Macroeconomics I - GRIPSjulen/teaching/macro1_13/todai_macro1_s13_ln11.pdf · Julen Esteban-Pretel...

Lecture 11

Macroeconomics IUniversity of Tokyo

Complete Markets 1I: Sequential TradingLS, Chapter 8

Julen Esteban-PretelNational Graduate Institute for Policy Studies

Lecture 11: Complete Markets 1I: Sequential TradingCore Macro I - Spring 2013

Household Wealth§ From the Arrow-Debreu economy define:

•

•

§ The AD equilibrium market clearing condition with equality implies

2

qt� (s� ) � q

0� (s� )

q0t (st),

�it(st) ⇥

T�

�=t

�

s� |stqt� (s

� )[ci� (s� )� yi� (s

� )].

I�

i=1

�it(st) = 0, �t, st.


§ In the sequential trading economy:

• markets open every period,• one-period-ahead state-contingent claims (Arrow securities) are traded,

• trades for st+1-contingent goods are only traded at a particular date t history st.

Figure 11.1§ Let - be the quantity of Arrow securities bought/sold by

agent i in node st.– Note that history st+1=(st+1|st), hence in time t+1 agent i collects

- be the price in node st of a security that pays 1 unit of

consumption in t+1 contingent on the realization of st+1,

Figure 11.1

Sequential Trading

3

�Qt(st+1|st)

�ait+1(st+1|st)

�ait+1(st+1).


Sequential Markets Household problem

4

§ The problem of the household who is at node st is:

(11.2)

s.t. for all τ ≥ t and all sτ in period τ,

(11.1)

condition to rule out Ponzi schemes.

(11.3)

max{ci� (s� ),a�+1(s�+1,s� )}��=t

��

�=t

�

s�

��u(⇥ci� (s� ))⇥� (s� |s0)

⇥ci� (s� ) +�

s�+1

⇥ai�+1(s�+1, s� )⇥Q� (s�+1|s� ) � yi� (s� ) + ⇥ai� (s� ),

ci� (s� ) � 0,

�ai� (s� ) given,


No-Ponzi Schemes and Natural Debt Limit§ The no-Ponzi scheme condition is the transversality condition:

where

§ A sufficient condition is the natural debt limit:

where

§ The natural debt limit is the max. value of debt that agent i can repay starting from time t history st, assuming that her cons. is 0 from t onwards.

§ is large enough so that (11.6) does not bind, but small enough that the transversality condition is satisfied.

5

Ait(st) =

��

�=t

�

s� |stqt� (s

� )yi� (s� ).

(11.4)

(11.5)

(11.6)

(11.7)

Ai

t+1(st+1)

�ait+1(st+1, st) ⇥ �Ait+1(st+1)

limT�⇥

�

sT+1

q0T+1(sT+1)⇥aiT+1(sT+1, sT) = 0,

q0T+1(sT+1) = �QT(sT+1|sT) · �QT�1(sT|sT�1) · · · �Q0(s1|s0).


Equilibrium with Arrow Securities§ Def: A wealth distrib. is a vector satisfying

§ Def: A sequential markets equilibrium is an initial distribution of wealth

an allocation , and pricing kernels

, such that:

(i) Optimization: for all i, given and the pricing kernels, the allocation solves the household’s maximization problem:

(ii) Market clearing: For all t and all st,- Goods market:

- Securities market:

6

��at(st) = {�ait(st)}Ii=1�

i

⇥ait(st) = 0.

⌅�⇤cit(st),⇤ait+1(st+1, st)

⇥Ii=1

⇧�t=0⇥

�Qt(st+1|st⇤�t=0

s.t. for all t ≥ 0 and all st in period t,

⇥cit(st) +�

st+1

⇥ait+1(st+1, st)⇥Qt(st+1|st) � yit(st) + ⇥ait(st),

cit(st) � 0, ait+1(st+1, st) ⇥ �Ait+1(st+1).

max{cit(st),at+1(st+1,st)}�t=0

��

t=0

�

st

�t⇥(st)u[cit(st)] (11.8)

(11.2)

(11.3), (11.6)

(11.9)

(11.10)

�ai0(s0)

��a0(s0),

�Ii=1⇥cit(st) =

�Ii=1 y

it(st),

�Ii=1 ⇥ait+1(st+1, st) = 0


Solving the Sequential Market Equilibrium§ Let and be the Lagrange multipliers on (11.2) and (11.6).

§ We can write the Lagrangian (for a given initial wealth ) as:

§ FOC:

for all st+1, t and st.§ In the optimum hence combining (11.12) and (11.13):

7

�it(st) � it(s

t; st+1)

Li =

��

t=0

�

st

⇤�tu(⇥cit(st))⇥t(st|s0)

+�it(st)⇤yit(st) + ⇥ait(st)�⇥cit(st)�

�

st+1

⇥ait+1(st+1, st)⇥Qt(st+1|st)⌅

�cit(st) :

{�ait+1(st+1, st)}st+1 :

+� it(st; st+1)

�Ai

t+1(st+1) + ⇤ait+1(st+1)

⇥ ⌅,

�tu�(�cit(st))⇤t(st|s0)� ⇥it(st) = 0,

��it(st)�Qt(st+1|st) + ⇥ it(st; st+1) + �it+1(st+1, s

t) = 0,

(11.11)

(11.12)

(11.13)

� it(st; st+1) = 0,

�Qt(st+1|st) = �u�(�cit+1(st+1))u�(�cit(st))

⇥t(st+1|st), for all st+1, t, st. (11.14)

�ai0(s0)


Arrow-Debreu and Sequential Markets§ Summarize the sequential market equilibrium (SME) with a list

and the Arrow-Debreu equilibrium (ADE) with a list

§ Equivalence Theorem:

(I) If (c,q) is an ADE, then there exists a SME, , with

for all t, all st and all i.

(II) If is a SME, then there exists an ADE, (c,q), with

for all t, all st and all i.

§ Proof: LS 8.8.5.

8

⇤⇧c,⇧a, ⇧Q

⌅�⌃�⇧cit(st),⇧ait+1(st+1, st);⇧ai0(s0), Ait+1(st+1)

⇥Ii=1, ⇧Qt(st+1|st)

⌥�t=0,

(c, q) �⌅�cit(st)⇥Ii=1,⇤q0t (s

t)⇧�t=0.

�⇤c,⇤a, ⇤Q

⇥cit(st) = �cit(st)

�⇤c,⇤a, ⇤Q

⇥�cit(st) = cit(s

t)

(11.15)

(11.16)


Primer on Asset Pricing§ Add two-period Arrow securities. The price at st of a two-period security

promising to pay one unit at st+2 is

since you can duplicate this security by a sequence of one-period Arrow securities.

§ The price at st of a τ-t period security promising to pay one unit at sτ is

§ These are redundant securities. Introducing them into the Arrow economy does not change the equilibrium allocation.

9

qt� (s� ) = �Qt(st+1|st) · �Qt+1(st+2|st+1) · · · �Q��1(s� |s��1).

qtt+2(st+2) = �Qt(st+1|st) · �Qt+1(st+2|st+1). (11.17)

(11.18)


Primer on Asset Pricing (cont.)§ You can price any security given the prices of Arrow securities.

§ What is the price at st of a security that promises to pay dτ(sτ) uts at all sτ|st?

§ What is the pr. at st of a one-period security whose random payoff is ωτ(st+1)?

§ What is the pr. at st of a one-period bond (payoff is ωτ(st+1)=1 for all st+1)?

§ Recall the hh’s problem FOC in the AD economy:

Hence:

§ Define the one-period stoch. discount factor at st as:

§ Then, letting be the one-period gross return on the asset, we have

10

�

��t

�

s� |stqt� (s

� )d� (s� ).

ptt(st) ��

st+1

qtt+1(st+1)�(st+1).

�

st+1

qtt+1(st+1).

Rt+1 � �(st+1)/ptt(st)

1= Et [mt+1Rt+1] .

(11.19)

(11.20)

(11.21)

(11.22)

�tu�(cit(st))⇥t(st) = µiq0t (s

t).

qt� (s� ) = ��t

u�(ci� (s� ))u�(cit(st))

⇥(s� |st).

mt+1 � �u�(cit+1(s

t+1))u�(cit(st))

.


Recursive Formulation§ Assume now that the stochastic process {st} is Markov and let π(s’|s) denote

the transition probabilities of the Markov chain, hence:

•

•

• for t > τ.

§ It can be easily shown (LS p.231) that in this environment the pricing kernel in the SME is only a function of the current state:

§ It can also be shown (LS p.232) that the natural debt limit exhibits history independence:

11

Prob(st+1 = s�|st = s) = �(s�|s) and Prob(s0 = s) = �0(s),

�t(st) = �(st|st�1)�(st�1|st�2) . . .�(s1|s0)�0(s0),

�t(st|s� ) = �(st|st�1)�(st�1|st�2) . . .�(s�+1|s� ),

�Qt(st+1|st) = Q(st+1|st).

Ai

t(st) = Ai(st).

(11.23)(11.24)

(11.25)

(11.26)

(11.27)


Recursive Formulation Problem§ The states variables are:

• Household’s wealth: • Current realization of st.

§ The control variables are:• Current consumption: • Next period’s assets:

§ We seek the pair of optimal pol. functions hi(a,s), gi(a,s,s’) such that:

§ The Bellman equation for the household i is:

12

vi(a, s) = max

c,⌅a(s�)

�u(c) + �

⇤

s�

vi[⌅a(s�), s�]⇥(s�|s)

⇥

c+�

s�

⇥a(s�)Q(s�|s) � yi(s) + a

c � 0, ��a(s�) ⇥ Ai(s�), ⇤s�.

s.t.

cit

ait,

ait+1(st+1)

cit = hi(ait, st), a

it+1(st+1) = gi(ait, st, st+1). (11.28)

(11.29)

(11.30)

(11.31)


Solving Recursive Formulation Problem§ We can write the unconstrained Bellman equation as:

§ FOC:

§ The Benveniste-Scheinkman formula implies

§ Hence in equilibrium:

where

13

vi(a, s) = max⌃a(s�)

⇤u

�yi(s) + a�

⇧

s�

⌃a(s�)Q(s�|s)⇥

+ �⇧

s�

vi[⌃a(s�), s�]⇥(s�|s)⌅

Q(st+1|st) =�u�(cit+1)⇥(st+1|st)

u�(cit),

(11.32)

(11.33)

(11.34)

(11.35)

cit = hi(ait, st), and c

it+1 = hi(ait+1(st+1), st+1) = hi(gi(ait, st, st+1), st+1),

Q(s�|s)u�(c) = �vi�(�a�(s), s�)⇥(s�|s)

vi�(a, s) = u�(c)


Recursive Competitive Equilibrium

14

§ Def: A recursive competitive equilibrium is an initial distribution of wealth

a pricing kernel Q(s’|s), sets of value functions , and decision

rules , such that:

(i) Optimization: for all i, given and the pricing kernel, the decision rules solve the household’s problem:

(ii) Market clearing: For all realizations of the cons. and asset portfolios implied by the decision rules satisfy:

- Goods market:

- Asset market:

�a0,

�vi(a, s)⇥Ii=1�hi(a, s), gi(a, s, s�)⇥Ii=1

ai0

s.t.

vi(a, s) = max

c,⌅a(s�)

�u(c) + �

⇤

s�

vi[⌅a(s�), s�]⇥(s�|s)

⇥

c+�

s�

⇥a(s�)Q(s�|s) � yi(s) + a

c ⇤ 0, ��a(s�) ⇥ Ai(s�), ⌅s�.

�Ii=1 c

it =�Ii=1 y

it(st),

{st}�t=0,

�Ii=1 ⇥ait+1(s�) = 0 for all t and s�.

{�cit, {�ait+1(s�)}s�⇥i}t

(11.29)

(11.30)

(11.31)

(11.36)

(11.37)


πa

Figure 11.1

15

Figure 11.1: Commodity Space and Prob. Structure

sun rain

rainrainsun sun

t = 0

t = 1

t = 2(1, 1, 1)

sun

(1, 1)

Example:

• S = (1,0) = (sun, rain) (1, 0)

(1, 1, 0) (1, 0, 1) (1, 0, 0)

1

g

c

f

b

d e

a

πgπfπeπd

πb πc

�Q(d|b) �Q(e|b)

�ai(e)�ai(d)

�Q(b|a)

�ai(a)

�ai(b)

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