Lecture 6: Arrow-Debreu Pricing: Multiple States, Stochastic Discount Factor, Heterogeneity, Pareto Optimality Economics of Finance
School of Economics, UNSW
Back to our Arrow-Debreu Consumer’s Problem:
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• The problem:
• Choose cG,cB,c0,aG,aB
• to maximize • subject to
u(c0)+β[πG ·u(cG)+πB ·u(cB)] c0 +qG ·aG +qB ·aB =e0,
cG = aG + eG, cB =aB +eB.
• Form the Lagrangian
L=u(c0)+β[πG ·u(cG)+πB ·u(cB)] −λ0 [c0 +qG ·aG +qB ·aB −e0]
− λ1 [cG − aG − eG]
−λ2[cB −aB −eB]
Solving the Consumer’s Problem: (cont’d)
• Equate the partial derivatives of the Lagrangian to zero: • Partials w.r.t. cG,cB,c0,aG,aB
∂L/∂c0 =u′(c0)−λ0 =0 ∂L/∂cG =βπG ·u′(cG)−λ1 =0 ∂L/∂cB =βπB ·u′(cB)−λ2 =0 ∂L/∂aG =−λ0 ·qG +λ1 =0
∂L/∂aB = −λ0 · qB + λ2 = 0
• Partial w.r.t. the multipliers λ0, λ1, λ2 are just the constrains:
c0 +qG ·aG +qB ·aB −e0 =0 cG − aG − eG = 0 cB − aB − eB = 0
Solving the Consumer’s Problem: (cont’d)
• Expressing atomic prices as functions of consumption allocations:
• The values of the multipliers:
λ0 = u′ (c0)
λ1 = βπG · u′ (cG) λ2 =βπB ·u′(cB)
• The prices of the atomic (Arrow-Debreu) securities: q =λ1 =βπ u′(cG)
G λ G u′ (c ) 00
q =λ2 =βπ u′(cB)
B λ Bu′(c) 00
The Prices of Atomic (Arrow-Debreu) Securities
• Combine the solution to the consumer’s problem with the market clearing conditions:
• Atomic prices as functions of consumption allocations: q =λ1 =βπ u′(cG)
G λ G u′ (c ) 00
q =λ2 =βπ u′(cB) B λ Bu′(c)
• Market clearing conditions:
c0 =e0;cG =eG;cB =eB.
• The prices of the atomic (Arrow-Debreu) securities: q =βπ u′(eG)
G G u′ (e0)
q =βπ u′(eB) B B u′ (e0)
Is there any trade of atomic (Arrow-Debreu) securities possible in this economy?
• Remember constraints
c0 + qG · aG + qB · aB = e0,
cG = eG + aG,
cB = eB + aB
• and market clearing
c0 =e0;cG =eG;cB =eB.
• gives us aG = aB = 0 in this equilibrium
• Since all agents are the same in this economy (represented by one representative agent) no trade is possible!
Arrow-Debreu Consumer’s Problem: Multiple States
• The setting:
• two periods 0 and 1;
• multiple states in period 1 indexed by s1
• set of all possible states in period 1 is S1, so that s1 ∈ S,
e.g., S1 = {Good, Fair, Bad} • The problem:
• Choose c0,cs1,as1, for all s1 ∈ S1
• to maximize
• subject to
u(c0)+β πs1 ·u(cs1) s1 ∈S1
c0 + qs1 · as1 = e0, s1 ∈S1
cs1 =as1 +es1, foralls1 ∈S1
Solving the Consumer’s Problem
• Form the Lagrangian L=u(c0)+β πs1 ·u(cs1)−
−λc+q ·a −e−λ (c −a −e)
00 s1s10 s1s1s1s1 s1 ∈S1 s1 ∈S1
• Equate the partial derivatives of the Lagrangian to zero: • Partials w.r.t. c0, cs1 , as1 , for all s1 ∈ S1
∂L/∂c0 =u′(c0)−λ0 =0
∂L/∂cs1 =βπs1 ·u′(cs1)−λs1 =0, foralls1 ∈S1 ∂L/∂as1 =−λ0·qs1 +λs1 =0, foralls1 ∈S1
• Partial w.r.t. the multipliers λ0 , λs1 are just the constrains: c0 + qs1 · as1 − e0 = 0
cs1 −as1 −es1 =0, foralls1 ∈S1.
The Prices of Atomic (Arrow-Debreu) Securities
• Expressing atomic prices as functions of consumption allocations:
q =λs1 =βπ u′(cs1),foralls ∈S. s1 λ s1u′(c) 1 1
• Combine with the market clearing conditions: c0 =e0;cs1 =es1, foralls1 ∈S1.
• The prices of the atomic (Arrow-Debreu) securities:
q =βπ u′(es1),foralls ∈S. s1 s1 u′(e0) 1 1
Stochastic Discount Factor
• The prices of the atomic (Arrow-Debreu) securities: q =βπ u′(es1),foralls ∈S.
s1 s1 u′(e0) 1 1
• The discount factor of a specific period is the sum of all
atomic security prices in this period
• The stochastic discount factor, m1, is a random variable
• its value is unknown at t = 0;
• its value at time 1 is m = βu′(es1), if state s is realized; s1 u′(e0) 1
df(1)=q =βπ u′(es1)
where E[·] is the expectation operator.
s1 s1 u′(e0) s1 ∈S1
• Then the discount factor is
df(1)=βπ u′(es1)=π m =E[m]
s1 u′(e0) s1 s1 1 s1 ∈S1
Forward atomic prices and risk neutral probabilities
• The (spot) prices of the atomic (Arrow-Debreu) securities: q =βπ u′(es1),foralls ∈S.
s1 s1 u′(e0) 1 1
• The forward prices of the atomic (Arrow-Debreu) securities:
qs1 qs1 u′(es1) u′(es1)
fs1 =df(1)= q =πs1 u′(e) πs1 u′(e),∀s1∈S1.
• The forward prices are often called risk neutral probabilities
π s 1 = f s 1 ∀ s 1 ∈ S 1 .
If agents are risk neutral, their utility is linear u′ = const
s1 0s1∈S1 0
and fs1 simplifies to
π = π π = π , ∀ s ∈ S .
s1 s1 s1 s1 1 1 s1 ∈S1
Pricing state-contingent claims
• Using the atomic state prices, often called, pricing kernel: p = q · c,
q – row vector of atomic state prices or pricing kernel,
c – column vector of state-contingent payments • Using risk-neutral measure:
p = d f · E ( c ) ,
c – random variable, realised value depends on a state, E(·) – expectation taken with respect to risk-neutral measure using risk-neutral probabilities π
• Using stochastic discount factor:
p = E(m1c),
c – random variable, realised value depends on a state, m1 – stochastic discount factor,
E(·) – expectation taken with respect to physical probability measure using actual probabilities π
Heterogeneity
Homogeneity (representative agent) is clearly a simplified assumption.
• Heterogeneity either in endowments or in preferences (incl. β), or both is necessary for trade
• Consider K agents, each indexed by k;
• with utilities uk
• each agent k chooses optimal ck0 and cks1 given endowments ek0 and eks1 for all s1.
Consumers’ Problem
• Each agent k maximises expected utility, Uk, given by Uk=ukck+ βkπ·ukck
0 s1 s1 s1 ∈S1
expected discounted future utility
• subject to period-0 constraint
c k0 + q s 1 · a ks 1 = e k0 ,
• and a series of period-1 constraints for every possible state:
cks1 = aks1 + eks1 , for all s1 ∈ S1 • Market clearing (now makes more sense)
ck0 =ek0; cks1 =eks1,∀s1 k=1 k=1 k=1 k=1
Market clearing conditions
Homogeneous consumers (representative agent):
c0 (s0) = e0 (s0); c1 (s1) = e1 (s1),∀s1 ∈ S1.
• Goods from endowment are non-storable and there is no trade, so consume all you can.
Heterogeneous consumers:
ck0 (s0) = ek0 (s0); cks1 = eks1, ∀s1 k=1 k=1 k=1 k=1
• Trade is possible
• The total number of goods from all endowments in each
time-state must equal the total number of goods consumed.
• Agents may use atomic (Arrow-Debreu) securities to shift
consumption between time-states, but all endowment must be consumed jointly in the respective time-state.
Characterisation of the Equilibrium
• From the first order conditions the prices of the atomic (Arrow-Debreu) securities
uk′ck k s1
q s 1 = β π s 1 u k ′ c k0 =
uk′ek +ak
= β πs1
uk′ ek0−qs1·aks1 s1 ∈S1
for all k and s1 ∈ S.
• also impose market clearing which implies that
a ks 1 = 0 , ∀ s 1 ∈ S . k=1
Example: Heterogeneous Consumers
• Consider a world in which there are two periods: 0 and 1.
• In period 1 there are two possible states of nature: a good weather state (G) and a bad weather state (B). That is: s1 ∈ S1 = {G, B}. They are equally probable, i.e.,
πG =πB =1/2
• There are two consumers in this economy.
• Their preferences over apples are exactly the same and are
given by the following expected utility function: 1ck+βπ lnck.
20 s1 s1 s1 ∈S1
where subscript k = 1, 2 denotes consumers. • The consumer’s time discount factor β = 0.9.
Example: Heterogeneous Endowment
The consumers are identical in every way (e.g. utility function and discount factor) except in their endowments which are given in the table below:
Consumer 1 Consumer 2
Endowments t=0 t=1
4 4 2 4 2 1
There is some inequality: consumer 1 has better endowments in both states.
• What is the equilibrium condition?
• Equilibrium price and trading volume? • Welfare gain from free trade?
Solving for Equilibrium
Equilibrium prices (same and taken for both consumers) q =βkπ uk′(cs1)
s1 s1 uk′(c0)
q =βπ 1/c1G,q =βπ 1/c2G ⇒c1 =c2
G G1/2G G1/2GG q =βπ 1/c1B,q =βπ 1/c2B ⇒c1 =c2
B B1/2B B1/2BB Clearing conditions: c1G + c2G = 6; c1B + c2B = 3 Equilibrium consumption: c1G = c2G = 3; c1B = c2B = 1.5 Equilibrium trades: a1G = −a2G = −1; a1B = −a2B = −0.5
= 2.9768 > U1 = 2.9357 autarky
= 2.3768 > U2 = 2.3119 autarky
Gains from trade
Consumers mutually benefit from trade. Why?
Consumption smoothing is an important driving force.
How does discount factor, βk, affect the allocation in this case?
Example: two symmetric agents
• two agents (1,2); two periods, two states (A,B) in period 1 πA =πB =1/2
• same preferences (u and β)
• same endowments in period 0 e10 = e20 = e0
• different (but “symmetrical”) endowments in period 1: e1A