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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
Macro II – Chapter 5 – Macro and Finance 2
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)
Macro II – Chapter 5 – Macro and Finance 3
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
Macro II – Chapter 5 – Macro and Finance 4
2.1 The physical setting
2.1.1 Preferences and endowments
• π(s′|s) is a Markov chain with initial distribution π0(s)
• Prob(st+1 = s′|st = s) = π(s′|s) and Prob(s0 = s) = πo(s)
• a sequence of probability measures π(st) on histories st = [st, st−1, ..., s0] is given by
π(st) = π(st|st−1)π(st−1|st−2)...π(s1|s0)π0(s0) (1)
and conditional probability is given by
π(st|s0) = π(st|st−1)π(st−1|st−2)...π(s1|s0) (2)
Macro II – Chapter 5 – Macro and Finance 5
• 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
Macro II – Chapter 5 – Macro and Finance 6
• 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) = +∞
Macro II – Chapter 5 – Macro and Finance 7
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
Macro II – Chapter 5 – Macro and Finance 8
• 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)
Macro II – Chapter 5 – Macro and Finance 9
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
Macro II – Chapter 5 – Macro and Finance 10
• 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)
Macro II – Chapter 5 – Macro and Finance 11
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)
Macro II – Chapter 5 – Macro and Finance 12
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.
Macro II – Chapter 5 – Macro and Finance 13
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)
Macro II – Chapter 5 – Macro and Finance 14
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)
Macro II – Chapter 5 – Macro and Finance 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).
Macro II – Chapter 5 – Macro and Finance 16
• 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].
Macro II – Chapter 5 – Macro and Finance 17
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
Macro II – Chapter 5 – Macro and Finance 18
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
Macro II – Chapter 5 – Macro and Finance 19
3 A Quantitative Model: Mehra & Prescott
3.1 Data
• See Table and Figures
Macro II – Chapter 5 – Macro and Finance 20
Macro II – Chapter 5 – Macro and Finance 21
Macro II – Chapter 5 – Macro and Finance 22
Macro II – Chapter 5 – Macro and Finance 23
Macro II – Chapter 5 – Macro and Finance 24
• 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.
Macro II – Chapter 5 – Macro and Finance 25
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.
Macro II – Chapter 5 – Macro and Finance 26
Equilibrium
Proposition 3 Define A = [aij], aij = βφijλ1−αj and assume limm−→∞Am = 0. Then,
a Debreu competitive equilibrium exists.
Macro II – Chapter 5 – Macro and Finance 27
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.
Macro II – Chapter 5 – Macro and Finance 28
• 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
Macro II – Chapter 5 – Macro and Finance 29
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
Macro II – Chapter 5 – Macro and Finance 30
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 .
Macro II – Chapter 5 – Macro and Finance 31
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%)
Macro II – Chapter 5 – Macro and Finance 32
• 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.
Macro II – Chapter 5 – Macro and Finance 33
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
Macro II – Chapter 5 – Macro and Finance 34
• 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
Macro II – Chapter 5 – Macro and Finance 35
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)
Macro II – Chapter 5 – Macro and Finance 36
• 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.
Macro II – Chapter 5 – Macro and Finance 37
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
Macro II – Chapter 5 – Macro and Finance 38
• 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
Macro II – Chapter 5 – Macro and Finance 39
• The solution is that F = .00001 with
α ξ β? ρ.82 .23 .99 .99
Macro II – Chapter 5 – Macro and Finance 40
• 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
Macro II – Chapter 5 – Macro and Finance 41
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
Macro II – Chapter 5 – Macro and Finance 42
Macro II – Chapter 5 – Macro and Finance 43
• 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.
Macro II – Chapter 5 – Macro and Finance 44
• 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
Macro II – Chapter 5 – Macro and Finance 45
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
Macro II – Chapter 5 – Macro and Finance 46
• 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
Macro II – Chapter 5 – Macro and Finance 47
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)
]
Macro II – Chapter 5 – Macro and Finance 48
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
Macro II – Chapter 5 – Macro and Finance 49
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.
Macro II – Chapter 5 – Macro and Finance 50
Macro II – Chapter 5 – Macro and Finance 51
4.3 Some More Observations
Macro II – Chapter 5 – Macro and Finance 52