endogenous growth theory - yin-chi wang's...

Endogenous Growth Theory

Yin-Chi Wang

The Chinese University of Hong Kong

October, 2012

Yin-Chi Wang (CUHK) Endogenous Growth October, 2012 1 / 40

Neoclassical Exogenous Growth Theory

Solow-Swan growth modelExogenous growth model

Discrete-time model (dynamic programming)Continuous-time: The optimal growth model (Ramsey-Cass-Koopman)

Endogenous growth model

Romer (1986 JPE) - general/knowledge capitalLucas (1988 JME) - human capitalStokey (1988 JPE) - learning-by-doingRebelo (1991 JPE) - basic one & two sector modelsGrossman-Helpman (1991 book) & Aghion-Howitt (1992Econometrica) - innovation & technical progress with imperfectcompetition


The Optimal Growth Model: Ramsey-Cass-Koopman Households

The Optimal Growth Model: Ramsey-Cass-KoopmanRepresentative Households

In�nite life, price takersContinuously and uniformly distributed in [0, 1]Population grows at rate n: L (t) = L (0) ent , normalize L (0) = 1Representative household maximizes the lifetime utility:U =

R ∞0 e

�ρtu (c (t)) L (t) dtu0 (c) > 0, u00 (c) < 0, limc!0 u0 (c) = ∞, limc!∞ u0 (c) = 0Assume CRRA (constant relative risk averse) utility

u (c) =c1�σ � 11� σ

σ = 1 : u (c (t)) = ln c (t)σ = 0 : u (c (t)) = c (t) (linear utility fn., consumptions in di¤erentperiods are perfect substitutesσ = 1 : U =R ∞0 min

nc (0) L (0) , e�ρc (1) L (1) , ..., e�ρT c (T ) L (T ) ...

o.


The Optimal Growth Model: Ramsey-Cass-Koopman Households

Representative Households

Each member provides 1 unit of labor inelastically and earn a wageincome w (t)

Own capital K (t) and enjoy capital rate of return r (t). K (0) given

Household�s budget constraint

K (t) = w (t) L (t) + r (t)K (t)� C (t)

Rewrite the budget constraint in terms of representative agent�s budgetconstraint (k � K

L , c �CL )

kk =

KK � n =

wk + r �

ck � n =)

k = w + (r � n) k � c (1)


The Optimal Growth Model: Ramsey-Cass-Koopman Firms

Representative Firms

Rent capital from households and pay rental rate r (t)

Hide labor and pay wage rate w (t)

Production function: F (K , L) satisfying

CRTSFK > 0,FKK < 0,FL > 0,FLL < 0Inada conditions: limK!0 FK = ∞, limK!∞ FK = 0,limL!0 FL = ∞, limL!∞ FL = 0

Firms pay the capital depreciation at rate δ (can have households paythe depreciation)


The Optimal Growth Model: Ramsey-Cass-Koopman Optimization

Optimization

Dynamic Programming

Calculus of variationThe methods we use nowadays were developed in 1950s

Discrete-time: Dynamic programming (Bellman)

Continuous-time: Maximum principle of optimal control (Pontriagain)

Set up the Hamiltonian function (present-value or current-value)We will use current-value Hamiltonian

References

Barro & Sala-i-Martin Appendix A.3 (present value)More details: Intrilligator (1971) ch13



Optimization �Representative Agent

Rewrite the representative household�s lifetime utility as

U =Z ∞

0e�ρtu (c (t)) L (t) dt

=Z ∞

0e�ρtu (c (t)) entdt

=Z ∞

0e�(ρ�n)tu (c (t)) dt

Step 1: Set up the current-value Hamiltonian (λ is the costate variable)

H (c , k,λ, t) = u (c (t)) + λk

=c1�σ � 11� σ

+ λk



Optimization �Representative Agent

Step 2: Take �rst-order condition(s) wrt control variable(s)

c�σ � λ = 0 (2)

Step 3: Obtain the Euler equation(s) wrt state variable(s)

λ = (ρ� n) λ� ∂H∂k

(3)

= (ρ� n) λ� (r � n) λ = (ρ� r) λ

Step 4: Transversality condition (TVC)

limt!∞

λ (t) k (t) e�(ρ�n)t = 0 (4)



Optimization �Representative Firm

maxK ,L

Π = F (K , L)� (r + δ)K � wL

=)maxk

π = f (k)� (r + δ) k � w (5)

FOC:f 0 (k)| {z }MPK

= r + δ| {z }MC

(6)

Zero pro�t condition: �rms are price takers (competitive market). From(5)

FL = f (k)� kf 0 (k) = w (7)


The Optimal Growth Model: Ramsey-Cass-Koopman Equilibrium

Equilibrium

Endogenous variables: k, c , w , r , λ

Equations we have: (1), (2), (3), (6), (7)Try to simplify the system into 2 equations and 2 unknowns (k, c)

From (2), take log and di¤erentiate wrt time:

�σ ln c = lnλ

=)cc=�1σ

λ

λ=1σ(r � ρ)

Substitute (6) into the above equation

cc=1σ

�f 0 (k)� δ� ρ

�(8)

)c =

cσ

�f 0 (k)� δ� ρ

�(9)



Equilibrium

Now substituting (7) and (6) into the budget constraint yields

k = w + (r � n) k � c (10)

= f (k)� kf 0 (k) +�f 0 (k)� δ� n

�k � c

= f (k)� (n+ δ) k � c

Equations (10) and (9) govern the system.

Use (10) and (9) to solve for (k�, c�)Once (k�, c�) is solved, use (7) and (6) to solve for w� and r�

Then use (2) to solve for λ�

Di¤erential

equations�

Particular solution: steady stateHomogeneous solution: transitional dynamics



Dynamic Equilibrium Path

Golden rule is dynamically ine¢ cient (over-accumulation)



Social Planner

maxH (c , k,λ, t) = u (c (t)) + λk

=c1�σ � 11� σ

+ λk

wherek = f (k)� (n+ δ) k � c

Other steps are similar to those in the representative agent�s problem.From FOC and the Euler equation, we immediately obtain

cc=

f 0 (k)� δ� ρ

σkk

=f (k)k

� (n+ δ)� ck

with the TVClimt!∞

λ (t) k (t) e�(ρ�n)t = 0



Problems of the exogenous growth models

1 Long-run steady state2 Growth-accounting - exogenous technical progress accounts for 70%of output growth

3 Lack of strong evidence in global convergence4 Failure to explain widened growth disparities5 Long-run imbalancedness between capital and labor6 Policy does not matter unless it can a¤ect the rate of technologicaladvancement

==> Birth of the endogenous growth theory


Endogenous Growth Theory

Development of Endogenous Growth Theory

Is the endogenous growth theory new?

Seminal work before 1980s:

von Neumann (1937, translated 1945/46 RES) - linear production& balanced growthSolow (1956, at the end of his seminal paper) - IRS & sustainedgrowthPitchford (1960 ER) - DRS with sustained growthShell (1966 AERP&P) - inventive activity & growthWan (1970 REStud) - learning by doing & growth

Creators of the new waves:

Romer (1986 JPE) - general/knowledge capitalLucas (1988 JME) - human capitalStokey (1988 JPE) - learning-by-doingRebelo (1991 JPE) - basic one & two sector modelsGrossman-Helpman (1991 book) & Aghion-Howitt (1992Econometrica) - innovation & technical progress with imperfectcompetition


Endogenous Growth Theory One-Sector Endogenous Growth Model

One-Sector Endogenous Growth Model

Key

Marginal products of reproducible factors are bounded below by aconstant, thus requiring the production function be CRS or IRS inreproducible factors

Trick:

In perfectly competitive equilibrium, the model must be consistent withzero pro�t conditions, thus requiring CRS in privately provided factors



AK Model

The setting on the household side is same as in the RCK model(assume n = 0 and K0 given)

Representative �rm�s production function

Y = ALk =) y = Ak

The marginal product of capital per capita is constant (no diminishingreturn, key assumption)



Optimization

Constrained social planner: take externalities exogenously whenoptimizingHamiltonian

maxH (c , k,λ, t) = c1�σ � 11� σ

+ λk

wherek = Ak � δk � c

FOC and the Euler equation:

c�σ = λ

λ = ρλ� λ [A� δ]

The TVC islimt!∞

λ (t) k (t) e�ρt = 0

By taking log to c�σ = λ and substitute λλ = ρ+ δ� A, the

endogenous growth rate is obtained:

θ =cc=A� (ρ+ δ)

σ



Main feature

1 Common growth for c , k and y2 Growth rate is increasing in A and decreasing in ρ, δ and σ

ρ decreases, growth rate increases (di¤erent from Solow-Swan andRCK model)

3 Non-congergence4 Any policy a¤ecting the level of A has a growth e¤ect5 No transitional dynamics6 CE,PO



One-Sector Endogenous Growth Model: Romer (1986)

Romer observed that there exist external economies of scale,spillovers/externalities in industries

Caballero & Lyons (1990, 1992)Burside (1996), Basu & Fernald (1997)

The setting on the household side is same as in the RCK model(assume n = 0 and K0 given)



One-Sector Endogenous Growth Model: Romer (1986)

Representative �rm�s production function

Y = F (K , K , L)

K and L are inputs by the �rmK is the "social capital" in the economy (uncompensated spillovers),taken as given by �rms8<: F1 > 0 > F11, limK!0 F1 = ∞, limK!∞ F1 = 0,

F3 > 0 > F33, limL!0 F3 = ∞, limL!∞ F3 = 0,F2 > 0 > F22.

F is CRTS in K and L but IRS in K , K and LRewrite in terms of per capita production

y = F�KL, K , 1

�= f (k, K )



Optimization �Constrained Social Planner

Constrained social planner: take externalities exogenously whenoptimizing

Hamiltonian

maxH (c , k,λ, t) = c1�σ � 11� σ

+ λk

wherek = f (k, K )� δk � c

FOC and the Euler equation:

c�σ = λ

λ = ρλ� λ [f1 (k, K )� δ]

The TVC islimt!∞

λ (t) k (t) e�ρt = 0



Optimization �Constrained Social Planner (cont.)

The growth rate of c can be solved as

cc=�1σ

λ

λ=1σ[f1 (k, K )� δ� ρ]

Now look at f (k, K )Assume that f (k, K ) = Ak1�αK α

K = Lk in equilibriumf1 (k, K ) = A (1� α) k�αK α= A (1� α) k�α (Lk)α= A (1� α) Lα

Substitute the above equation into cc to obtain

θ = cc =

1σ [A (1� α) Lα � δ� ρ]

cc is greater than 0 if A (1� α) Lα > δ+ ρ

From capital law of motion

kk=f (k, K )k

� δ� ck= ALα � δ� c

k

From y = f (k, K ) = ALαk, yy =kk



Main features

Growth rate θ is increasing in A and L, and decreasing in α, δ, σ and ρ

ρ decreases, growth rate increases (di¤erent from Solow-Swan andRCK model)1σ increases (intertemporal elasticity of substitution), growth rateincreases

Why? People are more willing to substitute consumption across time(i.e. more willing to save now)

Any policy a¤ecting A has growth e¤ectScale e¤ect: L increases, growth rate increases

Under doubt:�

Kremer (1993): Yes!Cross-sectional data: No (ex. India)

To avoid this, assume f (k , k) where k = k in equilibrium

Non-convergence

No transitional dynamics

CE is not the same as POYin-Chi Wang (CUHK) Endogenous Growth October, 2012 24 / 40


CE vs PO

If we substitute K = K into the production function before di¤erentiation,we have y = ALαk

Growth rate: θ� = cc =

1σ [AL

α � δ� ρ] > θ

Under investment in equilibrium due to the free-rider problem

Can we remedy this ine¢ ciency using a Pigovian policy?

See Barro and Sala-i-Matin (1992)

Production subsidy: α1�α (r + δ) k (τ� = α

1+α )

Trick: let θ (τ) = θ� and solve for τ

Factor price subsidy: fk1�β � δ (optimal price of investment

q� = 1� α)



IRS Model with Uncompensated Positive Spillover: Romer(1986)

Setup: y = f (k) = Ak1�αK α+γ, γ > 0

Aggregate capital K = k in equilibrium

Equilibrium paths:

No longer common growth: need to distinguish di¤erent rates of growth

θc =cc=

�1� α

σ

�Akγ � ρ

σ

θk =kk= Akγ � c

k

k is growing over time. At that time Romer (1986) cannot solve theproblemSolved by Xie (1991): use transformation: z = k

cLater on we will see the same technique is being used in morecomplicated growth models



Barro (1990): Public Capital

Setup: y = f (k) = Ak1�αG β (public capital G = τy in equilibrium)

Endogenous growth rate:

θ =1σ

h(1� α)A

11�α τ

α1�α (1� τ)� (ρ+ δ)

iwhich is maximized at τ� = β

When τ increase, G increases and θ increases (growth enhancingcapital)

τ should be taxed at the output elasticity of government public capital



Asymptotic AK Model

Jones and Manuelli (1992)

Production function (per capita)

y = Ak + Bkα, 0 < α < 1

It can be solved that

cc=1σ

hA+ Bαkα�1 � (ρ+ δ)

iDerivations and Graph


Endogenous Growth Theory Two-sector Endogenous Growth Model

Physical and Human Capital: Lucas (JME 1988)

Two models:1 One-sector model with physical and human capital, evaluate theneoclassical models with data

2 Two goods system: human capital and growth

Model 1 Setup (CRS)

The representative household

U =Z ∞

0e�ρt 1

1� σ

hc (t)1�σ � 1

iN (t) dt

s.t. N (t)C (t) + K (t) = A (t)K (t)β N (t)1�β

Growth rate of A (t) : µSolve for growth rates, saving rate...etc.



Physical and Human Capital: Lucas (JME 1988)

Model Two Setup (IRS)

Same assumption on household preferencesE¤ective labor force: Ne =

R ∞0 u (h)N (h) hdh, u (h) : working time,

N (h) : workers with skill level h, N =R ∞0 N (h) dh

External e¤ect of human capital (average skill level):

ha =R ∞0 u(h)N (h)hdhR ∞

0 N (h)dhAssume all workers are identical: Ne = uhN, ha = hOutput: F (K ,Ne ) = AK (t)β [u (t) h (t)N (t)]1�β ha (t)

γ , A isconstant and γ > 0 (externality, uncompensated spillover from peers)Resource constraint:N (t)C (t) + K (t) = AK (t)β [u (t) h (t)N (t)]1�β ha (t)

γ



Physical and Human Capital with IRS: Lucas (JME 1988)

Growth of human capital:

h (t) = h (t)ζ G (1� u (t)) , with G 0 > 0,G (0) = 0

Easy to see that h(t)h(t) = h (t)ζ�1 G (1� u (t))

Assume linear technology ζ = 1 (no dimisnishing return)[Uzawa-Rosenformulaton]

h (t)h (t)

= h (t) δ (1� u (t))

Hence the maximum h growth rate is δTVCs: limt!∞ e�ρt θ1 (t)K (t) = 0, limt!∞ e�ρt θ2 (t)H (t) = 0



Optimization

H (K , h, θ1, θ2, c , u, t) =N

1� σ

�c1�σ � 1

�+θ1

hAK β [uhN ]1�β haγ �NC

i+θ2 [δ (1� u)]

Compare the solution of the optimal path and the equilibrium path.

CE is not equal to PO (externality)

Similar to Romer�s case, people under-invest in CE due to thefree-rider problem; one can remedy the ine¢ ciency by an educationsubsidy



Transitional dynamics

Lucas believes that k and h should move together along the saddlepath toward the BGP, which has been shown incorrect by Xie (1994)and Benhabib-Perli (1994).

There may be dynamic indeterminacy in the sense that there is acontinuum of transition paths converging to the unique BGP,depending crucially on how strong the positive externality andintertemporal substitution are.

The dynamical system in (u, z1, z2) (z1 = ck , z2 =

yk ) is complex �3 x

3 without a brock recursive structure (Routh Theorem)



Learning-by-doing and Growth: Lucas (Econometrica 1993)

Liberty ship (same blue print): Searle (1945) and Rapping (1965)identify 12-24% and 11-29%Setup

Final good output: y = F (n, z) = Anzξ

Experience accumulation: z = G (n, z) = nzξ

Key:

experiences grow over timemore employment is better for both production and accumulatingexperiences

Results:

z (t) = [z1�ξ0 + (1� ξ)

R t0 n (τ) dτ]

11�ξ

Let n = n. Then y (t) = An[z1�ξ0 + (1� ξ)

R t0 n (τ) dτ]

ξ1�ξ

Rate of productivity: µ (t) � d ln(z ξ)dt = ξnzξ�1 ! ξ/ [(1� ξ) t] if

z0 ! 0



Main Findings

Main Findings

1 Presence of scale e¤ect: dµd n > 0

2 Based on Rapping (1965), ξ = 0.2, so dµdt = 0.25

3 Rapid decay in learning means that making a miracle requires continualemergence of new innovation for new products



Problems

Problems:

1 Scale e¤ect not empirically supported =>

Young (1991): bounded learningYoung (1998): removal of the scale e¤ect

2 Continual emergence of new products =>

1 Product ladder models:2 Romer (1990): intermediate goods broadening3 Aghion-Howitt (1992): ex ante perfect competition & ex postmonopoly in R&D

4 Grossman-Helpman (1991): monopolistic competition with zeropro�t ex post

5 Laing-Palivos-Wang (2002): continual development of newproduct in the presence of search frictions with zero pro�t ex ante

6 Endogenous basket models:7 Stokey (1988, 1995): LBD and new goods



Bond-Wang-Yip (JET 1996): General Two-Sector Model

Questions asked:

Why should we treat H as the main driving force?

Why should we believe in monotone transition?

Why should we focus on the primal instead of the dual?

Keys to the model

General technologies

General functional forms with CRS

GE trade theory

Intertemporal no-arbitrage

Polarization Theorem



Model Setup

Notation:

C ,K ,H,X ,Y - levels

c = C/H, k = K/H, x = X/H, y = Y /H

kx = (sK )/(uH), ky = [(1� s)K ]/[(1� u)H ]

p = px/py = λ/µ (goods are numeraire)

Representative household:

U =Z ∞

0e�ρt c (t)

1�σ

1� σdt

s.t. K = xH � δK � C , x = uf (kx ) , K (0) = K0 > 0 given

H = yH � ηH, y = (1� u) g (ky ) , H (0) = H0 > 0 given



Optimization and Equilibrium

Stolper-Samuelson Theorem

Rybczynski Theorem

Idea: By using intertemporal no-arbitrage condition and the property thaton the BGP, relative prices are constant. Reduce the system into p, c , andk. Then the authors are able to derive a block recursive structure in(p, c , k) which simpli�es the system greatly. Then the authors show thatno matter the ranking in the factor intensity is, the system always has aunique equilibrium and the equilibrium is saddle-path stable.



Key features

Polarization between p and (c , k) ==> saddle-path stability

Distortionary taxes such that price and physical factor intensity measuresare reversed ==> instability or dynamic indeterminacy

Extensions:

Factor taxation

Dynamic Heckscher-Ohlin

Dynamic sector shifts & economic transition

Public vs. private sector

Formal vs. informal sector


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