Review: From Solow to Ramsey to Endogenous Growth Ak model: A simple model of endogenous long-run growth: Akh model: endogenous growth with human capital Akh and Ak equivalence Growth Theory: Review Lecture 1, Endogenous Growth Economic Policy in Development 2, Part 2 March 2009 Lecture 1, Endogenous Growth 1/28 Economic Policy in Development 2, Part 2
28
Embed
Growth Theory: Review - Economics | University of … From Solow to Ramsey to Endogenous Growth Ak model: A simple model of endogenous long-run growth: Akh model: endogenous growth
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Growth Theory: ReviewLecture 1, Endogenous Growth
Economic Policy in Development 2, Part 2
March 2009
Lecture 1, Endogenous Growth 1/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Outline
Review: From Solow to Ramsey to Endogenous GrowthReview: Solow ModelReview: Ramsey Model
Ak model: A simple model of endogenous long-run growth:SetupGrowth rates are constantGrowth rates of c, k and y are the sameBalanced growth path theorem
Akh model: endogenous growth with human capitalSetupSocial Planner’s problemEuler equationOptimal k/h ratio and consumption growth rate
Akh and Ak equivalence
Lecture 1, Endogenous Growth 2/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Review: Solow ModelReview: Ramsey Model
Review: Solow Model 3
Solow Model
◮ Exogenously given savings rate s◮ Production function
Lecture 1, Endogenous Growth 18/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
SetupSocial Planner’s problemEuler equationOptimal k/h ratio and consumption growth rate
Social Planner’s problem 19
Assume u(c) = c1−σ
1−σ
Assume F (k , h) = Akαh1−α
With functional forms, the planner’s problem becomes
max(kt+1,ct)
∞
t=0
∞∑
t=0
βt c1−σt
1 − σ
s.t.
ct + kt+1 + ht+1 = Akαt h1−α
t + (1 − δ)kt + (1 − δ)ht , for all t
k0, h0 > 0 given
Lecture 1, Endogenous Growth 19/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
SetupSocial Planner’s problemEuler equationOptimal k/h ratio and consumption growth rate
Euler equations 20
Since there are 2 types of capital, we have 2 Euler eqnsEuler equation for physical capital
ct+1
ct= [β(1 + Fk (kt+1, ht+1) − δ)]1/σ
ct+1
ct= [β(1 + Aα
(
ht+1
kt+1
)1−α
− δ)]1/σ
Euler equation for human capital
ct+1
ct= [β(1 + Fh(kt+1, ht+1) − δ)]1/σ
ct+1
ct= [β(1 + A(1 − α)
(
kt+1
ht+1
)α
− δ)]1/σ
Lecture 1, Endogenous Growth 20/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
SetupSocial Planner’s problemEuler equationOptimal k/h ratio and consumption growth rate
Rate of return condition and k/h ratio 21
Combining Euler equations, we get
Fh(kt+1, ht+1) = Fk (kt+1, ht+1)
or,
Aα
(
ht+1
kt+1
)1−α
= A(1 − α)
(
kt+1
ht+1
)α
Hence, the optimal ratio of physical to human capital is given by
kt+1
ht+1=
α
1 − α
Notice that it is constant. Hence, they grow at the same rate.
Lecture 1, Endogenous Growth 21/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
SetupSocial Planner’s problemEuler equationOptimal k/h ratio and consumption growth rate
Growth rate of consumption 22
Using the k/h ratio in either one Euler equation, we find
ct+1
ct= [β(1 + A(1 − α)
(
kt+1
ht+1
)α
− δ)]1/σ
ct+1
ct= [β(1 + A(1 − α)
(
α
1 − α
)α
− δ)]1/σ
ct+1
ct= [β(1 + A(1 − α)1−ααα
− δ)]1/σ
Lecture 1, Endogenous Growth 22/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Existence of BGP and Growth rates 23
Using the k/h ratio in laws of motion
We found kt+1ht+1
= α1−α
Therefore,
ht+1 =1 − α
αkt+1 =
1 − α
α(ikt + (1 − δ)kt )
Also,
ht+1 = iht + (1 − δ)ht = iht + (1 − δ)1 − α
αkt
Hence,
iht =1 − α
αikt
Lecture 1, Endogenous Growth 23/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Existence of BGP and Growth rates 24
By redefining the variables as follows, this model isequivalent to an Ak modelThis is very useful since we know the conditions on parametersfor the Ak model so that a BGP exists and we know theproperties in terms of growth rates.
Let it = 1α ikt , A = A(1 − α)1−ααα and kt = 1
αkt
Then we can rewrite the resource constraint and laws of motionas follows*
ct + it = Akt
kt+1 = it + (1 − δ)kt
Lecture 1, Endogenous Growth 24/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Existence of BGP and Growth rates 25
The social planner’s problem becomes
max(kt+1,ct)
∞
t=0
∞∑
t=0
βt c1−σt
1 − σ
s.t.
ct + kt+1 = Akt + (1 − δ)kt , for all t
k0 > 0 given
This is just an Ak model. Using the conditions on parametersfrom the Theorem... we find
Lecture 1, Endogenous Growth 25/28 Economic Policy in Development 2, Part 2
Review: From Solow to Ramsey to Endogenous GrowthAk model: A simple model of endogenous long-run growth:
Akh model: endogenous growth with human capitalAkh and Ak equivalence
Balanced growth path theorem for Akh model 26
Akh Model: TheoremConsider the social planner’s problem with linear technologyf (k) = Ak and CEIS preferences. Suppose (β, σ, A, δ, α) satisfy