Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Robust Comparative Statics in Large Dynamic
Economies
Daron Acemoglu (MIT)
and
Martin K. Jensen (U. B’ham)
April 2011
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
1 Introduction
2 Large Dynamic Economies
3 Equilibrium Comparative Statics
4 The Answers
5 Conclusion
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
This paper is about equilibrium comparative statics in large
dynamic economies.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
This paper is about equilibrium comparative statics in large
dynamic economies.
Large=a non-atomic measure space of agents, here taken to
be [0, 1]
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
This paper is about equilibrium comparative statics in large
dynamic economies.
Large=a non-atomic measure space of agents, here taken to
be [0, 1]
Infinite horizon, standard stochastic dynamic programming
problems for each agent.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
This paper is about equilibrium comparative statics in large
dynamic economies.
Large=a non-atomic measure space of agents, here taken to
be [0, 1]
Infinite horizon, standard stochastic dynamic programming
problems for each agent.
Economic interaction through certain “market variables”
(typically prices).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
This paper is about equilibrium comparative statics in large
dynamic economies.
Large=a non-atomic measure space of agents, here taken to
be [0, 1]
Infinite horizon, standard stochastic dynamic programming
problems for each agent.
Economic interaction through certain “market variables”
(typically prices).
Shocks idiosyncratic. No aggregate uncertainty/risk ⇒
market variables are deterministic sequences.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
This paper is about equilibrium comparative statics in large
dynamic economies.
Large=a non-atomic measure space of agents, here taken to
be [0, 1]
Infinite horizon, standard stochastic dynamic programming
problems for each agent.
Economic interaction through certain “market variables”
(typically prices).
Shocks idiosyncratic. No aggregate uncertainty/risk ⇒
market variables are deterministic sequences.
An example is the Bewley-Aiyagari model (Bewley (1986) and
Aiyagari (1994)).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
Why should we (and macroeconomists) care ?
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
Why should we (and macroeconomists) care ?
Because it has become clear through such papars as Aiyagari
(1994) and Benabou (1996) that dynamic+stochastic DOES
NOT equal “dynamic with some noise”. Stochastic dynamics
have non-trivial, often quite suprising consequences.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
Why should we (and macroeconomists) care ?
Because it has become clear through such papars as Aiyagari
(1994) and Benabou (1996) that dynamic+stochastic DOES
NOT equal “dynamic with some noise”. Stochastic dynamics
have non-trivial, often quite suprising consequences.
This paper is about comparative statics. At present there
really are no such results (save for in supermodular cases).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Introduction
Why should we (and macroeconomists) care ?
Because it has become clear through such papars as Aiyagari
(1994) and Benabou (1996) that dynamic+stochastic DOES
NOT equal “dynamic with some noise”. Stochastic dynamics
have non-trivial, often quite suprising consequences.
This paper is about comparative statics. At present there
really are no such results (save for in supermodular cases).
My talk will be very non-technical. I’ll spend a lot of energy
describing the basic model. Therefore, I’m going to go
straight at it !
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
Economy populated by a contiuum of agents I = [0, 1].
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
Economy populated by a contiuum of agents I = [0, 1].
Agent i ∈ I’s preferences are repr. by:
Ui (xi ; Q, ai ) = E0 [
∞
X
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(1)
t=0
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
Economy populated by a contiuum of agents I = [0, 1].
Agent i ∈ I’s preferences are repr. by:
Ui (xi ; Q, ai ) = E0 [
∞
X
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(1)
t=0
... which is maximized subject to:
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
(2)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
Economy populated by a contiuum of agents I = [0, 1].
Agent i ∈ I’s preferences are repr. by:
Ui (xi ; Q, ai ) = E0 [
∞
X
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(1)
t=0
... which is maximized subject to:
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
(2)
Here xi ≡ (xi,t )∞
t=1 is a strategy (a sequence of random
variables/history dependent measurable maps).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
M
(zi,t )∞
t=0 , zi,t ∈ Zi ⊆ R , is a Markov process. It is assumed
that this process has a unique invariant distribution µzi .
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
M
(zi,t )∞
t=0 , zi,t ∈ Zi ⊆ R , is a Markov process. It is assumed
that this process has a unique invariant distribution µzi .
A special case of this is when zi,t is i.i.d. as in Bewley (1986)
and Aiyagari (1994).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
M
(zi,t )∞
t=0 , zi,t ∈ Zi ⊆ R , is a Markov process. It is assumed
that this process has a unique invariant distribution µzi .
A special case of this is when zi,t is i.i.d. as in Bewley (1986)
and Aiyagari (1994).
Q
Just to repeat, let Zit ≡ tm=0 Zi denote the set of (possible)
histories at date t. Then a strategy at date t: xi,t is a
measurable map xi,t : Zit → Xi (here Xi ⊆ RN is the strategy
set).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Agents
M
(zi,t )∞
t=0 , zi,t ∈ Zi ⊆ R , is a Markov process. It is assumed
that this process has a unique invariant distribution µzi .
A special case of this is when zi,t is i.i.d. as in Bewley (1986)
and Aiyagari (1994).
Q
Just to repeat, let Zit ≡ tm=0 Zi denote the set of (possible)
histories at date t. Then a strategy at date t: xi,t is a
measurable map xi,t : Zit → Xi (here Xi ⊆ RN is the strategy
set).
So economically, xi,t is like a “state-dependent contingency
plan”: Given a specific realized history of shocks zit−1 ∈ Zit−1
at date t − 1 (and given knowledge of the probabilities of
future events), the agent will choose xi,t = xi,t (zit−1 ).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Economy
The sequence of market variables/aggregates
Q = (Q0 , Q1 , Q2 , . . .), Qt ∈ R is deterministic. All interaction
between the agents happens through Q. In words: There is no
aggregate uncertainty (risk is purely idiosyncratic).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Economy
The sequence of market variables/aggregates
Q = (Q0 , Q1 , Q2 , . . .), Qt ∈ R is deterministic. All interaction
between the agents happens through Q. In words: There is no
aggregate uncertainty (risk is purely idiosyncratic).
How does this come about ? Well, the most common case is
where Qt is the average/aggregate:
Z
Qt =
xi,t di
(3)
[0,1]
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Economy
The sequence of market variables/aggregates
Q = (Q0 , Q1 , Q2 , . . .), Qt ∈ R is deterministic. All interaction
between the agents happens through Q. In words: There is no
aggregate uncertainty (risk is purely idiosyncratic).
How does this come about ? Well, the most common case is
where Qt is the average/aggregate:
Z
Qt =
xi,t di
(3)
[0,1]
For example in the Bewley-Aiyagari model, xi,t is savings of
agent i at date t, and Qt is then aggregate savings/capital.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Economy
The sequence of market variables/aggregates
Q = (Q0 , Q1 , Q2 , . . .), Qt ∈ R is deterministic. All interaction
between the agents happens through Q. In words: There is no
aggregate uncertainty (risk is purely idiosyncratic).
How does this come about ? Well, the most common case is
where Qt is the average/aggregate:
Z
Qt =
xi,t di
(3)
[0,1]
For example in the Bewley-Aiyagari model, xi,t is savings of
agent i at date t, and Qt is then aggregate savings/capital.
Qt then enters consumers’ problems because capital
determines prices (wt = W (Qt ) and Rt = R(Qt ), real wage
and interest rate).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Rerun and Comparative Statics
Let’s take the basic problem again, for agent i ∈ [0, 1] to
maximize:
∞
X
Ui (xi ; Q, ai ) = E0 [
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(4)
t=0
s.t.
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
(5)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Rerun and Comparative Statics
Let’s take the basic problem again, for agent i ∈ [0, 1] to
maximize:
∞
X
Ui (xi ; Q, ai ) = E0 [
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(4)
t=0
s.t.
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
(5)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Rerun and Comparative Statics
Let’s take the basic problem again, for agent i ∈ [0, 1] to
maximize:
∞
X
Ui (xi ; Q, ai ) = E0 [
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(4)
t=0
s.t.
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
(5)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Rerun and Comparative Statics
Let’s take the basic problem again, for agent i ∈ [0, 1] to
maximize:
∞
X
Ui (xi ; Q, ai ) = E0 [
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(4)
t=0
s.t.
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
(5)
ai (could be a vector or a just a scalar) are exogenous variables
with respect to which we wish to do comparative statics.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Rerun and Comparative Statics
Let’s take the basic problem again, for agent i ∈ [0, 1] to
maximize:
∞
X
Ui (xi ; Q, ai ) = E0 [
β t ui (xi,t , xi,t+1 , zi,t , Qt , ai )].
(4)
t=0
s.t.
xi,t+1 ∈ Γi (xi,t ; zi,t , Qt , ai ) , t = 0, 1, 2, . . .
(5)
ai (could be a vector or a just a scalar) are exogenous variables
with respect to which we wish to do comparative statics.
So, roughly speaking, if we change ai , i ∈ I how does this
affect the equilibrium ? Of course, we must define what we
mean by an equilibrium first. And here I’ll only speak of
stationary equilibrium.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Stationary Equilibrium
Definition
(Stationary Equilibrium) Let the exogenous parameters
a ≡ (ai )i∈I be given. A stationary equilibrium (given a) is a list
{Q ∗ , (xi∗ )i∈I } such that:
1
For each agent i ∈ I, the stationary sequence of random
variables (xi∗ , xi∗ , xi∗ , . . .) is an optimal strategy when the
agent takes the stationary sequences of market aggregates
(Q ∗ , Q ∗ , Q ∗ , . . .), and the exogenous parameters ai as given.
2
The market aggregates “clear” (at all dates), i.e.,
Q ∗ = H((xi∗ )i∈I )
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Stationary Equilibrium: Technical Remarks
In a second I’ll mention assumptions under which a SE exists
(these are very weak, no concavity required for example).
Technically, in “the background” you should see transition
correspondences:
xi,t+1 ∈ Fi (xi,t , zi,t ; Qt , ai )
Think of this as a multivalued stochastic transformation.
Then existence becomes a fixed point problem (searching over
Q’s so that aggregates clear). We also approach comparative
statics questions from this fixed point perspective.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Questions
(“Plain Vanilla”) The simplest comparative statics question
that we answer is this: Imagine that we change a. Then how
does Q ∗ (equilibrium aggregate) change in response ? How do
strategies change ?
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Questions
(“Plain Vanilla”) The simplest comparative statics question
that we answer is this: Imagine that we change a. Then how
does Q ∗ (equilibrium aggregate) change in response ? How do
strategies change ?
(“Uncertainty”) A deeper one is this: Imagine that we change
some features of the (zi,t )’s (e.g., replace with one that firstor second-order stochastically dominates, or “mean preserving
spread”). How does Q ∗ change in response ? How do
strategies change ?
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
The Questions
(“Distribution of Agents”) Last but not least (but this is in a
separate paper), we will seek to address: Take a distribution
of agents and change that distribution. How do Q ∗ , xi∗ , etc.
change in response. Not time to explain in any detail, but I
hope you may be able to sense what this is about.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Existing literature does not have a whole lot to say about
comparative statics in this kind of models. In fact, we know of
no one who has done anything on this except for in the
supermodular case (returned to below).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Existing literature does not have a whole lot to say about
comparative statics in this kind of models. In fact, we know of
no one who has done anything on this except for in the
supermodular case (returned to below).
The motivation for our research is, essentially, that this could
be very useful for macroeconomics. It makes comparative
statics easy in a class of models which is, otherwise impossibly
hard to deal with qualitatively.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Assumptions
NOTE: Attention is restricted to the case where the Xi ’s are
one-dimensional here.
I won’t speak about standard stuff like ui continuous, Γi
upper hemi-continuous, Xi compact, β < 1, and so on.
[secures existence btw.]
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Assumptions
NOTE: Attention is restricted to the case where the Xi ’s are
one-dimensional here.
I won’t speak about standard stuff like ui continuous, Γi
upper hemi-continuous, Xi compact, β < 1, and so on.
[secures existence btw.]
Remember that ui = ui (xi,t , xi,t−1 , Qt , ai ). We assume that
2 u ≥ 0 (supermodularity in own strategies). We also need
D12
i
that Γi is ascending.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Assumptions
NOTE: Attention is restricted to the case where the Xi ’s are
one-dimensional here.
I won’t speak about standard stuff like ui continuous, Γi
upper hemi-continuous, Xi compact, β < 1, and so on.
[secures existence btw.]
Remember that ui = ui (xi,t , xi,t−1 , Qt , ai ). We assume that
2 u ≥ 0 (supermodularity in own strategies). We also need
D12
i
that Γi is ascending.
2 u , D 2 u ≥ 0.
We consider “positive shocks” meaning that D14
i
24 i
This isn’t really an assumption, if you don’t say something
about the shocks, you obviously cannot say anything (you
need to specify what kind of exogenous changes you’re
looking at!).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Assumptions
What we don’t assume is almost as important as what we do
assume (D12 ui ≥ 0 is the main one).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Assumptions
What we don’t assume is almost as important as what we do
assume (D12 ui ≥ 0 is the main one).
This does not turn the model into a supermodular game of
any sorts ! In particular, and crucially, we make no
assumptions on how Q enters ui and Γi .
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Assumptions
What we don’t assume is almost as important as what we do
assume (D12 ui ≥ 0 is the main one).
This does not turn the model into a supermodular game of
any sorts ! In particular, and crucially, we make no
assumptions on how Q enters ui and Γi .
This is dreadfully important because in economic models, that
is exactly what we usually can’t say anything about (just think
about R = R(Qt ) and W = W (Qt ) entering as in the Bewley
economy).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Main Theorem, Plain Vanilla
Theorem
(Main Theorem on Comparative Statics of the Aggregate)
An increase in ai (for all players or any subset) will lead to an
increase in the smallest and largest Q in stationary equilibrium.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Main Theorem, Noise Statics
ui (xi,t , xi,t−1 , Q, zi,t , ai,t ) exhibits increasing differences in
2 u , D 2 u ≥ 0).
(xi,t , xi,t−1 ) and zi,t (D13
i
23 i
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Main Theorem, Noise Statics
ui (xi,t , xi,t−1 , Q, zi,t , ai,t ) exhibits increasing differences in
2 u , D 2 u ≥ 0).
(xi,t , xi,t−1 ) and zi,t (D13
i
23 i
Theorem
A first-order stochastic dominance increase in the stationary
distribution of zi,t for all i (or any subset hereof), will lead to an
increase in the smallest and largest Q in stationary equilibrium.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
We also have a version of the previous result that addresses
mean-preserving spreads (and one that looks at second order
stochastic dominance changes).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
More Results (some of which are not really done yet!)
We’ve also got “individual comparative statics” results (how
does a change in the ai ’s or noise process affect the xi ’s ?).
I’ll speak a little about those if time permits.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
More Results (some of which are not really done yet!)
We’ve also got “individual comparative statics” results (how
does a change in the ai ’s or noise process affect the xi ’s ?).
I’ll speak a little about those if time permits.
Finally, results on changes in distribution.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Going Macro :-)
For this class of models we get results that are based on
extremely simply conditions (verifying that some cross-partial
derivatives are positive).
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Going Macro :-)
For this class of models we get results that are based on
extremely simply conditions (verifying that some cross-partial
derivatives are positive).
Looking at macroeconomic modeling where these kinds of
models are used, people always take “the simulation path”.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Going Macro :-)
For this class of models we get results that are based on
extremely simply conditions (verifying that some cross-partial
derivatives are positive).
Looking at macroeconomic modeling where these kinds of
models are used, people always take “the simulation path”.
Of course we don’t make simulation worthless, it yields
quantitative results. But we do establish qualitative results.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Going Macro :-)
For this class of models we get results that are based on
extremely simply conditions (verifying that some cross-partial
derivatives are positive).
Looking at macroeconomic modeling where these kinds of
models are used, people always take “the simulation path”.
Of course we don’t make simulation worthless, it yields
quantitative results. But we do establish qualitative results.
And qualitative result are nice to have also because they show
which quantitative conclusions are robust and which are not.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies
Outline
Introduction
Large Dynamic Economies
Equilibrium Comparative Statics
The Answers
Conclusion
Going Macro :-)
For this class of models we get results that are based on
extremely simply conditions (verifying that some cross-partial
derivatives are positive).
Looking at macroeconomic modeling where these kinds of
models are used, people always take “the simulation path”.
Of course we don’t make simulation worthless, it yields
quantitative results. But we do establish qualitative results.
And qualitative result are nice to have also because they show
which quantitative conclusions are robust and which are not.
From a more theoretical point of view, the math is quite
wonderful to work with, and the field seems to be quite open.
Daron Acemoglu (MIT) and Martin K. Jensen (U. B’ham)
Robust Comparative Statics in Large Dynamic Economies