Problem set 3 - exercise 1
The solution proposed in the TA session is theoretically correct since the problem states that there is a single agent
in economy (which is dierent from assuming that the economy is populated by many identical agents). However,
in general, when we think to a recursive equilibrium we think to an economy populated by many agents.
I talked to Roberto and we agreed that the denition proposed in class, even if theoretically correct, is not
optimal to learn how to dene a recursive competitive equilibrium in the most general cases and is likely to create
some confusion. A better denition of a recursive competitive equilibrium is the following:
A recursive competitive equilibrium is a set of functions {V (a, A) , g (a, A) , c (a, A) , G (a, A) , R (A)} such that
1. V (a, A) solves the consumer's functional equation
n
0 0 o
V (a, A) =max u (c) + βV a , A
c
0
s.t.c + a = aR (A) + ω
0
A = G (A)
and a = g (a, A), c = c (a, A) are the associated policy function
2. Consistency is satised
0
G (A) = g (A, A)
Notice that since I have dened the value function in term of consumption, in the denition I have to specify
an additional policy function c (a, A), as correctly pointed out by Roberto.
An additional remark: in a recursive equilibium we do not impose a No-Ponzi game condition because the
restriction imposed to derive a value function implicitely rule out Ponzi schemes (lecture notes explain this point).
Probles set 3 - exercise 2
If you remember, we have
m
1
βX
2
=
πs
.
I − qa
q i=s
ω1s + a
If we want to solve for the competitive equilibrium, since in the economy there is a single agent, market clearing
implies a = 0 and the price of the bond in equilibrium needs to be
q=β
m
X
πs
i=s
2
I
ω1s
Problem set 3 - exercise 3
I need to make some clarication about the role of complete markets. Complete markets allow agents to insure
their consumption against idiosyncratic risk but not against aggregate risk.
For example, assume that there are two agents and two states of the world (s1 , s2 ). Agents are identical apart
from their endowments: agent 1 is endowed with (1, 0) and agent 2 is endowed with(0, 1). Notice that the aggregate
endowment of the economy is 1 whatever the state. This is an example where agents face idiosyncratic risk but
there is no aggregate risk in the economy.
In the exercise we have a single agent in the economy, and as such the risk he faces correspond to the aggregate
risk in the economy. Market clearing then requires that n1s ωs = cs and c0 = I and the price of the assets need to
adjust for market clearing to be satised.
.
βπs
I
= qs
ω1s n1s
βπs I = qs ωs − βπs I
qs ωs = 2βπs I
1
qs = βπs
2I
ωs
Notice that to buy a bond corresponds to buy an unit of each asset, which implies that the price of a bond is
q=
X
qs = β
m
X
i=s
πs
2
I
ω1s
exactly the same as before.
The point is that in an economy with only one agent, even if there are complete markets, there is no possibility
to hedge risk, and the equilibrium allocation does not change with respect to the case with incomplete markets.
2