ECO 402 Fall 2013 Prof. Erdinç Economic Growth The Solow Model.

ECO 402

Fall 2013

Prof. Erdinç

Economic Growth

The Solow Model

The Neoclassical Growth modelSolow (1956) and Swan (1956)

• Simple dynamic general equilibrium model of growth

Output produced using aggregate production function Y = F (K , L ), satisfying:

A1. positive, but diminishing returns

FK >0, FKK<0 and FL>0, FLL<0

A2. constant returns to scale (CRS)

0 allfor ),,(),( LKFLKF

Neoclassical Production Function

Production Function in Intensive Form

• Under CRS, can write production function

)1,(.),(LK

FLYLKFY

• Alternatively, can write in intensive form:

y = f ( k )

- where per capita y = Y/L and k = K/L

Exercise: Given that Y=L f(k), show:

FK = f’(k) and FKK= f’’(k)/L .

Competitive Economy

• Representative firm maximises profits and take price as given (perfect competition)

• Inputs paid by their marginal products:

r = FK and w = FL

– inputs (factor payments) exhaust all output:

wL + rK = Y

– general property of CRS functions (Euler’s THM)

A3: The Production Function F(K,L) satisfies the Inada Conditions

0),(lim and ),(lim 0 LKFLKF KKKK

0),(lim and ),(lim 0 LKFLKF LLLL

Note: As f’(k)=FK have that

0)('lim and )('lim 0 kfkf kk

Production Functions satisfying A1, A2 and A3 often called Neo-Classical Production Functions

Technological Progress

= change in the production function Ft

),( LKFY tt

),(),(.1 LKFBLKF tt Hicks-Neutral T.P.

))(,(),(.2 LAKFLKF tt Labour augmenting (Harrod-Neutral) T.P.

)),((),(.3 LKCFLKF tt Capital augmenting (Solow-Neutral) T.P.

A4: Technical progress is labour augmenting

gtt

tt

eAA

LAKFLKF

0

and

))(,(),(

Note: For Cobb-Douglas case three forms of technical progress equivalent:

ttt

tttt

DAB

LKDLAKLKBLKF

)1(

)1()1()1(

when

)()(),(

Under CRS, can rewrite production function in intensive form in terms of effective labour units

)(kfy

-note: drop time subscript to for notational ease

- Exercise: Show that

KKK FALkfFkf )('' and )('

AL

Kk

AL

Y and y where

A5: Labour force grows at a constant rate n

ntt eLL 0

A6: Dynamics of capital stock:

KIKdt

dK

net investment = gross investment - depreciation

– capital depreciates at constant rate

Model Dynamics

• National Income Identity

Y = C + I + G + NX• Assume no government (G = 0) and closed

economy (NX = 0)• Simplifying assumption: households save constant

fraction of income with savings rate 0 s 1 I = S = sY

• Substitute in equation of motion of capital:

… closing the model

KALKsFKsYK ),(

Fundamental Equation of Solow-Swan model

kgnsykdt

dk)(

)()(

d

lnd

d

lnd

d

lnd

d

lnd

lnlnlnln :

gnk

sygn

K

sYng

K

K

k

k

t

L

t

A

t

K

t

k

LAKkAL

Kk

Proof

Steady State

0y 0

ck

0)( ** kgnksf

Definition: Variables of interest grow at constant rate (balanced growth path or BGP)

• at steady state:

ky syiy ,,

k

sksy

kgni evenbreak

** kgnsk

*k

Solow Diagram: Steady State

ss

Existence of Steady State

• From previous diagram, existence of a (non-zero) steady state can only be guaranteed for all values of n,g and if

0)('lim and )('lim 0 kfkf kk

- satisfied from Inada Conditions (A3).

Transitional Dynamics

• If , then savings/investment exceeds “depreciation”, thus

• If , then savings/investment lower than “depreciation”, thus

• By continuity, concavity, and given that f(k) satisfies the INADA conditions, there must exists an unique

*kk

00

k

kgk k

*kk

*** )()( kgnksfthatsuchk

00

k

kgk k

Properties of Steady State1. In steady state, per capita variables grow at the rate g, and aggregate variables grow at rate (g + n) Proof:

StateSteady in 0

loglogloglog

loglogloglog

as

dt

Ld

dt

Ad

dt

Kd

dt

kdg

gngdt

kd

dt

Ld

dt

Ad

dt

Kdg

AL

Kk

k

kK

2. Changes in s, n, or will affect the levels of y* and k*, but not the growth rates of these variables.

Prediction: In Steady State, GDP per worker will be higher in countries where the rate of investment is high and where the population growth rate is low - but neither factor should explain differences in the growth rate of GDP per worker.

- Specifically, y* and k* will increase as s increases, and decrease as either n or increase

Policies to Promote Growth

1. Are we saving enough? Too much or too little?

2. What policies may change the savings rate?

3. How should we allocate savings between physical and human capital?

4. What policies could generate faster technological progress?

Golden Rule• Definition: (Golden Rule) It is the saving rate

that maximises consumption in the steady-state.• We can use the rule to evaluate if we are saving

too much, too little or about right.

• Given we can use

to find .

)()('0)()(

)()()()1(max

***

*

**

****

gnkfs

kgn

s

k

k

kf

s

c

kgnkfkfsc

GR

s

,*GRk ** )()( GRGR kgnksf

GRs

Golden Rule and Dynamic Inefficiency

)()(' * gnkf GR

• If our savings rate is given by then our savings rate is optimal and

• If then we must be under-saving• If then we must be over-saving

• Check why this is the case!

GRs

)()(' * gnkf GR

)()(' * gnkf GR

Is Golden Rule attained in the US? Is it Dynamically Efficient?

• Let us check: Three Facts about the US Economya) The capital stock is about 2.5 times the GDPb) About 10% of GDP is used to replace depreciating capitalc)

OR

Capital income is 30% of GDP: Note alpha also measures the elasticity of output with respect to capital!

yk 5.2

yk 1.0

ykkf 3.0).(' 3.0)(

).(. '

kf

kkf

y

kMPk

Is Golden Rule attained in the US? Is it Dynamically Efficient?

Since

US real GDP grows on average at 3% per year, i.e.Hence, US economy is under-saving because

04.05.2

1.0

y

y

k

k

12.05.2

3.0. k

k MPy

y

k

kMP

03.0 gn

)( gnMPk

Changes in the savings rate

• Suppose that initially the economy is in the steady state:

• If s increases, then

• Capital stock per efficiency unit of labour grows until it reaches a new steady-state

• Along the transition growth in output per capita is higher than g.

*1

*1 )()( kgnksf

0)()( *1

*1

kkgnksf