Homework 3, due in class on November 23rd
Jonathan Heathcote
October 25th, 2005
Consider the two agent planner’s problem studied by Kocherlakota.
Suppose the two agents receive stochastic endowment streams, such that
the aggregate endowment is 1 in every period. Endowments cannot be stored
or disposed of. Each period, agent 1’s endowment is drawn from a set S=
{0.5−ε, 0.5,0.5+ε}.Denote the elements of this set s1,s
2and s3.Let stdenote
the draw from Sat date t. Let st=(s0,s
1,s
2, ...st)∈Stdenote the history of
the economy up to and including date t. Denote agent 1’s endowment y(st).
Thus agent 2’s endowment is 1−y(st).Let the date zero probability of history
stbe denoted π(st).Assume that π(st|st−1)=π(st|st−1)(i.e. productivity
shocks are first order Markov). Assume that transition probabilities are given
by the mat rix:
Γ=⎛
⎝
q1−q0
1−p
2p1−p
2
01−qq
⎞
⎠
where the element Γi,j is equal to π(sj|si),and p∈(0,1) ,q∈(0,1) .
Assume that preferences for agent 1 are given by PtβtPstπ(st)log(c(st)).
Preferences for agent 2 are similar. Note that in equilibrium, the consumption
of the type two agent is given by 1−c(st). Assume that the planner cares about
both types equally.
1. Assume that half of the mass of the ergodic distribution across Simplied
by the matrix Γis on s2.Given this assumption, derive an expression for
qas a function of p. Thus there are three independent parameters in the
model: : β, ε, and p
2. Assume, to start, that agents are not allowed to reject allocations proposed
by the planner. Describe the planner’s problem, and characterize the
solution to it.
3. Now suppose that at any date, agents have the option of rejecting the
allocations proposed by the planner and reverting to permanent autarky.
Once in autarky, allocations are given by c(st)=y(st)∀t, st.Define Vi(st)
to be the value for agent iof reverting to autarky at tgiven st.
(a) Suppose β=0.96,p=0.9,and ε=0.25.For these parameter values,
compute the set of possible values for Vi(st).
1
