Macroeconomics 1

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Macroeconomics 1

• Static general equilibrium.

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Beyond partial equilibrium a simple
macroeconomic model

• The set up
• A standard 3 goods model good, labour, money,
• The equations fixed wage w equilibrium
• D(p, R) M/p Q
• R pQ
• S(p/w) Q
• g(Q) lt E_ , g f -1
• Walrasian same equations, w,
• The decison making production units firm i
  hires one worker and produces a(i), a(i) gt a(j)
• The decision process
• Firms decide simultaneously
• To hire one worker
• To post a reservation wage

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Strong rational expectations equilibrium

• The  eductive process 
• In both cases, it is enough to guess Q.
• Results in the fixed wage model
• Consider h S(F(Q)/w))
• F market clearing price when Q is production.
• Rationalizable equilibria in (h(Q_), Q_)
• Local Strong rationality
• e(S)/e(D) lt m, in .
• m inverse of marginal propensity to consume.
• Intuition
• Effect strategic substitutability Muth e(S)/e(D)
• m Keynesian multiplier (effect str.complementariti
  es).

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Eductive stability with flexible wage

• An example of a  rationalizable equilibrium 
  which is not walrasian.
• Rule out this to obtain
• e(S)/e(D) lt 4m/l
• If the fixed wage equilibrium is SR in an
  interval of w, then the W equilibrium is SR

E
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On the generality of results dimensionality.

• From
• the 3 goods models
• to the n2 goods models.
• A necessary condition
• the eigen value of highest modulus of (?S)
  (?Z)-1(I-A) is smaller than one
• Sufficient condition under gross
  substitutability.
• A(b) bA
• If (?S) (?Z)-1(I-A(b)) has a spectral radius
  smaller than one for some b, then the radius
  decreases for larger b

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Introducing infinite horizon models.

• The model
• Reduced form one step forward-looking, memory
  one, multi dimensional.
• Cx(t1) x(t)Dx(t-1)0, matrices or numbers.
• Game theoretical flesh
• x(t) Dx(t-1) B ?Z(w(t))x(e,t1, w(t))dw(t).
• Solutions
• Perfect foresight trajectories.
• Perfect foresight dynamics of growth rates,
   extended  growth rates.
• Example if g(t)x(t)/x(t-1), g(t)-cg(t1)g(t)
  d,
• If x(t)B(t)x(t-1), B(t1)-(GB(t)I)-1D.
• Based on Evans-Guesnerie (2003, 2005),
  Gauthier(2002, 2003, 2005), Desgranges-Gauthier
  (2003).

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Standard expectational criteria in infinite
horizon models.

• The Standard expectational criteria.
• Determinacy of
• trajectories (C0,1), of the long term extended
  growth rate.
• Iterative Expectational Stability
• Belief (perceived law of motion) the growth
  rate is ge,
• Realisation the growth rate is gke, klt1.
• On  extended growth rates 
• Absence of sunspot equilibrium
• Without memory x, x, Markov matrix
• Can be defined for  extended growth rates .
• Reasonable learning rule adaptive learning on
  growth rates that detect cycles of order 2.
• The equivalence theorem
• The four criteria are equivalent in the one
  dimensional case
• The first three are equivalent in the
  multi-dimensional case.
• They pick up the saddle path solution

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The concepts in a simple monetary economy.

• A monetary, cashless simple economy.
• No production, continuum identical agents, manna
  (one unit each period) infinite horizon,
• FOC (1i(t))(P(t)/P(t1))(1/?)U(C(t1))/U(C(t
  )
•  Wicksellian  rule
• i(t,m)
• Inflation target ? P(t)/P(t-1)), 1 ?(?)
  ?/?.
• The starting point.
• equilibrium E?, 1/?, no trade.
• V(E), set of trajectories, (inflation)  close 
  to the equilibrium trajectories.
• The mental process (in peoples mind).
• For each social state of belief, (identical point
  beliefs, special), compute the outcome at all
  periods
• If the set so generated is within V(E), go on.
• Come back on what is known in infinite horizon
  OLG type.

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Monetary models

• A new keynesian model
• (simplified, without intrinsic noise)
• ?(t)bE(?(t1))lx(t)
• x(t)f(i(t)-E(?(t1))E(x(t1))
• i(t)a ?(t-1)cx(t).
• Fits the above framework 2-dimensional, memory
  one, one step forward looking model.
• What does the  eductive  viewpoint suggest for
  the Taylor rule
• Apparently, since it is justified by determinacy,
  the above local equivalence results supports the
  usual conclusion,
• IE-Stability is only necessary cdt / truly
   eductive  stability.
• Additional difficulties reflect that agents may
  be hetrogenous or have hetrogenous expectations.
  This may make a serious difference., in the sense
  of strengthening the cdt.EG (2005)
• OLG interpretation expectational coordination not
  appropriate here..

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The cashless simple economy.

• The above equilibrium is determinate
• E?, 1/?, no trade, another equilibrium meets
• ?(1 ?(P(t)/P(t-1))P(t)P(t1)
• dP(t)dP(t1), determinacy/ standard cdts.
• The equivalence theorem fails when one views exp.
  Coord. from the inf. horizon lenses rather than
  from the OLG lenses.
• In an OLG model, P(t) is close to P, for t,
  P(t-1) still closer.
• In an infinite horizon model, the above fact,
  does not imply anything for P(t-1), indep. /
  beliefs on the rest of the horizon.
• Backward induction, as used above, is relevant
  for determinacy, but not /IE stability argument /
  prev. theorem.
• A necessary condition for  eductive  stability.
  
• The equilibrium EP(1),P(t), P(t1),
• Nbd of beliefsV(E) Inflation between ?e,
  ?-e,
• Beliefs determine first period inflation
  P(1)/P(0), then,
• Is the trajectory in V(E) ?

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A tentative assesment of  eductive  stability
in a cashless economy.

• A necessary condition for  eductive  stability.
  
• The computation.
• Case ? e, homogenous beliefs
• Changed in planned consumption t,
• (dC(t)/C)(dC(1)/C) (/?)?d1?(P(s)/P(s-1))((
  P(s)/P(s1))
• An envelope argument fixes the (individual)
  sequence
• And equilibrium implies dC(1)0
• The condition ?v
• 1/?ltvlt 1/?(1/(2?-1)).
• More on  eductive  stability
• Conjecturesufficient for  eductive  stability
  with C2 top.
• Stricter conditions than for determinacy.
• Strong conditions on v.
• What about adaptive learning, global
   eductive  stability..

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Beyond the prototype one-dimensional model.

• Sequential decisions,
• Multi-dimensional issues,
• Back to incomplete information.

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On the generality of results dimensionality

• The Muth model
• Conditions for a local SREE
• S 1 / D 1lt
• The Muth model with two crops
• S(1, p(1), p(2)), S(2, p(1), p(2)), S12 S21
• D(1, p(1)), D(2, p(2))
• Eductive stability
• k S 1 / D 1 S 2 / D 2
• S12 /?(D 1 D 2) lt 1-k,
• 1-k measures stability on each market, normalized
  cross effects small is OK.

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Observing is good for guessing

• A sequential version of the model
• Drop simultaneity
• Half of the farmers plant (Fall wheat)
• Later half of the farmers plant, after observing
  (Spring wheat)
• Results
• Clt2B is the condition for  eductive  stability.
• Care in the proof
• CltTB with T periods of observation !
• More care ...

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Provisionnal conclusions.

• What matters ?
• Low supply elasticity
• high demand elasticity
• A less grandiose story
• but economic flesh,
• weak reactions of actions to expectations.
• Robustness.
• Randomness confirms the conclusion.
• If demand and supply are not linear, the local
  version is robust, S ltD 
• Cross competition fragilizes expect. Stability
• Observation possibilities (sequentiality)
  improves it.
• Comparing with evolutive learning.

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On the generality of results dimensionality

• N-dimensional version of a Muth like setting.
• X f(X(e)),
• X F(?iX(i,e)), X G(?i? (i,e)),
• Equilibrium Xf(X)
• X f(X) (df)(X(e)-X) (A(X))(X(e)-X)
• Uniqueness obtains if I-A(X) has a positive
  determinant..
• Local IE Stability (unformal  eductive 
  argument)
• X(e) in V(X),
• X in X A(X) V(X),
• then X(e) in X A (X) V(X),
• then X in X A (X)2V(X),
• lim. A n 0, all eigenvalues of A modulus lt1

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On the generality of results dimensionality

• N-dimensional version of a Muth like setting.
• X f(X(e)), X F(?iX(i,e)), X G(?i?
  (i,e)),
• Equilibrium Xf(X)
• X f(X) ?(A(i,X))(X(i)-X)di
• LSR Whatever ? (i,e) in V(X), then it is CK
  XX
• Statement
• A?(A(i))di, semi-simple, B the eigenvector basis
  of A
• n(i) the norm of A(i) induced by the Euclidean
  norm on B
• ?(n(i))di lt1 is a sufficient condition for LSR.
• LSR implies IE stability ?(A(i))di lt ?
  (A(i))di
• SR more demanding, reflects the heterogeneity of
  expectations
• One-dimensional case.
• X ?(a(i)x(i))di
• IE Stability a!?(a(i))di! lt1, LSR
  a?!(a(i)!)di lt1

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On the generality of results dimensionality

• Sequential setting
• a representative sample takes decision in period
  1 and 2
• A?(A(i))di, semi-simple, B the eigenvector basis
  of A
• n(i) the norm of A(i) induced / the Euclidean
  norm on B
• Results
• ?(N(i))di lt2 is a sufficient cond./ LSR 2d period
• ?(N(i))di lt1 is a suf.cond./ LSR over the two
  periods
• ?(N(i))di lt2 is a sufficient LSR over the two
  periods if all eigenvalues of A have a negative
  real part.
• Comments.

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Extensions and comments.

• Extension1 An uncertain version of the model
• Demand is uncertain.A-Bp ?
• Initial CK restriction
•  Eductive  stability conditions are the same
  (risk neutral farmers) Cvgce as fast.
• Extension 2 Non-linear version
• LSR in the non-linear version
• Supply elasticity lt demand elasticity
• Comments results, economic intuition
• Cobweb in peoples mind.
• High supply elasticity bad
• Demand elasticity..