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ECON 581. Introduction toArrow-Debreu Pricing and Complete
Markets
Instructor: Dmytro Hryshko
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Arrow-Debreu economy
General equilibrium, exchange economy
Static (all trades done at period 0) but multi-period
No restrictions on preferences
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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}
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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
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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
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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.
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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))]
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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
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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
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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
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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.)
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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.
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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.
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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.
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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))]
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Suggested readings for this lecture
Danthine, J. P., Donaldson, J. (2005), IntermediateFinancial Theory, Elsevier Academic Press. Chapter 8.
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