + n)kt−1, see figure
1.
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Linearization of (6) around the steady state k̄ (see appendix to IDM) gives:
µ
¶
1−γ γ
1−δ
s
k̄
kt ≈
+
γ k̄γ−1 kt−1 + s
1+n 1+n
1+n
which is a linear AR equation of the type we use throughout the course.
(9)
If the autoregressive coefficient is between 0 and 1 (9) is consistent with dynamic stability in the neighborhood of the steady state capital intensity k̄.
Frequently macro economic models are formulated as non-linear models. If
they are also dynamic, they are often made operational by similar linearization
techniques.
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0.8
(δ+ n)kt−1
s kt−1γ
0
→
→
k̄
← ←
4
Figure 1: Solow growth model: IIlustration of equation (7) for the case of
δ + n = 0.165) and s = 0.25 and γ = 0.5.
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The real business cycle model
The first equation of the RBC model is the macro production function (1),
augmented by a technology variable At
γ
Yt = Kt (AtNt)1−γ , 0 < γ < 1,
(10)
The hallmark of the RBC is that a model for technological progress is introduced:
ln At = gAt + as,t,
0 < gA < 1,
(11)
where the random part is given by the autoregressive equation:
as,t = αas,t−1 + εa,t,
0≤α<1
where εa,t is an unpredictable technology shock.
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(12)
The RBC model makes use of the saving equation (4), and the capital evolution
equation (3), which we however simplify by setting δ = 1, meaning that capital
equipment lasts for only one period, so that
Kt = St−1.
(13)
It is the infallible mark of RBC models that the labour marked is assumed to be
in equilibrium in each period. Labour supply, NtS , is a function of the relative
wage wt/w̄t:
wt
S
> 0.
(14)
Nt = N̄ ( ) ,
w̄t
where w̄t is the steady state wage and is the labour supply elasticity. When
the wage is equal to the steady state wage, labour supply is also equal to its
steady state value.
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Labour demand is obtained by assuming optimizing behaviour by a macro producer, meaning that the wage rate is equal to the marginal product of labour
in each time period.
As shown in IDM, chapter 2.8.4 (or in IAM ch 19.4) the following dynamic
equation for the logarithm of the output gap, (ln Yt − ln Ȳt), can be derived:
γ(1 + )
(1 − γ)(1 + )
(15)
(ln Yt−1 − ln Ȳt−1) +
as,t.
1+γ
1+γ
Hence we have the important result that the macro economy will be characterized by periods of booms (positive output-gap), and troughs (negative
output-gap). This is implied even if the initial situation is characterized by
ln Yt = ln Ȳt, because there is always a new technology shock in period t + 1.
(ln Yt − ln Ȳt) =
10
0.03
0.02
0.01
0.00
−0.01
−0.02
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Figure 2: A simultated series of the logarthm of the output gap using equation
(15) and (12) with ω = 0.1, = 4 and γ = 0.5. The standard deviation of the
technology shock variable εt is set to 0.01.
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The explanation of macro dynamics in the RBC model is that there is a flow
of technology shocks, that are propagated into persistent deviations from the
steady state by saving and investment dynamics, and by workers willingness to
supply more labour in good times, and to substitute work by leisure in economic
downturns.
Unlike the Keynesian income-expenditure model, periods with below capacity
output, and below average recorded employment, is an equilibrium phenomenon
in the RBC model–it is a theory of equilibrium business cycles.
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