chapter 5 macroeconomics and finance -...

Macro II – Chapter 5 – Macro and Finance 1

Chapter 5

Macroeconomics and Finance

• Main references :

- L. Ljundqvist and T. Sargent, Chapter 7

- Mehra and Prescott 1985 JME paper

- Jerman 1998 JME paper

- J. Greenwood and B. Jovanovic, ‘‘The

Information-Technology Revolution and the Stock

Market", AER, 1999

- Kocherlakota 1996 survey in the JEL


1 Introduction

• In this chapter, I want to

1. show how to compute asset prices in general equilibrium

2. discuss of the some quantitative properties of asset prices in (simple) GE models

3. show an application to the US stock market in the 70s (if I have time)


2 Asset Prices in General Equilibrium

• I describe here the competitive equilibrium of a pure exchange infinite horizon economy

with stochastic Markov endowments.

• This is a basic setting for studying risk sharing, asset pricing, consumption.

• 2 different market structures

1. Arrow-Debreu structure with complete markets in dated contingent claims all traded at

period 0

2. recursive structure with complete one-period Arrow securities.

• The 2 have different asset structures but identical consumption allocations


• Trading occurs after s0 has been observed.

• the probability of state (history) st conditional on being in state (history) sτ at date τ is

π(st|sτ) = π(st|st−1)π(st−1|st−2)...π(sτ+1|sτ) (3)

• (because of Markov property, π(st|sτ) does not depend on history sτ−1


• Households: i = 1, ..., I . Each owns a stochastic endowment of one good yit = yi(st), and

st is publicly observable

• Each household purchase a history-dependant consumption plan ci = cit(s

t)∞t=0

• Household objective

U(ci) =

∞∑t=0

∑st

βtu[cit(s

t)]π(st|s0) = E0

∞∑t=0

βtu[cit(s

t)] (4)

• u has all nice properties, including limc↓0 u′(c) = +∞


2.1.2 Complete markets

• Household trade dated state-contingent claims to consumption

• q0t (s

t) = price of a claim on time-t consumption, contingent on history st, in terms of a

numeraire not specified

• the BC is∞∑t=0

∑st

q0t (s

t)cit(s

t) =

∞∑t=0

∑st

q0t (s

t)yi(st) (5)

• Hh problem: choose ci to maximize (4) s.t. (5)

• Notice that one can collapse the problem into a problem with a single budget constraint

because of complete markets


• let µi be the Lagrange multiplier of this constraint, FOC:

∂U(ci)

∂cit(s

t)= µiq0

t (st) (6)

and with the specification (4) of preferences,

∂U(ci)

∂cit(s

t)= βtu′[ci

t(st)]π(st|s0) (7)

one gets

βtu′[cit(s

t)]π(st|s0) = µiq0t (s

t) (8)


Definition 1 A price system is a sequence of functions q0t (s

t)∞t=0. An allocation

is a list of sequences of functions cit(s

t)∞t=0, one for each i. A feasible allocation

satisfies ∑i

yi(st) ≥∑

i

cit(s

t) (9)

Definition 2 A competitive equilibrium is a feasible allocation and price system

such that the allocation solves each household problem


• Notice that (8) impliesu′[ci

t(st)]

u′[cjt(s

t)]=

µi

µj(10)

which means thats ratios of marginal utilities between pairs of agents are constant across all

sates and dates.

• An equilibrium allocation solves (10), (9),and (5). Note that (10) implies

cit(s

t) = u′−1

u′[c1

t (st)]

µi

µ1

(11)

and substituting into feasibility condition (9) at equality gives∑i

u′−1

u′[c1

t (st)]

µi

µ1

=

∑i

yi(st) (12)

• the RHS of (12) does not depend on the entire history st, but only on current state st,

therefore the LHS, therefore c1t (s

t). Then, from (11), it is also the case for all cit(s

t). One then

has the following proposition

Proposition 1 The competitive equilibrium allocation is not history dependent; cit(s

t) =

ci(st)


2.1.3 Equilibrium pricing function

• Let ci, i = 1, ...I b an equilibrium allocation. Then (6) or (8) gives the price system q0t (s

t)

as a function of the allocation to Hh i, for any i.

• The price system is a stochastic process

• Because the units of the price system are arbitrary, one can normalized one of the multipliers

at any positive value. I set µ1 = u′[c1(s0)], so that q00(s0) = 1, i.e. the price system is in units

of time-0 goods. (one has therefore µi = u′[ci(s0)] for all i)


2.1.4 Examples: Risk sharing

• suppose u(c) = (1− γ)−1c1−γ, γ > 0 (CRRA). Then (10) implies

cit = cj

t

(µi

µj

)−1γ

(13)

• time-t elements of consumption allocations to distinct agents are constant fractions of one

another.

• The individual consumption is perfectly correlated with the aggregate endowment or aggre-

gate consumption.

• The fractions assigned to each individual are independent of the realization of st.

• There is extensive cross-time cross-state consumption smoothing.


2.2 Asset pricing

2.2.1 Pricing Redundant Assets

• Let d(st)∞t=0 be a stream of claims on time t, state st consumption, where d(st) is a

measurable function of st. The price of an asset entitling the owner to this stream must be

a00 =

∞∑t=0

∑st

q0t (s

t)d(st) (14)

(this can be understood as an arbitrage equation)

2.2.2 Riskless Consol

• A riskless consol offers for sure one unit of consumption at each period, i.e. dt(st) = 1 for

all t and st. The price is

a00 =

∞∑t=0

∑st

q0t (s

t)


2.2.3 Riskless strips

• Consider a sequence of strips of returns on the riskless consol. The time-t strip is the return

process dτ = 1 if τ = t ≥ 0, and 0 otherwise. The price of time-t strip at 0 is∑st

q0t (s

t)

2.2.4 Tail assets

• Consider the stream of dividends d(st)t≥0

• For τ ≥ 1, suppose that we strip off the first τ − 1 periods of the dividend and want to get

the time-0 value of the dividend stream d(st)t≥τ .

• Let a0τ(s

τ) be the time-0 price of an asset that entitles the dividend stream d(st)t≥τ if

history sτ is realized:

a0τ(s

τ) =∑t≥τ

∑st:sτ=sτ

q0t (s

t)d(st) (15)


• Let us convert this price into units of time τ , state sτ by dividing by q0τ(s

τ):

aττ(s

τ) =a0

τ(sτ)

q0τ(s

τ)=

∑t≥τ

∑st:sτ=sτ

q0t (s

t)

q0τ(s

τ)d(st) (16)

• Notice that for all consumers i

qτt (s

t) =q0t (st)

q0τ (sτ )

=βtu′[cit(s

t)]π(st)

βτu′[ciτ (sτ )]π(sτ )

= βt−τ u′[cit(st)]

u′[ciτ (sτ )]π(st|sτ)

(17)

• Here qτt (s

t) is the price of one unit of consumption delivered at time t, state st in terms of

the date-τ , state-sτ consumption good.

• The price at t for the tail asset is

aττ(s

τ) =∑t≥τ

∑st:sτ=sτ

qτt (s

t)d(st) (18)

• This tail asset formula is useful if one wants to create in a model a time series of equity

prices: an equity purchased at time τ entitles the owner to the dividends from time τ forward,

and the price is given by (18).


• Note: The relative price is (17) is not history dependent, given Proposition 1. This is stated

in the following proposition:

Proposition 2 The equilibrium price of date-t ≥ 0, state-st consumption good ex-

pressed in terms of date τ (0 ≤ τ ≤ t), state sτ consumption good is not history

dependent: qτt (s

t) = qjt (s

k) for j, k ≥ 0 such that t − τ = k − j and [st, st−1, . . . , sτ ] =

[sk, sk−1, . . . , sj].


2.2.5 Pricing One Period Returns

• The one-period version of equation (17) is

qττ+1(s

τ+1) = βu′(ci

τ+1)

u′(ciτ)

π(sτ+1|sτ)

• The RHS is the one-period pricing kernel at time τ .

• The price at time τ in state sτ of a claim to a random payoff ω(sτ+1) is given, using the

pricing kernel, bypτ

τ(sτ) =

∑sτ+1

qττ+1(s

τ+1)ω(sτ+1)

= Eτ

[βu′(cτ+1)

u′(cτ ) ω(sτ+1)] (19)

where superscripts i and dependence to sτ have been deleted.

• Let denote the one-period gross return on the asset by Rτ+1 = ω(sτ+1)/pττ(s

τ). Then, for

any asset, equation (19) implies

1 = Eτ

[β

u′(cτ+1)

u′(cτ)Rτ+1

](20)

• The term mτ+1 = βu′(cτ+1)u′(cτ ) is a stochastic discount factor. Equation (20) can be understood


as a restriction on the conditional moments of returns and mτ+1.

• Applying the law of iterated expectations to equation (20), one gets the unconditional

moments restrection:

1 = E

[β

u′(cτ+1)

u′(cτ)Rτ+1

](21)

2.3 A Recursive Formulation: Arrow Securities

• One introduce another market structure that preserves the equilibrium allocation from our

competitive equilibrium. This setting also preserves the one-period asset-pricing formula (19).

• Arrow (1964): one-period securities are enough to implement complete markets, provided

that new one-period markets are reopened for trading each period

• See Ljundqvist and Sargent for a formal proof


3 A Quantitative Model: Mehra & Prescott

3.1 Data

• See Table and Figures


• The risk premium is high (6.18 %), as the s.d. of real returns is 5.67% for riskless asset and

16.54% for risky asset.

•Mehra and Prescott have proposed a relatively simple endowment economy to quantitatively

reproduce this fact.


3.2 A Pure Exchange Economy

Environment

• representative agent, E0

∑∞t=0 βtu(ct); 0 < β < 1,

u(c; α) =c1−α − 1

1− α

• One productive unit (a tree) ; gives yt units of a perishable good. This tree is an equity

share that is competitively traded, and yt is its dividend.

• the growth rate of yt is stochastic: yt+1 = xt+1yt

• x is markov: xt+1 ∈ λ1, ..., λn, Prob(xt+1 = λj|xt = λi) = φij. Is is assumed that this

markov chain is ergodic, and that λi > 0, y0 > 0.

• yt is observed at the beginning of the period and securities are traded ex-dividend.


Equilibrium

Proposition 3 Define A = [aij], aij = βφijλ1−αj and assume limm−→∞Am = 0. Then,

a Debreu competitive equilibrium exists.


Pricing

• In this economy, the ex dividend price of a security with dividends dt is

Pt = Et

[ ∞∑s=t+1

βs−tu′(ys)

u′(yt)ds

]• For the equity, given the functional forms and the fact that d = y,

P et = P e(yt, xt) = Et

[ ∞∑s=t+1

βs−tyαt

yαs

ys

]• (yt, xt) is a sufficient description of the past history. It defines the state of the economy.

P et = Et

β

u′(yt+1)

u′(yt)(P e

t+1 + yt+1)

• Given that ys = yt× xt+1× xt+2 · · · × xs, P e

t is homogenous of degree 1 in yt, which is the

current endowment of consumption good.

• Given that equilibrium values of the economy are time invariant functions of (yt, xt), the

subscript t can be dropped. The state can be written as (c, i), where yt = c and xt = λi.


• With these notations, the price of an equity satisfies

P e(c, i) = β∑n

j=1 φij︸︷︷︸ (λjc)−α︸︷︷︸ [P e(λjc, j) + cλj]︸︷︷︸ cα︸︷︷︸

i ii iii iv

withi: probability of state j knowing iii: inverse of marginal utility of tomorrow consumption

in state jiii: tomorrow price + dividend in state jiv: marginal utility of today consumption

• Given that P e is homogenous of degree 1 in c, we can write

P e(c, i) = wic

where wi is a constant. Then the pricing equation becomes

wi = βn∑

j=1

φijλ(1−α)j (wj + 1) ∀ i = 1, ..., n

• This is a system of n linear equations in n unknowns (the wi) ; this has a unique positive

solution when a competitive equilibrium exists ; we can derive prices


Prices

• The return of an equity if current state is c, i and next state j is

reij =

P e(λjc, j) + λjc− P e(c, i)

P e(c, i)=

λj(wj + 1)

wi− 1

and the equity expected return is, conditional on state i:

Rei =

n∑j=1

φijreij

• Let us also consider a riskless security that pays 1 unit of good in each state, i.e. di = 1 ∀i.

The price P f of this asset is

P fi = P f(c, i) = β

n∑j=1

φiju′(λjc)

u′(c)dj = β

n∑j=1

φijλ−αj

and Rfi = 1/P f

i − 1

• Now we can compute expected returns w.r.t. the stationary distribution

• Let π ∈ Rn be the vector of stationary probabilities of the markov chain: π is such that

π = φ′π


with∑n

i=1 πi = 1 and φ = φij

• Then we define the expected returns as

Re =∑n

i=1 πiRei

Rf =∑n

i=1 πiRfi

and the risk premium is given by Re −Rf .


3.3 Results

• preference parameters: α and β

• Technology parameters φij, λi : it is assumed that λ takes two values: λ1 = 1 + µ + δ

and λ2 = 1 + µ− δ, and φ11 = φ22 = φ, φ12 = φ21 = 1− φ.

• For US data over 1889-1978, consumption growth = .018, consumption growth s.d. = .036,

consumption growth serial correlation = -.14 ; µ = .018, δ = .036, φ = .43

• Then, we search for (α, β) so that the average risk free rate and the equity risk premium

are reproduced.

• α = people’s willingness to substitute consumption between successive yearly time period

; not greater that 10.

• β ∈]0, 1[

• The model cannot reproduce a equity premium of more that .35%, while it is 6 in the data

(given that the risk free rate is .8%)


• There is therefore an Equity Premium Puzzle

• A large literature has been devoted to this question

• See Kocherlakota 1996 for a nice survey

• Here I present a quantitative solution of this (quantitative) puzzle, as proposed by Jerman

1998

3.4 A Possible Resolution of the Puzzle

• Jerman (1998, JME) proposed a model with production, capital, habit formation and K

adjustment costs that is quantitatively satisfactory.


3.4.1 The Model

• Firms

max Et

∞∑k=0

βkΛt+k

Λt[At+kF (Kt+k, Xt+kNt+k)− wt+kNt+k − It+k]

where βkΛt+kΛt

is the MRS of the owners of the firm.

Kt+1 = (1− δ)Kt + φ

(It

Kt

)Kt

• φ(·) < 1 is a positive concave function that models adjustment costs on capital ; the

shadow price of one installed unit of capital, q, differs from the price of one new unit of

capital (Tobin’s q)

• Firms are financed by retained earnings, and dividends are given by

Dt = AtF (Kt, XtNt)− wtNt − It


• Hh

max Et

∞∑k=0

βku(ct+k)

s.t. wtNt + a′t(Vat + Da

t ) ≥ Ct + a′t+1Vat (Λt)

• at is a vector of financial assets held at t and chosen at t − 1. this vector contains the

representative firm, + possibly other assets. V a is the vector of asset prices and Da the vector

of dividends payments.

• The Hh also face a time constraint : Nt + Lt = 1, and we assume habit persistence :

u = u(ct − αct−1)

• Market equilibrium: AtF (Kt, XtNt) = Ct + It.

• Shocks are to the technology A


3.4.2 Model Solution

• If we use a log-linear approximation to solve the model, the expected returns will be the

same for all agents ; no possibility to account for the risk premium

• The model is solved by log-linearization, but asset prices are computed in a second round

using lognormal pricing formulas (see Hansen & Singleton, 1983)

• The model solution can be written

st = Mst−1 + εt (22)

where ε could be a multivariate normal iid shock (In this model, it is univariate).

• Then we use basic asset pricing formula: a claim on future payment Dt+k(st+k) has a value

Vt(st) = βkEt

[Λt+k(st+kDt+k(st+k)

Λt(st)

](23)


• If Λ and D are lognormal, with distribution given by (22), then the risk free rate ca be

computed from (23) with k = 1 and D(st+1) = 1 :

E(Rt,t+1(st)) =γβ? exp

12(var(Etλt+1 − λt)− var(λt+1 − Etλt+1)

where λt = Λt

Xt, Xt = γXt−1, β? = βγt

• The return on equity is given by

Rdt,t+1(st, st+1) =

Vt+1(st+1) + Dt+1(st+1)

Vt(st)

• Jerman uses simulations to find the unconditional mean.


3.4.3 Quantitative predictions

Calibration

• easy part :

1. long run restrictions: Cobb-Douglas elasticity on labor = .64, γ = 1.005 (per quarter),

δ = .025

2. Productivity shocks : A follows a AR(1) process, with persistence .95 or 1, with s.d. of

the innovation which is such that postwar US gdp s.d. is reproduced

3. Risk aversion:c1−τ

1− ττ = 5


• Difficult part :

1. habit formation parameter α

2. K adjustment cost elasticity ξ

3. time preference β

4. shock persistence ρ

• Let θ1 = [α, β, ξ, ρ]′. θ1 is chosen (estimated) to minimize

F = (θ2 − f (θ1))′Ω(θ2 − f (θ1))

where θ2 is a vector of moments to match, f (θ1) is the vector of corresponding moments

generated by the model and Ω a weighting matrix. (Simulated Method of Moments)

• θ2 =(s.d. of c growth/s.d. of y growth, s.d. of i growth/s.d. of y, mean risk free rate,

equity premium), Ω is identity


• The solution is that F = .00001 with

α ξ β? ρ.82 .23 .99 .99


• The simulation results are

Table 1: Simulation results, Jerman 1998

σ∆cσ∆y

σ∆iσ∆y

E(rf) E(re − rf)

Data .51 2.65 .80 6.18Standard RBC .77 1.54 4.26 .02Standard RBC + τ = 10 .78 1.53 3.36 .26Habit persistence .33 3 4.2 .03K adj. costs 1.14 .68 3.91 .67Benchmark .51 2.65 .8 6.18


4 The Information Technology Revolution and the Stock Mar-

ket

4.1 Motivations

• Here we show that a simple model of asset pricing can account for the late 60’s early 70’s

drop in U.S. (and OECD) market capitalization


• Puzzling phenomenon: The market value of U.S. equity relative to GDP plunged in 1973.

Greenwood & Jovanovic story:

1. The market declined in the late 1960’s because installed firms would have to give way

to IT, while IT firms were not yet listed

2. IT innovators boosted the stock market’s value only in the 1980’s.

• The main assumptions are

1. The IT revolution was heralded in 1973, or perhaps in stages during 1968-1974.

2. The IT revolution favored new firms.


• Reasons for which the IT revolution did favor new firms:

1. Awareness and skill: the manager of an old firm may not know what the new technology

offers or may be unable to implement it.

2. Vintage capital: An old firm’s human and physical capital is tied to its current practices,

and may not easily convert to new technology.

3. Vested interests: management and workers in an older firm may resist new technol-

ogy because it devalues their skills. In doing so, they harm the interests of the firms

shareholders.

• Let’s put down a formal model


4.2 A Simple Model

4.2.1 Fundamentals

• Consider a Lucas 1978 economy: exchange economy, many infinitely-lived identical agents,

equally many infinitely-lived trees.

• Preferences∑∞

t=0 βtU(yt), perfect foresight

• A tree promises a stream of dividends dt, its date-0 price is

P0 =

∞∑t=0

βt

[U ′(yt)

U ′(y0)

]dt

• We assume that a tree yield 1 unit of output in each period, forever, and that is the only

source of income for a representative agent, so that

P0 =

∞∑t=0

βt

[U ′(1)

U ′(1)

]=

1

1− β


• Assume that some unexpected news arrives at t = 0 (“IT revolution”)

• The news is “some of the existing trees will die in the future, say at period T , and they will

be replaced by more productive trees”.

• Formally: a fraction x of the existing trees will die at period T . They will be replaced by

equally many new trees yielding forever 1 + z per period.

• The type of a tree (dying at period T or living forever) is announced at period 0, and new

trees are not traded before period T . At T , the owner ship of those new trees will be allocated

equally among agents.

• Per capita output is given by

yt =

1 for t ≤ T − 11 + xz for t ≥ T


4.2.2 Asset Prices

• Before period T , two types of trees are traded on the stock market:

1. type-1 trees (that dies at T ) : price

P1t =1− βT−t

1− β

2. type-2 trees (that lasts forever) : price

P2t = P1t +

∞∑τ=T−t

βτ

[U ′(1 + xz)

U ′(1)

]

P2t = P1t +βT−t

1− β

[U ′(1 + xz)

U ′(1)

]


4.2.3 Stock Market Dynamics

• Before T , the stock market value is a weighted average of the two types of trees

Pt = xP1t + (1− x)P2t = P1t + (1− x)βT−t

1− β

[U ′(1 + xz)

U ′(1)

]• After T , the stock market value is

Pt =1 + xz

1− β


Comment :

• Both x and z act to lower P before T , while T raises P . Why?

1. Pt is decreasing in x: some trees are expected to be replaced by trees which are not yet

in the market portfolio.

2. A rise in x raises the interest rate, and then decreases P .

3. P decreases in z because of an interest rate effect.

4. A rise in T rises P1t: trees live longer.

• In T , the stock price increases permanently to Pt = 1+xz1−β

• The dynamics of P/GDP is depicted on the figure.

• The empirical counterpart of this figure is the first figure.


4.3 Some More Observations

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