IV.The Solow Growth Model(continued)
(III)Shocks and Policies
The Solow model can be interpreted also as a primitive Real Business Cycle(RBC) model.We can u the model to predict the respon of the economy to productivity or taste shocks,or to shocks in government policies.
Productivity(or Taste)Shocks
Suppo output is given by
Y t=A t F(K t,L t)
or in intensive form
y t=A t f(k t)
where A t denotes total factor productivity.
Consider a permanent negative shock in productivity.The G(k)andγ(k)functions shift down,as illustrated
in Figure4.The new steady state is lower.The economy transits slowly from the old steady state to the new.
If instead the shock is transitory,the shift in G(k)andγ(k)is also temporary.Initially, capital and output fall towards the ow steady state.But when productivity reverts to the initial level,capital and output start to grow back towards the old high steady state.
The effect of a productivity shock on k t and y t is illustrated in Figure5.The solid lines correspond to a transitory shock,the dashed lines correspond to a permanent shock.
Unproductive Government Spending
Let us now introduce a government in the competitive market economy.The government spends resources without contributing to production or capital accumulation.
The resource constraint of the economy now becomes
c t+i t+g t=y t=f(k t),
where g t denotes government consumption.It follows that the dynamics of capital are given by
k t+1−k t=f(k t)−(δ+n)k t−c t−g t.
Government spending isfinanced with proportional income taxation,at rateτ≥0.The government thus absorbs a fractionτof aggregate output:
圣诞歌曲g t=τy t.
Disposable income for the reprentative houhold is(1−τ)y t.We continue to assume that consumption and ivestment absorb fractions1−s and s of disposable income:
c t=(1−s)(y t−g t)
i t=s(y t−g t).
Combining the above,we conclude that the dynamics of capital are now given by
γt=k t+1−k t
k t
=s(1−τ)φ(k t)−(δ+n).
whereφ(k)≡f(k)/k.Given s and k t,the growth rateγt decreas withτ.
A steady state exists for anyτ∈[0,1)and is given by
k∗=φ−1
δ+n
s(1−τ)
.
Given s,k∗decreas withτ.
Productive Government Spending
汉译英在线Suppo now that the production is given by
y t=f(k t,g t)=kα
t
gβt,
whereα>0,β>0,andα+β<1.Government spending can thus be interpreted as infrastructure or other productive rvices.The resources constraint is
c t+i t+g t=y t=f(k t,g t).
We assume again that government spending isfinanced with proportional income taxa-tion at rateτ,and that private consumption and investment are fractions1−s and s of disposable houhold income:
七下英语作业本答案g t=τy t
c t=(1−s)(y t−g t)
i t=s(y t−g t).学考查询成绩入口2020
Substituting g t=τy t into y t=kα
t
gβt and solving for y t,we infer
y t=k
α
1−β
t
τβ1−β≡k a
t
会计英语
τb
where a≡α/(1−β)and b≡β/(1−β).
We conclude that the growth rate is given by
γt=k t+1−k t
k t
=s(1−τ)τb k a−1
t
−(δ+n).
The steady state is
k∗=
s(1−τ)τb
δ+n
1/(1−α)
.
Consider the rateτthat maximizes either k∗,orγt for any given k t.This is given by
d
dτ
[(1−τ)τb]=0⇔bτb−1−(1+b)τb=0⇔τ=b/(1+b)=β.
That is,the growth-maximizationτequals the elasticity of production with respect to
government rvices.The more productive government rvices are,the higher their
“optimal”provision.
(IV)Continuous Time and Convergence Rate
The Solow Model in Continuous Time
Recall that the basic growth equation in the discrete-time Solow model is
k t+1−k t
k t
=γ(k t)≡sφ(k t)−(δ+n).
We would expect a similar condition to hold under continuous time.We verify this below.
The resource constraint of the economy is
C+I=Y=F(K,L).
In per-capita terms,
c+i=y=f(k).
Population growth is now given by
˙L别再犹豫国语版全集41
L
=n
and the law of motion for aggregate capital is
˙K=I−δK.
Let k≡K/L.Then,
˙k k =
˙K
K
−
˙L
L
.
Substituting from the above,we infer
˙k=i−(δ+n)k.
qk
Combining this with
i=sy=sf(k),
we conclude
˙k=sf(k)−(δ+n)k.
Equivalently,the growth rate of the economy is given by
˙k
k
unesco=γ(k)≡sφ(k)−(δ+n).(1) The functionγ(k)thus gives the growth rate of the economy in the Solow model,whether time is discrete or continuous.
Log-linearization and the Convergence Rate
Define z≡ln k−ln k∗.We can rewrite the growth equation(1)as
˙z=Γ(z),
雅思写作评分标准where
Γ(z)≡γ(k∗e z)≡sφ(k∗e z)−(δ+n).
Note thatΓ(z)is defined for all z∈R.By definition of k∗,Γ(0)=sφ(k∗)−(δ+n)=0. Similarly,Γ(z)>0for all z<0,Γ(z)<0for all z>0.Finally,Γ (z)=sφ (k∗e z)k∗e z< 0for all z∈R.
We next(log)linearize˙z=Γ(z)around z=0(first-order Taylor-ries approximation ofΓ(z)around z=0):
˙z=Γ(0)+Γ (0)·z
or equivalently
˙z=λz
where we substitutedΓ(0)=0and letλ≡Γ (0).
Straightforward algebra gives
Γ (z)=sφ (k∗e z)k∗e z<0
φ (k)=f (k)k−f(k)
k2
=−
1−
f (k)k
f(k)
f(k)
k2
sf(k∗)=(δ+n)k∗
We infer
Γ (0)=−(1−εK)(δ+n)<0
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