growth theory: revie€¦ · to say about savings/investment decisions → 1sgm lecture 1.2,...

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

Growth Theory: ReviewLecture 1.2, Endogenous Growth

Topics in Growth, Part 2

June 11, 2007

Lecture 1.2, Endogenous Growth 1/28 Topics in Growth, Part 2



Outline

Review: From Solow to Ramsey to Endogenous GrowthReview: Solow ModelReview: 1SGM

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




Review: Solow ModelReview: 1SGM

Review: Solow Model 3

Solow Model

◮ Exogenously given savings rate s◮ Production function

◮ Assumptions:

→ A1: Constant returns to scale (CRS)F (λK , λL) = λF (K , L)

→ A2: Marginal products positive and diminishingFK > 0, FL > 0, FKK < 0, FLL < 0

◮ From CRS (A1), we can write F in per capita termsf (k) = F (k , 1). Then (A2) becomes

FK > 0 ⇔ f ′(k) > 0FKK < 0 ⇔ f ′′(k) < 0

◮ For example, f (k) = Akα with α < 1Lecture 1.2, Endogenous Growth 3/28 Topics in Growth, Part 2




Review: Solow Model → Results 4

Solow Model

◮ Unless there is exogenous technological change(At+1 = (1 + g)At , g > 0), the economy converges to asteady state in per capita variables.

◮ No long-run growth except due to technological change.

◮ Golden Rule steady state level of consumption requiress = α

Investments result from choices: the Solow model has nothingto say about savings/investment decisions → 1SGM





Review: 1SGM 5

1SGM

◮ Households choose how much to save and how much toconsume → dynamics results from this choice.

◮ Production function: same as for the Solow model◮ Assumptions (Lecture 3):

→ A1: Constant returns to scale (CRS)→ A2: Marginal products positive and diminishing

◮ From CRS (A1), we can write F in per capita termsf (k) = F (k , 1). Then (A2) becomes

FK > 0 ⇔ f ′(k) > 0FKK < 0 ⇔ f ′′(k) < 0

◮ For example, f (k) = Akα with α < 1





Review: 1SGM → Results 6

1SGM

◮ Unless there is exogenous technological change(At+1 = (1 + g)At , g > 0), the economy converges to asteady state in per capita variables (g = 0).

◮ No long-run growth except due to technological change.◮ Modified Golden Rule steady state level of consumption

and capital are smaller than Golden Rule. This is due toimpatience of households.

◮ If government consumption is financed with taxes oncapital gains, HH save less and the steady state (c∗τk ) and(k∗τk ) are lower than with LS taxes.

How does long-run growth occur endogenously?




SetupGrowth rates are constantGrowth rates of c, k and y are the sameBalanced growth path theorem

A simple model of endogenous long-run growth 7

Ak Model

◮ Take 1SGM but change 1 assumption: α = 1

◮ Thus the production function becomes: f (k) = Ak (orF (K , L) = AK , i.e. labor does not enter into the productionfunction)

◮ That is, f ′(k) = A > 0 → used in Euler equation◮ That is, f ′′(k) = 0. There are no more diminishing marginal

returns to capital. It is constant returns to capital alone!

How does long-run growth occur endogenously?





Growth rate of consumption and the Euler equation 8

Ak Model

◮ The Euler equation becomes:

ct+1

ct= [β(f ′(kt+1) + 1 − δ)]1/σ

ct+1

ct= [β(A + 1 − δ)]1/σ = 1 + γc

◮ There is no steady state in this model

◮ But clearly, consumption grows at a constant rate—

FOREVER!!!





Growth rate of consumption and the Euler equation 9

Ak Model: Assumption 1

ct+1

ct= [β(A + 1 − δ)]1/σ = 1 + γc

◮ Assumption 1: γc > 0

This requires: β(A + 1 − δ) > 1





Growth rate of capital and the resource constraint 10

Ak Model: The growth rate of capital is constant

◮ The resource constraint can be written as:

ct + kt+1 = (A + 1 − δ)kt

Dividing both sides by kt

ct

kt+

kt+1

kt= A + 1 − δ





Growth rate of capital and the resource constraint 11

Ak Model: The growth rate of capital is constant

ct

kt+

kt+1

kt= A + 1 − δ

◮ Suppose the growth rate of k is increasing over time

→ctkt

must be decreasing over time→ violates transversality condition

◮ Suppose the growth rate of k is decreasing over time

→ctkt

must be increasing over time→ drives k and in turn c to 0, cannot be optimal

◮ Thus, kt+1kt

= 1 + γk is constant





Equal growth rates 12

Ak Model: Equal growth rates γc = γk = γy

ct

kt+ (1 + γk ) = A + 1 − δ

◮ Thus the ratio of consumption to capital is constant

◮ Thus, capital and consumption must grow at the same rate

◮ Since yt = Akt , output per capita must be growing at thesame rate as well

Thus, γ = γc = γk = γy = [β(A + 1 − δ)]1/σ − 1





Conditions for a BGP to exist 13

Ak Model: Assumption 2

ct

kt+ (1 + γk ) = A + 1 − δ

ct

kt+ [β(A + 1 − δ)]1/σ = A + 1 − δ

ct

kt= (A + 1 − δ)(1 − β

1σ (A + 1 − δ)

1σ−1)

◮ Assumption 2:

β1σ (A + 1 − δ)

1σ−1 < 1





Conditions for a BGP to exist 14

Ak Model: TheoremConsider the social planner’s problem with linear technologyf (k) = Ak and CEIS preferences. Suppose (β, σ, A, δ) satisfy

β(A + 1 − δ) > 1 > β1σ (A + 1 − δ)

1σ−1

Then the economy exhibits a balanced growth path wherecapital, output and consumption all grow at a constant rategiven by

kt+1

kt=

yt+1

yt=

ct+1

ct= 1 + γ = [β(A + 1 − δ)]1/σ

The growth rate is increasing in A and β and it is decreasing inδ and σ.




SetupSocial Planner’s problemEuler equationOptimal k/h ratio and consumption growth rate

Setup: Production function 15

Production function with human capital

Yt = F (Kt , Ht) = F (Kt , htLt)

◮ F : Neoclassical production function:

→ A1: Constant returns to scale (CRS)F (λK , λH) = λF (K , H)

→ A2: Marginal products positive and diminishingFK > 0, FH > 0, FKK < 0, FHH < 0

→ Use CRS , write F in per capita terms F (K ,H)L = F (k , h)

→ For example, f (k , h) = Akαh1−α with α < 1

◮ Ht = htLt is effective labor.





Setup: Investments and Resource constraint 16

Capital-type specific investment

ikt : investment in physical capital

iht : investment in human capital

Resource constraintTotal output can be used for consumption or investment ineither type of capital.

ct + ikt + iht = F (kt , ht)





Setup: Investments and depreciation 17

Laws of motion

kt+1 = ikt + (1 − δ)kt

ht+1 = iht + (1 − δ)ht

Resource constraint can be written as

ct + kt+1 + ht+1 = F (kt , ht) + (1 − δ)kt + (1 − δ)ht





Social Planner’s problem 18

With the usual utility function, Social planner’s problemcan be written as

max(kt+1,ct)

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + kt+1 + ht+1 = F (kt , ht) + (1 − δ)kt + (1 − δ)ht ,

for all t = 0, 1, 2, ...

k0, h0 > 0 given





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





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





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.





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




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





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





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




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

β(A + 1 − δ) > 1 > β1σ (A + 1 − δ)

1σ−1

β(A(1−α)1−ααα + 1− δ) > 1 > β1σ (A(1−α)1−ααα + 1− δ)

1σ−1

and suppose h0 = 1−αα k0.Then the economy exhibits a

balanced growth path where capital, output and consumptionall grow at a constant rate given by

kt+1

kt=

kt+1

kt=

ht+1

ht=

yt+1

yt=

ct+1

ct= 1 + γ = [β(A + 1 − δ)]1/σ




Concluding remarks 27

◮ This model exhibits long run growth, even though someform of labor is taken into account.

◮ This is because the QUALITY of labor is taken intoaccount.

◮ This quality, human capital, is accumulable.

◮ Therefore the diminishing marginal returns don’t kick in.Both factors grow simultaneously and therefore allow theeconomy to grow FOREVER.




Think about other production functions 28

1. Capital and land enter the production function,Y = F (K ,L)

→ land is not accumulable, it is fixed→ decreasing marginal products kick in → steady state

→ land is accumulable, e.g. US expansion East to West→ economy expands

2. Capital and population/labor enter the production function,

Y = F (K , L) → labor is not accumulable, it is fixed→ decreasing marginal products kick in → steady state

Y = F (K , N) → population is accumulable→ endogenous fertility models → economy expands


growth theory: revie€¦ · to say about savings/investment decisions → 1sgm lecture 1.2,...

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