econ 581. introduction to arrow-debreu pricing and ... · introduction to arrow-debreu pricing and...

ECON 581. Introduction toArrow-Debreu Pricing and Complete

Markets

Instructor: Dmytro Hryshko

1 / 28

Arrow-Debreu economy

General equilibrium, exchange economy

Static (all trades done at period 0) but multi-period

No restrictions on preferences

2 / 28

Basic setting

Two dates: 0, and 1. This set-up, however, is fullygeneralizable to multiple periods.

S possible states of nature at date 1, indexed bys = 1, 2, . . . , S with the corresponding probabilities π(s).

One perishable (=non storable) consumption good

I agents, indexed i = 1, . . . , I, with preferences

ui0(ci0) + βiS∑s=1

π(s)ui(ci1(s))

Agent i’s endowment is described by the vector{yi0, (yi1(s))s=1,2,...,S}

3 / 28

Traded securities

Arrow-Debreu securities (AD) (contingent claims): securityfor state s date 1 priced at time 0 at q0

1(s) promisesdelivery of one unit of commodity tomorrow (at date 1) ifstate s is realized and nothing otherwise

Thus, individual i’s consumption in state s will equal herholdings of AD securities for state s, date 1

4 / 28

Agent’s problem. Competitive equilibrium setting

maxci0,c

i1(1),...,ci1(S)

ui0(ci0) + βiS∑s=1

π(s)ui(ci1(s))

s.t. (P)

ci0 +

S∑s=1

q01(s)ci1(s) ≤ yi0 +

S∑s=1

q01(s)yi1(s)

ci0, ci1(1), . . . , ci1(S) ≥ 0

5 / 28

Definition of the equilibrium

Equilibrium is a set of contingent claim prices

q01(1), q0

1(2), . . . , q01(S)

such that:

1 at those prices ci0, ci1(1), . . . , ci1(S) solve problem (P) for all

i’s, and

2

I∑i=1

ci0 =

I∑i=1

yi0,

I∑i=1

ci1(s) =

I∑i=1

yi1(s), for each s = 1, 2, . . . , S.

6 / 28

Competitive equilibrium and Pareto optimalityillustrated

Agents Endowments Preferencest=0 t=1

s=1 s=2

Agent 1 10 1 2 12c

10 + 0.9

[13 ln(c1

1(1)) + 23 ln(c1

1(2))]

Agent 2 5 4 6 12c

20 + 0.9

[13 ln(c2

1(1)) + 23 ln(c2

1(2))]

7 / 28

Maximization problemsAgent 1:

max{c10,c11(1),c11(2)}≥0

1

2

10 + 1 · q01(1) + 2 · q01(2)− c11(1) · q01(1)− c11(2) · q01(2)︸ ︷︷ ︸=c10

+ 0.9︸︷︷︸

=β1

[1

3ln(c11(1)) +

2

3ln(c11(2))

]

Agent 2:

max{c20,c21(1),c21(2)}≥0

1

2

5 + 4 · q01(1) + 6 · q01(2)− c21(1) · q01(1)− c21(2) · q01(2)︸ ︷︷ ︸=c20

+ 0.9︸︷︷︸

=β2

[1

3ln(c21(1)) +

2

3ln(c21(2))

]8 / 28

Optimum

Optimality conditions:

Agent 1:

c11(1) :

q01(1)2 = 0.9 · 1

3 ·1

c11(1)

c11(2) :

q01(2)2 = 0.9 · 2

3 ·1

c11(2)

Agent 2:

c21(1) :

q01(1)2 = 0.9 · 1

3 ·1

c21(1)

c21(2) :

q01(2)2 = 0.9 · 2

3 ·1

c21(2)

Feasibility conditions:

c11(1) + c2

1(1) = 5

c11(2) + c2

1(2) = 8

c11(1) = c2

1(1) = 2.5

c11(2) = c2

1(2) = 4.9 / 28

Prices of AD securitiesOptimality conditions can be expressed as

q01(s) =

0.9 · π(s) · 1ci1(s)

1/2, s, i = 1, 2, or

q01(s) =

β · π(s) · ∂ui

∂ci1(s)

∂ui0∂ci0

, s, i = 1, 2.

That is,

today’s price of the tomorrow’s good if state s is realized

price of the today’s good=

MUi1(s)

MUi0

q01(1) = 2 · 0.9 · 1

3· 1

c11(1)

= 2 · 0.9 · 1

3· 1

2.5= 0.24

q01(2) = 2 · 0.9 · 2

3· 1

c11(2)

= 2 · 0.9 · 2

3· 1

4= 0.30

10 / 28

Notes on AD prices

Prices reflect probabilities, and marginal rates ofsubstitution and therefore relative scarcities of the goods(total quantities of goods differ in different states)

If date 1 marginal utilities were constant (linear, riskneutral preferences), the goods endowments wouldn’tinfluence the AD prices, which would be then exactlyproportional to the state probabilities

11 / 28

Post-trade equilibrium consumptions

t=0 t=1 utilitys=1 s=2

Agent 1 9.04 2.5 4 5.62

Agent 2 5.96 2.5 4 4.09

Total 15 5 8

c10 = 10 + 1 · 0.24 + 2 · 0.3− 2.5 · 0.24− 4 · 0.3 = 9.04

c20 = 5 + 4 · 0.24 + 6 · 0.3− 2.5 · 0.24− 4 · 0.3 = 5.96

12 / 28

Is the equilibrium Pareto optimal? (If yes, it is impossible torearrange the allocation of consumptions so that the utility ofone agent is higher without reducing the utility of the otheragent.)

13 / 28

Pareto problem

max{c10,c11(1),c11(2)}≥0

u1(c10, c

11(1), c1

1(2)) + λu2(c20, c

21(1), c2

1(2))

s.t.

c10 + c2

0 = 15 c11(1) + c2

1(1) = 5 c11(2) + c2

1(2) = 8

c10, c

11(1), c1

1(2), c20, c

21(1), c2

1(2) ≥ 0

FOCs:

u10

u20

=u1

1(1)

u21(1)

=u1

1(2)

u21(2)

= λ.

14 / 28

In terms of our example, the first 3 equalities are

1/2

1/2=

0.9 · 13 ·

1c11(1)

0.9 · 13 ·

1c21(1)

=0.9 · 2

3 ·1

c11(2)

0.9 · 23 ·

1c21(2)

In our example, competitive equilibrium corresponds to thePareto optimum with equal weighting of the two agents’utilities, λ = 1.

15 / 28

Incomplete markets=less AD securities than statesAssume that only state-1 date-1 AD security is available.

Agent 1 :

max{c10,c11(1)}≥0

1

2

10 + 1 · q01(1)− c11(1) · q01(1)︸ ︷︷ ︸=c10

+ 0.9︸︷︷︸

=β1

[1

3ln(c11(1)) +

2

3ln(2)

]Agent 2 :

max{c20,c21(1)}≥0

1

2

5 + 4 · q01(1)− c21(1) · q01(1)︸ ︷︷ ︸=c20

+ 0.9︸︷︷︸

=β2

[1

3ln(c21(1)) +

2

3ln(6)

]16 / 28

Post-trade allocation

FOCs and the feasibility condition implyc1

1(1) = c21(1) = (1 + 4)/2 = 2.5.

t=0 t=1 utilitys=1 s=2

Agent 1 9.64 2.5 2 5.51<5.62

Agent 2 5.36 2.5 6 4.03<4.09

Total 15 5 8

The market with AD securities for each state, called completemarket, is Pareto superior to the incomplete market.

17 / 28

Risk sharing. New endowment matrix

Agents Endowments Preferencest=0 t=1

s=1 s=2

Agent 1 4 1 5 12c

10 + 0.9

[12 ln(c1

1(1)) + 12 ln(c1

1(2))]

Agent 2 4 5 1 12c

20 + 0.9

[12 ln(c2

1(1)) + 12 ln(c2

1(2))]

18 / 28

Table 1: No trade

Agents Utilities Expected utility in period 1t=1

s=1 s=2

Agent 1 ln(1) ln(5) 12 ln(1) + 1

2 ln(5)=0.8047

Agent 2 ln(5) ln(1) 12 ln(5) + 1

2 ln(1)=0.8047

19 / 28

Table 2: Trade under complete markets

Agents Utilities Expected utility in period 1t=1

s=1 s=2

Agent 1 ln(3) ln(3) 12 ln(3) + 1

2 ln(3)=1.099

Agent 2 ln(3) ln(3) 12 ln(3) + 1

2 ln(3)=1.099

Both agents are perfectly insured=no variation intomorrow’s consumption regardless of the realized state ofnature.

This happens because the aggregate endowment in state 1and 2 is the same (=6), that is there’s no aggregate risk.

20 / 28

Notes on Pareto optimal allocations

λ =u1

0

u20

=u1

1(1)

u21(1)

=u1

1(2)

u21(2)

⇔ u11(1)

u11(2)

=u2

1(1)

u21(2)

If one of the two agents is fully insured—no variation in her date 1consumption (MRS=1)—the other must be as well

More generally, if the MRS are to differ from 1, given that they mustbe equal between the agents, the low consumption-high MU statemust be the same for both agents and similarly for the highconsumption-low MU state. Impossible when there’s no aggregaterisk, hence individuals are perfectly insured in the absence ofaggregate risk.

If there is aggregate risk, however, the above reasoning also impliesthat, at a Pareto optimum, it is shared “proportionately” amongagents with same risk tolerance.

If agents are differentially risk averse, in a Pareto optimal allocationthe less risk averse will typically provide some insurance services tothe more risk averse.

More generally, optimal risk sharing dictates that the agent mosttolerant of risk bears a disproportionate share of it.

21 / 28

Notes on Pareto optimal allocations

λ =u1

0

u20

=u1

1(1)

u21(1)

=u1

1(2)

u21(2)

⇔ u11(1)

u11(2)

=u2

1(1)

u21(2)

If one of the two agents is fully insured—no variation in her date 1consumption (MRS=1)—the other must be as well

More generally, if the MRS are to differ from 1, given that they mustbe equal between the agents, the low consumption-high MU statemust be the same for both agents and similarly for the highconsumption-low MU state. Impossible when there’s no aggregaterisk, hence individuals are perfectly insured in the absence ofaggregate risk.

If there is aggregate risk, however, the above reasoning also impliesthat, at a Pareto optimum, it is shared “proportionately” amongagents with same risk tolerance.

If agents are differentially risk averse, in a Pareto optimal allocationthe less risk averse will typically provide some insurance services tothe more risk averse.

More generally, optimal risk sharing dictates that the agent mosttolerant of risk bears a disproportionate share of it.

21 / 28

Notes on Pareto optimal allocations

λ =u1

0

u20

=u1

1(1)

u21(1)

=u1

1(2)

u21(2)

⇔ u11(1)

u11(2)

=u2

1(1)

u21(2)

If one of the two agents is fully insured—no variation in her date 1consumption (MRS=1)—the other must be as well

More generally, if the MRS are to differ from 1, given that they mustbe equal between the agents, the low consumption-high MU statemust be the same for both agents and similarly for the highconsumption-low MU state. Impossible when there’s no aggregaterisk, hence individuals are perfectly insured in the absence ofaggregate risk.

If there is aggregate risk, however, the above reasoning also impliesthat, at a Pareto optimum, it is shared “proportionately” amongagents with same risk tolerance.

If agents are differentially risk averse, in a Pareto optimal allocationthe less risk averse will typically provide some insurance services tothe more risk averse.

More generally, optimal risk sharing dictates that the agent mosttolerant of risk bears a disproportionate share of it.

21 / 28

Notes on Pareto optimal allocations

λ =u1

0

u20

=u1

1(1)

u21(1)

=u1

1(2)

u21(2)

⇔ u11(1)

u11(2)

=u2

1(1)

u21(2)

If one of the two agents is fully insured—no variation in her date 1consumption (MRS=1)—the other must be as well

More generally, if the MRS are to differ from 1, given that they mustbe equal between the agents, the low consumption-high MU statemust be the same for both agents and similarly for the highconsumption-low MU state. Impossible when there’s no aggregaterisk, hence individuals are perfectly insured in the absence ofaggregate risk.

If there is aggregate risk, however, the above reasoning also impliesthat, at a Pareto optimum, it is shared “proportionately” amongagents with same risk tolerance.

If agents are differentially risk averse, in a Pareto optimal allocationthe less risk averse will typically provide some insurance services tothe more risk averse.

More generally, optimal risk sharing dictates that the agent mosttolerant of risk bears a disproportionate share of it.

21 / 28

Notes on Pareto optimal allocations

λ =u1

0

u20

=u1

1(1)

u21(1)

=u1

1(2)

u21(2)

⇔ u11(1)

u11(2)

=u2

1(1)

u21(2)

If one of the two agents is fully insured—no variation in her date 1consumption (MRS=1)—the other must be as well

More generally, if the MRS are to differ from 1, given that they mustbe equal between the agents, the low consumption-high MU statemust be the same for both agents and similarly for the highconsumption-low MU state. Impossible when there’s no aggregaterisk, hence individuals are perfectly insured in the absence ofaggregate risk.

If there is aggregate risk, however, the above reasoning also impliesthat, at a Pareto optimum, it is shared “proportionately” amongagents with same risk tolerance.

If agents are differentially risk averse, in a Pareto optimal allocationthe less risk averse will typically provide some insurance services tothe more risk averse.

More generally, optimal risk sharing dictates that the agent mosttolerant of risk bears a disproportionate share of it.

21 / 28

CRRA preferences

Let ui0 and ui(ci1(s)) be CRRA, and assume homogeneous timediscounting factors:

u(c) =c1−ρ

1− ρ, ρ > 0, ρ 6= 1

u(c) = log c, ρ = 1.

22 / 28

AD prices with homogeneous CRRA preferences

Recall FOCs:

q01(s)(ci0)−ρ = π(s)β(ci1(s))−ρ, s = 1, . . . , S, i = 1, . . . , I

⇒ ci1(s) =

[π(s)β

q01(s)

]1/ρ

ci0, s = 1, . . . , S, i = 1, . . . , I

⇒∑i

ci1(s)︸ ︷︷ ︸=y1(s)

=

[π(s)β

q01(s)

]1/ρ∑i

ci0︸ ︷︷ ︸=y0

, s = 1, . . . , S, i = 1, . . . , I

⇒ q01(s) = π(s)β

[y1(s)

y0

]−ρ, s = 1, . . . , S,

where y0 and y1(s) are total, economy-wide, endowments atdate 0, and date 1, state s, respectively.

23 / 28

The existence of the representative consumer. CRRApreferences

q01(s) = π(s)β

[y1(s)

y0

]−ρ, s = 1, . . . , S

Notice that the economy with the representative consumer whoowns the economy-wide endowments at each date-state willresult into the same equilibrium vector of prices and aggregateconsumption as a decentralized economy populated byconsumers with

identical time discount factors and

identical CRRA preferences.

24 / 28

Equilibrium consumption levels: CRRA preferencesπ(s)β(ci1(s))−ρ

(ci0)−ρ= q0

1(s) =π(s)β(cj1(s))−ρ

(cj0)−ρ

⇒ ci1(s)

ci0=y1(s)

y0=cj1(s)

cj0

⇒ ci1(s)

y1(s)=ci0y0

andcj1(s)

y1(s)=cj0y0

Furthermore,

π(s)β(ci1(s))−ρ

π(s′)β(ci1(s′))−ρ=q0

1(s)

q01(s′)

=π(s)β(cj1(s))−ρ

π(s′)β(cj1(s′))−ρ

⇒ ci1(s)

ci1(s′)=q0

1(s)

q01(s′)

=y1(s)

y1(s′)=cj1(s)

cj1(s′)

⇒ ci1(s)

y1(s)=ci1(s′)

y1(s′)and

cj1(s)

y1(s)=cj1(s′)

y1(s′)25 / 28

Summary

Any agent i’s consumption is a constant share κi of date 1total endowment regardless of the state.

Any agent i’s date 0 consumption share in total output isthe same as her date 1 share.

Agent i’s share in aggregate consumption/wealth is theagent’s share of the aggregate wealth on date 0, evaluatedat equilibrium Arrow-Debreu prices.

26 / 28

Individual consumption share: CRRA preferences

yi0 +

S∑s=1

q01(s)yi1(s) = ci0 +

S∑s=1

q01(s)ci1(s)

= κiy0 +

S∑s=1

q01(s)κiy1(s)

= κi

[y0 +

S∑s=1

q01(s)y1(s)

]

κi =yi0 +

∑Ss=1 q

01(s)yi1(s)

y0 +∑S

s=1 q01(s)y1(s)

You can further simplify the above expression by plugging inthe AD prices.

27 / 28

Suggested readings for this lecture

Danthine, J. P., Donaldson, J. (2005), IntermediateFinancial Theory, Elsevier Academic Press. Chapter 8.

28 / 28