Macroeconomics: A Dynamic General Equilibrium
Approach
Mausumi Das
Lecture Notes, DSE
February 2-22, 2016
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Modern Macroeconomics: the Dynamic General
Equilibrium (DGE) Approach
As we have stated before, modern macroeconomics is based on a
dynamic general equilibrium approach which postulates that
Economic agents are continuously optimizing/re-optimizing subject to
their constraints and subject to their information set up. They optimize
not only over their current choice variables but also the choices that
would be realized in future.
All agents have rational expectations: thus their ex ante optimal future
choices would ex post turn out to be less than optimal if and only if
their information set was incomplete and/or there are some random
elements in the economy which cannot be anticipated perfectly.
The optimal choice of all agents are then mediated through the
markets to produce an outcome for the macroeconomy.
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Modern Macroeconomics: DGE Approach (Contd.)
This approach is ‘dynamic’because agents are making choices over
variables that relate to both present and future.
This approach is ‘equilibrium’because the outcome for the
macro-economy is the aggregation of individuals’‘equilibrium’
behaviour.
This approach is ‘general equilibrium’because it simultaneously
takes into account the optimal behaviour of diiferent types of agents
in di¤erent markets and ensures that all markets clear.
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DSG Approach vis-a-vis Traditional Macroeconomics
The need to build macro-models based on internally-consistent,
dynamic optimization exercises of rational agents arose once it was
realized that ad-hoc micro founations for the aggregative system may
not be consistent with one another.
This begs the following question: Why do we need such optimization
based micro-founded framework at all?
Why cannot we just take the aggregative equations as a
representation of the macro-economy and try to estimate various
parameters, using aggregative data?
After all, if we are ultimately interested in knowing how the
macroeconomy would respond to various kinds of policy shocks, all
that we need to do is to econometrically estimate the parameters of
the aggregative system.
Then from the estimated parameter values or coe¢ cients, we can
predict the implcations of various policy changes.
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DSG Approach vis-a-vis Traditional Macroeconomics:
(Contd.)
Indeed, this is exactly how macroeconomic analysis was conducted
traditionally!
As we have already seen, traditional macroeconomics was based on
some aggregative behavioural relationship (e.g., Keyensian Savings
Function - which postulates a relationship between aggregate income
and aggregate savings; Phillips Curve - which posits a relationship
between umployment rate and in‡ation rate).
Often one would construct detailed behavioural equations for the
macroeconomy and would try to estimate the parameters of these
equations using time series data.
To be sure some of these equations would be dynamic in nature.
But optimization over time was not considered to be important or
even relevant. Indeed, the concept of optimization itself - either by
households or …rms or even government - was rather alien in the …eld
Macroeconomics.
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Lucas Critique: Optimization Comes to the Fore
The need to build macro models based explicitly on agents’
optimization exercises came from the so-called Lucas Critique.
Lucas (1976) argued that aggregative macro models which are
estimated to predict outcomes of economic policy changes are useless
simply because the estimated parameters themselves may depend on
the existing policies.
As the policy changes, these coe¢ cients themsleves would change,
thereby generating wrong predictions!
His solution was to build macroecnomic models with clear and
speci…c microeconomic foundations - models that are explicitly based
on agents’optimization exercises.
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Lucas Critique: Optimization Comes to the Fore (Contd.)
Such models will enable us to di¤erentiate between true parameters
- primitives like tastes, technology etc - which are independent of the
government policies, and variables that treated as exogenous by the
agents but are actually endogenous and are in‡uenced by government
policies.
Moreover such models would take into account agents’expectations
about government policies.
Predictions based on such microfounded models would be more
accurate than the aggregative models which club all the true
parameters as well as other policy-related parameters together.
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How Does Micro-foundation Help? An Example
Let us see exactly what Lucas critique means in the context of a
simple example.
Consider the Keynesian savings function, speci…ed as an aggregative
relationship:
St = α 1 + α 2 Yt + e t
An aggregative macro model would take the above behavioural
relationship as given and would estimate the coe¢ cients α1 and α2
from data.
We have already provided a micro-foundation for this kind of
Keynesian Consumption/Savings function.
Let us re-visit that exercise.
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Micro-foundation of Keynesian Savings Function:
We assume that the economy consists of a …nite number (H) of
identical households. We can then talk in terms of a ‘representative’
household.
Let us de…ne a 2-period utility maximization problem of the
representative household as:
Max. log(ct ) + β log(ct +1 )
fct ,ct +1 g
subject to,
(i) Pt ct + st
(ii)
Pte+1 ct +1
= yt ;
= (1 + rte+1 )st + yte+1 .
From (i) and (ii) we can eliminate St to derive the life-time budget
constraint of the household as:
P e ct + 1
yte+1
P t ct + t + 1 e
= yt +
( 1 + rt + 1 )
(1 + rte+1 )
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Micro-foundation of Keynesian Savings Function: (Contd.)
From the FONCs:
ct + 1
= (1 + rte+1 )
βct
Pt
Pte+1
.
Solving we get:
P t ct =
yte+1
1
yt +
(1 + β )
(1 + rte+1 )
Thus
yte+1
1
β
yt
(1 + β )
(1 + β) (1 + rte+1 )
Aggregating over all households:
Yte+1
β
1
St =
Yt
(1 + β )
(1 + β) (1 + rte+1 )
st =
Notice that an aggregative model would equate
h e i
Y t +1
1
to α1 .
(1 + β ) (1 +r e )
t +1
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β
(1 + β )
to α2 and
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Micro-foundation of Keynesian Savings Function: (Contd.)
While the coe¢ cient α2 is indeed based on true parameters
(primitives) and would therefore be una¤ected by policy changes,
coe¢ cient α1 is not.
Any policy that changes the household’s expectation about its future
income or future rate of interest rate would a¤ect α1 .
Thus predicting outcomes of such a policy based on the estimated
values of the aggregative equations would be wrong.
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Modern Macroeconomics: DGE Approach
The Lucas critique and the consequent logical need to develop a
uni…ed micro-founded macroeconomic framework which would allow
us to accurately predict the macroeconomic outcomes in response to
any external shock (policy-driven or otherwise) led to emergence of
the modern dynamic general equilibrium approach.
As before, there are two variants of modern DGE-based approach:
One is based on the assumption of perfect markets (the
Neoclassical/RBC school). As is expected, this school is critical of any
policy intervention, in particular, monetary policy interventions.
The other one allows for some market imperfections (the
New-Keynesian school). Again, true to their ideological underpinning,
this school argues for active policy intervention.
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Modern Macroeconomics: DGE Approach (Contd.)
However, both frameworks are similar in two fundamental aspects:
Agents optimize over in…nte horizon; and
Agents are forward looking, i.e., when they optimize over future
variable they base their expectations on all available information including information about (future) government policies. In other
words, agents have rational expectations.
We now develop the choice-theoretic frameworks for households and
…rms under the DGE approach.
As before, we shall assume that the economy is populated by H
identical households so that we can talk in terms of a representative
household.
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Household’s Choice Problem under Perfect Markets:
In…nite Horizon
Let us examine the consumption-savings choices of the representative
household over in…nite horizon when markets are perfect.
To simplify the analysis, we shall only focus on the consumption
choice of the household and ignore the labour-leisure choice (for the
time being).
At any point of time the household is endowed with one unit of
labour - which it supplies inelastically to the market.
We shall also ignore prices and the concomitant role of money and
focus only on the ‘real’variables.
Let at denote the asset stock of the household at the beginning of
period t.
Then Income of the household at time t: yt = wt + rt at .
We shall assume that savings of an household in any period are
invested in various forms of assets (all assets have the same return),
which augments the household’s asset stock in the next period.
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Household’s Choice Problem: In…nite Horizon (Contd.)
If we do not allow intra-household borrowing, then the representative
household h’s problem would given by:
∞
Max. ∞
∞
fcth gt =0 ,fath+1 gt =0
∑ βt u
cth ; u 0 > 0; u 00 < 0
t =0
subject to
(i) cth
(ii) ath+1
5 wt + rt ath for all t = 0;
= wt + rt ath + (1 δ)ath cth ; ath = 0 for all t = 0; a0h given
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Household’s Choice Problem: In…nite Horizon (Contd.)
Notice that that the household is solving this problem at time 0.
Therefore, in order to solve this problem the households would have
to have some expectation about the entire time paths of wt and rt
from t = 0 to t ! ∞.
We shall however assume that households’have rational expectations.
In this model with complete information and no uncertainty, rational
expectation is equivalent to perfect foresight. We shall use these two
terms here interchangeably.
By virtue of the assumption of rational expectations/perfect foresight,
the agents can correctly guess all the future values of the market
wage rate and rental rate, but they still treat them as exogenous.
As atomistic agents, they belive that their action cannot in‡uence
the values of these ‘market’variables.
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Household’s Choice Problem: In…nite Horizon (Contd.)
∞
Notice that once we choose our consumption time path cth t =0 , the
∞
corresponding time path of the asset level ath+1 t =0 would
automatically get determined from the constraint functions (and vice
versa).
So in e¤ect in this constrained optimization problem, we only have to
choose one set of variables directly. We call them the control
∞
variables. Let our control variable for this problem be cth t =0 .
We can always treat c0 , c1 , c2 ,......as independent variables and solve
the problem using the standard Lagrangean method.
The only problem is that there are now in…nite number of such choice
variables (c0 , c1 , c2 ,....., c∞ ) as well as in…nite number of constraints
(one for each time period from t = 0, 1, 2....., ∞) and things can get
quite intractable.
Instead, we shall employ a di¤erent method - called Dynamic
Programming - which simpli…es the solution process and reduces it to
a univariate problem.
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Dynamic Optimization in Discrete Time: Dynamic
Programming
Consider the following canonical discrete time dynamic optimization
problem:
∞
Max.
∞
subject to
(i) yt
(ii) xt +1
∑ βt Ũ (t, xt , yt )
fxt +1 gt =0 ,fyt gt∞=0 t =0
2 G̃ (t, xt ) for all t = 0;
= f˜ (t, xt , yt ); xt 2 X for all t = 0; x0 given.
Here yt is the control variable; xt is the state variable; Ũ represents
the instantaneous payo¤ function.
(i) speci…es what values the control variable yt is allowed to take (the
feasible set), given the value of xt at time t;
(ii) speci…es evolution of the state variable as a function of previous
period’s state and control variables (state transition equation).
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Dynamic Programming (Contd.)
It is often convenient to use the state transition equation given by (ii)
to eliminate the control variable and write the dynamic programming
problem in terms of the state variable alone:
∞
∑ βt U (t, xt , xt +1 )
Max.
∞
f x t +1 g t =0 t = 0
subject to
(i) xt +1 2 G (t, xt ) for all t = 0; x0 given.
We are going to focus on stationary dynamic programming problems,
where time (t) does not appear as an independent argument either in
the objective function of in the constraint function:
∞
Max.
∞
∑ βt U (xt , xt +1 )
f x t +1 g t =0 t = 0
subject to
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(i) xt +1 2 G (xt ) for all t = 0; x0 given.
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Stationary Dynamic Programming: Value Function &
Policy Function
Ideally we should be able to solve the above stationary dyanamic
programming problem by employing the Lagrange method. Let
∞
xt +1 t =0 denote such a solution.
We can then write the maximised value of the objective function as a
function of the parameters alone, in particular as a function of x0 :
∞
V (x0 )
Max.
∞
=
∑ βt U (xt , xt +1 ) ;
f x t +1 g t =0 t = 0
∞
βt U (xt , xt +1 ) .
t =0
xt +1 2 G (xt ) for all t = 0;
∑
The maximized value of the objective function is called the value
function.
The function V (x0 ) represents the value function of the dynamic
programming problem at time 0.
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Value Function & Policy Function (Contd.)
Suppose we were to repeat this exercise again the next period i.,e. at
t = 1.
Now of course the time period t = 1 will be counted as the initial
point and the corresponding initial value of the state variable will be
x1 .
Let τ denote the new time subscript which counts time from t = 1 to
∞. By construction then, τ t 1.
When we set the new optimization exercise (relevant for
t = 1, 2...., ∞) in terms of τ it looks exactly similar. In particular, the
new value function will be given by:
∞
V (x1 )
Max.
∞
=
∑ β τ U (xτ , xτ +1 ) ;
f x τ +1 g τ =0 τ = 0
∞
β τ U (xτ , xτ +1 ) .
τ =0
xτ +1 2 G (xτ ) for all τ = 0;
∑
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Value Function & Policy Function (Contd.)
Noting the relationship between t and τ, we can immediately see that
the two value functions are related in the following way:
∞
V (x0 ) =
∑ βt U (xt , xt +1 )
t =0
∞
= U (x0 , x1 ) + β ∑ βt
1
U (xt , xt +1 )
t =1
∞
= U (x0 , x1 ) + β
∑ β τ U (xτ , xτ +1 )
τ =0
= U (x0 , x1 ) + βV (x1 ).
The above relationship is the basic functional equation in dynamic
programming which relates two successive value functions recursively.
It is called the Bellman Equation. It breaks down the ini…nite
horizon dynamic optimization problem into a two-stage problem:
what is optimal today (x1 );
what is the optimal continuation path (V (x1 )).
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Value Function & Policy Function (Contd.)
Since the above functional relationship holds for any two successive
values of the state variable,we can write the Bellman Equation more
generally as:
V (x ) = Max [U (x, x̃ ) + βV (x̃ )] for all x 2 X .
x̃ 2G (x )
(1)
The maximizer of the right hand side of equation (2) is called a
policy function:
x̃ = π (x ),
which solves the RHS of the Bellman Equation above.
If we knew the value function V (.) and were it di¤erentiable, we
could have easily found the policy function by solving the following
FONC (called the Euler Equation):
x̃ :
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∂U (x, x̃ )
+ βV 0 (x̃ ) = 0.
∂x̃
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(2)
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Value Function & Policy Function (Contd.)
Unfortunately, the value function is not known.
In fact we do not even know whether it exists; if yes then whether it
is unique, whether it is continuous, whether it is di¤erentiable etc.
A lot of theorems in Dynamic Programming go into establishing
conditions under which a value exists; is unique and has all the nice
properties (continuity, di¤erentibility and others).
For now, without going into futher details, we shall simply assume
that all these conditions are satis…ed for our problem.
In other words, we shall assume that for our problem the value
function exists and is well-behaved (even though we do not know
its precise form).
Once the existence of the value function is established, we can then
solve the FONC (3) (the Euler Equation) to get the policy function.
But there is still one hurdle: what is the value V 0 (x̃ )?
Here the Envelope Theorem comes to our rescue.
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Value Function & Policy Function (Contd.)
Recall that V (x̃ ) is nothing but the value function for the next period
where x̃ is next period’s initial value of the state variable (which is
given - from next period’s perspective).
Since the Bellman equation is de…ned for all x 2 X , we therefore get
a similar relationship between x̃ and its subsequent state value (x̂):
V (x̃ ) = Max [U (x̃, x̂ ) + βV (x̂ )] .
x̂ 2G (x̃ )
Then applying Envelope Theorem:
V 0 (x̃ ) =
∂U (x̃, x̂ )
.
∂x̃
(3)
Combining the Euler Equation (3) and the Envelope Condition (4),
we get the following equation:
∂U (x, x̃ )
∂U (x̃, x̂ )
+β
= 0 for all x 2 X .
∂x̃
∂x̃
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Value Function & Policy Function (Contd.)
Replacing x, x̃, x̂ by their suitable time subscripts:
∂U (xt , xt +1 )
∂U (xt +1 , xt +2 )
+β
= 0; xt given.
∂xt +1
∂xt +1
(4)
Equation (5) is a di¤erence equation which we should be able to solve
to derive the time path of the state variable xt (and consequently that
of the control variable yt ).
Since it is a di¤erence equation of order 2, apart from the initial
condition, we need another boundary condition.
Typically such a boundary condition is provided by the following
Transversality condition (TVC):
lim βt
t !∞
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∂U (xt , xt +1 )
xt = 0.
∂xt
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(5)
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Transversality Condition and its Interpretation:
The TVC is to be read as a complementary slackness condition in the
following way:
∂U (xt ,x t +1 )
as t ! ∞, if βt
> 0, then xt = 0;
∂x t
∂U (xt ,xt +1 )
on the other hand, as t ! ∞, if xt > 0, then βt
=0
∂x t
∂U (x ,x
)
t t +1
In interpreting the TVC, notice that βt
captures the
∂xt
marginal increment in the pay-o¤ function associated with an increase
in the current stock, or its shadow price.
The TVC states that if the shadow price is positive then at the
terminal date, agents will not leave any stock unused (or would leave
any postive stock at the end of the period); on the other hand, if any
stock indeed remains unused at the terminal date, then it must be the
case that its shadow valuation is zero.
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Stationary Dynamic Programming: Existence &
Uniqueness of Value Function
We now provide some su¢ cient conditions for the Value function of
the above stationary dynamic programming problem to exist, to be
twice continuously di¤erentiable, to be concave etc.
We just state the theorems here without proof. Proofs can be found
in Acemoglu (2009).
1
2
Let G (x ) be non-empty-valued, compact and continuous in all x 2 X
where X is a compact subset of <. Also let U : XG ! < is
continuous, where XG = f(xt , xt +1 ) 2 X X : xt +1 2 G (xt )g . Then
there exits a unique and continuous function V : X ! < that solves
the stationary dynamic programming problem speci…ed earlier.
Let us further assume that U : XG ! < is conacave and is
continuously di¤erentiable on the interior of its domain XG . Then the
unique value function de…ned above is strictly concave and is
di¤erentiable.
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Non-Stationary Dynamic Programming: Existence &
Uniqueness of Value Function
Even when the dynamic programming problem is non-stationary, we
can …nd analogous su¢ cient conditions that will ensure the existence,
uniqueness, concavity and di¤erentiability of the corresponding value
function.
Then we can proceed exactly as above to write down the Bellman
equation that relates the value functions of two successive time
periods and then solve for the optimal policy function from the
corresponding Euler Equation and the Envelope condition.
All the economic problems that we would be looking at in this course
will satisfy these su¢ ciency properties.
So we shall stop bothering about this su¢ ceny condition from now on
and focus on applying the dynamic programming technique to the
economic problems at hand.
Interested students can look up Acemoglu (2009): Introduction to
Modern Economic Growth, Chapter 6, for the theorems and proofs.
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Back to Household’s Choice Problem: In…nite Horizon
Recall that we had speci…ed the representative household’s
optimization problem under in…nite horizon as:
∞
Max. ∞
∞
fcth gt =0 ,fath+1 gt =0
∑ βt u
cth ; u 0 > 0; u 00 < 0
t =0
subject to
(i) cth
(ii) ath+1
5 wt + rt ath for all t = 0;
= wt + rt ath + (1 δ)ath cth ; ath = 0 for all t = 0; a0h given
However in specifying the problem, we assumed that there is no
intra-household borrowing.
This assumption of no borrowing is too strong, and we do not really
need it for the results that follow.
So let us relax that assumption to allow households to borrow from
one another if they so wish.
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Household’s Choice Problem: In…nite Horizon (Contd.)
Allowing for intra-household borrowings means that constraint (i)
would no longer hold. A household can now consume beyond its
current income at any point of time - by borrowing from others.
Allowing for intra-household borrowings also means that a household
now has two forms of assets that it can invest its savings into:
1
2
physical capital (kth );
…nancial capital, i.e., lending to other households (lth
bth ).
Let the gross interest rate on …nancial assets be denoted by (1 + r̂t ) .
Let physical capital depreciate over time at a constant rate δ. Then
the gross interest rate on investment in physical capital is given by
( rt + 1 δ ) .
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Household’s Choice Problem: In…nite Horizon (Contd.)
Arbitrage in the asset market ensures that in equlibrium two interest
rates are the same :
1 + r̂t = 1 + rt
δ ) r̂t = rt
δ.
Thus we can de…ne the total asset stock held by the household in
period t as ath kth + lth .
Notice that lth < 0 would imply that the household is a net borrower.
Hence the aggregate budget constraint of the household is now given
by:
cth + sth = wt + r̂t ath , where sth ath+1 ath .
Re-writing to eliminate sth :
ath+1 = wt + (1 + r̂t )ath
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cth .
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Household’s Choice Problem: Ponzi Game
But allowing for intra-household borrowing brings in the possibility of
households’playing a Ponzi game, as explained below.
Consider the following plan by a household:
Suppose in period 0, the household borrows a huge amount b̄ - which
would allow him to maintain a very high level of consumption at all
subsequent points of time. Thus
b0 = b̄.
In the next period (period 1)he pays back his period 0 debt with
interest by borrowing again (presumably from a di¤erent lender). Thus
his period 1 borrowing would be:
b1 = (1 + r̂0 )b0 .
In period 2 he again pays back his period 1 debt with interest by
borrowing afresh:
b2 = (1 + r̂1 )b1 = (1 + r̂1 )(1 + r̂0 )b0 .
and so on.
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Household’s Choice Problem: Ponzi Game (Contd.)
Notice that proceeding this way, the household e¤ectively never pays
back its initial loan b̄; he is simply rolling it over period after period.
In the process he is able to perpetually maintain an arbitrarily high
level of consumption (over and above his current income).
His debt however grows at the rate r̂t :
bt +1 = (1 + r̂t )bt
which implies that lim ath ' lim bth ! ∞.
t !∞
t !∞
This kind scheme is called a Ponzi …nance scheme.
If a household is allowed to play such a Ponzi game, then the
household’s budget constraint becomes meaningless. There is
e¤ectively no budget constraint for the household any more; it can
maintain any arbitrarily high consumption path by playing a Ponzi
game.
To rule this out, we impose an additional constraint on the
household’s optimization problem - called the No-Ponzi Game
Condition.
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Household’s Choice Problem: No-Ponzi Game Condition
One Version of No-Ponzi Game (NPG) Condition:
ath
= 0.
t !∞ (1 + r̂0 )(1 + r̂1 )......(1 + r̂t )
lim
This No-Ponzi Game condition states that as t ! ∞, the present
discounted value of an household’s asset must be non-negative.
Notice that the above condition rules out Ponzi …nance scheme for
sure.
If you play Ponzi game then lim ath '
lim b h , when the latter term
t !∞
t !∞ t
is growing at the rate (1 + r̂t ).
For simplicity, let us assume interest rate is constant at some r̄ . Then
bth = (1 + r̄ )t b̄.
Plugging this in the LHS of the NPG condition above:
ath
( bth )
(1 + r̄ )t b̄
' lim
= lim
=
t
t
t !∞ (1 + r̄ )
t !∞ (1 + r̄ )
t !∞ (1 + r̄ )t
lim
b̄ < 0.
This surely violates the NPG condition speci…ed above.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Choice Problem: No-Ponzi Game Condition
(Contd.)
At the same time the NPG condition speci…ed above is lenient enough
to allow for some borrowing.
In fact the condition even permits perpetual borrowing as long as
borrowing grows at a rate less than the corresponding interest rate.
To see this, suppose the household’s borrowing is growing at some
rate ḡ < r̄ such that
bth = (1 + ḡ )t b̄.
Plugging this in the LHS of the NPG condition above:
ath
( bth )
(1 + ḡ )t b̄
'
lim
=
lim
=
t !∞ (1 + r̄ )t
t !∞ (1 + r̄ )t
t !∞ (1 + r̄ )t
lim
Notice that ḡ < r̄ implies that the term
fraction and as t ! ∞,
Das (Lecture Notes, DSE)
1 +ḡ
1 +r̄
t
1 + ḡ
1 + r̄
b̄ lim
t !∞
1 + ḡ
1 + r̄
t
is a positive
! 0.
DGE Approach
February 2-22, 2016
36 / 105
.
Household’s Choice Problem: No-Ponzi Game Condition
(Contd.)
Since b̄ is …nite, this implies that in this case
ath
! 0.
t !∞ (1 + r̄ )t
lim
In other words, the NPG condition is now satis…ed at the margin!
Economically, this kind of borrowing behaviour implies that the debt
of the agent is not exploding and the agent must have started
repaying at least some part of it (though not all) from his own pocket!
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Choice Problem - Revisited:
After imposing the No-Ponzi Game condition, the household’
optimization problem now becomes:
∞
Max. ∞
∞
fcth gt =0 ,fath+1 gt =0
∑ βt u
cth
t =0
subject to
(i) ath+1 = wt + (1 + r̂t )ath
cth ; ath 2 < for all t = 0; a0h given.
(ii) The NPG condition.
Here cth is the control variable and ath is the state variable.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Choice Problem - Revisited: (Contd.)
We can now apply the dynamic programming technique to solve the
household’s choice problem.
First let us use constraint (i) to eliminate the control variable and
write the above dynamic programming problem in terms of the state
variable alone:
∞
Max.∞ ∑ βt u
fath+1 gt =0 t =0
n
wt + (1 + r̂t )ath
ath+1
o
Corresponding Bellman equation relating V (a0h ) and V (a1h ) is given
by:
h n
o
i
V (a0h ) = Max u
w0 + (1 + r̂0 )a0h a1h
+ βV (a1h ) .
fa1h g
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Bellman Equation
More generally, we can write the Bellman equation for any two time
periods t and t + 1 as:
h n
o
i
V (ath ) = Max u
wt + (1 + r̂t )ath ath+1
+ βV (ath+1 ) .
fath+1 g
Maximising the RHS above with respect to ath+1 , from the FONC:
n
o
u0
wt + (1 + r̂t )ath ath+1
= βV 0 (ath+1 )
(6)
Notice that V (ath+1 ) and V (ath+2 ) would be related through a similar
Bellman equation:
h n
o
i
V (ath+1 ) = Max u
wt +1 + (1 + r̂t +1 )ath+1 ath+2
+ βV (ath+2 ) .
fath+2 g
Applying Envelope Theorem on the latter:
n
o
V 0 (ath+1 ) = u 0
wt +1 + r̂t +1 ath+1 ath+2
.(1 + r̂t +1 ).
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Optimal Solutions
Combining (5) and (6):
n
o
u0
wt + r̂t ath ath+1
n
o
= βu 0 wt +1 + r̂t +1 ath+1 ath+2 (1 + r̂t +1 ).
The above equation implicitely de…nes a 2nd order di¤erence equation
is ath .
However we can easily convert it into a 2
di¤erence equations in the following way.
Das (Lecture Notes, DSE)
DGE Approach
2 system of …rst order
February 2-22, 2016
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Household’s Problem: Optimal Solutions (Contd.)
Noting that the terms inside the u 0 (.) functions are nothing but cth
and cth+1 respectively, we can write the above equation as:
u 0 cth = βu 0 cth+1 (1 + r̂t +1 ).
(8)
We also have the constraint function:
ath+1 = wt + (1 + r̂t )ath
cth ; a0h given.
Equations (7) and (8) represents a 2 2 system of di¤erence
equations which implicitly de…nes the ‘optimal’trajectories cth
∞
and ath+1 t =0 .
(9)
∞
t =0
The two boundary conditons are given by the initial condition a0h , and
the NPG condition.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Optimal Solution Path to Household’s Problem: An
Example
Let us look at an explicitly characterisation of the household’s optimal
paths for a speci…c example.
Suppose
u (c ) = log c
Let us also assume that wt = w̄ and rt = r̄ for all t .
Then we can immediately get two di¤erence equations characterizing
the optimal trajectories for the household as:
cth+1 = β(1 + r̄ )cth
(10)
and
ath+1 = w̄ + (1 + r̄ )ath
cth ; a0h given.
(11)
The two equations along with the two boundary conditons can be
solved explicitly to derive the time paths of cth and ath .
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Optimal Solutions (Contd.)
Equation (9) is a linear autonomous di¤erence equation, which can be
directly solved (by iterating backwards) to get the optimal
consumption path as:
cth = βt (1 + r̄ )t c0h .
(12)
However,we still cannot completely characterise the optimal path
because we still do not know the optimal value of c0h . (Recall that c0h
is not given; it is to be chosen optimally).
Here the NPG condition comes in handy in identifying the optimal c0h .
Note that the NPG condition in this case is given by:
ath
= 0.
t !∞ (1 + r̄ )t
lim
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Role of the NPG Condition
Now let us take the budget constraint of the household at any future
date T > 0:
aTh +1 = w̄ + (1 + r̄ )aTh
cTh .
Iterating backwards,
aTh +1 = w̄ + (1 + r̄ )aTh
cTh
h
= w̄ + (1 + r̄ ) w̄ + (1 + r̄ )aTh
1
cTh
1
= ....
T
=
∑
w̄ (1 + r̄ )T
t
t =0
T
∑
i
cth (1 + r̄ )T
cTh
t
+ (1 + r̄ )T +1 a0h .
t =0
Rearranging terms:
T
aTh +1
=
∑
(1 + r )T
t =0
Das (Lecture Notes, DSE)
w̄
(1 + r̄ )t
+ (1 + r̄ )a0h
DGE Approach
T
∑
t =0
cth
(1 + r̄ )t
February 2-22, 2016
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Household’s Problem: Role of the NPG Condition (Contd.)
Now let T ! ∞. Then applying the NPG condition to the LHS, we
get:
∞
∞
w̄
ch
∑ (1 + r̄ )t + (1 + r̄ )a0h ∑ (1 +t r̄ )t = 0
t =0
t =0
∞
i.e.,
∑
t =0
cth
(1 + r̄ )t
5
∞
∑
t =0
w̄
(1 + r̄ )t
+ (1 + r̄ )a0h .
(13)
Equation (12) represents the lifetime budget constraint of the
household. It states that when the NPG condition is satis…ed, then
the discounted life-time consumption stream of the household cannot
exceed the sum-total of its discounted life-time wage earnings and the
returns on its initial wealth holding.
It is easy to see that even though we have speci…ed the NPG
condition in the form of an inequality, the households would always
satisfy it at the margin such that it holds with strict equality.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Role of the NPG Condition (Contd.)
Given that equation (12) holds with strict equality, we can now
identify the optimal value of c0h .
We had already derived the optimal time path of cth as:
cth = βt (1 + r̄ )t c0h .
Using this in equation (12) above, we get:
∞
∑
t =0
∞
)
∑
β
t
=
w̄
(1 + r̄ )t
∑
t =0
c0h =
t =0
) c0h = (1
Das (Lecture Notes, DSE)
∞
βt (1 + r̄ )t c0h
(1 + r̄ )t
β)
"
"
∞
∑
t =0
∞
∑
t =0
+ (1 + r̄ )a0h
#
w̄
(1 + r̄ )t
+ (1 + r̄ )a0h
w̄
(1 + r̄ )t
+ (1 + r̄ )a0h .
DGE Approach
#
February 2-22, 2016
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Household’s Problem: NPG vis-a-vis TVC
So for this particular example, we have been able to explicitly solve
for the optimal consumption path of the households.
But there is a problem that we still need to sort out.
Recall that while discussing the dynamic programming problem we
had speci…ed a transversality condition (TVC) as one of our boundary
condition (Refer to equation (5) speci…ed earlier).
Then in de…ning the household’s problem with intra-household
borrowing, we have introduced the NPG condition as another
boundary condition.
So we now have a problem of plenty: for a 2 2 dynamic system, it
seems that we have three boundary conditions!!!
Between the TVC and the NPG condition, which one should we use
to characterise the solution?
As it turns out, along the optimal path the NPG condition and the
TVC become equivalent.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: NPG vis-a-vis TVC (Contd.)
To see this, let us take a closer look at the TVC as had been speci…ed
earlier in equation (5)
In the context of the current problem, this transversality condition
would be given as follows (verify this):
lim βt u 0 (cth )(1 + r̂t )ath = 0
t !∞
For our speci…c example with log utility and constant factor prices,
this condition reduces to
1
lim βt (1 + r̄ )ath = 0
t !∞ c h
t
Now given the solution path of cth , we can further simplify the above
condition to:
1
(1 + r̄ )ath = 0
lim βt t
h
t
t !∞
β (1 + r̄ ) c0
ath
=0.
t !∞ (1 + r̄ )t
) lim
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
49 / 105
Household’s Problem: NPG vis-a-vis TVC (Contd.)
But this is nothing but our earlier NPG condition - now holding with
strict equality!
Thus when the household is on its optimal path, the NPG condition
and the Transversality condition become equivalent - except that the
NPG condition now must hold with equality.
So in identifying the optimal trajectories, we could use either of them
as the relevant boundary condition.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Heterogenous Agents
So far we have assumed that all households are identical so that we
could carry out the analysis in terms of a representative agent.
But if all households are identical in every respect, then allowing for
intra-household borrowing and the consequent NPG condition does
not make sense. In any case one side of the borrowing/lending market
will always be missing and hence no borrowing or lending will ever
take place.
All these conditions make sense only if households are heterogenous.
Let us now extend the framework to allow for heterogenous
households.
From now on, we shall assume that households di¤er in terms of their
intial wealth.
However, we shall continue to assume that preference-wise they are
all identical.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Heterogenous Agents (Contd.)
Let us now go back to our earlier example of log utility and constant
factor returns.
We have already seen that for any household with an initial wealth
level of a0h will have the following optimal consumption path:
cth = βt (1 + r̄ )t c0h .
where
c0h = (1
β)
"
∞
∑
t =0
w̄
(1 + r̄ )t
#
+ (1 + r̄ )a0h .
Notice that the rate of growth of consumption along the optimal path
is given by β(1 + r̄ ) 1, which is independent of the initial wealth
level (or even the accumulated wage income!).
Thus along the optimal path, consumption of all households grow at
the same rate - irrespective of their initial wealth.
The initial wealth only determines the level of optimal consumption:
higher initial wealth means higher level of consumption.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Heterogenous Agents (Contd.)
This is a striking result because it tells us that the initial wealth has
no growth e¤ect, only level e¤ect.
It also tells us that when all households are following their respective
optimal trajectories, the initial inequality in consumption will be
maintained perpetually.
What about the asset stock?
Note that we can solve for the time path of ath (given a0h ) by solving
the following dynamic equation:
ath+1 = w̄ + (1 + r̄ )ath
[ β(1 + r̄ )]t c0h
This is a di¤erence equation which linear but non-autonomous;
solving this would require more elaborate technique than mere
backward induction.
We shall come back to this equation once we discuss the methods of
solving such non-autonomous di¤erence equations.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Household’s Problem: Heterogenous Agents (Contd.)
All these results were of course derived under the assumption of
constant factor returns. When factor returns (wt and rt ) are changing
over time, the consumption growth rate itself will change.
We can easily generalize the results to such non-autonomous cases
but to see what is happenning to the growth rate of consumption in
such cases, we shall have to derive the precise time path of rt , which
means we shall have to discuss the production side story.
Before we move on to the production side story, notice that in
deriving all these results, we have also made use of the log utility,
which we know is special.
Can we generalize these results to other utility functions as well?
It turns out, all the results will go through for a broad class of utility
functions called the CRRA variety:
u (c ) =
Das (Lecture Notes, DSE)
c1 σ
; σ 6= 1.
1 σ
DGE Approach
February 2-22, 2016
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Household’s Problem: Heterogenous Agents (Contd.)
This utility function has several interesting characteristics:
1
2
3
It is associated with constant elasticity of marginal utility:
cu 00 (c )
=σ
u 0 (c )
It is associated with constant relative risk aversion (as de…ned by the
cu 00 (c )
=σ
Arrow-Pratt measure of relative risk aversion): 0
u (c )
It is associated with constant elasticity of substitution between
d ctc+t 1 / ctc+t 1
1
current and future consumption:
=
0
0
u (c )
u (c )
σ
d u 0 (tc+t )1 / u 0 (tc+t )1
Exercise: Assume that wages and interest rates are constant and use
the dynamic programming technique to derive the dynamic equation
for the optimal consumption path of an agent with an initial asset
stock of a0h , when his utility function is of the CRRA variety, as
de…ned above.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Production Side Story: A Micro-founded Dynamic Theory
of Firms’Choices?
We typically do not think of the …rm’s choice problem as a dynamic
one. In a perfectly competitive set up, a …rm is a blackbox: it does
not own any factors of production and merely decides how much
labour to employ and how much capital to hire in every period so as
to maximise its current pro…t (taking all prices as given).
As long as such hiring decisions do not a¤ect future pro…ts, setting
the optimization problem in a dynamic framework (i.e., optimizing
over multiple time periods) does not bring in any extra insight over
the static optimization problem.
The …rms will have meaningful dynamic choices if and only if the
…rms own the capital stock that they employ and part of the current
pro…t can be invested in augmenting their capital stock which a¤ects
their future pro…tability.
We now turn to the optimal choices of …rm in such a dynamic setting.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour:
Once again we ignore the price dynamics and assume that the price
level remains constant at unity.
Thus all variables that we consider below are expressed in terms of
their real values.
Consider a …rm that which produced a …nal commodity in every
period using a production function
Yt = F (Nt , Kt ).
The usual diminishing marginal product proprties and CRS property
are assumed to hold.
At time t, the capital stock available to the …rm is given. However it
can augment its capital stock over time by investing an amount It
(out of its current pro…t) which augments the capital stock in the
next period:
Kt +1 = It + (1
Das (Lecture Notes, DSE)
δ)Kt for all t = 0, 1, 2, ....∞.
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour: (Contd.)
Assuming a competitive market structure (such that the …rm is a
price taker), its instantaneous net pay-o¤/pro…t at any time period t
is given by:
π t = F (Nt , Kt ) wt Nt It .
Let r̂ be the given time-invariant net rate of interest in the economy.
The …rm is believed to be in…nitely-lived. So the present discounted
value of its sum of net pro…t from time 0 to ∞ is given by:
∞
V =
∑
t =0
∞
F (Nt , Kt ) wt Nt
πt
=
∑
t
(1 + r̂ )
(1 + r̂ )t
t =0
It
.
(15)
The …rm maximizes (15) subject to its period by period capacity
augmentation equation (given by (14)), to determine its optimal level
of employment in every period (Nt ) and its optimal level of
investment in every period (It ).
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour: (Contd.)
However, since choosing current level of investment is equivalent to
choosing next period’s capital stock, we can use the capacity
augmentation equation (given by (14)) to eliminate It and write down
the optimization problem of the …rm as an unconstraint problem:
∞
Max
∞
V =
∞
f N t g 0 , f K t +1 g 0
∑
F (Nt , Kt )
t =0
wt Nt Kt +1 + (1
(1 + r̂ )t
δ)Kt
; K0 given.
Corresponding Bellman equation relating V (K0 ) and V (K1 ):
V (K0 ) = Max
fL 0 ,K 1 g
F (K0 , N0 )
w0 N0
fK1
(1
δ)K0 g +
V (K1 )
.
(1 + r̂ )
More generally:
V (Kt ) = Max
fL t ,K t +1 g
Das (Lecture Notes, DSE)
F (Kt , Nt )
wt Nt
DGE Approach
fKt +1
(1
δ)Kt g +
February 2-22, 2016
V (Kt +1 )
(1 + r̂ )
59 / 105
A Dynamic Theory of Firm Behaviour: (Contd.)
This yields the following set of …rst order conditions:
∂V (Kt )
∂Nt
∂V (Kt )
(ii)
∂Kt +1
(i)
= 0 ) FN (Nt , Kt )
= 0)
wt = 0
∂V (Kt +1 )
= (1 + r̂ )
∂Kt +1
If the value function exists and is di¤erentiable, we can once again
apply the envelope theorem to the value function de…ned for period
t + 1, (i.e., V (Kt +1 )) to get:
∂V (Kt +1 )
= FK (Kt +1 , Nt +1 ) + (1
∂Kt +1
Das (Lecture Notes, DSE)
DGE Approach
δ)
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour: (Contd.)
This tells us that the dynamically optimizing …rm would choose an
employment level and an investment level in every period such that
FN (Nt , Kt ) = wt ;
FK (Nt +1 , Kt +1 )
δ = r̂ ) FK (Nt +1 , Kt +1 ) = r .
These decision rules look exactly analogous to the decisions that will
undertaken by a …rm in a static optimization framework where it only
maximises its current pro…t period after period.
Thus adding a dynamic framework does not add anything extra
to the producer’s side of the story; their optimal decision
making rules remain identical to the static story.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour: (Contd.)
This equivalence arises because the marginal bene…ts and the marginal
costs assocaited with both the exercises are precisely the same:
In the dynamic set up, an act of investment generates more pro…t
tomorrow by generating more output; hence the associated net
marginal bene…t is captured by the correponding MPK δ. The
relative cost on the other hand is measured by r̂ , since the extra pro…t
appearing tomorrow will be discounted at the rate r̂ .
In the static set up, an additional unit of capital currently employed
generates more pro…t today by generating more output; hence the
associated net marginal bene…t is once again captured by the
correponding MPK . The relative cost on the other hand is again
measured by r , since this is the rental price that the …rm has to pay to
the capital-owners (households).
Since both the bene…ts and the costs appear in the same time period
(tomorrow - for the dynamic problem; today - for the static problem)
and since the bene…ts and costs associated with the two frameworks
are also identical, it is not suprising that they produce identical
optimal decision rules.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour: (Contd.)
To make the dynamic story of …rm behaviour look di¤erent from the
corresponding static story, we must bring in some additional
intertemporal linkages, e.g., a cost that is incurred today but the
bene…t is reaped only tomorrow.
Introducing an adjustment cost of investment serves this purpose.
So let us now augment the dynamic model of the …rm to take into
account some adjustment costs of investment.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour: Adjustment Costs
As before, consider a representative …rm which takes all its input
prices as given. It has access to a production technology that uses
labour and capital as inputs:
Yt = F (Kt , Nt )
The …rm hires labour from the labour market at the market wage rate
wt. But it owns the capital stock that it employs.
The stock of capital owned by the …rm can be augmented over time
by investing a part of the pro…t.
However the investment process is now subject to adjustment costs.
Adding new machines is disruptive to the production process and
leads to loss of revenue.
These adjustment costs are convex: they are low when the level of
investment is low; but rise steeply as the level of investment rises.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
For convenience we shall assume a quadratic cost function:
C (It ) = bIt2
At any point of time the net pay-o¤/pro…t of the …rm is:
π t = F (Kt , Nt )
wt Nt
bIt2
It
The dynamic optimization problem of the representative …rm can then
be written as:
∞
Max.
∞
fIt gt∞=0 ,fL t gt =0 ,fK t +1 gt∞=0
∑
t =0
π (Kt , Nt , It )
(1 + r̂ )t
subject to
(i) Kt +1
Kt = It
δKt ; Kt 2 < for all t = 0; K0 given.
Here It and Nt are control variables and Kt is the state variable.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
We can use constraint (i) to eliminate the control variable It and
write the above dynamic programming problem in terms of the Kt
and Nt and Kt +1 alone:
∞
Max.
∞
fL t gt =0 ,fK t +1 gt∞=0
1
∑ (1 + r̂ )t [F (Kt , Nt )
wt Nt
t =0
b fKt +1
(1
δ)Kt g2
fKt +1
(1
δ)Kt g]
Corresponding Bellman equation relating V (K0 ) and V (K1 ):
V (K0 ) =
Max [F (K0 , N0 )
fN 0 ,K 1 g
fK1
Das (Lecture Notes, DSE)
DGE Approach
w0 N0
(1
b fK1
(1
δ)K0 g] +
δ)K0 g2
V (K1 )
].
(1 + r )
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
More generally:
V (Kt ) =
Max [F (Kt , Nt )
fKt +1
b fKt +1
wt Nt
fN t ,K t +1 g
(1
δ)Kt g] +
(1
δ)Kt g2
V (Kt +1 )
].
(1 + r̂ )
From the FONCs:
∂V (Kt )
∂Nt
∂V (Kt )
∂Kt +1
Das (Lecture Notes, DSE)
DGE Approach
= 0;
(17)
= 0.
(18)
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
Simplifying:
FN (Kt , Nt ) = wt ;
(19)
and
1 ∂V (Kt +1 )
(1 + r̂ ) ∂Kt +1
i.e., It
Das (Lecture Notes, DSE)
= 2b fKt +1
=
(1
δ)Kt g + 1
1
1 ∂V (Kt +1 )
2b (1 + r̂ ) ∂Kt +1
DGE Approach
1
February 2-22, 2016
(20)
68 / 105
A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
∂V (K
)
The term ∂K t +t +1 1 in the above equation, which is the derivative of
the value function, measures the marginal valuation of an unit
addition of capital stock in terms of the entire maximised stream of
pro…ts.
In other words, this terms measures the shadow price of investment at
time t. We shall denote this by qt .
Thus equation (20) can be written as:
It =
1
qt
2b (1 + r̂ )
1
(21)
Interpretation?
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
Assuming that the value function exists and is di¤erentiable, we can
once again apply the envelope theorem to the value function de…ned
for period t + 1, (i.e., V (Kt +1 )) to get:
qt
∂V (Kt +1 )
∂Kt +1
= FK (Kt +1 , Nt +1 ) + (1
δ) [2b fKt +2
(1
δ)Kt +1 g + 1]
Plugging this value in equation (21), we once again get an implicit
2nd order di¤erence equation in Kt , which will determine the optimal
investment plan and therefore the optimal capital trajectory of the
…rm, subject to the given initial value of the capital stock (K0 ) and
the associated Transversality condition.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
Notice that from equation (21):
It T 0 according as qt T (1 + r̂ )
i.e., FK (Kt +1 , Nt +1 )
δ + (1
δ) [2b fKt +2
(1
δ)Kt +1 g] T r̂ .
Thus it is no longer optimal for …rms to invest in capital stock as long
the net return (FK (Kt +1 , Nt +1 ) δ) is greater than the cost (r̂ ).
Indeed to induce …rms to investment, the marginal bene…t from
investment now has to be su¢ ciently high to cover the associated
adjustment cost. This explain the presence of an additional term in
the LHS of the above equation.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
As before, it might be easier to deal with the dynamics if we convert
the 2nd order di¤erence equation in a 2 2 system of …rst order
di¤erence equations. This can be done in the following way.
∂V (K )
Replace ∂K t +t +1 1 in equation (20) by the above expression. Further,
note that fKt +2 (1 δ)Kt +1 g It +1 , while
Nt +1 : FN (Kt +1 , Nt +1 ) = wt +1 ) Nt +1 = f (Kt +1 , wt +1 ). Using all
these in the equation (20), we get the following system of di¤erence
equation in the control variable (It ) and the state variable (Kt ):
(1 + r̂ ) [2bIt + 1] = FK (Kt +1 , f (Kt +1 , wt +1 )) + (1
Kt +1 = It + (1
δ) [2bIt +1 + 1]
δ)Kt
These two equations along with the initial condition and the TVC will
completely characterise the optimal investment path for the …rm.
Exercise: Use equation (15) to write down the TVC for this
particular investment problem of the …rm.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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A Dynamic Theory of Firm Behaviour with Adjustment
Costs: (Contd.)
Notice that the above di¤erence equation is still implicit. We cannot
derive the explicit solution and the precise optimal investment path
unless we assume some speci…c production function.
Later we shall look at some speci…c examples and characterise the
precise optimal paths in the context of those examples.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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Bringing the Households and Firms Together: The General
Equilibrium Set Up
So far we have looked at the households’problem and the …rms’
problem in isolation.
Both sets of agents were assumed to be ‘atomistic’, who take all the
market variables as exogenously given.
But in the aggregate economy, the market variables are not
exogenous; they are determined precisely by the aggregate actions of
the households and the …rms.
So we now consider the general equilibrium set up where the
households’and the …rms’actions - mediated through the market generates some aggregative behaviour for the entire macroeconomy.
The corresponding solution for the aggregate economy will be called
the ‘decentralized’or ‘market’equilibrium solution (as opposed to a
contrasting case where production is centralized under a social
planner).
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium in the Decentralized Economy:
Recall that we have solved the production side story under two
alternative of assumptions - one where investment does not entail any
adjustment costs and another one where investment is associated
with some adjusment costs.
We have seen that in the …rst case, setting the …rms’problem in a
dynamic framework (where the …rms own the capital stock and carrry
out the act of investment) yields results which are equivalent to the
results that we would obtain in a static framework where …rms do not
own the capital stock and simply rent them in from the households.
Since these two exercises are equivalent, when we consider the
problem without adjustment costs, we shall simply revert back to the
assumption that all capital stocks are owned by the households; …rms
merely rent them in period by period to maximise their static
period-by-period pro…t.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium in the Decentralized Economy:
(Contd.)
In the presence of adjustment costs, the dynamic set up generates
results di¤erent from the static one.
So in discussing the general equilibrium outcome for this case, we
shall retain the assumption that capital stocks are owned by the …rms
and investment are carried out by …rms.
However, this begs the following question: what happens to the
accumulated pro…ts of the …rms? How do we bring it back to the
circular ‡ow of income such that it eventually goes back to the
households - to be consumed or saved?
Here we shall assume that even though the …rms are carrying out the
act of investment to maximise the pro…ts, ultimately the households
are the owners of the …rms because they own shares of the …rms.
Thus the accumulated pro…ts of the …rms ‡ow back to the households
in the form of dividend income.
Hence we’ll have to suitably modify the households’problem.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium in the Decentralized Economy:
(Contd.)
We now look at both these general equilibrium cases one by one.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs
Let us quickly revisit the household and …rm speci…cations for this
case:
We have H single-membered households which are identical in terms of
preferences but di¤er in terms of their initial asset holdings;
Each household is endowed with one unit of labour - which it supplies
inelastically to the market in every period;
Households are atomistic and take the market wage rate (wt ) and
market the interest rate (rt ) (and the corresponding net interest rate,
r̂t = rt δ) as given. But they are endowed with perfect foresight - so
they can correctly guess the entire stream of current & future wage
=∞
t =∞
rates fwt gtt =
0 , as well as the current & future interest rates frt gt =0 .
The households own the entire labour and the capital stock in the
economy. In addition, they also hold loans against one another.
Each household maximises its lifetime utility subject to its period by
period budget constraint.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
On the production side:
There are M identical …rms endowed with a technology to produce the
…nal commodity.
The technology uses capital and labour as inputs; it exhibits
diminishing returns with respect to each of the inputs; it is also CRS in
both the inputs.
The …rms do not own any capital all labour; they hire labour and
capital from the market to carry out production in each period.
The …rms operate under perfect competetion; they take the market
wage rate (wt ) and market the interest rate (rt ) as given.
The …rms optimally decide about how much labour/capital to employ
in every period so as to maximise its period-by-period pro…t.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
For expositional simplicity, we shall assume speci…c functional forms
for the utility function and the production function. Accordingly, let
u (c ) = log c
and
Yt = F (Kt , Nt ) = (Kt )α (Nt )1
Das (Lecture Notes, DSE)
DGE Approach
α
; 0 < α < 1.
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
The optimization problem of a household h with an initial asset
holding of a0h is given by:
∞
Max. ∞
∞
fcth gt =0 ,fath+1 gt =0
∑ βt log
cth
t =0
subject to
ath+1 = wt + (1 + r̂t )ath
cth ; ath = 0 for all t = 0; a0h given.
Characterization of the optimal paths:
cth+1 = β(1 + r̂t +1 )cth ;
ath+1 = wt + (1 + r̂t )ath
(22)
cth ;
(23)
ath
= 0 (NPG/TVC).
t !∞ (1 + r̂0 )(1 + r̂1 )......(1 + r̂t )
a0h given; lim
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
While one can solve for the optimal paths for each household, we are
more interested in tracking the aggregate economy.
For this purpose, de…ne per capita consumption and per capita asset
holding in this economy as:
H
H
∑ ath
∑ cth
ct
h =1
; at
H
h =1
.
H
Recall that households hold their assets in the form of either physical
capital or …nancial capital (loans) such that
H
∑
at
Das (Lecture Notes, DSE)
h =1
H
H
H
∑ (kth + lth )
ath
=
h =1
H
DGE Approach
∑
=
H
∑ lth
kth
h =1
H
+
h =1
H
.
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
Since one household’s lending is another household’s borrowing, on
H
∑ lth
the aggregate
h =1
H
= 0. Thus,
H
∑ ath
= h =1
kt ,
H
H
where kt denotes the per capita capital stock in the economy.
Notice that the individual optimal transition equations (22 & 23) can
be used to derive the transition equations for the per capita
consumption and per capita capital stock of the economy in the
following way:
at
H
∑
ct + 1
h =1
H
∑ kth
H
∑
cth+1
h =1
H
Das (Lecture Notes, DSE)
=
H
β(1 + r̂t +1 )cth
h =1
H
DGE Approach
∑ cth+1
= β(1 + r̂t +1 ) h =1
H
= β(1 + r̂t
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
On the other hand,
H
H
∑ ath+1
kt +1 = at +1
H
∑ wt
H
=
wt + (1 + r̂t )ath
cth
h =1
H
∑ ath
+ (1 + r̂t ) h =1
H
H
= wt + (1 + r̂t )kt ct .
=
h =1
h =1
∑
H
H
∑ cth
h =1
H
Finally, the individual boundary conditions can also be aggregated
over all H households to get the boundary conditions for kt as:
kt
=0
t !∞ (1 + r̂0 )(1 + r̂1 )......(1 + r̂t )
k0 given; lim
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
We have now derived the transition equations of the per capita
consumption and per capita capital stock for the aggregative
economy - except that we still do not know the precise values of the
market wage rate (wt ) and the net interest rate ( r̂t = rt δ).
These factor prices are determined in the market by the demand and
supply of labour and capital respectively.
At any time period t, total supply of capital (coming from all the
households) is given by:
KtS =
H
∑ kth = H.kt
h =1
Likewise, total supply of labour (coming from all the households) is
given by:
NtS = H
The demand for these factors on the other hand comes from the …rms.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
At any point of time t, the pro…t maximization problem of a …rm i is
given by:
Max. (Kti )α (Nti )1 α wt Nti rt Kti .
N ti ,K ti
Corresponding FONCs:
(1
α)(Kti )α (Nti )
α
= wt
) (1
α(Kti )α
1
(Nti )1
α
Kti
Nti
α
= wt
(24)
= rt
) α
Das (Lecture Notes, DSE)
α)
DGE Approach
Kti
Nti
α 1
= rt
(25)
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General Equilibrium: No Adjustment Costs (Contd.)
Since all …rms are endowed with identical technologies and face the
same market-determined factor prices, they all employ the same
amount of capital and labour, so that the aggregate demand for
labour and capital respectively are given by:
KtD
NtD
M
=
∑ Kti = M.Kti
i =1
M
=
∑ Nti = M.Nti
i =1
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
Equilibrium in the factor market requires that:
KtS = KtD and NtS = NtD .
In other words, factor market clearing conditions are given by:
= M.Kti
H = M.Nti
H.kt
Writing in ratio terms, factor market clearing condition requires that:
kt =
Kti
Nti
(26)
At every point of time t, for any historically given value of per capita
capital stock (kt ) owned by the households, the above equality is
ensured by the full ‡exibility of wt and rt .
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
Substituting
K ti
N ti
by kt in equations (24) and (25), we get:
(1
α) (kt )α = wt
α (kt )α
1
(27)
= rt
(28)
Given kt , the wt and rt adjust in every period to maintain the above
two equalities.
Thus we have precisely identi…ed the market determined values of wt
and rt in every period as a function of the historically given per capita
capita stock (which is also the equilibrium capital-labour ratio
employed by each …rm).
We now use these information to completely characterise the dynamic
paths of per capita consumption and per capita capital stock for the
aggregative economy.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium: No Adjustment Costs (Contd.)
Recall the dynamic equations for ct and kt :
ct +1 = β(1 + r̂t +1 )ct ;
kt +1 = wt + (1 + r̂t )kt
ct .
Noting that r̂t = rt δ, and replacing the market clearing values of
wt and rt derived above, we get:
h
i
ct +1 = β 1 + α (kt +1 )α 1 δ ct ;
kt +1
=
(1
h
α) (kt )α + 1 + α (kt )α
) kt +1 = (kt )α + (1
δ) kt
1
ct .
i
δ kt
ct
These two equations along with the two boundary conditions
completely characterize the evolution of per capita consumption and
per capita capital stock for this decentralized economy.
We shall come back to the precise description of these time paths
later.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium with a Convex Adjustment Cost for
Investment:
Let us now turn to characterization of the general equilibrium when
investment is associated with a convex adjustment cost.
We have indicated earlier that in this case the households’problem
will have to be suitably modi…ed.
Let us revisit the household side of the story for this case:
We have H single-membered households which are identical in terms of
preferences but di¤er in terms of their initial asset holdings;
Each household is endowed with one unit of labour - which it supplies
inelastically to the market in every period;
The households do not own the capital stock directly; nor do they
undertake investments in physical capital.
However, households hold shares of the …rms which allow them to earn
some dividend income in every period. In addition, they also hold loans
against one another.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
Household Side Story:
Assume that the …ms’equity prices remain constant over time normalized to unity.
Then the budhet equation of household h in any period t is given by:
cth + sth = wt + r̂t ath ,
where the asset stock of the household, ath = lth + nth such that the
household holds its assets either in the form of intra-household loans
(lth ) or in the form of equity holdings over …rms (nth ).
In equilibrium, the rate of return from both assets must be the same
(otherwise, households will hold only one form of asset - whichever
gives them higher return); the common rate of return is denoted by r̂t
here.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
Household Side Story:
The savings of the households are spent in buying new assets (new
loans and/or new shares), which augments the asset stock of the
household over time:
sth = ath+1
ath = 4lth + 4nth .
Replacing sth by ath+1 ath in the budget equation of the household
and simplifying, we get the period by period burget constraint of the
household as:
ath+1 = wt + (1 + r̂t ) ath
cth ; a0h given.
Each household maximises its lifetime utility subject to its period by
period budget constraint.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
93 / 105
General Equilibrium with Adjustment Costs: (Contd.)
On the production side:
There are M identical …rms endowed with a technology to produce the
…nal commodity.
The technology uses capital and labour as inputs.
The …rms employ labour services provided by the households, but they
own the capital stock that they employ. They also carry out investment
activiies that augment the capital stock in every period.
Investment is subjet to an adjustment cost: C (It )
Firms distribute part of their pro…ts in every period as dividends (dt ) to
the existing shareholders. It …nances the new investment (as well as
the adjustment cost) from the retained pro…ts as well as by issueing
new shares.
At any point of time the net pay-o¤ of …rm i is therefore given by:
π it = F (Kti , Nti )
wt Nti
dt nti + nti +1
C (Iti )
Iti
The …rms maximise the discounted sum of their lifetime pay-o¤s.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
For expositional simplicity, we shall again assume speci…c functional
forms.
Accordingly, let
u (c ) = log c;
Yt = F (Kt , Nt ) = (Kt )α (Nt )1
α
; 0 < α < 1;
C (I ) = bIt2 .
We shall also assume that there is 100% depreciation of the existing
capital stock such that
δ = 1.
Notice that 100% depreciation implies,
It
Das (Lecture Notes, DSE)
Kt +1 .
DGE Approach
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
As before, the optimization problem of a household h with an initial
asset holding of a0h is given by:
∞
Max. ∞
∞
fcth gt =0 ,fath+1 gt =0
∑ βt log
cth
t =0
subject to
ath+1 = wt + (1 + r̂t )ath
cth ; ath = 0 for all t = 0; a0h given.
Characterization of the optimal paths:
cth+1 = β(1 + r̂t +1 )cth ;
ath+1 = wt + (1 + r̂t )ath
Das (Lecture Notes, DSE)
DGE Approach
(29)
cth ;
(30)
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
Once again we are going to focus on the aggregate economy, in
particular, on the dynamics of the per capita (or average)
consumption and per capita (or average) asset stock:
H
H
∑ ath
∑ cth
h =1
ct
h =1
; at
H
H
.
Recall that now households hold their assets in the form of either
loans to other households or equility holdings of …rms. As before the
loans held by various households on the aggregate cancel each other
so that we have,
H
H
∑ lth
at
Das (Lecture Notes, DSE)
h =1
H
H
∑ nth
+
h =1
H
DGE Approach
∑ nth
=
h =1
H
nt .
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
As before, aggregating over all households, the dynamics of ct and nt
can be derived as:
ct +1 = β(1 + r̂t )ct ;
nt +1 = wt + (1 + r̂t )nt
ct .
These along with the corresponding boundary conditions will
determine the time path of ct and nt .
However, what we still do not know:
1
the equilibrium values of wt and r̂t ;
2
the evolution of the physical capital formation (kt ).
For these, we will have to turn to the …rms’problem.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
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General Equilibrium with Adjustment Costs: (Contd.)
At any point of time t, the dynamic optimization problem of a …rm i
is given by:
∞
1
i
i
Max.
∑
t [F (Kt , Nt )
∞
∞
∞
i
i
i
fNt gt =0 ,fK t +1 gt =0 ,fnt +1 gt =0 t =0 (1 + r̂ )
dt nti + nti +1
b fKti +1 g2
wt Nti
fKti +1 g]
Note that now there are two state variables for the …rm who values
are historically given: the existing capital stock that the …rm already
owns (K0i ) and the existing stock of shares that the …rm has already
issued (n0i ).
As before one can write down the corresponding Bellman equation in
terms of the value function:
V (Kti , nti ) =
Das (Lecture Notes, DSE)
Max
[F (Kti , Nti )
b fKti +1 g2
Kti +1 +
fNti ,K ti +1 ,nti +1 g
DGE Approach
wt Nti
dt nti + nti +1
V (Kti +1 , nti +1 )
].
(1 + r̂ )
February 2-22, 2016
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Equilibrium with Adjustment Costs: (Contd.)
The …rm now has three choice variables: Nti , Kti +1 ,nti +1 .
The corresponding FONCs are given by:
∂V (Kti , nti )
∂Nti
∂V (Kti , nti )
∂Kti +1
∂V (Kti , nti )
∂nti +1
= 0;
(31)
= 0;
(32)
= 0.
(33)
Simplifying:
FN (Kt , Nt ) = wt ;
(34)
1 ∂V (Kti +1 , nti +1 )
= 2b fKti +1 g + 1
i
(1 + r̂ )
∂Kt +1
(35)
1+
Das (Lecture Notes, DSE)
1 ∂V (Kti +1 , nti +1 )
= 0;
(1 + r̂ )
∂nti +1
DGE Approach
February 2-22, 2016
(36)
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General Equilibrium with Adjustment Costs: (Contd.)
Now applying Envelope Theorem to the next period’s value function:
∂V (Kti +1 , nti +1 )
=
∂nti +1
dt + 1
∂V (Kti +1 , nti +1 )
= FK (Kti +1 , Nti +1 )
∂Kti +1
Using these envelope conditions in the FONCs and noting that for the
given production function, FK (K , N ) = αK α 1 N 1 α and
FN (K , N ) = (1 α) K α N α , we get the follow
ing dynamic equations for …rm i :
wt = (1
Iti =
Das (Lecture Notes, DSE)
1
2b
"
α) Kti
α
Nti
α
;
(37)
dt +1 = (1 + r̂ );
α 1
Nti +1
(α) Kti +1
(1 + r̂ )
DGE Approach
1 α
1
#
February 2-22, 2016
(38)
(39)
101 / 105
General Equilibrium with Adjustment Costs: (Contd.)
Aggregating over all …rm, we get the dynamic equation for capital
formation in the economy as:
"
#
M (α) (Kt +1 )α 1 (Nt +1 )1 α
It =
1
(40)
2b
(1 + r̂ )
"
#
M (α) (Kt +1 )α 1 (Nt +1 )1 α
=
1 [from (38)] (41)
2b
dt + 1
Noting that all …rms are identical, average capital stock in the
economy is given by:
Kt
kt =
M
Using the above equation and equation the demand and supply of
labour, the dynamic equation for evolution of capital stock in this
economy can be derived as:
"
#
1 α (kt +1 )α 1
it
kt +1 =
1
(42)
2b
dt + 1
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
102 / 105
General Equilibrium: What have we learnt so far?
Notice that the general equilibrium analysis for both the cases
(without and with adjustment costs) specify the dynamic equations
characterizing the evolution of the capital stock of this economy.
Given that total population/labour force is constant at H, this will
also govern the evolution of the per capita as well as aggregate
output in this economy.
In other words, through the general equilibrium analysis, we have
actually characterized the growth path for the economy under
alternative assumptions about investment costs.
This brings us directly to the realm of economic growth.
Notice however that such growth paths would be relevant only for a
perfectly competetive market economy populated by rational agents
with complete information and no uncertainty.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
103 / 105
General Equilibrium: What have we learnt so far?
Are these the only possible characterization of the growth path of an
economy?
To put it di¤erently, could there be alternative growth paths
assocaited with alternative speci…cation of the macroeconomy (say,
with imperfect markets or incomplete information or uncertainty)?
To answer these questions, we shall have to get into a detailed
discussion of various theories of economic growth.
We shall stop at this point and come back to the speci…c issues
related economic growth at the third module of the course.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
104 / 105
DGE Approach: Reference
Reference for Dynamic Programming Technique:
Daron Acemoglu (2009): Introduction to Modern Economic
Growth; Princeton University Press, chapter 6.
Reference for DGE approach to Macroeconomics:
Michael Wickens (2008): Macroeconomic Theory: A Dynamic
General Equilibrium Approach, Princeton University Press, chapters
1& 2.
Statutory Warning: I do not follow any particular textbook word by
word. The references are to be treated only as broad guidebooks,
complementary to the lecture notes.
Das (Lecture Notes, DSE)
DGE Approach
February 2-22, 2016
105 / 105