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Two Agent Planner Problem: Stochastic Endowments and Incentive Compatibility, Assignments of Economics

A two agent planner problem where agents receive stochastic endowments, and the planner aims to find constrained efficient allocations while respecting incentive compatibility constraints. The problem involves calculating transition probabilities, agent preferences, and deriving expressions for the planner's problem and the agents' value functions. The document also discusses the use of a grid and iterative methods to find optimal decision rules and value functions.

Typology: Assignments

Pre 2010

Uploaded on 11/08/2009

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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=
{0.5ε, 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 1y(st).Let the date zero probability of history
stbe denoted π(st).Assume that π(st|st1)=π(st|st1)(i.e. productivity
shocks are first order Markov). Assume that transition probabilities are given
by the mat rix:
Γ=
q1q0
1p
2p1p
2
01qq
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 1c(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
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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 s 1 , s 2 and s 3. Let st denote the draw from S at date t. Let st^ = (s 0 , s 1 , s 2 , ...st) ∈ St^ denote 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 st^ be 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 matrix:

Γ =

q 1 − q 0 1 −p 2 p^

1 −p 2 0 1 − q q

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

P

t β

t P st^ π(s

t) 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 S implied by the matrix Γ is on s 2. Given this assumption, derive an expression for q as 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 i of reverting to autarky at t given 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).

(b) Formulate (as a Lagrangian) the planner’s problem now that the planner has to worry about the commitment problem. Use the Marcet and Marimon trick to formulate the problem in terms of the sum of the past values of the multipliers on the participation constraints. Take first order conditions. Now you will solve the model recursively, as we outlined in class. The state variable will be xt =

z(st−^1 ), st

where z(st−^1 ) is the ratio of the sum of the values of multipliers for the history st−^1. Since neither t nor st^ is a state variable in the recursive formulation, we can simply write x = (z− 1 , s). (c) Create a grid on z− 1. Let Z = (z 1 ,z 2 , ...zn) denote the n points on this grid. To start, set n = 11. To space the grid points “evenly” so that zi < 1 , i ≤ 5 , z 6 = 1, zj > 1 , j ≥ 6 , use an exponential spacing formula:

zi = zi− 1

μ zn z 1

¶ (^) n− (^11) 1 < i < n

How should one choose the end points for this grid? (hint: think about the possible range for the ratio of marginal utilities in autarky) (d) Now you need initial guesses for decision variables, for the law of motion of the state variables, and for the value functions Wi(x). Use the problem without enforcement constraints to produce these initial guesses given the parameter values above: W (^) i^0 (x), z^0 (x), c^0 (x), v i^0 (x) ∀x ∈ S × Z. (v i^0 (x) is the ratio of the multiplier on agent i’s incentive compatibility constraint to the sum of multipliers on this constraint).

  1. Recall that the goal is to solve for constrained efficient allocations when the planner has to respect incentive compatibility constraints. You will do this by iterating on Wi(x), z(x), c(x), vi(x) as follows:

(a) Take x 1 the first point on the grid over X = Z × S. Check whether agent 1 has an incentive to default at this grid point. If he does set solve numerically for c^1 (x 1 ) and z^1 (x 1 ) to satisfy (i) the first order condition for c, and (ii) the incentive compatibility constraint for agent 1 with equality, where current period utility is given by c^1 (x 1 ) and continuation utility for each possible s^0 is given by W 10 (z^1 (x 1 ), s^0 ). Note that z^1 (x 1 ) does not necessarily lie on the grid on Z. Thus you will want to interpolate (linearly) to evaluate the function W 10 (., s^0 ) in between grid points. (b) If agent 1 does not want to default, check incentive compatibility for agent 2. If agent 2 wants to default, solve for c^1 (x 1 ) and z^1 (x 1 ) following an analogous procedure to the one above. (c) If neither agent wants to default, z^1 (x 1 ) ≡ z^1 (z− 1 = z 1 , s = s 1 ) = z 1 , v 11 (x 1 ) = v 21 (x 1 ) = 0, and c^1 (x 1 ) satisfies the FOC for c. (d) Compute and store the new guess for the value functions

W 11 (x 1 ) = u(c^1 (x 1 )) + β

X

s^0

π(s^0 |s 1 )W 10 (z^1 (x 1 ), s^0 )