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Lecture 4: Infinite-horizon dynamics
• Introduction
• Dynamic models with finite time horizon
• Infinite horizon: dynastic models
• Infinite horizon: overlapping generations models of
exchange economies
• Infinite horizon: overlapping generations models with
production
• Remarks
Aim of lecture 4
• Explicitating dynamic behavior of consumers and
producers
• Highlighting main issues related to models with infinite
horizon models
• Highlighting main differences between dynastic and OLG
models of infinite horizon
Introduction
• Recapitulating lectures 1-3
– Competitive equilibrium can be represented as welfare optimum
with welfare weights that are such that budgets hold (Negishi)
– This facilitates direct link to welfare analysis, among other things
the welfare gains from (tax) reforms
– Other formats, such as open economy format, full format, and
CGE format are useful for the application of general equilibrium
theory to specific issues
– Within CGE format, macro-economic mechanisms, including
savings rules can be included by closure rules, but care should be
taken that the closure rule does not become the dominant
mechanism in the model
– Within all formats, direct and indirect taxes can be introduced
Introduction (continued)
• Models discussed so far had no explicit time subscript
• However, commodity classification can distinguish
between different dates of delivery
• Consumer and producer decisions can then be given a
dynamic interpretation, but time remains implicit
• Now: time dependence is made explicit and specific
dynamics are specified
• First: consider models with finite time horizon
Dynamic models with finite time horizon
• Dynamics
– Number and composition of commodities: new products entering
the economy via endogenous or exogenous processes
– Number and composition of agents
• Social classes
• Age cohorts
• Occupational groups
– Some of these changes can be represented through migration
(lecture 1), some are changes in the characteristics of the agents
themselves
– Entry and exit of producers: are all producers potentially active in
all periods or not?
Dynamics in producer decisions
• Goods produced in t+1 cannot serve as inputs for production in t
• Production process must be separable over time and can be
represented as:
Fjt ( y jt , k j ,t 1 , k jt )  0
k jt  k jt
• Where y jt is the net supply, k jt is the stock of capital goods used
as inputs, and k j ,t 1 stands for the stocks of capital goods that
will made available as inputs for the production in time t+1
• Capital goods are goods that can be transferred through time.
They can be produced, increase at exogenous rates or remain
fixed over time
Dynamics in producer decisions (continued)
• Optimization problem of producer:
max k ,k 0, y , all t  t pt y jt
jt
j ,t 1
jt
subject to
Fjt ( y jt , k j ,k 1 , k jt )  0
(  jt )
k jt  k jt
( jt ),
where k j1 is given
• As in lecture 3, additional restriction and variable is added
to recover price; in this case price of capital jt
• Storage is interpreted as a productive activity, transforming
goods at time t to goods at time t+1
Dynamics in producer decisions (continued)
• The objective of the firm is to maximize the sum of its profits
over the finite time horizon.
• This does not rule out negative receipts at specific points in
time
• It enables firms are to borrow to invest in expanding
production capacity in later periods
• Note: there is no interest rate in the model: the ratio pkt pk ,t 1
represents discounting as well as relative scarcity of each of
the commodities in the model
Dynamics in consumer decisions
• Consumer maximizes intertemporal utility function subject to
intertemporal budget
max xt 0 u ( x1 ,..., xT )
subject to
 t pt xt   t ptt
• Separability of utility over time not obvious, but is often
imposed (“no regret”)
ut ( xt , ut 1 )
• Common specification of intertemporal utility function
assumes that the discount factor is constant over time for
consumer i. More general specifications can be used that
maintain the “no regret” property (viz. chapter 2, section 2.2.5)
Dynamics in consumer decisions (continued)
• Intertemporal budget:
t pt xit  t ptit   jij  jt


• This can be rewritten to explicitate net savings Sit and net
accumulation of worth Vit :
pt xit  Sit  ptit   j  ij  jt
Vi ,t 1  Vit  Sit ,
with Vi1  0
Dynamics in consumer decisions (continued)
• Prices in budgets are discounted at commodity-specific
rates, just as in producer case
• Differences between prices in different time periods cannot
be interpreted as interest rate nor as rate of discount by
individuals (viz prices in different countries and exchange
rate)
Dynamic implications of market demand
equilibrium conditions
• In additional to intertemporal budgets and intertemporal profits, at each time t,
commodity balances must hold
• This implies that at each time t, there are no net aggregate savings or lendings:
savings and lendings act as transfers between agents
• This also implies that there can be no accumulation of aggregate net worth over
time, and no debt inherited by the economy as a whole:
V
i
i ,t 1
 i Vit  i Sit , where i Sit  0
• Prices adjust to restore equilibrium between demand and supply at every time t:
the “rate of interest” that restores balance between savings and lendings is
implicitly accounted for by these commodity-specific prices
Finite horizon dynamics in CGE
• CGE model for T-period equilibrium has exactly the same structure as CGE
model of Chapter 3, except for commodity classification
• Dynamics with respect to production enter by specifying investment
functions for replacement of existing capital goods and new capital goods.
Disaggregation of capital into different vintages is possible
• Investment can also be set exogenously as part of public consumption, in
which case some closure rule is needed
• Consumer dynamics enter by specifying savings functions that may or may
not be derived from intertemporal optimization by consumers. If investment
is set exogenously, consumer savings may also be assumed to be the
adjusting variable, and no behavioral equation is then specified
• Population dynamics can be made endogenous by specifying functional
relations between economic variables and population growth, e.g. by basing
migration decisions on wage differentials
Infinite horizon models
• Dynastic model
• OLG model
t=1
t=2
t=3
x1
x2
x3
x1
x2
x3
t=4
t=1
t=2
t=3
x10
x11
x12
x22
x32
t=4
Difficulties in going from finite to infinite horizon
• Number of commodities goes to infinity
• Number of prices goes to infinity
• Number of agents goes to infinity (OLG)
• In production economies: value of production and
hence of consumption can become infinite
Infinite horizon: dynastic model
• intertemporal welfare program for given welfare weights:
max  i  i ui1
subject to
uit  Wi ( xit , ui ,t 1 )
F ( yt , kt 1 , kt )  0
kt  kt
 i xit  yt  0,
for given k1
Dynastic model (continued)
• Finite and constant number of dynasties over the whole
horizon of the model
• Infinite horizon may cause solution of program to be
unbounded (production can become infinite)
• Therefore, impose additional constraints to ensure
boundedness of dynastic utility function (CD1)
• A stronger assumption is to assume CD1 with the additional
requirement of equal discount rate for all consumers.
• Then, the dynastic utility function can be written as:
Wi ( xit , ui ,t 1 )  ui ( xit )   ui,t 1
• Note: if discount rates differ between consumers, then
eventually, only the most patient one will consume
Dynastic model (continued)
• With constant and identical discount rate for all consumers,
difficulties associated with infinite horizon can be avoided:
– determine expenditure as difference between full expenditure and
consumer surplus, using the extended (homogeneous of degree
one) utility function ui ( xi , ni )  niui  xi ni  with ni  1 as in
lecture 1.
– And using the derived intertemporal utility function, the dynastic
model can be reformulated as a dynamic program:
V ( , kt , n)  max xit 0,all i,kt 1 0  i i ui ( xit , ni )  V ( , kt 1 , n)
subject to
F ( i xit , kt 1 , kt )  0
Dynastic models: multiplicity and indeterminacy
• If the shadowprices on capital use are unique, then they are
a continuous function of welfare weights and there may be
several regular equilibria, but their number is finite
• when period-specific distortions are introduced, then the
number of equations and unknowns becomes infinite and it
may happen that there is a continuum of equilibria
• This is referred to as indeterminacy
– Geanakoplos and Polemarchakis (1991) suggest this gives room
for policy intervention (e.g. through institution with infinite
horizon budget)
– Indeterminacy makes it impossible to determine the value of stocks
without having to impose some arbitrary valuation at time T
Dynastic models: dynamics and steady states
• Steady state: using the optimal values of k and  as initial values
will reproduce the same k and a new  , proportional to the previous
one by the same factor  .
• In steady state of dynastic models, welfare weights are not such that
budgets hold, but such that aggregate capital stocks and prices of
capital remain constant (upto scaling)
• Convergence to steady state:
– global asymptotic convergence not natural property of trajectory
– Single man-made capital good that is also consumed (Ramsey model): unique
steady state and convergence
– multicommodity case: convergence depends on discount rate, curvature of
utility and transformation function
Dynastic models: real business cycles
• Research in RBC concentrates on reproducing moments of
observed business fluctuations in macro-economic variables
• using stochastic dynastic models with agents having rational
expectations
• models are simple: one representative dynastic consumer,
one producer
• stochastic element is introduced in transformation function
Dynastic models: rbc models (continued)
• RBC literature uses stochastic version of dynastic model:


t 1
V (k1 , 1 )  max xt 0,kt 1 0,t 1,2,..., E  (  ) u ( xt ) 
 t 1

subject to
F ( xt , kt 1 , kt , t )  0
• For given k1 and the distribution of the scalar error term  t , and
where E[] denote the expectation operator.
• Usually,  t is specified according to the autocorrelated process
 t   t 1   t , where  t follows some stochastic distribution
that is independent across time periods and  is a non-negative
autocorrelation term.
Dynastic models: rbc models (continued)
• Under appropriate assumptions on the way the disturbance enters
the transformation function, it is possible to represent the program
as a single-period dynamic programming formulation:
V (kt ,  t )  max xt  0, kt 1 0 u ( xt )   E V (kt 1 ,  t 1 ) 
subject to
F ( xt , kt 1 , kt ,  t )  0
for given kt ,  t
• The basic problem is that E V (kt 1 , t 1 ) is not known, and only an
approximation of it can be obtained from a finite data set
• Working with first-order conditions directly is not possible since
the number of equations would be infinite
Dynastic models: rbc models (continued)
• Solving problem for finite horizon, finite number of states
• Solving problem for finite horizon, continuum of states
– Sampling from the continuum of states, approximation of the value function in
the neighborhood of the steady state, or of the distribution
• Solving problem for infinite horizon, finite number of states
– Approximate value function, deriving policy functions from infinite number
of first order conditions, approximate model itself
• Solving the model for infinite horizon and infinite number of states
– Approximate the model itself or the Euler equations (first order conditions).
But: convergence of Euler equations is not guaranteed (Judd, 1998).
Dynastic models (concluding)
• Dynastic models are natural counterpart of welfare
program
• Strength of dynastic models
– Possibility to reformulate program in dynamic programming
format
– Efficiency of the solution
• Weaknesses of dynastic models
– Generations alive at time t do not possess the resources available at
that time
– Assumptions on discount rate unrealistic
Full format representation of OLG model
• Infinite number of consumers and budgets
• Infinite number of goods and prices
max xt 1 , xt 0,t 1,2,....  t 0 t u ( xtt , xtt1 )

t
t
subject to
xtt 1  xtt  tt 1  tt
( pt )
pt xtt  pt 1 xtt1  pttt  pt 1tt1
( t )
OLG models: excess demand representation
• OLG model is started in year t=1, with consumer born in
t=0 coming in with a claim M0 and with commodity
endowment 10. His consumption x00 is a given parameter
in his optimization problem:
max x0 0 u ( x00 , x10 )
1
subject to
p1 x10  M 0  p110
• Claim M0 is a financial claim not backed by commodities
that can be sold to new generations
Walras’ law will
not hold over any finite number of periods
OLG models: claims
• Different types of claims:
0
0
– M 0 is equal to p1m, where m is a given vector expressed in terms
of commodities (a real claim); this is the claim used in Chapter 8
0
– M is a fixed amount (a nominal claim)
0
– M is enforced by a government through a direct tax levied on the
generation born in t=1, which in turn will have a claim on the next
generation
OLG models: equilibrium with real claims
• The allocation xtt* , xtt*1 supported by the price vectors pt*  0, t  1, 2,... which
are bounded for any finite t, is an OLG pure exchange equilibrium if
consumers solve:
max x0 0 u ( x00 , x10 )
max xt , xt
t
1
subject to
subject to
p1 x10  p1m0  p110
• and all markets clear:
xtt 1  xtt  tt 1  tt 
t 1
t
t
u
(
x
,
x
t
t
1 )
0
pt  0
pt xtt  pt 1 xtt1  pttt  pt 1tt1
OLG models: efficiency and indeterminacy
• Inefficiency of OLG models stems from the possibility to shift
debt to the future indefinitely
• Efficiency can be ensured by ensuring that incomes converge to
zero at infinity, since then, shifting the burden to the future is
not possible . In exchange economy, assuming prices go to zero
in infinity establishes zero income at infinity
• indeterminacy:
– infinite number of prices, so equilibrium problem is in infinite
dimensional space
– existence proof depends on arbitrary set consumption in period T
– in one-commodity economy with two-period lived consumers, there is
no indeterminacy
OLG models: dynamics and steady-state
• As in the case of dynastic models, path followed through time may diverge,
converge, cycle, or be chaotic
• Dynamics of pure exchange model are described by time invariant excess
demand function z( pt 1 , pt , pt 1 )  a( pt 1, pt )  c( pt , pt 1 ) where c( pt , pt 1 ) is the
net demand by the old, while a( pt 1 , pt ) is the net demand by the young
• A steady state is a price vector p  0 and a discount factor   (0,1] such that
a  p  , p   c  p,  p   0  p  0 which by homogeneity of a () is
equivalent to a  p,  p   c  p,  p   0  p  0
OLG models: steady state characterization
• Rewriting of condition yields: (1   ) pa( p,  p)  0
• This implies that steady state can be of two types
– if   1, pa( p,  p)  0 .
The value of the claim is zero
real steady state
– if   1, there is no restriction on pa( p,  p) and there may be
savings
monetary steady state.
• If   1 , the steady state describes a path with prices
converging to zero, hence, steady state is efficient
• In monetary steady state, convergence to a steady state is
problematic
OLG models with production
• Introduction of production allows savings to be specified by
buying and selling capital stock
• Discussion on inefficiency can be extended to production
case
• The same holds for indeterminacy
• In OLG with production, there are no nominal claims, so
every steady state is a real steady state
– if   1, prices go to zero and the steady state is efficient
– if   1 , then prices rise over time, and the steady state is inefficient
– if   1 , then py  0 . Efficiency cannot be assured
OLG models (concluding)
• OLG models are natural counterpart of competitive equilibria
• Strength of OLG models
– Welfare of future generations does not depend on benevolence of
single planner as in dynastic models
– Increases in population size and shifts in composition of population
can be easily accommodated
– Generations that live in the present have full control over present
resources
• Weaknesses
– No possibility to represent marriages and no explicit relation between
ancestors and children
– Efficiency needs to be ensured explicitly by additional assumptions
Discussion of main theorems
•
Proposition 8.2 (existence of dynastic Negishi equilibrium)
1.
2.
3.
4.
5.
6.
Proving (8.8) is a standard convex program, using assumed boundedness of
production (PD1:3), and utility (CD1:5). Convexity is shown and closedness
follows from Lucas/Stokey
Since it is a standard convex program, the value function is bounded and
continuous
Then, perturbation theorem and envelope theorem can be used to further
characterize value function
For Negishi update, budget deficits need to be calculated.
Using boundedness of value function and assumed strict concavity of Wi,
budget deficits can be expressed as function of finite number of parameters
Construct fixed point mapping as in Proposition 3.1
Discussion of main theorems (continued)
•
Proposition 8.5 (existence of equilibrium of pure exchange OLG model with
claims)
1.
2.
Prove equilibrium in truncated model, along lines of proposition 3.3
Extending T to infinity
a)
b)
c)
d)
e)
f)
g)
h)
i)
Since prices cannot be normalized over infinite time horizon, reformulate model with
consumer-specific normalizations
From (1) it follows that there is at least one optimal value for consumption in period T by
consumer born in T
As T increases, the number of optimal values decreases, and with that, the number of optimal
trajectories decreases
To prove that there exists an optimal solution at infinity, define price vectors that are the
optimal prices for t<T, and arbitrary normalized prices for t>T. This implies construction of a
sequence of prices
The set of all possible trajectories and corresponding price sequences is compact (Tychonoff
theorem: the infinite-dimensional product of compact sets is compact)
Therefore, the set of all possible trajectories forms a nonincreasing sequence of non-empty
sets and the set is nonempty at infinity
Recover prices from the consumer-specific prices
Show that consumer demand is continuous in infinite time using A.6.2.
Therefore, consumer demand converges as as the set of all possible trajectories converges for
T to infinity
Discussion of main theorems (continued)
•
Proposition 8.7 (existence of equilibrium of OLG model with
production
1.
2.
As in 8.5, first prove equilibrium in truncated model
Then, extend T to infinity
a)
b)
c)
d)
By same reasoning as in 8.5, there is at least one optimal value for consumption
and one optimal value of capital stock in period T, and the set of all possible
trajectories forms a nonincreasing sequence of non-empty sets and the set is
nonempty at infinity
Recover prices from the consumer-specific prices. Here, an additional problem
is that welfare may become unbounded. Boundedness of the value of initial
capital stock is proved by contradiction: if it were unbounded, the income of
consumer 0 would be infinite, and since his utility function is non-satiated by
assumption, his demand would be infinite. This cannot be an equilibrium and
therefore, the value of initial capital stocks must be bounded
Show that consumer demand is continuous in infinite time using A.6.2. as in 8.5
Therefore, consumer demand converges as as the set of all possible trajectories
converges for T to infinity
Remarks
• Four important differences between dynastic and OLG
models
– in the dynastic model, generations do not overlap
– each generation is altruistic
– in the OLG model, generations have to buy capital but receive
endowments freely, whereas in the dynastic model, consumers
receive capital and possibly endowments as an explicit gift
– in the dynastic model, the individual consumer can borrow from
every other agent in the economy; in the OLG model, the young
cannot borrow from the old, since they cannot repay them later
Remarks (continued)
• It is possible to rewrite a dynastic model of infinite horizon as an infinite
number of consumers who live for one period only and care for the welfare of
their immediate descendants:
uit  max xit 0,vi ,t 1 0 Wi ( xit , vi ,t 1 )
subject to
 i xit  yt   i it
pt xit  it vi ,t 1   t kit  ptit ,
where it is the perceived cost of the utility level  i,t+1 desired for the descendant,
 t is the value of capital and kit is his capital stock at the beginning of period t
• If dynasties no longer care for their descendants, they are no longer linked, the
wealth constraint needs to be split and welfare weights given to each dynasty.
• If this occurs an infinite number of times, we are in the OLG framework
• The inclusion of altruism in OLG models introduces a dynastic feature, since
it links generations.
Remarks (continued)
• In e.g. applications of dynamic models to environmental issues, one
would like to impose a social discount rate that is lower than that of
individuals
• In OLG models, the interpretation as a dynastic model shows that
welfare weights of the individuals living at time t are determined
endogenously. Hence, the discount factor  t 1  t is also endogenous.
• In dynastic models, every dynasty has an explicit discount rate, and
imposing a low discount rate, equal across dynasties, is possible in
principle, but the value of such an exercise for policy purposes is
questionable