Production on OLG Models
Jesús Fernández-Villaverde
University of Pennsylvania
February 12, 2016
Jesús Fernández-Villaverde (PENN)
Production on OLG Models
February 12, 2016
1 / 18
Production
Introducing Production in an OLG Model
Are the properties of OLG models consequence of the absence of
production?
First explored by Diamond (1965).
We want to have models with production for policy purposes.
Tradition of Auerbach and Kotliko¤ (1987).
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Production on OLG Models
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Setup
Demographics
Individuals live for two periods.
Ntt : number of young people in period t.
Ntt
1
: number of old people at period t.
Normalize the size of the initial old generation to 1: N00 = 1.
People do not die early, Ntt = Ntt+1 .
Population grows at constant rate n:
Ntt = (1 + n )t N00 = (1 + n )t
The total population at period t:
Ntt
Jesús Fernández-Villaverde (PENN)
1
+ Ntt = (1 + n)t 1 +
Production on OLG Models
1
1+n
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Setup
Preferences
Preferences over consumption streams given by:
u (ctt , ctt+1 ) = U (ctt ) + βU (ctt+1 )
U is strictly increasing, strictly concave, twice continuously
di¤erentiable and satis…es the Inada conditions.
All individuals are assumed to be purely sel…sh and have no bequest
motives whatsoever.
The initial old generation has preferences
u (c10 ) = U (c10 )
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Production on OLG Models
February 12, 2016
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Setup
Endowments
Each individual of generation t 1 has as endowments one unit of
time to work when young and no endowment when old.
How we can generalize it?
1
Life cycle pro…le of productivity.
2
Leisure in the utility function.
Hence the labor force in period t is of size Ntt with maximal labor
supply of 1 Ntt .
Each member of the initial old generation is endowed with capital
stock (1 + n )k̄1 > 0.
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Production on OLG Models
February 12, 2016
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Setup
Firms
Constant returns to scale technology:
Yt = F (Kt , Lt )
Pro…ts are zero in equilibrium and we do not have to specify
ownership of …rms.
Single, representative …rm that behaves competitively in that it takes
as given the rental prices of factor inputs (rt , wt ) and the price for its
output.
De…ning the capital-labor ratio kt =
yt =
Kt
Lt ,
Yt
F (Kt , Lt )
=
=F
Lt
Lt
we have:
Kt
,1
Lt
= f (kt )
We assume that f is twice continuously di¤erentiable, strictly
concave, and satis…es the Inada conditions.
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Production on OLG Models
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Setup
Timing
1
At the beginning of period t, production takes place with labor of
generation t and capital saved by the now old generation t 1 from
the previous period. The young generation earns a wage wt .
2
At the end of period t, the young generation decides how much of the
wage income to consume, ctt , and how much to save for tomorrow,
stt . The saving occurs in form of physical capital, which is the only
asset in this economy.
3
At the beginning of period t + 1, production takes place with labor of
generation t + 1 and the saved capital of the now old generation t.
The return on savings equals rt +1 δ, the real interest rate from
period t to t + 1.
4
At the end of period t + 1 generation t consumes its savings plus
interest rate, i.e. ctt+1 = (1 + rt +1 δ)stt and then dies.
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Production on OLG Models
February 12, 2016
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Setup
Sequential Markets Equilibrium I
Given k̄1 , a sequential markets equilibrium is allocations for households
ĉ10 , f(ĉtt , ĉtt+1 , ŝtt )gt∞=1 , allocations for the …rm f(K̂t , L̂t )gt∞=1 and prices
f(r̂t , ŵt )gt∞=1 such that:
1
For all t
1, given (ŵt , r̂t +1 ), (ĉtt , ĉtt+1 , ŝtt ) solves
max
ctt ,ctt+1 0,stt
U (ctt ) + βU (ctt+1 )
s.t. ctt + stt
ctt+1
2
(1 + r̂t +1
ŵt
δ)stt
Given k̄1 and r̂1 , ĉ10 solves
max U (c10 )
c10 0
s.t. c10
Jesús Fernández-Villaverde (PENN)
(1 + r̂1
Production on OLG Models
δ)k̄1
February 12, 2016
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Setup
Sequential Markets Equilibrium II
3
For all t
1, given (r̂t , ŵt ), (K̂t , L̂t ) solves
max F (Kt , Lt )
K t ,L t 0
4
For all t
r̂t Kt
ŵt Lt
1:
1
(Goods Market) Ntt ĉtt + Ntt 1 ĉtt 1 + K̂t +1
2
(Asset Market) Ntt ŝtt = K̂t +1 .
3
(Labor Market) Ntt = L̂t .
Jesús Fernández-Villaverde (PENN)
Production on OLG Models
(1
δ)K̂t = F (K̂t , L̂t ).
February 12, 2016
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Setup
Stationary Equilibrium
A steady state (or stationary equilibrium) is (k̄, s̄, c̄1 , c̄2 , r̄ , w̄ ) such that
the sequences ĉ10 , f(ĉtt , ĉtt+1 , ŝtt )gt∞=1 , f(K̂t , L̂t )gt∞=1 and f(r̂t , ŵt )gt∞=1 ,
de…ned by
ĉtt
ĉtt 1
ŝtt
r̂t
ŵt
K̂t
L̂t
=
=
=
=
=
=
=
c̄1
c̄2
s̄
r̄
w̄
k̄
Ntt
Ntt
are an equilibrium, for given initial condition k̄1 = k̄.
Jesús Fernández-Villaverde (PENN)
Production on OLG Models
February 12, 2016
10 / 18
Setup
Saving Equals Investment
Investment:
K̂t +1
(1
Ntt ĉtt + Ntt
δ)K̂t = F (K̂t , L̂t )
1 t 1
ĉt
Saving:
Ntt ŝtt
|{z}
savings of the young
Also:
Ntt
1 t 1
1 ŝt 1
Ntt 11 ŝtt 11
| {z }
dissavings of the old
= (1
δ)K̂t
Hence,
K̂t +1
(1
δ)K̂t = Ntt ŝtt
(1
δ)K̂t
or our asset market equilibrium condition
Ntt ŝtt = K̂t +1
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Production on OLG Models
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Setup
Characterizating Equilibrium
Characterizing the equilibrium in an OLG model with production is
di¢ cult.
However, we can prove in general existence of equilibrium.
We can have multiplicity of equilibria, even without money.
Moreover, we may even have chaotic dynamics.
Welfare theorems break down.
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Production on OLG Models
February 12, 2016
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Setup
Optimality of Allocations I
Consider …rst steady state equilibria.
Let c1 , c2 be the steady state consumption levels when young and
old, respectively, and k be the steady state capital labor ratio.
Consider the goods market clearing (or resource constraint)
Ntt ĉtt + Ntt
1 t 1
ĉt
+ K̂t +1
(1
δ)K̂t = F (K̂t , L̂t )
Divide by Ntt = L̂t to obtain
ĉtt +
ĉtt 1
+ (1 + n)k̂t +1
1+n
(1
δ)k̂t = f (kt )
Use the steady state allocations to obtain
c1 +
Jesús Fernández-Villaverde (PENN)
c2
+ (1 + n )k
1+n
(1
Production on OLG Models
δ )k = f (k )
February 12, 2016
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Setup
Optimality of Allocations II
c
De…ne c = c1 + 1 +2 n to be total (per worker) consumption in the
steady state. We have that
c = f (k )
(n + δ )k
Now suppose that the steady state equilibrium satis…es:
f 0 (k )
δ<n
something that may or may not hold, depending on functional forms
and parameter values.
This steady state is not Pareto optimal: the equilibrium is
dynamically ine¢ cient.
Jesús Fernández-Villaverde (PENN)
Production on OLG Models
February 12, 2016
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Setup
Intuition
If f 0 (k ) δ < n, it is possible to decrease the capital stock per
worker marginally, and the e¤ect on per capita consumption is
dc
= f 0 (k )
dk
(n + δ ) < 0
so that a marginal decrease of the capital stock leads to higher
available overall consumption.
An allocation is ine¢ cient if the interest rate (in the steady state) is
smaller than the population growth rate, that is, if we are in the
Samuelson case.
Jesús Fernández-Villaverde (PENN)
Production on OLG Models
February 12, 2016
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Setup
General Result
Theorem
Cass (1972), Balasko and Shell (1980) . A feasible allocation is Pareto
optimal if and only if
∞
t
(1 + rτ +1 δ )
= +∞
t =1 τ =1 (1 + nτ +1 )
∑∏
As an obvious corollary, alluded to before we have that a steady state
equilibrium is Pareto optimal (or dynamically e¢ cient) if and only if
f 0 (k )
δ
n
With technological progress:
f 0 (k )
Jesús Fernández-Villaverde (PENN)
δ
n+g
Production on OLG Models
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Setup
Empirical Relevance
Dynamic ine¢ ciency is not purely an academic matter.
Some reasonable numbers: U.S. population growth n
1%, g
2%.
Is rate of return higher or lower than 3%?
Abel, Mankiw, Summers, and Zeckhauser (1989) extend result to an
economy with uncertainty.
Su¢ cient condition for dynamic e¢ ciency: net capital income exceeds
investment.
U.S., net capital income investment
income 19% of GDP.
Jesús Fernández-Villaverde (PENN)
17% of GDP, net capital
Production on OLG Models
February 12, 2016
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Setup
Policy Implications
If the competitive equilibrium of the economy features dynamic
ine¢ ciency, its citizens save more than is socially optimal.
Hence, we need government programs that reduce national saving:
Tax on capital.
An unfunded, or pay-as-you-go social security system.
Having government debt.
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Production on OLG Models
February 12, 2016
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