# 1 A Pure Exchange Economy with House- hold Heterogeneity ```Economics 702, Dirk Krueger
1
A Pure Exchange Economy with Household Heterogeneity
Consider a stochastic pure exchange economy where the current state of
the economy is described by st 2 S = fs1 ; : : : ; sM g: Event histories are
denoted by st and the initial node s0 is …xed. Probabilities of event histories
are given by t (st ): There are 2 di&curren;erent types of households with equal
mass normalized to 1. Households potentially di&curren;er in their endowment
stream feit (st )g; their initial asset position ai0 and their time discount factors
i
i 2 (0; 1): Preferences for each household over consumption allocations c =
i t
fct (s )g are given by
i
i
u (c ) =
1 X
X
( i )t
t (s
t
)U (cit (st )):
t=0 st 2S t
where U (:) is strictly increasing and strictly concave.
1. Suppose that S = fs1 ; s2 g and that
matrix
t (s
t
) is Markov with transition
1
(s0 js) =
1
where ; 2 [0; 1] are parameters. For which parameter combinations
( ; ) is the associated invariant distribution
(a) Unique?
(b) Satis…es
= (0:8; 0:2)?
2. Suppose that households can trade a full set of Arrow securities.
P De…ne
a sequential markets equilibrium, for arbitrary fai0 gi21;2 with i ai0 = 0:
3. De…ne a recursive competitive equilibrium.
4. Suppose that the aggregate endowment satis…es, for t
et (st ) = 2(1 + g)t
1
0
with g &gt; 0; and that
1
=
2
as well as
U (c) =
c1
1
1
:
The individual endowments satisfy eit (st ) &gt; 0 for all i; t; st ; but no
further assumptions are given, of course apart from
2
X
eit (st ) = et (st ) = 2(1 + g)t :
i=1
Characterize as fully as possible the sequential market equilibrium consumption allocations and the prices of Arrow securities.
5. Compute the risk-free interest rate in this economy.
6. Now suppose that households cannot borrow, that is, impose ait+1 (st+1 )
0 and also assume that 1 &gt; 1 &gt; 2 and a10 = a20 = 0 and
eit (st ) = (1 + g)t
for both i = 1; 2: Repeat questions 4. and 5.
2
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