Transcript Dias nummer 1 - Syddansk Universitet

The Overlapping
Generations Model
(Romer chapter 2, Part B)
By Ole Hagen Jørgensen,
[email protected]
4/10 2006
Introduction
I will teach three lectures:
1. One lecture: on the basic OLG model in Romer (2001), chapter 2, part B
2. One lecture: on a recently developed solution method for the OLG model:
3. One lecture: on the Real Business Cycle literature (RBC)


My own research actually applies RBC-techniques for solving OLG models!
There will be a presentation in PowerPoint on each subject - Therefore, you
will receive 3 handouts (or download from Blackboard or www.cebr.dk/oj)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Intergenerational issues
Motivation for life-cycle approach to economic dynamics
1.
Life-cycle aspects of human behavior are important to study…

We model explicitly the different periods of life
2.
Distribution of welfare over generations

How the choices of one generation can affect the succeeding generation

How different exogenous shocks to the economy affects different
generations (demographic shocks, productivity shocks)
3.
Intergenerational transfers

Purpose: If the market equilibrium allocates consumption unevenly
across generations there may be a scope for redistribution.

How:
Taxes and benefits

Case:
Pensions, education, health
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
The model
The OLG model will be presented according to the following outline:
1.
Description of the economy
A.
Basic assumptions
B.
Demographics
C.
Household utility
D.
Life-cycle consumption
E.
Firms
F.
Resources
2.
Dynamics of the economy
A.
Household utility maximization
B.
Capital accumulation and Steady State
3.
4.
Case study
Efficiency and welfare
A.
Dynamic efficiency
B.
Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Basic assumptions
Overall assumptions about the economy:
 Time is discrete
 There is one good, to be consumed or saved/invested
 The economy “lives” on forever (no last generation)
 Individuals have finite lifetime (finite horizon)
 Infinite number of agents
 Closed economy
 Perfect competition
 Absence of externalities
 No government sector (could be included easily)
 No uncertainty (perfect foresight)
Of course the economy has more detailed characteristics – we turn to
those when relevant…
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Demographics


The life-cycle of generations
The lifetime is divided into two periods: young and old
 When individuals are young they work
 When individuals are old they are retired
time period
generation
0
-1
old (C2t )
0
young (C1t )
1
1
2
old (C2t1 )
young (C1t1 )
2
old (C2t2 )
young(C1t2 )
We keep track of generation 0 denoted with t...



One period therefore amounts to a half lifetime
Who are alive at the same time? (vertical box)
We want to inspect the behavior of one specific generation (horizontal box)
 We trace generation “0” that is born at time t=0 and is old in t+1
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Demographics


Assumptions on the demographic structure of the economy
Could be modeled in great detail
 Different sexes
 Survival probabilities
 Different skills by different people

Very simple assumption in this model
 Fixed growth rate of the population over generational periods:
L t 
1 n
L t1
(Equivalent to the continuous time variant L
t nL t )
 where Lt is the number of individuals born at time t,
 where n is the growth rate of the population
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Household utility

Individuals derive utility only from consumption in their two periods of life
Ut Ut 
C1t , C2t1 

Two factors determine how individuals decide to divide consumption over
time in a risk-free (certain/perfect foresight) environment
1.
2.
The consumption “smoothing” motive, captured by the term ρ
The consumption “fluctuation” motive, captured by the term θ
Ut 

C 1
1t
1
1
1 C 2t1
1
 1
We discuss each in turn…
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Household utility

The consumption “smoothing” motive
 Individuals generally like to smooth (evenly divide) their consumption
over periods
 The degree of impatience towards consuming today is captured by the
discount rate, ρ
 The discount rate of future consumption is generally 1/(1+ρ) so that
household utility can be represented in present value terms as:
1
Ut C1t 1
C 1
 2t
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Household utility
Consumption “fluctuation” motive
 Uncertain environment
 In an uncertain environment you might be risk averse, and might not
be willing to shift consumption very freely over time.
 Say, if you decide to smooth consumption 50/50 over your two
periods, and if you are uncertain about how your consumption will
vary your tend to stick to the safer level in each period.
 If you expect the interest rate to increase in the next period, you
would get a higher lifetime consumption if you shift some units of
consumption. If you are risk averse you would rather stick to the safer
levels of consumption – you then miss out on the extra consumption
 θ measures the degree of consumption risk aversion
 Certain environment
 In this case there is no risk (perfect foresight)
 You can still appreciate stable consumption levels in each period, so
the parameter θ then measures the degree to which you like stable
and consumption
 Again, if you expect the interest rate to increase in the future period,
and you prefer consumption not to fluctuate – you will then not take
advantage of the potentially higher lifetime consumption.
 θ measures the degree of consumption fluctuation aversion
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Household utility

The utility function:
 utility from consumption
 features intertemporal consumption smoothing motive, ρ
 features consumption fluctuation motive, θ
1
1
1t
1 C 2t1
Ut  C1


1
 1




We divide by (1-θ) to ensure positive marginal utility in case θ>1
Note: for ρ>0, second period utility is valued less than first period utility
We assume that ρ>-1: weight on second period consumption>0.
Also,
assume  0, n 0, 1
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Life-cycle consumption

People live for two periods, as adults and as old, and they need to consume in
each period

Adults (workers)
 The adults work, consume, and save:
C1t wt A t St
where

A t 
1 g
A t1
Old (retirees)
 The elderly are retired, and consume (they do not work, but live of their
savings and interest earnings)
C2t1 
1 rt1 
St

Intertemporal budget constraint (IBC)
C 2t1
C1t 1
wt A t
r 
t
1
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Firms


Firms use two factors in production: labor and capital
 Firms pay the wage rate, wt , to the labor, Lt, supplied by workers
 Firms rent capital, Kt, from retirees at a rental price of rt
The production function is generally:
Y t F
Kt , A t L t 

Due to CRS we can restate the capital in efficiency units, where: k t
F
K, ALF
K
AL
,
AL
AL
F
K
AL
,1
f
k
 AKLt
t
t
 The wage rate is the marginal product of labor in production:
wt f
k t k t f 
kt
 Return to capital is defined by marginal product of capital in production
(assume no capital depreciation, δ=0)
 Total Returns:
rt f 
k t 
R t 1 rt f 
k t 
1 
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Resources (economy-wide)


We have seen the consumption budget constraint for the household (IBC)
There is also a constraint on consumption for the economy as a whole: society
cannot prioritize over more than is actually produced (closed economy)
 In each period people save and consume (to save is to invest):
Y t Ct I t
 Resource constraint (RC): Ct L t C1t L t1 C2t
I t Kt1 Kt
 In efficiency units:
Y t Ct I t
Y t Kt L t C1t L t1 C2t Kt1
Yt
A t1 L t1
1
y
1
n
1
g t


A
Kt
t
1 L t
1
1
 A K t
L
t
1 t
1
A
Lt
t
1 L t
1
C1t A L tL1 C2t
t
1 t
1
1
1
1
1n
k
k

c

c
t
t
1
1t
1
g
1
n
1
g
1
n
1
n
1
g 2t






1
yt k t 
1 n
1 g
k t1 c1t 1
c

n 2t
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Description of the economy
A.
B.
C.
D.
E.
F.
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
The model
The OLG model will be presented according to the following outline:
1.
Description of the economy
A.
Basic assumptions
B.
Demographics
C.
Household utility
D.
Life-cycle consumption
E.
Firms
F.
Resources
2.
Dynamics of the economy
A.
Household utility maximization
B.
Capital accumulation and Steady State
3.
4.
Case study
Efficiency and welfare
A.
Dynamic efficiency
B.
Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamics of the economy
A. Household utility maximization
B. Capital accumulation and Steady State
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Household utility maximization

Simple intertemporal utility maximization:
1
C 1t , C 2t
st.

Ut

C 1t

Ut

C 2t
1


C
1t
1
C  1
1
 2t
1
1t
1 C 2t1
Ut  C1


1
 1
max
C1t 1r1 C2t1 A t wt
t
1

1 
C 2t1
C 1t


IBC

C 1t

IBC

C 2t
1
C2t+1
Slope of Utility function (MRS)
Slope of IBC = -(1+rt+1)
C1t
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
1
1
1
r t
1

1 rt1 
Household utility maximization

Optimal consumption allocation depends on
1. The consumption “smoothing” motive, ρ
2. The consumption “fluctuation” motive, θ
3. The future interest rate, rt+1

The intertemporal optimality condition (Euler equation)
or

C 2t1
C 1t

1
r t1
1

C2t1 
1
r t1
1

1/ 
1/ 
C1t
This equation is all we need derive through maximization!! The rest of
the solution of the model is only based on simple math (insert/reduce)
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Household utility maximization

To find first and second period consumption, C1t and C2t+1, we just
insert the Euler equation into IBC:
C1t 1r1 C2t1 A t wt
t
1
C1t 1r1
t
1
- reduce
1
r t1
1

1/ 
C1t
1/
r t1 1/1
1
1
A t wt
C1t A t wt
1/
1
C1t

1/
1
1/
r t1 1/
1
1
A t wt
C1t 
1 st 
A t wt
- The savings rate:

st 
rt1 
r t1 1/
1
1/
r t1 1/
1
1
insert the optimal C*1t into The Euler equation to derive C*2t+1:
C2t1 st A t wt 
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r t1 1/
1
1/
r t1 1/
1
1
A t wt
Household utility maximization



We need to derive how much people save, because this determines our
consumption in the future.
As such, the intertemporal structure of this model evolves around savings
(recall that today’s savings is equal tomorrows capital stock)
Savings can simply derived from first period consumption:
St wt A t C1t
or
St st 
rt1 
A t wt
lets analyze the dynamics of the savings rate for alt. parameter assumptions…
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Household utility maximization
The dynamics of the savings rate


Substitution effect:
Income effect:
st 
rt1 

1/
 savings rate increases
 savings rate decreases
C2
r t1 
1
1/
r t1 1/
1
1
Special case, θ=1:
st 
rt1 st 

relative price change
purchasing power
1
2

No consumption impatience, ρ=0:
st 
rt1 st 
1
2
half of your lifetime income is:
1) consumed in period one,
2) and saved for period two
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C’2
-(1+r’)
C2
-(1+r)
C’1 C1
C1
Household utility maximization
Summary of households intertemporal choice
The Euler equation:
C 2t1
C 1t

1
r t1
1

1/ 
 rt+1 ↑:
2. period consumption becomes relatively more preferable
 ρ
1. period consumption becomes relatively more preferable

↑:
θ ↓:
For a given change in (1+rt+1)/(1+ ρ) Consumption is shifted
more freely over periods (larger increase in C2t+1/C1)
 Inverse elasticity of intertemporal substitution is constant (CRRS):

MRS C1,C2 

MRS C1,C2
C
C 2
C2
C1
 MRS C1,C2  
1
(your rate of marginal substitution, MRS, changes more when your
consumption “fluctuation” aversion, θ, is low)
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Household utility maximization
Summary of the household’s intertemporal choice

The household maximization problem boiled down to deriving the
savings rate, st(rt+1).
st 
rt1 
r t1 1/
1
1/
r t1 1/
1
1
this concludes the section on household utility maximization – we’ll move
on to study the economy’s capital accumulation…
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Dynamics of the economy
A. Household utility maximization
B. Capital accumulation and Steady State
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Capital accumulation and Steady State
The equation of motion for the capital stock
Current Savings Current Investments st 
rt1 
A t wt L t
Next Period Capital Stock Kt1
Next period’s capital stock is the current period’s investments!
Kt1 st 
rt1 
A t wt L t
Thus: we know the level of next period’s capital stock in this period…
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Capital accumulation and Steady State

We can now derive the dynamics of the economy
(the way the capital stock evolves over time – thus, also all other variables)

Transform the expression for the motion of the capital stock into efficiency
units to derive the Balanced Growth Path: (divide by A t1 L t1)
Kt1 st 
rt1 
A t wt L t
K t1
A t1 L t1

s t r t1 A t w t
A t1 L t1
1
k t1  1n
s
r 1
wt
g t t
1

Insert the general expressions for rt+1 and wt:


1
k t1  1n
s f
k t1  f
k t k t f 
kt 
1
g t

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Capital accumulation and Steady State
The general case

If we don’t know exactly how factor returns are determined relative to kt
we obtain this expression for the evolvement of the economy:


1
k t1  1n
s f
k t1  f
k t k t f 
kt 
1
g t


The economy evolves over time, and households want to save/invest to
generate the capital stock that will provide them with the highest
possible utility. When they reach this capital stock they will keep their
savings at the level that will re-generate this particular capital stock in
all future periods. Equilibrium condition for the economy to be in its long
run equilibrium:
kt+1 = kt
k
k
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0
Capital accumulation and Steady State
The Steady State capital stock can be determined as illustrated:
 Given well-behaved preferences, and given Cobb-Douglas technology
 We will discuss the more general case later!...
kt+1
45o
k*
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kt
Capital accumulation and Steady State
1. Existence

Determined by Inada conditions, where the slope of kt+1 is
approaching 0 for lim. kt ∞, and approaching ∞ for lim. kt 0.
This is related to the decreasing marginal product of capital
through the production function and the requirement that kt+1=kt
2. Uniqueness

Also determined by the Inada conditions. Hence, the slope is falling
for k getting larger and larger – therefore the motion of capital can
only cross the 45-degree line once.
3. Stability

Determined through inspection of the phase diagram:
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Capital accumulation and Steady State
Convergence to Steady State – two cases
A. Initial over-accumulation:
A too high capital stock is caused by too
much savings by households (too high for
utility to be maximized over intertemporal
consumption allocation). If utility could be
higher by changing the consumption
allocation then the current level of capital
is not sustainable and is not compatible
with household utility maximization.
Consequently, people will start saving less,
spending more in the current period –
total savings fall – investments fall – the
capital stock decreases until savings has
reached its optimal level compatible with
household preferences and utility
maximizing…
B. Initial under-accumulation:
Opposite: People will save more to
maximize utility – the capital stock rises…
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
kt+1
45o
kB
k*
kA
kt
Capital accumulation and Steady State

To summarize: If we do not know the relationship between the factor
returns, rt+1 and wt and the level of capital, kt, then the savings can shift
up and down. As such, the path of the capital stock can also shift up
and down.

This is determined through the expression for the path of the economy:
the equation of motion of the capital stock


1
k t1  1n
s f
k t1  f
k t k t f 
kt 
g t
1
Different versions of the relationship between the capital stock and
factor returns can be illustrated graphically…
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Capital accumulation and Steady State
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Capital accumulation and Steady State
Summary of the section on Capital accumulation:
We have determined the path of the economy in two ways:

Analytically
- through the expression for the dynamic evolution of capital, kt+1(kt)


Graphically
- through the steady state condition for capital, kt+1=kt
Recall key relationships
when the capital stock has been derived, all other variables in the model
can be determined!
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamics of the economy
A.
B.
Household utility maximization
Capital accumulation and Steady State
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The model
The OLG model will be presented according to the following outline:
1.
Description of the economy
A.
Basic assumptions
B.
Demographics
C.
Household utility
D.
Life-cycle consumption
E.
Firms
F.
Resources
2.
Dynamics of the economy
A.
Household utility maximization
B.
Capital accumulation and Steady State
3.
4.
Case study
Efficiency and welfare
A.
Dynamic efficiency
B.
Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Case study
The Diamond OLG model
Three assumptions:
1. Logarithmic utility:
2.
Cobb-Douglas technology:
1
Ut ln C1t 1
ln C2t1

yt k 
t
 wt 
1 
k
t
1
 rt  k 
t
3.
No capital depreciation:
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 0
Case study
Recall the general expression for the law of motion of capital (and the economy):


1
k t1  1n
s f
k t1  f
k t k t f 
kt 
1
g t

For assumed log utility insert the savings rate (for θ=1):
st 
rt1 
r t1 1/
1
1/
1/
 
r t1 
1
1

1
2


1
1
k t1  1n
f
k t k t f 
kt 
g 2

1
For assumed Cobb-Douglas technology insert the wage rate:

1
1
k t1  1n
1


k


t
1
g
2


k t1 
1
1
n1
g
2
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k
t
Case study

If we should draw curve for this fundamental difference equation then note
that  1 and we have again:
kt+1
45o
k*

Consequently we can derive a Steady State…
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kt
Case study
Comments on existence, uniqueness, and stability
k t1 
1
1
n1
g
2
1. Existence:
Is the slope positive and decreasing in k?
dkt1
dk
d 2 kt1
dk2

k
t
1
n1
g2

1
1
k
0
t
1
2

1  1n1g2 k 
0
t
2. Uniqueness:
Since the expression for kt+1 is unchanged for increasing values for kt, the
Inada conditions ensure that for low k’s the curve is very steep and for high
k’s the curve flattens
3. Stability:
The function is based on well-behaved preferences and Cobb-Douglas
technology so the analysis of the phase-diagram before also applies here:
thus stability
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Case study
Steady State

Remove subscripts:



k t1 
1
1
n1
g
2
k 
1
1
n1
g
2
k
t
1
1
This value for the Steady State capital stock can be calibrated and a
numerical estimate can be derived
Hence, all other variables can also be derived numerically (since they all
ultimately depend on the capital stock)
One could then make experiments with the model:

Change parameter values and trace the effects on variables

How would workers consumption change?

How would retirees consumption change?
lets do some sensitivity analyses…
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Case study
Experiments
Change parameter values and trace the effects on variables

How would workers’ consumption change?

How would retirees’ consumption change?
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Case study
Sensitivity analyses
1.
What if you get less impatient with your consumption (ρ 
)
2.
What if the growth rate of the population increases (n 
)
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Case study
Sensitivity analysis: A fall in ρ:
k 
1
1
n1
g
2
1
1
 k 
kt+1
45o
k*

k’
*
kt
Why: People want to consume less today and more tomorrow – this increases
savings – increases the long run capital stock per effective worker – increases
wages – decreases return to capital!
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Case study
Sensitivity analysis: A rise in n:
k

1
1
n1
g
2
1
1
 k 
kt+1
45o
k’

*
k*
kt
Why: There are now more workers to share the capital – capital/labor ratio
falls – wages fall – returns increase – savings fall – the long run level of
capital per effective worker falls!
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
The model
The OLG model will be presented according to the following outline:
1.
Description of the economy
A.
Basic assumptions
B.
Demographics
C.
Household utility
D.
Life-cycle consumption
E.
Firms
F.
Resources
2.
Dynamics of the economy
A.
Household utility maximization
B.
Capital accumulation and Steady State
3.
4.
Case study
Efficiency and welfare
A.
Dynamic efficiency
B.
Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare
A. Dynamic efficiency
B. Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare

We have determined the equilibrium capital stock, but we have to ask two
questions:
1. Is capital at its efficient level
(we need to derive the Golden Rule capital stock)
2. What can be done to achieve the optimal level of capital
(we need to consider policy)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare

We have determined the equilibrium capital stock, but we have to ask two
questions:
1. Is capital at its efficient level
(we need to derive the Golden Rule capital stock)
2. What can be done to achieve the optimal level of capital
(we need to consider policy)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamic efficiency

Is the level of long run capital, k*, optimal? How to determined this:
 The capital stock should be at a level consistent with maximum utility
 Utility is maximized over lifetime consumption, so when lifetime
consumption is maximized – so must welfare!
 When is consumption maximized? Since everything in the model
depends on the level of kt then which level must kt have?

We have available resources for allocation to worker’s and retirees’
consumption (i.e. RC). Assume no productivity growth in this example:
1
yt k t 
1 n
1 g
k t1 c1t 1
c

n 2t

Maximize consumption w.r.t. kt in Steady State, where:
Ct L t C1t L t1 C2t 
Ct
AtL t
1
 LAt CL1t L At1LC 2t  ct c1 1
c
n 2
t t
t t
1
f
kk 
1 n
k c1 1
c
n 2
c f
knk
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamic efficiency

Maximize consumption w.r.t. k:
dc
dk
f 
kn 0
f 
k GR n

The Golden Rule capital stock, kGR, that maximizes utility can then be
derived if we now the expression for f(kGR)

Recall that:
c
1
Y K
A t Lt 
 y k 
t 
t

nk
f(k)
Golden Rule capital stock:
1
k 
GR n
1
1
k GR 
n 
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
cMAX
y
I
kGR
k
Dynamic efficiency

Key issue: is the market solution for k, which is k*, equal to kGR?

If yes, then welfare is maximized automatically!

If no, then is the allocation at least Pareto efficient?

Compare k* to kGR: Is it possible that k* ≠ kGR. Check for k* > kGR:

Recall that for log-utility, zero capital depreciation, and g=0:
k  k GR
1 

1 n
2 


1 

n
1 n
1
1
1
1
 
n
1
1
1
1
2 
1
for 0   1, n 0,  0, g 0, 1


It is definitely possible that k  k GR .
If k  k GR is the allocation then efficient or inefficient (graphically)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamic efficiency
1.
2.
If the Golden Rule capital stock, kGR, is lower than the market determined
capital stock, k*, is k* then an efficient allocation?

YES: because if current workers were somehow forced to save more (to
increase the capital stock) then they would have to give up current
consumption in order for future generations to better of (welfare function
should value future generations’ utility higher than current generations’)
If the Golden Rule capital stock, kGR, is higher than the market determined
capital stock, k*, is k* then an efficient allocation? NO: everybody gains…
nk
c
f(k)
Lost consumption potential
y
cMAX
Lost consumption potential I
k*
kGR
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k*
k
Efficiency and welfare

We have determined the equilibrium capital stock, but we have to ask two
questions:
1. Is capital at its efficient level
(we need to derive the Golden Rule capital stock)
2. What can be done to achieve the optimal level of capital
(we need to consider policy)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare

We have determined the equilibrium capital stock, but we have to ask two
questions:
1. Is capital at its efficient level
(we need to derive the Golden Rule capital stock)
2. What can be done to achieve the optimal level of capital
(we need to consider policy)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamic efficiency
Think of dynamic in-efficiency in two ways:
1. Pareto efficiency: If we are in an equilibrium where the government can
redistribute from young to old so both generations, and all succeeding
generations, are better of – then the current equilibrium could clearly not
be Pareto efficient!
2. Savings vs. transfers: If the real interest rate is lower than the
population growth rate – then it would be more efficient to take 1 unit of
consumption from the current young and transfer the 1 unit to the old.
Since the current old generation is (1+n) times smaller than the current
young the 1 unit from the young can actually be divided to the young so
they each get (1+n) units. If this goes on for ever through all generations,
that is a way for young to give up one unit of consumption and in turn get
(1+n) units in old age. The return on savings would be (1+r), so if r<n, it
would actually be more efficient (all generations would get more lifetime
utility) to permanently transfer units from young to old instead of saving
through capital investments. The government could facilitate this transfer,
and thus bring savings down to the Golden Rule level of savings.
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Dynamic efficiency
Government policy: redistribution
1.
2.
3.
4.
5.
Government transfers x units of
consumption from workers
Retirees receive x units, but the
size of the old generations is
smaller by (1+n), so the will
receive (1+n)*x units
The level of x is determined by
the Government
This “arrangement” will (must) go
on forever…
It is clear that workers save less,
so k* falls over time, and the
Government has fixed x so
eventually: k*=kGR
Time
Generation
0
-1
C2t
1
2
3
(1+n)*x
0
C1t
-x
1
C2t+1
(1+n)*x
C1t+1
-x
2
C2t+2
(1+n)*x
C1t+2
-x
3
C2t+3
(1+n)*x
C1t+3
-x
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
4
C2t+4
Dynamic efficiency
Two sources of financing of consumption in old age:
1. Savings
2. Transfers




We
We
We
We

know
know
know
know
that
that
that
that
savings will yield a return of: (1+r)
transferring x=1 units of income will yield a return of (1+n)
if there is dynamic inefficiency we have r<n
in this situation transfers will yield higher return than savings
Consequently:
If we have dynamic inefficiency, transfers will be more
efficient than savings – thus the government can
improve on the decentralized equilibrium!!
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare

We have determined the equilibrium capital stock, but we have to ask two
questions:
1. Is capital at its efficient level
(we need to derive the Golden Rule capital stock)
2. What can be done to achieve the optimal level of capital
(we need to consider policy)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare
A. Dynamic efficiency
B. Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare
A. Dynamic efficiency
B. Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare
Government participation can be incorporated in various ways?
 Through the Government budget (see Romer, 2001:section 2.12)
 Through a pension system, e.g. Pay-As-You-Go
(I will deal with a pension system in my next lecture)
Bottom-line: We can incorporate several different mechanisms that is able to
handle intergenerational transfers
More one this in my next lecture!!
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Efficiency and welfare
A. Dynamic efficiency
B. Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
The model
The OLG model will be presented according to the following outline:
1.
Description of the economy
A.
Basic assumptions
B.
Demographics
C.
Household utility
D.
Life-cycle consumption
E.
Firms
F.
Resources
2.
Dynamics of the economy
A.
Household utility maximization
B.
Capital accumulation and Steady State
3.
4.
Case study
Efficiency and welfare
A.
Dynamic efficiency
B.
Government policy
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
Concluding remarks





Intergenerational aspects
A model of life-cycle optimization
Key dynamic variable of the model is the capital stock per effective worker
Importance of population dynamics
Importance of productivity
Important properties
Note that for the economy to be in Steady State:

What is the growth rate for of variables?
k
k
0 hence:
y
y
0
Kt k t A t L t
ln Kt ln k t ln A t ln L t

ln X t

t
ln X t

Xt



t

X
t

 XX

ln K t

t
K
K
K
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
K
ln kt

ln A t

ln L t




t

t

t

k
A
L
A L
k
g n
Concluding remarks


For national income:
Y t yt A t L t
y
A
L
y A L
Y
Y

Y
Y
g n
For income per worker, relative to productivity growth (not per effective worker)
Y
Y
L



Y
L
Y
0 AA
Y
L

Y
L
 AA
y per worker
y

 kk AA LL

g
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
y per worker
y

y per effective worker
y
0
Concluding remarks
Potential extensions of the model (NEXT LECTURE):
1. Government
2. Pensions (PAYG)
3. Endogenous retirement
4. Endogenous labor supply
5. Endogenous population (fertility) growth
6. Bequests
7. Survival rates
Potential extensions of the solution method (NEXT LECTURE)
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
The next lecture
An analytical solution method for transitional dynamics of the OLG model
I will show you a recently developed method to solve for the transitional dynamic
of the model to a new steady state:


Changing parameters in the model above only gives the new values in the new
steady state, while the new method derives the dynamics of the transition
towards the new steady state
This solution method is on the frontier of research on OLG models and has
never been taught in a lecture before…
Copyright 2006 © CEBR, Copenhagen - www.cebr.dk
The next lecture
Which path does the economy
follow to the new steady state?
k
k2
k1



Transitional dynamics
t=0
t=j
t
Shock to nt (negative)
Key issue: A different capital stock for different generations
 different wages and interest rates  a shock produces
unequal intergenerational risk sharing
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