EXGA 2101 - Advance Macroeconomics

Lecture 2

The ( )n g k+ + δ line has a slope of ( )n g+ + δ

( ) ( ) cf ' k n g 0s

∗∗ ∂< + + δ ∴ <

∂

( )n g k+ + δ

( )f k

( )sf k

( )n g k+ + δ

( )f k

( )sf k

( ) ( ) cf ' k n g 0s

∗∗ ∂> + + δ ∴ >

∂

2/1

( )n g k+ + δ

( )f k

( )sf k

( ) ( )f ' k n g ∗ = + + δ

marginal in s →no effect on c ∆

consumption is at maximum along the balanced growth path.

This is the golden rule of capital stock. k∗

Impact On Output (Long Run)

( )y kf ' k s s

∗ ∗∗∂

=∂ ∂

∂ where ( )k f s,n,g,∗ = δ

to determine y s

∗∂∂

we need to first determine ks

∗∂∂

2/2

on the balanced growth path

( ) ( )k 0

sf k n g k

•

∗ ∗

=

∴ = + + δ

Derive both sides

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( )

k ksf ' k f k n gs s

k kf k n g sf ' ks s

kf k n g sf ' ks

f kks n g sf ' k

∗ ∗∗ ∗

∗ ∗∗ ∗

∗∗ ∗

∗∗

∗

∂ ∂+ = + + δ

∂ ∂∂ ∂

= + + δ −∂ ∂

∂ ⎡ ⎤= + + δ −⎣ ⎦∂

∂⇒ =

∂ ⎡ ⎤+ + δ −⎣ ⎦

Substitute this into the expression for y s

∗∂∂

above,

2/3

( )y kf ' k s s

∗ ∗∗∂ ∂

=∂ ∂

( ) ( )( ) ( )

f ky f ' ks n g sf ' k

∗∗∗

∗

∂=

∂ ⎡ ⎤+ + δ −⎣ ⎦

1. Multiply both sides with s y∗

to convert LHS to an elasticity of

income w.r.t. savings

2. Use ( ) ( )sf k n g k∗ ∗= + + δ to substitute for s

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )

( )

f ks y s f ' ksy y n g sf ' k

n g k f ' ks ysy n g k f ' k

f k n gf k

∗∗∗

∗ ∗ ∗

∗ ∗∗

∗ ∗ ∗∗

∗

∂=

∂ ⎡ ⎤+ + δ −⎣ ⎦

+ + δ∂=

∂ ⎡ ⎤+ + δ⎢ ⎥+ + δ −⎢ ⎥⎣ ⎦

cancel ( )n g+ + δ from numerator and denominator

( )( )

( )( )

( )( )

K

K

k f ' kf ks y

sy k f ' k1

f k

ks ysy 1 k

∗ ∗

∗∗

∗ ∗ ∗

∗

∗∗

∗ ∗

∂=

∂ ⎡−⎢ ⎥

⎢ ⎥⎣ ⎦

α∂=

∂ − α

⎤

where ( ) ( )( )K

k f ' kk

f k

∗ ∗∗

∗α =

elasticity of output wrt capital at k k∗= is balanced growth path.

2/4

If markets are competitive and there are no externalities, capital earns its

marginal product.

∴Amount received by capital = ( )k f ' k∗ ∗

Share of capital income out of total = ( )

( ) ( )K

k f ' kk

f k

∗ ∗∗

∗= α

In most countries ( )K1k3

∗α =

( )( )

K

K

1ks y 131s 2y 11 k 3

∗∗

∗ ∗

α∂∴ = =

∂ −− α=

∴if savings rate 10%, output per worker ↑ ↑ 5%.

2/5

Speed of Convergence

Focus on k rather than y for simplicity. Determine speed k approaches k∗

( )k k k• •= i.e. is determined by k. k

•

When , is zero. k k∗= k•

Using a first order Taylor series approximation,

( ) ( )k kk k

k k k|•

•∗

∗

⎛ ⎞⎜ ⎟∂

−⎜ ⎟∂ =⎜ ⎟⎝ ⎠

k

Recall that ( ) ( )k sf k n g k•

∗ ∗= − + + δ

( ) ( ) ( )k ksf ' k n g

k k k|•

∗∗

∂= − + +

∂ =δ

since ( )( )

n g ks

f k

∗

∗

+ + δ=

2/6

( ) ( )( ) ( ) ( )

( )( ) ( )

( )( )( )K

k k n g kf ' k n g

k f kk k

k f ' k = 1 n g

f k

= k 1 n g

|•∗

∗∗ ∗

∗ ∗

∗

∗

∂ + + δ= −

∂ =

⎡ ⎤⎢ ⎥− + + δ⎢ ⎥⎣ ⎦

α − + + δ

+ + δ

Substitute this into the equation for k•

( ) ( )( )( ) ( )( )Kk t 1 k n g k t k•

∗ ∗− − α + + δ −

In the area of the balanced growth path, capital per unit effective labor

converges towards k at a speed ∗ α distance from k∗

Using ( )( )(K1 k n g∗λ − −α + + δ) the path of k is

( ) ( )tk 0 kk t k e

−λ∗−∗−

2/7

y approaches y∗ at the same rate as k approaches k∗

( ) ( )ty 0 yy t y e

−λ∗−∗−

Calibration using examples:

If ( )n g 6%+ + δ ≈ p.a. and 13α ≈

( )( )( ) ( )( )K11 k n g 1 6 43

∗ %∴ −α + + δ = − =

k & y move 4% of the remaining distance towards k∗ and each year.

It takes 18 years to get halfway to their balanced growth path values.

Overall impact of a is savings rate is modest and occurs relatively

slowly.

y∗

∆

2/8

Key Issues In Growth Theory

Variations in output per worker is possibly due to

1. differences in K L

2. differences in A (effectiveness of labor)

Finding that only growth in A can lead to permanent growth in output per

worker because:

1. if ∆ in output attribute solely to capital, then required difference in

capital is too large. Actual difference in k is much smaller.

2. if A is ignored then this implies large variation in rate of return on

capital – No evidence of such difference in rates of return.

2/9

Chapter 2

Infinite-Horizon Model

Ramsey-Cass-Koopmans

Similar to Solow

But dynamics of macro variables determined by decisions at the

microeconomic level

Saving rate is endogenous

Firms rent capital and hire labor

Households sumpply L, hold K, consume & save

Assumptions:

Firms Y=F(K,AL)

Competitive factor markets

Competitive output markets

A is exogenous and grows at rate g

Firms maximize and owned by households π

2/10

Households:

Large numbers

Size of households grows at rate n

Each member of households supplies 1 uynit of laor at every point in

time

Initial capital holdings ( )K 0H

where K(0) initial amount of capital, H

number of households

Assume no depreciation 0∴δ =

Households divide income between C & S to maximize utility

Household utility function

( )( ) ( )tt 0

L tU e u C t

H∞ −ρ=

= ∫ dt

2/11

C(t): consumption of each member of household

u(0): instantaneous utility function gives each member’s utility at a

given date

L(t): number of household

( )( ) ( )L tu C t

H∴ is the household’s total instantaneous utility at t

ρ= discount rate

Instantaneous utility function

( )( ) ( )

( )

1C tu C t

10, n 1 0

−θ

=− θ

θ > ρ − − − θ >

Also known as the CRRA utility

2/12

Or Constant Relative Risk Aversion

θ= coefficient of relative risk aversion

θ= determines household’s willingness to shift consumption between

different periods. The smaller the θ the more willing the household is

to allow consumption to vary over time

if θ→0, large swings in C to take advantage of small difference

between discount rate and the rate of return.

The Behaviours Of Households And Firms

Firms: Employ Capital & Labor

Pay them their marginal products – sell the output

Production function has constant returns

Firms earn zero profits

( ) ( )F K,ALf ' k

K∂

=∂

= marginal product of capital

real rate of return on capital equals its earnings per unit of time

( ) ( )( )r t f ' k t=

real wage per unit effective labor equals marginal product of effective

labor ( ) ( )( ) ( ) ( )( )w t f k t k t f ' k t= −

marginal product of labor (as opposed to effective labor) is A(t)w(t)

representative a household takes the path of r and w as given

2/13

Households Maximization Problem

Budget constraint is that the present value (PV) of its lifetime

consumption cannot exceed its initial wealth plus the PV of its lifetime

labor income.

R(t) = the real interest rate at time t

R may vary over time

( ) ( )t

0R t r d

τ== τ∫ τ

household has ( )L tH

members

Labor income = ( ) ( ) ( )L tA t w t

H

Consumption expenditure = ( ) ( )L tC t

H

2/14

Household Budget Constraint

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )R t R tt 0 t 0

L t K 0 L te C t dt e A t w t d

H H Hlifetime consumption initial labor income wealth

∞ ∞− −= =

≤ +∫ ∫ t

Budget Constraint is rewritten :

( ) ( ) ( ) ( ) ( ) ( ) ( )R t n g t R t n g t

t 0 t 0e c t e dt K 0 e w t e dt

small cconsumption per unit effective labor

∞ ∞− + − += =

≤ +

↑

∫ ∫

2/15

this expression may be simplified.

If ( )K s = capital holdings at time s

Then

( ) ( )R s K slim e 0s H

− ≥→∞

Which states that the PV of household asset holding cannot be negative in

the limit.

Households objective function is reduced to

( )1tt 0

c tU B e dt

1

−θ∞ −β=

=− θ∫

where

( ) ( )

( )

1 L 0B A 0

Hand

n 1 g

discount raten = labor growth

= coefficient of relative risk aversiong = technical growth

−θ≡

β ≡ ρ − − − θ

ρ =

θ

2/16

Household Behaviour

Household must choose path of c(t) to maximize lifetime utility subject to

budget constraint combining he objective function & budget constraint,

set up the Langrange:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

R t n g t R t n g ttt 0 t 0 t 0

c tL B e dt k 0 e e w t dt e e c t dt

1 Utility Function income consumption

−θ∞ ∞ ∞− − + − − +−β= = =

⎡ ⎤= + λ + −⎣ ⎦− θ∫ ∫ ∫

2/17

Solving leads to:

( )( )

( )

( )( )

( )

c t r t n gc t

c t r tc t

•

•

− − −β=

θ

−ρ −β=

θ

This is the Euler equation for the maximization problem.

2/18