Intro to Arrow-Debreu Pricing and Complete Markets

32 Pages·2014·0.27 MB·English

by  Instructor: Dmytro Hryshko

ECON 581. Introduction to
Arrow-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 fully
generalizable to multiple periods.
S possible states of nature at date 1, indexed by
s = 1,2,...,S with the corresponding probabilities π(s).
One perishable (=non storable) consumption good
I agents, indexed i = 1,...,I, with preferences
S
(cid:88)
ui(ci)+βi π(s)ui(ci(s))
0 0 1
s=1
Agent i’s endowment is described by the vector
{yi,(yi(s)) }
0 1 s=1,2,...,S
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Traded securities
Arrow-Debreu securities (AD) (contingent claims): security
for state s date 1 priced at time 0 at q0(s) promises
1
delivery of one unit of commodity tomorrow (at date 1) if
state s is realized and nothing otherwise
Thus, individual i’s consumption in state s will equal her
holdings of AD securities for state s, date 1
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Agent’s problem. Competitive equilibrium setting
S
(cid:88)
max ui(ci)+βi π(s)ui(ci(s))
0 0 1
ci,ci(1),...,ci(S)
0 1 1 s=1
s.t. (P)
S S
(cid:88) (cid:88)
ci + q0(s)ci(s) ≤ yi + q0(s)yi(s)
0 1 1 0 1 1
s=1 s=1
ci,ci(1),...,ci(S) ≥ 0
0 1 1
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Definition of the equilibrium
Equilibrium is a set of contingent claim prices
q0(1),q0(2),...,q0(S)
1 1 1
such that:
1 at those prices ci0,ci1(1),...,ci1(S) solve problem (P) for all
i’s, and
2
I I I I
(cid:88) (cid:88) (cid:88) (cid:88)
ci = yi, ci(s) = yi(s), for each s = 1,2,...,S.
0 0 1 1
i=1 i=1 i=1 i=1
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Competitive equilibrium and Pareto optimality
illustrated
Agents Endowments Preferences
t=0 t=1
s=1 s=2
Agent 1 10 1 2 1c1+0.9(cid:2)1 ln(c1(1))+ 2 ln(c1(2))(cid:3)
2 0 3 1 3 1
Agent 2 5 4 6 1c2+0.9(cid:2)1 ln(c2(1))+ 2 ln(c2(2))(cid:3)
2 0 3 1 3 1
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Maximization problems
Agent 1:
 
1
max 10+1·q0(1)+2·q0(2)−c1(1)·q0(1)−c1(2)·q0(2)
{c10,c11(1),c11(2)}≥02(cid:124) 1 1 (cid:123)(cid:122)1 1 1 1 (cid:125)
=c1
0
(cid:20) (cid:21)
1 2
+ 0.9 ln(c1(1))+ ln(c1(2))
(cid:124)(cid:123)(cid:122)(cid:125) 3 1 3 1
=β1
Agent 2:
 
1
max 5+4·q0(1)+6·q0(2)−c2(1)·q0(1)−c2(2)·q0(2)
{c20,c21(1),c21(2)}≥02(cid:124) 1 1 (cid:123)(cid:122)1 1 1 1 (cid:125)
=c2
0
(cid:20) (cid:21)
1 2
+ 0.9 ln(c2(1))+ ln(c2(2))
(cid:124)(cid:123)(cid:122)(cid:125) 3 1 3 1
=β2 8/28
Optimum
Optimality conditions:

c1(1) : q10(1) = 0.9· 1 · 1
Agent 1: 1 2 3 c11(1)
c1(2) : q10(2) = 0.9· 2 · 1
1 2 3 c1(2)
1

c2(1) : q10(1) = 0.9· 1 · 1
Agent 2: 1 2 3 c21(1)
c2(2) : q10(2) = 0.9· 2 · 1
1 2 3 c2(2)
1
Feasibility conditions:
c1(1)+c2(1) = 5
1 1
c1(2)+c2(2) = 8
1 1
c1(1) = c2(1) = 2.5
1 1
c1(2) = c2(2) = 4.
1 1
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Prices of AD securities
Optimality conditions can be expressed as
0.9·π(s)· 1
q0(s) = ci1(s), s,i = 1,2, or
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β·π(s)· ∂ui
q0(s) = ∂ci1(s), s,i = 1,2.
1 ∂ui
0
∂ci
0
That is,
today’s price of the tomorrow’s good if state s is realized MUi(s)
= 1
price of the today’s good MUi
0
1 1 1 1
q0(1) = 2·0.9· · = 2·0.9· · = 0.24
1 3 c1(1) 3 2.5
1
2 1 2 1
q0(2) = 2·0.9· · = 2·0.9· · = 0.30
1 3 c1(2) 3 4
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