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Solow model
2504: Macroeconomics (461142U017)
Aarhus Universitet
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The Solow Growth Model
2504 Macroeconomics Similan Rujiwattanapong
September 13, 2019
1 Introduction
The Solow model is a dynamic model which features production and capital accumulation. The key features are the diminishing marginal returnsfrom the accumulable factor (capital) and an exogenously given savings rate. 1 The latter feature departs from the Gottfries textbook and assumes little behaviour from the household side. The advantage of this model is that dynamics out of steady state are simplified by the exogenous savings rate and convenient to analyse.
2 The Solow Model
We make the following assumptions:
Single-good economy
Time is discrete
Perfect competition in all markets (in contrast to the models in Gottfries)
Constant and exogenous savings rate,s∈(0,1)
Capital depreciates at rateδ > 0
Constant and exogenous growth raten≥0 of labour input (Nt), i. Nt+1=Nt(1 +n)
With perfectly competitive labour market, we have thatNtdenotes both population and employment
Constant and exogenous growth rateg≥0 of technology (Et), i.e+1= Et(1 +g)
Neoclassical production functionYt=F(Kt, EtNt) where 1 As the savings rate is bounded above by unity, an endogenous and increasing savings rate cannot be the driver of sustained endogenous growth.
F(0,·) =F(·,0) = 0, i. each input is essential for production
Fj≥0 forj={K, EN}i. positive marginal products
Fjj<0 forj={K, EN}, i. decreasing marginal products
Homogenous of degree one inKandEN, i. CRS in capital and labour
limj→∞Fj = 0,limj→ 0 +Fj =∞forj={K, EN}(Inada condi- tions).
We will denote variables measured in per effective worker unit using lower case letters:
yt=
Yt EtNt
= F
( K
t EtNt
, 1
)
≡f(kt)
kt =
Kt EtNt
This will be convenient later since these variables are constant on the balanced growth path (in steady state).
2 Capital accumulation
We have previously cleared the labour market. For the goods market in aclosed economy, output equals its final use, i. Yt=Ct+Itwith savings defined as Yt−Ct=sYt. Therefore, we also have that gross investments equal savings, i.e=sYt. The law of motion for the stock of capital in general equilibrium is given by
Kt+1 = Kt+It−δKt = (1−δ)Kt+sYt = (1−δ)Kt+sF(Kt, EtNt)
We can rewrite this in terms of per effective worker units
kt+1 =
Kt+ Et+1Nt+
=
(1−δ)Kt+sF(Kt, EtNt) Et+1Nt+
=
(1−δ)Kt EtNt
EtNt Et+1Nt+
+
sF(Kt, EtNt) EtNt
EtNt Et+1Nt+
=
(1−δ)Kt EtNt
EtNt Et(1 +g)Nt(1 +n)
+sF
(
Kt EtNt
, 1
)
EtNt Et(1 +g)Nt(1 +n)
= (1−δ)kt
1
(1 +g) (1 +n)
+sF(kt,1)
1
(1 +g) (1 +n)
= (1−δ)kt
1
(1 +g) (1 +n)
+sf(kt)
1
(1 +g) (1 +n)
Figure 1:
Figure 2:
line (k[(1 +g) (1 +n)−(1−δ)]) denotes the gross investments per effective worker unit that are required for the stock of capital per effective worker unit to remain constant. 3 The curvesf(k) shows the actual gross investments per effective worker unit. We are in a steady state whenever these two lines cross,
3 Intuition: We have thatkt= Kt EtNt growth rates in technology (Et),g, and work- force (Nt),n, increases the growth rate of the denominator. For the ratioto stay constant, the numerator, i. the stock of capital, must grow faster which requires high investments. Similarly, a higher depreciation rate requires higher gross investments for the ratio to remain constant.
and the capital per effective worker unit will remain constant. One of the intersections is atk∗= 0 whenf(0) = 0. This is an unstable resting point where there is no capital to begin with (poverty trap). But it is unstable because if we suddenly have a little bit of capital we willmove away from this point. However, the curves also cross at a uniquek∗>0. This is due tof′′(k)<0, limk→ 0 +f′(k) =∞, and limk→∞f′(k) = 0. The first Inada condition makes sure that the slope ofsf(k) is greater than that ofk[(1 +g) (1 +n)−(1−δ)] for smallk(k→ 0 +) and the latter Inada condition insures that the slope of k[(1 +g) (1 +n)−(1−δ)] is greater than that ofsf(k) for sufficiently large k(k→ ∞). Given thatf(k) is continuous andf(0) = 0, the existence of a steady state with positivek∗>0 is established. Uniqueness follows from the strict concavity, i.e′′(k)<. The steady state withk∗>0 is a stable resting point because whenever capital is positive, we will in the long run converge to this level. Since a steady state withk∗=y∗= 0 is not relevant empirically we focus on the steady state withk∗>0.
The steady state has the following properties:
Stock of capital per effective worker unit is constant (k=k∗)
Output per effective worker unit and consumption per effective worker unit are constant (y∗=f(k∗) andc∗= (1−s)y∗)
The capital-labour ratio grows at rateg: Kt Nt
=ktEt=k∗Et
andEtgrows at rateg.
- GDP (output) per worker grows at rategin the steady state asNYtt = F(Kt,EtNt) Nt =Et
F(Kt,EtNt) EtNt =Etf(k
∗) =Ety∗
- GDP and the stock of capital grow at rate (1 +n) (1+g)−1 =n+g+ng≈ n+gas
Yt = EtNtf(k∗) Kt = EtNtk∗
2 Dynamics
The dynamic adjustment and the steady state equilibrium can be illustrated in Figure 3. Only the steady state withk∗>0 is relevant as it is a stable steady state. Moreover,k= 0 is not empirically relevant.
Recall that the two lines have intuitive interpretations. sf(k) is the actual gross investments per effective worker unit, andk[(1 +g) (1 +n)−(1−δ)] is
grow faster and eventually catch up with richer economies given that they all have similar structures, i. similarn, g, δ, f(k) ands. Importantly, we have shown that the economy will eventually reach a steady state and ifk 0 >0 we have that limt→∞kt=k∗>0. To further analyse convergence, note that we can approximate the growthrate around the steady state using a first-order Taylor approximation.
gk(kt) ≈ gk(k∗) + ∂gk(kt) ∂kt
∣
∣
∣
∣
kt=k∗
(kt−k∗)
= 0 +
s (1 +g) (1 +n)
(
k∗f′(k∗)−f(k∗) (k∗) 2
)
(kt−k∗)
=
s (1 +g) (1 +n)
(
k∗f′(k∗)−f(k∗) (k∗)
)(
kt−k∗ k∗
)
This can be used to approximate growth rates empirically. Turning to growth in GDP per effective worker unit we can write this as a function ofkt
gy =
yt+1−yt yt
=
f(kt+1)−f(kt) f(kt)
=
f
(
(1−δ)kt(1+g)(1+ 1 n)+sf(kt)(1+g)(1+ 1 n)
)
−f(kt) f(kt)
which around the steady state can be approximated, again, using a first-order Taylor approximation.
gy(kt) ≈ gy(k∗) +
∂gy(kt) ∂kt
∣
∣
∣
∣
kt=k∗
(kt−k∗)
= 0 +
f′(k∗) f(k∗)
(
(1−δ) +sf′(k∗) (1 +g) (1 +n)
− 1
)
(kt−k∗)
=
(
(1−δ) +sf′(k∗) (1 +g) (1 +n)
− 1
)
f′(k∗) (kt−k∗) f(k∗)
Now we use the fact that
yt = f(kt) yt ≈ f(k∗) +f′(k∗) (kt−k∗) =y∗+f′(k∗) (kt−k∗)
⇒
yt−y∗ y∗
=
f′(k∗) (kt−k∗) y∗
=
f′(k∗) (kt−k∗) f(k∗)
to obtain
gy(yt)≈
(
(1−δ) +sf′(k∗) (1 +g) (1 +n)
− 1
)
yt−y∗ y∗
where
(
(1−δ)+sf′(k∗) (1+g)(1+n) − 1
)
=
(
sf′(k∗)−[(1+g)(1+n)−(1−δ)] (1+g)(1+n)
)
<0 since the slope of
sf(k) is smaller than that ofk[(1 +g)(1 +n)−(1−δ)] when evaluated at the steady state.
This means:
A larger negative gap between GDP and steady state GDP implies a higher growth rate (conditional convergence). It is conditional in the sense that g, n, sandδwhich may differ across countries (and, therefore, different steady states) must be taken into account.
A lower GDP for a given steady state GDP implies a higher growth rate (absolute convergence). Two economies with similar structure (same g, n, sandδ) but differentk 0 andy 0 will eventually converge to the same levels ofk∗andy∗.
3 Further analyses
We now analyse what happens when there is a change to an exogenous variable.
3 Increase in the savings rate
Figure 4:
We consider the effects of a permanent increase in the savings rate (s). From Figure 4, an increase instilts up the actual gross investment curvesf(k), which increases the steady state level of capital per effective worker unit. Since the old and new steady states have the same properties, an increase in the savings rate only temporarily increases the growth rate. 4 We can show mathematically
4 Recall that
gk(kt) =
[ (1−δ) (1 +g) (1 +n) − 1
]
+s f(kt) kt
1 (1 +g) (1 +n)
Figure 5:
Figure 6:
Growth increases in the short and medium run but is unaffected in the long run (steady state)
Consumption unambiguously drops in the short run
Ambiguous effect on consumption in the long run (depends on whether the savings rate is below or above the golden rule level). But the longrun consumption (per effective worker unit) increases from the initialdrop due to the endogenous capital accumulation.
Our analysis implies that ifk 0 > kGRa proper reduction in the savings rate will imply that consumption increases in all periods and thus unambigously increases welfare.
3 Other analysis
Similarly we could analyse the effects of changes inn, gandδ. Further, we could also analyse what happens in response to an exogenous change inK, NorE. Try to do this yourself.
4 How to include technology ‘E’?
Kaldor (1963) listed 6 stylised facts about economic growth (fits data reasonably well at least for developed economies):
Per capita output grows over time and its growth rate does not tend to diminish
Physical capital per worker grows over time
The rate of return to capital is nearly constant
The ratio of physical capital to output is nearly constant
The shares of labour and physical capital in national income are nearly constant
The growth rate of output per worker differs substantially across countries
The Solow model above withYt=F(Kt, EtNt) is consistent with facts 1-5 and has nothing to say about fact 6. To see this note that in the steady state we have that
- Per capita output is
Yt Nt
=
F(Kt, EtNt) Nt =Et
F(Kt, EtNt) EtNt =Etf(kt) =Etf(k∗)
and grows at rateg >0.
References
[1] Barro, Robert J & Sala-i-Martin, Xavier (2004). Economic growth (2nd ed). MIT, Cambridge, Mass.; London.
[2] Kaldor, Nicholas (1963). “Capital Accumulation and Economic Growth.” In Friedrich A. Lutz and Douglas C. Hague, eds., Proceedings of a Conference Held by the International Economics Association. London: Macmillan.
[3] Uzawa, Hirofumi (1961). “Neutral Inventions and the Stability of Growth Equilibrium.” Review of Economic Studies, 28, February, 117–124.
Solow model
Kursus: 2504: Macroeconomics (461142U017)
Universitet: Aarhus Universitet
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