Macroeconomics and finance

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1. Temporary equilibrium, expectations, perfect foresightequilibrium

2. Risk and finance

3. The medium run: Models of inflation

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Temporary equilibrium

1. The time frame: history, expectations, description of resourceallocation

2. Resource allocation during the period: competitiveequilibrium...

3. Expectations: error and revision.

4. Dynamics: overlapping generations; going from one period tothe next.

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Simple setup

Aggregation and study of a two periods model, with one asset(‘money’: numeraire, no dividends), one non storable consumptiongood. Certain expectations.

Typical agent

max U i (C it , C i

t+1)

p1C it + B i

t = p1Y it + B i

t−1

pet+1C i

t+1 = pet+1Y ie

t+1 + B it

C it , C i

t+1 ≥ 0

If debt is not allowed, the B’s are constrained to be non negative.

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Eliminating Ct+1: max U i

(C it ,Y

iet+1 +

B it

pet+1

)ptC

it + B i

t = ptYit + B i

t−1

Eliminating Bt :

max U i

(C it , Y ie

t+1 +ptY

it + B i

t−1 − ptCit

pet+1

)

with, in case of borrowing constraint

B it = p1Y i

t + B it−1 − ptC

it ≥ 0.

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-

6C2

B0

p2+ p1

p2Y1 + Y2

B0

p1+ Y1

B0

p1+ Y1 +

p2

p1Y2

C1

Figure 1: The demand for savings

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Temporary competitive equilibrium

Definition temporary competitive equilibrium at date 1: value ofprice pt such that supply equals demand, both for the consumptiongood and for the asset. ∑

i

[C it − Y i

t ] = 0

∑i

[B it − B i

t−1] = 0

Walras’ law.

Behavior of demand when the good price (the asset is thenumeraire) goes either to zero, or to infinity.

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Existence of an equilibrium

All the history is given and fixed: to avoid unnecessary notations, itis omitted from the arguments. The only explicit argument of theexpectations is with respect to the current endogenous variable.

pet+1 = ψ(pt ,history)

Behavior of demand when the price of the good (the asset is thenumeraire) goes either to zero, or to infinity.

max U i

(C it , Y ie

t+1 +ptY

it + B i

t−1 − ptCit

pet+1

)

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How price expectations vary with current price

Sufficient conditions to warrant the existence of an equilibrium are

Price expectations are independent of current price;

Real balance effect in economies without credit.

Difference in required conditions for economies without or withcredit: assumptions bear on a single consumer or on all consumers.Equilibrium p∗t function of the history.

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Temporary equilibrium dynamics

Explicit dynamic structure: infinite horizon agents or overlappinggenerations.

Asset accumulation.

Formation of expectations: pet+1 = ψ(pt , pt−1, . . . , pt−T ).

No error in the long run: in case of constant prices in the past, theagents forecast a future price equal to the past ones.

Dynamical system: stationary points, convergence.

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A different point of view: perfect foresight intertemporalequilibrium

max U(C y ,C o)ptC

yt + Bt = ptY

y

pt+1C ot+1 = pt+1Y o + Bt

At the initial date 1, the old consumer has a quantity B0 ofnominal asset, and consumes all her wealth, i.e. Y o + B0/p1.

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Intertemporal equilibrium

Definition 1 : A perfect foresight intertemporal equilibriumwith nominal asset is a sequence (pt ,C

yt ,C

ot ), t = 1, . . . , with

(strictly) positive prices, which satisfies:

1.C yt + C o

t = Y y + Y o ,

2. For t ≥ 1, (C yt ,C

ot+1) maximizes the program of the consumer

born at date t, given (pt , pt+1). C v1 is equal to B0/p1 + Y o .

Walras’ law, nominal assets.

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Finding the equilibria

{max U(C y ,C o)

C y +pt+1

ptC o = Y y +

pt+1

ptY o

Let zy and zo be the excess demand functions coming out of theprogram:

zy

(pt

pt+1− 1

)= C y

t − Y y = −B

pt

zo

(pt

pt+1− 1

)= C o

t+1 − Y o = − pt

pt+1zy

(pt

pt+1− 1

).

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Supply curve

The range of values of (zy , zo), when the price ratiopt/pt+1 = 1 + ρt+1 varies, is the supply curve of the consumer.

zy goes to infinity when pt/pt+1 tends to 0 (a finite zy wouldimply C o = Y o and C y finite, something incompatible with theequality of the marginal rate of substitution with the price ratio).Similarly, zo tends to infinity when pt/pt+1 tends to infinity.

Samuelson case:

U ′o(Y y ,Y o)

U ′y (Y y ,Y o)=

1

1 + ρ> 1.

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-

6

E

0

Co − Y o

Cy − Y y

Figure 2: Equilibria in the overlapping generations model

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Finding the equilibria: continued

Gross real interest rate: 1 + ρt+1 = pt/pt+1. In the plan (C y ,C v ),−(1 + ρ) is the slope of the budget line.

The equality between supply and demand at date t is

zy (ρt+1) + zo(ρt) = 0.

Finite difference equation with initial condition zo(ρ0) = B0/p1 (ofsame sign as B0).

A priori, there exists a continuum of equilibria, depending on theinitial value ρ0.

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Stationary equilibria

Fixed points of the difference equation

zy (ρ∗) + zo(ρ∗) = 0,

at the intersection of the supply curve with the second bissector.

Since zy (ρ∗) + zo(ρ∗) = −ρ∗zy (ρ∗),

1. Autarky of every generation : zy = zo = 0. The equilibriumgross interest rate is ρ. The real aggregate quantity ofnominal assets is null: B = 0.

2. Transfers between generations, B 6= 0, interest rate ρ∗ = 0,golden rule.

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Non stationary equilibria

In the Samuelson case, continuum of indeterminate equilibriaconverging to the autarkic equilibria.

Isolated (determinate) golden rule equilibrium.

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Expectations and learning: conclusion

The assumption of perfect foresight (or rational) expectations in adynamic model does restrict the shape of the expectation function.

However it does not always yield determinate isolated equilibria,while the situation is more favourable when one only looks forstationary equilibria. Indeed, in the latter case, the interaction ofpolicy with expectations will be of interest.

Learning is not a catch-all validation of the perfect foresightassumption. It may serve as a selection device (some potentialequilibria are unstable, others are stable), but all equilibria may beunstable!

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Finance

1. Arbitrage

2. Mean variance portfolio choice

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Assets

There are k = 1, . . . ,K assets.

A unit of asset k is defined by the received earnings (interest,dividends, possibly liquidation value at the end of the game) or duepayments in currency associated with its ownership in the variousstates of nature.

For asset k in state e this is denoted by ake = ak(e), positive for again, negative for a payment.

The matrix of general term ake , row index k, column index e, isdenoted a. Its dimension is (K × E ).

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Portfolios and marketsLet zk denote a (positive or negative) number of units of asset k.A portfolio is a column vector z = (zk)k=1,...,K , with possibly longand short positions.

Assets are exchanged on a (competitive) market: the selling priceis equal to the buying price, and is independent of the tradedquantity.

The price today of asset k is denoted pk , and the price vector p isa column vector of dimension K .

K∑k=1

pkzk = p′z .

The earnings accruing to the owner of z in state e are

cz(e) =K∑

k=1

zkak(e),

cz = z ′a.

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Arbitrage opportunities

Definition 1 : Arbitrage opportunity

An arbitrage opportunity is a portfolio z such that

z ′a ≥ 0 and z ′p ≤ 0,

or equivalently

K∑1

zkak(e) ≥ 0 for all e and z ′p ≤ 0,

with at least a strict inequality among the E + 1’s.

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Absence of arbitrage opportunities

Theorem 1 : State prices

A market is without arbitrage opportunities if and only if thereexists a vector q = (q(e))e∈E with all components strictly positivesuch that

pk =∑e

q(e)ak(e) for all k.

The vector q is called a state price (or Arrow-Debreu price) vector.Consequence: valuation by arbitrage (‘duplication’, ‘redundancy’)

Corollary 1 : valuation by state prices

Let q be a state price vector, z a portfolio. Then

p′z =∑e

q(e)cz(e) (1)

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Complete markets

Definition 2 : Complete markets

Markets are complete if for all c = (c(e)) in IRE there exists aportfolio z such that c = z ′a, i.e.

c(e) =K∑1

zk ak(e) for all e.

Theorem 2 :

1) Markets are complete if and only if a has rank E.

2) A complete market without arbitrage opportunities has a uniquestate price vector (q(e)), and any future income stream c has acurrent value given by the present value formula :∑

e

q(e)c(e).

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Complete and incomplete markets

Corollary 2 : Arrow Debreu prices

Let q be a state price vector. Consider the income stream made of$1 in state e and nothing in any other state.If there exists a portfolio replicating this income stream, its value isq(e).

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Risk adjusted probability

Risk-free asset between dates 0 and 1

1 =∑e

q(e)(1 + r).

Probability (??) distribution π on E :

π(e) = q(e)(1 + r).

Valuation of asset k :

pk =1

1 + r

∑e

π(e)ak(e)

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Mean variance portfolio choice: Plan

1. Uncertainty and the demand for assets

2. Mean-variance efficiency

3. Beyond mean variance

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A simple model: setup

Two dates t = 0, t = 1.

States of nature e, e = 1, . . . ,E , at date 1.

(Subjective?) probability of the realization of these states : πe ,πe > 0 and

∑Ee=1 πe = 1.

k = 0: risk free asset a0(e) = 1, for all e, i.e. a0 = 11ERisky assets k , k = 1, . . . ,K .

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Typical investor

Initial wealth ω0, which serves as the numerairePortfolio: (z0, z), where z = (zk)k=1,...,K .Budget constraint at date 0

p0z0 + p′z = ω0.

Random endowment available at date 1 ω(e).Consumption at date 1 is

cz(e) = ω(e) + z0 +∑

k=1,...,K

zkak(e)

Assumption :There are no redundant assets. The matrix (11E , a) is of full rank.

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Interest rate and prices

By definition the (net risk-free) interest rate r between date 0 anddate 1 satisfies

1

1 + r= p0.

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max Ev(c) = Ev(ω + z0 + z ′a) =E∑

e=1

πev

[ω(e) + z0 +

K∑k=1

zkak(e)

]p0z0 + p′z = ω0

First order conditions:Ev ′(c) = λp0

Ev ′(c)ak =E∑

e=1

πev ′

[ω(e) + z0 +

K∑k=1

zkak(e)

]ak(e) = λpk

Asset demand as a function of prices is the solution of the K + 2equations, made of the K + 1 first order conditions and of thebudget constraint, in the K + 2 unknowns (z0, z , λ).

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Mean variance efficiency

Assumption :The investor portfolio ranking is increasing in the expectation offinancial income z0 + Ez ′a and decreasing in its variance var(z ′a).

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min var(z ′a)z0 + Ez ′a ≥ Mp0z0 + p′z = ω0,

Homogeneity (of degree 2) in the couple (M, ω0).

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min var(z ′a)z0 + Ez ′a ≥ Mp0z0 + p′z = ω0,

Mean variance efficient portfolios:min var(x ′R)

x0(1 + r) + Ex ′R ≥ Mω0

x0 + 11′x = 1.

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x∗ = αΓ−1(E R − R011K )

There exists a fixed composition of risky assets x∗, such that allmean variance efficient portfolios are linear combinations of x∗ andof the risk free asset:Two funds theorem: All mean variance efficient portfolios aremade of two mutual funds, the risk free fund and a unique riskyfund.

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The medium run : inflation

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Dynamic stochastic general equilibrium models

Random shocks to generate trajectories that look like the observednational accounts.

Rational expectations; representative agents; often calibration andsolution by numerical techniques.

1. The classical or real business cycle model: households, firms,equilibrium

2. Monopolistic competition: households, firms, equilibrium

3. Forward looking price fixation

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Description of consumer’s behavior

The household has a competitive behavior: s/he considers prices,wages and interest rates as given.

max E0

∞∑t=0

βtU(Ct , Lt) = E0

∞∑t=0

βt

[C 1−σt

1− σ −L1+φt

1 + φ

]Budget constraints:

PtCt + QtBt = WtLt + Bt−1 + lump sum transfers net of taxes

End of game:lim

T→∞BT ≥ 0

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First order conditions

Intra period in logarithms

wt − pt = σct + φ`t labor supply

Between period t and t + 1

QtUC ,t

Pt= βEt

UC ,t+1

Pt+1Euler equation (1)

Log linearized

it = ρ+ Etπt+1 + σEt∆ct+1,or Euler equation (2)

ct = Etct+1 − 1σ [it − Etπt+1︸ ︷︷ ︸

rt

−ρ]

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Firms

Exogenous stochastic technical progress, no capital (or fixedcapital stock)Competitive representative firm with production function

Yt = AtN1−αt

or in logarithmyt = at + (1− α)nt

Maximization of profit PtYt −WtNt in each period, given pricesand wages

Wt

Pt= (1− α)AtN

−αt

which yields the labor demand schedule in logarithm

wt − pt = at − αnt + ln(1− α) labor demand

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Equilibrium allocation

Equalizing labor supply with labor demand, and good demand withproduction yields

wt − pt = at − αnt + ln(1− α) = σyt + φnt ,

where yt = at + (1− α)nt . A straightforward computation gives

nt =1− σ

σ(1− α) + φ+ αat +

ln(1− α)

σ(1− α) + φ+ α

yt =1 + φ

σ(1− α) + φ+ αat +

(1− α) ln(1− α)

σ(1− α) + φ+ α

Employment and output are determined independently of nominalquantities (dichotomy).

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Real interest rate

The Euler equation gives the real interest rate that supports theallocation:

rt = it − Etπt+1 = ρ+ σEt∆yt+1,

so that

rt = ρ+ σ1 + φ

σ(1− α) + φ+ αEt∆at+1

The real interest rate is determined by the real path ofconsumption (or output) together with the impatience of theagents.

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Price indeterminacy, or the path of assets

Rational expectations (or expectations are chosen to rationalize theequilibrium path)

In the absence of bank interventions (Qt = 1 or it = 0), thereseems to be no motive to hold nominal assets in the long run (theaggregate stock of assets has to be zero, and the general pricelevel indeterminate).

Short run models: fixed capital stock, no motives to hold nominalbalances (but only productivity shocks?).

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How to have unemployment and a non neutral monetarypolicy, at least in the short run

I Equilibria with quantity rationing, with slow adjustment ofnominal wages and prices (Walras tatonnement). Who setsprices? The only one with ‘true’ unemployment.

I Confusion between real and nominal shocks may give scopefor monetary policy.

I Equilibria with monopolistic competition (local monopolypower), underemployment compared with perfect competition.Some nominal inertia is added:

I Delayed arrival of informationI Menu costs in changing prices : the new Keynesian model

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Escaping from classical theoryThe first step is to introduce monopolistic competition.

Continuum of goods, designated with an index i which takes itsvalued in the interval [0, 1].

max E0

∞∑t=0

βtU(Ct , Lt)

where

Ct =

(∫ 1

0Ct(i)1−1/εdi

)ε/(ε−1), ε > 1

Budget constraints:∫ 1

0Pt(i)Ct(i)di + QtBt = WtLt + Bt−1 + lump sum transfers

Macroeconomic first order conditions are identical to the singlegood case.

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Firms

One firm per variety, all with identical production functions

Yt(i) = AtNt(i)1−α

Common aggregate productivity shock.Firms are local monopolists who take the aggregates (Ct ,Pt) asgiven, and face identical isoelastic demand schedules.They choose optimally their price, given the possible rigidities.

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Equilibrium with flexible prices

The only difference with the classical labor demand scheduledemand comes from the markup term : firms reduce their outputand labor demand to increase their profits.

The dichotomy still is present and there are no real effects ofmonetary policy.The main step is to introduce frictions, so that prices do not adjustinstantaneously to the shocks in the economy.

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The output gap

The natural level of output ynt is defined as the one that would

prevail under flexible prices and monopolistic competition.The output gap yt is the difference between output and its naturallevel yn

t .

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The final equations

The New Keynesian Phillips curve

πt = βEtπt+1 + kyt .

The dynamic IS equation (a rewriting of the Euler equation)

yt = Et yt+1 −1

σ[it − Etπt+1 − rnt ].

Two equations to determine the output gap and the inflation rate.

One needs to make precise the monetary policy to determine thenominal interest rate it and close the model, i.e.

it = ρ+ φππt + φy yt + vt

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