Lecture 1: Complete Markets

Florian Scheuer

1 Course Overview

• This course will focus on the determinants of consumption in dynamic economies

– most closely related to economic welfare

– important to study inequality

– biggest part of GDP

• But we’ll also address asset prices, investment later on

• This week, we’ll start with how consumption is determined under a complete set of(insurance) markets

• This is the benchmark against which we’ll later compare more restricted (and realistic)environments, with incomplete markets

• Understanding these models will require you to be familiar with some tools (notablydynamic programming), which are important in their own right, so we’ll spend sometime on those as well

2 Questions

• How does competition lead to prices and allocations?

• General equilibrium: all markets at the same time

• Are competitive allocations efficient?

• How to solve for a competitive equilibrium

• Introducing some of the notation that we’ll use later

• In order to focus on consumption and prices, for now we’ll ignore how goods areproduced and just assume people are “endowed” with them

1

3 Setup

• N households

• A single good

• Letci ≡

{ci

t

}∞

t=0

be the sequence of consumption for household i

• Preferences for household i:ui

(ci)

• Special case that we often use:∞

∑t=0

βtui

(ci

t

)• Possible generalizations:

– Non time-separable specifications (e.g. habit formation), for example

ui(cit, Si

t) = ui(cit − αSi

t)

where St measures average consumption in the past or just St = ct−1

– Many goods at each point in time (e.g. leisure and consumption)

– Other patterns besides exponential discounting (e.g. finite lifetime)

• Endowments:e ≡

{ei

t

}∞

t=0for i = 1, .., N

– Goods are not storable

– (storage is a way to convert t-goods into t + 1-goods; when we look at economieswith production we’ll call the available ways of converting one good into another“technologies”)

• An allocation isc ≡

{ci

t

}∞

t=0for i = 1, .., N

Definition 1. An allocation is feasible if it satisfies the resource constraint:

N

∑i=1

cit ≤

N

∑i=1

eit ∀t

• Markets

2

– Each date is a different good

– There is a price pt for consumption goods delivered at date t

– All these goods trade in a market at the beginning of time

– Normalize p0 ≡ 1 and denote the price vector by p ≡ {pt}∞t=0

4 Competitive Equilibrium

• Assume that households behave competitively, taking prices as given

• Each household faces the following problem:

maxci

ui

(ci)

s.t.∞

∑t=0

ptcit ≤

∞

∑t=0

pteit

(1)

Definition 2. A competitive equilibrium consists of:

1. A price vector p

2. An allocation c

such that

1. ci solves problem (1) for all i, taking p as given

2. Markets clear:N

∑i=1

cit =

N

∑i=1

eit ∀t

• Generalizations

– Many goods (instead of just one good per date)

– Many states of the world

– Going beyond exchange economies to economies with production

3

5 The First Welfare Theorem

Definition 3. A feasible allocation c is Pareto efficient (or Pareto optimal) if there is no otherfeasible allocation c such that

ui

(ci)≥ ui

(ci)

∀i

ui

(ci)> ui

(ci)

for some i

Proposition 1. (The First Welfare Theorem). Let c be a competitive equilibrium allocation. Thenc is Pareto efficient

Proof. Assume the contrary. Then, there is a feasible allocation c such that

ui

(ci)≥ ui

(ci)

∀i

ui

(cj)> ui

(cj)

for some j

This implies

∞

∑t=0

pt cit ≥

∞

∑t=0

ptcit ∀i

∞

∑t=0

pt cjt >

∞

∑t=0

ptcjt

(otherwise allocation c would not be chosen). Adding:

N

∑i=1

[∞

∑t=0

pt cit

]>

N

∑i=1

[∞

∑t=0

ptcit

]∞

∑t=0

pt

N

∑i=1

cit >

∞

∑t=0

pt

N

∑i=1

cit

∞

∑t=0

pt

N

∑i=1

eit >

∞

∑t=0

pt

N

∑i=1

eit

which is a contradiction. (The last step uses that in a competitive equilibrium ∑Ni=1 ci

t =

∑Ni=1 ei

t and any feasible allocation c satisfies ∑Ni=1 ei

t ≥ ∑Ni=1 ci

t for all t).

• Notice that the theorem requires very few assumptions:

– Local nonsatiation. Where do we use that?

– The value of the aggregate endowment is finite, i.e. ∑∞t=0 pt ∑N

i=1 eit < ∞. Where

did we use that?

– but not concavity, exponential discounting, etc.

4

• FWT limits the possible justifications for government intervention

1. Conditions of the theorem not met

(a) imperfect competition

(b) externalities

(c) incomplete markets

(d) the value of the aggregate endowment is infinite (OLG)

(e) etc.

2. Redistribution

• FWT is useful for finding competitive equilibria (sometimes computing Pareto Opti-mal allocations is easier than computing equilibria)

6 Example

• N = 2

• Endowment

eti =

{1 if i + t is even0 if i + t is odd

• Identical utility functions

u (c) =c1−σ

1− σ

with exponential discounting

6.1 Look for the competitive equilibrium directly

• Steps:

1. Solve household’s problem for arbitrary prices

2. Find prices such that markets clear

3. Replace the market-clearing prices in household problem to find equilibrium al-location

1. Household problem

5

• FOC for household i:1

βtu′(

cit

)− λi pt = 0

u′(

cit

)= λi pt

βt (2)

• Note that (2) implies an important property:

u′(ci

t)

u′(

cjt

) =λi pt

βt

λj ptβt

=λi

λj

so the ratios of marginal utilities across agents are constant over time. This is aquite general property of competitive models. What’s the logic?

– Each household equates its MRS to the relative price

– All households face the same relative prices

– Therefore all households’ MRS are equated

• Now rearrange (2):

u′(

cit

)= λi pt

βt(ci

t

)−σ= λi pt

βt (3)

cit =

(βt

λi pt

) 1σ

(4)

• Use budget constraint

∞

∑t=0

pt

(βt

λi pt

) 1σ

=∞

∑t=0

pteit(

λi)− 1

σ∞

∑t=0

p1− 1σ

t βtσ =

∞

∑t=0

pteit

λi =

∑∞t=0 p1− 1

σt β

tσ

∑∞t=0 ptei

t

σ

• Replace in (4)

cit =

∑∞s=0 psei

s

∑∞s=0 p1− 1

σs β

sσ

(βt

pt

) 1σ

(5)

1Since the utility function is concave and the budget set is convex, the FOC holds as long as we have aninterior solution. We typically (but not always) make assumptions such that this is indeed the case.

6

2. Market-clearing prices

• Market-clearing condition

c1t + c2

t = 1 ∀t(βt

pt

) 1σ

∑∞s=0 ps

(e1

s + e2s)

∑∞s=0 p1− 1

σs β

sσ

= 1

(βt

pt

) 1σ

∑∞s=0 ps

∑∞s=0 p1− 1

σs β

sσ

= 1

• Therefore(

βt

pt

) 1σ

must be constant

• Since we have normalized p0 = 1, then

pt = βt (6)

3. Allocation

• Replace prices (6) in household decision (5):

cit =

∑∞s=0 βsei

s

∑∞s=0 βs = (1− β)

∞

∑s=0

βseis

=

β1−β1−β2 if i = 11−β1−β2 if i = 2

6.2 Look for the equilibrium via the planner’s problem

• Finding a Pareto efficient allocation

maxc1,c2

∞

∑t=0

βtu(

c1t

)s.t.

c1t + c2

t ≤ 1∞

∑t=0

βtu(

c2t

)≥ u

(7)

7

• Solve by setting up a Lagrangian:2

L(

c1, c2, λt, µ)=

∞

∑t=0

βtu(

c1t

)− µ

[u−

∞

∑t=0

βtu(

1− c1t

)]

• Take FOC:

βt[u′(

c1t

)− µu′

(1− c1

t

)]= 0

u′(c1

t)

u′(1− c1

t) = µ

u′(c1

t)

u′(c2

t) = µ

so again we have the property that the ratio of marginal utilities is equated across time.

• Given that c1 + c2 is constant and equal to 1, we also have that the consumption ofeach agent is constant over time, which we also found to be true in the competitiveequilibrium

• Each value of µ will correspond to a different Pareto efficient allocation

• µ can be interpreted as either a multiplier or as the relative weight of household 2 inthe planner’s objective function.

• By trying all possible values of µ ∈ (0, ∞), we can trace out all the possible Paretoefficient allocations.3

• Thanks to the FWT, we know one of these is the competitive equilibrium.

• Why is this useful?

– If we establish properties that all efficient allocations satisfy, we know the com-petitive equilibrium satisfies them too

– Sometimes (for instance if there is a single household) there is a single efficientallocation; then we have found the equilibrium

– For this particular problem, solving the Planner’s problem doesn’t completelycharacterize the solution - we would need to find the value of µ such that eachhousehold exactly spends their budget

2We’ll talk more about what exactly this means, local & global conditions, necessary and sufficient condi-tions, etc.

3This is exactly equivalent if the problem is concave.

8

7 The Representative Agent

Proposition 2. Suppose we have a competitive equilibrium with a price vector p. Let

λi ≡ 1µi

where µi is the Lagrange multiplier on household i’s budget constraint, and define

uR (x) ≡ maxci

N

∑i=1

λiui (ci)

s.t.N

∑i=1

ci ≤ x

Then there exists a competitive equilibrium in an economy with just one agent with preferences

∞

∑t=0

βtuR (ct)

and the endowment is et = ∑Ni=1 ei

t where the price is p

Proof. By the envelope theorem:

∂uR (c)∂c

= λiu′i (ci)

The competitive equilibrium in the many-agents economy satisfies

βtu′i(

cit

)− µi pt = 0

βt 1λi

∂uR (ct)

∂ct− µi pt = 0

βt ∂uR (ct)

∂ct− pt = 0

which is the FOC for a competitive equilibrium in the representative agent economy. Marketclearing is satisfied at ct because, by definition of uR, ∑i ci

t = ct, and we have ∑i cit = et since

we started from a many-agents competitive equilibrium.

• The idea of the proof is to show that the SAME prices that arise in competitive equi-librium are the ones that would persuade the representative agent to consume theaggregate endowment in each history.

• This does NOT imply that equilibrium prices don’t depend on the distribution of en-dowments, because the preferences of the representative consumer could depend on

9

this, through the weights λi, which are endogenous to the equilibrium.

• We can establish a stronger result if we assume CRRA preferences

Proposition 3. Suppose we have a competitive equilibrium with a price vector pt and the utilityfunction is u (c) = c1−σ

1−σ . Then there exists a competitive equilibrium in an economy with just oneagent with preferences

∞

∑t=0

βtu (ct)

and the endowment is et = ∑Ni=1 ei

t where the price is pt

Proof. The competitive equilibrium in the many-agents economy satisfies

βtu′(

cit

)− µi pt = 0

βt(

cit

)−σ − µi pt = 0

cit =

(µi)− 1

σ

(βt

pt

) 1σ

ct =N

∑i=1

(µi)− 1

σ

(βt

pt

) 1σ

=

(βt

pt

) 1σ N

∑i=1

(µi)− 1

σ

βt (ct)−σ =

[N

∑i=1

(µi)− 1

σ

]−σ

pt

which is the FOC for consuming ct if the prices are pt. Again, market clearing is satisfied,since ct = et in the many-agents competitive equilibrium we started from, with multipliersµi.

8 Sequential Markets

• So far, we imagined that there is a market at the beginning of time where people tradet-dated goods with each other, and never trade again

• There is an alternative assumption on markets, which leads to the same results

• Suppose that instead of markets for every commodity trading at time 0, we have mar-kets opening up every period, but where you can only trade goods one period ahead

– ait+1: number of units of consumption that household i buys at time t to be deliv-

ered to them at time t + 1 (at+1 < 0 means that the household is borrowing)

10

– qt+1: price of a unit of consumption delivered at time t + 1, bought in period t

• Household’s dynamic budget constraint:

cit + qt+1ai

t+1 ≤ eit + ai

t

• Note: the numeraire is now the consumption good in each period

• We can also defineRt+1 ≡

1qt+1

and call Rt+1 the interest rate between periods t and t + 1.

• The budget constraint would become

ait+1 ≤ Rt+1

[ei

t + ait − ci

t

]• Note: this way of writing it reflects the following within-period timing assumption:

1. Get income and consume

2. Interest accrues

– An alternative assumption is the opposite:

1. Interest accrues

2. Get income and consumeai

t+1 = Rtait + ei

t − cit

– Nothing important depends on which way you do it but it’s easy to get mixed upbetween these two formulations.

• Relationship between time-0 prices pt+1 and time-t prices qt+1

– In period t, buy one unit of consumption delivered in t + 1: price is qt+1

– Alternatively, in period 0, buy one unit of consumption delivered in t + 1: priceis pt+1

– Then, sell this unit in period t: earn qt+1 units of consumption delivered in periodt

– Hence, in period 0, you can sell qt+1 units of consumption delivered in t at pricept: earn ptqt+1 in period 0

11

– By no arbitrage,−pt+1 + ptqt+1 = 0 ⇒ qt+1 =

pt+1

pt

• Incomplete way to formulate the household’s problem:

maxci,ai

∞

∑t=0

βtui

(ci

t

)s.t.

cit + qt+1ai

t+1 ≤ eit + ai

t

• Why is this incomplete? As stated, the problem has no solution, because the householdwants to choose ci

t = ∞ every period

• Any arbitrary consumption plan can be financed by borrowing and financing the debtby means of taking on further debt (a Ponzi game)

• To turn this problem into a problem that makes sense, we need to impose some lowerbound on at, so that the household cannot borrow an infinite amount

• Different ways to impose this limit lead to different problems.

• Examples:

– No borrowing:ai

t ≥ 0

– Exogenous debt limit B:ai

t ≥ −B

– Natural debt limit:

ait ≥ −

∞

∑k=t

pkpt

eik

where pt ≡ ∏ts=1 qs = q1 × · · · × qt so pk

pt= qt+1 × · · · × qk. Idea: by the same

logic as above to argue that qt+1 = pt+1/pt, the price of consumption deliveredin period k in terms of period t is pk/pt. Hence, the RHS is just the present-valueof the future endowment stream from the perspective of period t.

– No-Ponzi-game condition:limt→∞

ptat ≥ 0

Idea: cannot just keep rolling over the debt, need to repay “eventually.” But notthat this does not rule out perpetual indebtedness. For example, suppose the

12

interest rate is constant, so

qt =1R

=1

1 + r⇒ pt =

(1

1 + r

)t.

Then debt growing at constant rate g satisfies the No-Ponzi-game condition when-ever g < r. So the condition requires debt not to grow too fast in the long run.

– There are tight relationships between these borrowing limits (exercise!)

• We’ll impose the no-Ponzi-game condition, so the problem becomes:

maxci,ai

∞

∑t=0

βtui

(ci

t

)s.t.

ci + qt+1ait+1 ≤ ei

t + ait

limt→∞

ptat ≥ 0

Proposition 4. If {c, p} is a competitive equilibrium with date-0 markets, then letting

qt+1 =pt+1

pt

{c, q} is a competitive equilibrium with sequential markets.

Proof. The key step is to show that the budget sets of the household coincide in the twoformulations (left as an exercise).

• What’s the logic? Suppose I want to buy 1 claim for period t. How much does it costme?

– In period t− 1 I need to have qt consumption goods to buy it

– In period t− 2 I need to have qt−1qt consumption goods to buy qt goods for periodt− 1

– . . .

– In period 0 I need to have pt = q1 × · · · × qt to buy goods for period 1 and keepreinvesting those until I reach period t, and I’ll have achieved my objective

9 Adding Uncertainty

• Introduce the notation we use to describe uncertainty

• Discuss the notion of “complete markets”

13

– Time-0 trading implementation

– Sequential trading implementation

• Some results on efficient allocations

• Implications for asset prices

9.1 An Exchange Economy with Uncertainty

• Suppose at each date one of S possible states of the world is realized

• Notation:

– State at date t: st

– Histories: st = {s0, s1, ...st}. We can also write st ={

st−1, st}

– Endowments: ei (st)– Aggregate endowment: e

(st) ≡ ∑N

i=1 ei (st)– Probabilities: Pr

(st)

– Conditional probabilities: Pr(st|st−1) or Pr

(st|st−1)

– Think of an event tree, where each st represents a “node”

• Example:

– s ∈ {Rain, Sunshine}

– Pr(st = Rain|st−1) = p (iid case)

– Pr(st = Rain|st−1) = { pH if st−1 = Rain

pL if st−1 = Sunshine(non-iid case)

• More commodities!

– Each date/history combination is a different good

– Example: “consumption goods in period 3 in case of {Rain, Sunshine, Rain}”

• An allocation is: c = ci (st) for i = 1, 2, ..., N

• An allocation is feasible if

N

∑i=1

ci (st) ≤ N

∑i=1

ei (st) ∀st

14

• Household i gets utility

ui

(ci)=

∞

∑t=0

∑st

βt Pr(st) ui

(ci (st))

• This specification of preferences involves several assumptions at once:

– Time - separability

– Geometric discounting

– Expected utility representation

9.2 The Planner’s Problem

• Look for Pareto Efficient Allocations

maxc

∞

∑t=0

∑st

βt Pr(st) ui

(ci (st))

s.t.N

∑i=1

ci (st) ≤ N

∑i=1

ei (st) ∀st

∞

∑t=0

∑st

βt Pr(st) uj

(cj (st)) ≥ uj ∀j 6= i

• Use Lagrangian / welfare weights:

maxc

∞

∑t=0

∑st

βt Pr(st) [ N

∑i=1

λiui

(ci (st))]

s.t.N

∑i=1

ci (st) ≤ N

∑i=1

ei (st) ∀st

• FOC w.r.t. ci (st):βt Pr

(st) λiu′i

(ci (st))− µ

(st) = 0

• Take ratio of two different households:

u′i(ci (st))

u′j(cj (st)

) =λj

λi (8)

• Ratio of marginal utilities is constant across time and states of the world

15

• Rearrange:

ci (st) = (u′i)−1(

λ1

λi u′1(

c1 (st))) (9)

(I’m taking as a benchmark an arbitrary household 1)

• Sum over households:

N

∑i=1

ci (st) = N

∑i=1

(u′i)−1

(λ1

λi u′1(

c1 (st)))e(st) = N

∑i=1

(u′i)−1

(λ1

λi u′1(

c1 (st))) (10)

• Equation (10) defines a relationship between c1 (st) and the aggregate endowmente(st)

• In particular c1 (st) does NOT depend on the individual endowments, only on the sumand on the weights λ

• By equation (9), this is true of for every household!

• A form of full-insurance

• This is true of EVERY Pareto efficient allocation

• Therefore, by the FWT, it must be true of the competitive equilibrium

• Moreover, ci(st) does not depend on the history st conditional on the aggregate en-dowment e(st).

• I.e. in general, individual consumption should only depend on aggregate endowment,not on individual endowment:

ci(st) = gi(e(st))

for some increasing function gi

16

• Special case of CRRA preferences:

ci (st) = (λ1

λi

(c1 (st))−σ

)− 1σ

=

(λ1

λi

)− 1σ

c1 (st)N

∑i=1

ci (st) = c1 (st) N

∑i=1

(λ1

λi

)− 1σ

⇒ c1 (st) = e(st) N

∑i=1

(λ1

λi

) 1σ

so everyone’s consumption moves in proportion to the aggregate endowment.

• Consider another technology, with storage

f (Kt(st−1), st) + (1− δ)Kt(st−1) ≥ Kt+1(st) +N

∑i=1

cit(s

t)

• In this case, you can show that, in any Pareto optimum,

cit(s

t) = gi(C(st))

with C(st) = ∑i cit(s

t), so individual consumption only varies with aggregate con-sumption (the previous endowment economy was a special case where C(st) = e(st)).Idea: write the Pareto problem and consider the subproblem

U(C) = maxci

∑i

λiui(ci) s.t. ∑i

ci = C

and then solvemaxC,K

∑st,t

βtU(C(st))Pr(st)

s.t.f (Kt(st−1), st) + (1− δ)Kt(st−1) ≥ Kt+1(st) + C(st)

• In other words, even if there is heterogeneity and idiosyncratic risk, under perfectrisk sharing, we can think of the planning problem as one for a single representativeconsumer with preferences U(C)

17

9.3 Tests

• These properties of perfect risk-sharing can be tested (see the Townsend papers on thesyllabus). Regression

cit = αiCt + βiei

t + γiXit + εi

t

or in logs

• Then βi should be zero (more generally: once we condition on Ct, noting else shouldbe significant!)

• Moreover, αi should be 1 with CARA and identical risk aversions, and the same shouldhold with CRRA in the log-regression (exercise!)

• Data from southern Indian villages (ICRISAT), later also from Thailand (socio-economicsurvey), Ivory Coast (World Bank LSMS)

• High idiosyncratic income risk - especially because weather has different impact de-pending on each household’s source of income

• Insurance includes gifts, loan forgiveness, etc.

• Townsend (1994) runs regression separately for each household, i.e. time-series test

• for India, finds αi < 1, βi > 0, i.e. no perfect risk-sharing, but also no autarky

• α = 1 is rejected in favor of α < 1 in 16% of households

• Note: with 10 years of data, tests have low power. α = 0 is rejected in favor of α > 0for only 52% of households.

• β = 0 is rejected for β > 0 in 15% of households (and in favor of β < 0 in 13%)

• β = 1 is rejected for β < 1 for 87% of households

• For Thailand, finds β to be bigger in overall sample

• Some evidence that insurance (i.e., co-movement of individual ci with aggregate C) islower

– in Bangkok

– for entrepreneurs

– as we look at larger aggregates (Rashid, 1990, for Pakistan)

18

19

• Attanasio and Davis (1996): using US data, they reject insurance across education andage groups

• large increases in wage and earnings inequality led to increases in consumption in-equality, inconsistent with perfect risk sharing

9.4 Complete Time-0 Markets

• “Complete markets” or “complete Arrow-Debreu markets”: there is a market for eachpossible commodity

– Trade once for all possible contingent claims

– These are sometimes called “Arrow-Debreu securities”

– p(st): price (in terms of date-0 goods) of one unit of consumption for delivery if

history st is realized

• Household problem

max{ci(st)}∞

t=0

∞

∑t=0

∑st

βt Pr(st) ui

(ci (st))

s.t.∞

∑t=0

∑st

p(st) ci (st) ≤ ∞

∑t=0

∑st

p(st) ei (st)

(11)

• FOC w.r.t. ci (st):βt Pr

(st) u′

(ci (st))− µi p

(st) = 0

• Take ratios of two different households:

u′i(ci (st))

u′j(cj (st)

) =µi

µj (12)

so we have the feature that ratios of marginal utilities are equated across time andstates of the world

• (This feature has to be true by the FWT since it’s true for every Pareto efficient alloca-tion, this just confirms what we knew)

• Again, the property that ci (st) does not depend on ei (st) (only on e(st)) also holds;

the above tests apply

20

• If we set the welfare weights in the planning problem to

λi =1µi

where µi is what results in equilibrium, then equation (8) coincides with (12) and theplanner’s solution will coincide with the equilibrium

• (But we need to know the prices for this, i.e. until we don’t solve for prices we don’tknow the value of µi for household i’s problem)

9.5 Sequential Markets

• Suppose that instead of markets for every commodity (i.e. every history) trading attime 0, we have markets opening up every period, but where you can only trade claimscontingent on the following day’s realization

• Notation:

– ai (st, st+1): number of units of consumption that household i buys in history st

contingent on the realization of state st+1 in period t + 1

– These claims are sometimes called “Arrow securities”

– q(st, st+1

): price of a unit of consumption contingent on the realization of state

st+1 in period t + 1, bought in history st

• Household’s dynamic budget constraint:

ci (st)+ ∑st+1

q(st, st+1

)ai (st, st+1

)≤ ei (st)+ ai (st)

• Note: the numeraire is now the consumption good in each period

• Incomplete way to formulate the household’s problem:

maxci(st),ai(st)

∞

∑t=0

∑st

βt Pr(st) ui

(ci (st))

s.t.

ci (st)+ ∑st+1

q(st, st+1

)ai (st, st+1

)≤ ei (st)+ ai (st)

• Why is this incomplete? Just like in the case with no uncertainty, we need to add eithera debt limit (“natural”) or otherwise a no-Ponzi condition for this problem to makesense.

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Proposition 5. If {c, p} is a competitive equilibrium with date-0 markets, then letting

q(st, st+1

)=

p(st+1)

p (st)

{c, q} is a competitive equilibrium with sequential markets.

Proof. The key step is to show that the budget sets of the household coincide in the twoformulations (left as an exercise).

• What’s the logic? Suppose I want to buy 1 claim on history st. How much does it costme?

– In history st−1 I need to have q(st−1, st

)consumption goods to buy 1 claim on

history st ={

st−1, st}

– In history st−2 I need to have q(st−2, st−1

)q(st−1, st

)consumption goods to buy

q(st−1, st

)claims on history st−1 =

{st−2, st−1

}– . . .

– In period 0 I need to have p(st) = q

(s0, s1

)× q

(s1, s2

)× · · · × q

(st−1, st

)to buy

claims on history s1 ={

s0, s1}

and keep reinvesting those until I reach history st,and I’ll have achieved my objective

10 Asset Pricing with Complete Markets

• An asset is a stream of dividends d ={

d(st)}

• Examples:

– A consol / perpetuity:d(st) = d ∀st

– One year’s worth of car insurance (the period is one month):

d(st) = r

(st) I (t ≤ 12)

where r(st) is the value of the repairs that my car needs in history st

• Any asset can be replicated by Arrow-Debreu securities

• Therefore, if we denote the price at time zero of asset d by p0 (d), we must have

p0 (d) =∞

∑t=0

∑st

p(st) d

(st) (13)

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• Using the FOC for the household (for ANY household!)

βt Pr(st) u′i

(ci (st))− µi p

(st) = 0

p(st) = βt Pr

(st) u′i

(ci (st)) 1

µi

so for the first period:

1 = p(

s0)= u′i

(c(

s0)) 1

µi

µi = u′i(

c(

s0))

which implies

p(st) = βt Pr

(st) u′i

(c(st))

u′i (c (s0))

(14)

• This expression is known as the “stochastic discount factor”

• The price of an asset is therefore

p0 (d) =∞

∑t=0

∑st

βt Pr(st) u′i

(c(st))

u′i (c (s0))

d(st)

= E

[∞

∑t=0

βt u′i(c(st))

u′i (c (s0))

d(st)]

• An asset is worth more if it pays off in high-marginal-utility states of the world

• If markets were to reopen in history st, what would be the price of the asset in termsof history-st consumption goods?

– Price of an Arrow-Debreu security that pays in history st+k:

p(

st+k|st)=

βk Pr(st+k|st) u′i

(c(st+k))

u′i (c (st))

(notice that this implies p(st+k|st) = 0 if st+k is not a successor of history st)

– Price of asset d in history st:

pst (d) =∞

∑k=0

∑sk

βk Pr(st+k|st) u′i

(c(st+k))

u′i (c (st))

d(

st+k)

23

• One-period returns of holding asset d from history st up to history st+1:

R(st+t|st) = d

(st+1)+ pst+1 (d)

pst (d)

=d(st+1)+ ∑∞

k=0 ∑skβk+1 Pr(st+k+1|st)u′i(c(st+k+1))

u′i(c(st+1))d(st+k+1)

∑∞k=0 ∑sk

βk Pr(st+k|st)u′i(c(st+k))u′i(c(s

t))d(st+k

)• Let

m(

st+1)≡

βu′i(c(st+1))

u′i (c (st))

• Then

∑st+1

Pr(

st+1|st)

m(

st+1)

R(st+t|st) =

= ∑st+1

Pr(

st+1|st)

m(

st+1) d

(st+1)+ ∑∞

k=0 ∑skβk+1 Pr(st+k+1|st)u′i(c(st+k+1))

u′i(c(st+1))d(st+k+1)

∑∞k=0 ∑sk

βk Pr(st+k|st)u′i(c(st+k))u′i(c(s

t))d(st+k

) = 1

or, summarizing:E (m · R) = 1 (15)

• m is sometimes known as a pricing kernel

• Notice that equation (15) holds for ANY asset

• You will sometimes see equation (15) written as

u′i (ct) = βE[Ru′i (ct+1)

](16)

• Example: the risk-free rate

– Suppose

d(

st+k)=

{1 if k = 10 otherwise

– Then

pst (d) = ∑st+1

β Pr(st+1|st) u′i

(c(st+1))

u′i (c (st))

so the risk-free rate is

R =1

pst (d)=

u′i(c(st))

β ∑st+1 Pr (st+1|st) u′i (c (st+1))

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– This is just a special case of (16) for the case where R is deterministic so we cantake it out of the expectation operator.

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