Numerical MethodsSpring 2008

Henriksen

Problem set 8Due Thu March 27th

Consider a model economy with a continuum of individuals where each in-dividual faces idiosyncratic labor earnings risk. Each individual solves thefollowing program

max{ct,at+1}∞t=0

E0

∞∑t=0

βtu (ct)

subject toct + at+1 = (1 + r) at + wεt

where a is asset holdings, r is rate of return on assets, w is the marginalproduct of labor, c is consumption, ε is the realization of the individuals’productivity.

Labor productivity is governed by a Markov process

π(ε′ | ε

)= Pr

{εt+1 = ε′ | εt = ε

}.

We can measure the transition matrix from survey data. For this exerciseassume each period is one quarter and the transition matrix is

Π =

0.5 0.3 0.2 0.0 0.0 0.0 0.00.2 0.5 0.2 0.1 0.0 0.0 0.00.1 0.2 0.4 0.2 0.1 0.0 0.00.0 0.1 0.2 0.4 0.2 0.1 0.00.0 0.0 0.1 0.2 0.4 0.2 0.10.0 0.0 0.0 0.1 0.2 0.5 0.20.0 0.0 0.0 0.0 0.2 0.3 0.5

and

ε =(

0.25 0.50 0.75 1.00 1.25 1.50 1.75)′

The utility function can be parameterized as

u (ct) =c1−σt − 11 − σ

,

The production function is given the Cobb-Douglas specification

Yt = γKαt N1−α

t

1

whereK =

∑∀ε

∫ ∞

amin

af (ε, a) da.

The factor prices are

rt = αγ

(Kt

Nt

)α−1

− δ,

wt = (1 − α) γ

(Kt

Nt

)α

The technology and preference parameters can also be calibrated from data.Here we assume we have found the following measurements: α = .35, β =.98, δ = 0.025, σ = 2, and γ = 5.

1. Reformulate this problem as a dynamic programming problem, i.e.write up the Bellman equation. What are the control variable(s),endogenous and exogenous state variable(s)?

2. Without loss of generality, normalize the size of the total populationto 1. Assume each individual inelastically supplies one unit of labor.Compute the stationary labor supply by iterating on the Markov tran-sition matrix and multiplying by the productivity levels.

3. Compute the stationary equilibrium of the model. The equilibriumcontains a type distribution of agents, equilibrium interest rates andwage levels, value function, and optimal decision rules.

(a) Compute the individual’s decision rules for a′ (a, ε) and c (a, ε)using value function iteration and piecewise-linear interpolation.As starting values you might take 1+r = 1.0135 and w = 25.3794and that the grid over the endogenous state variable is boundedby amin = 220 and a = 450.

(b) Compute the stationary distribution of individual asset holdingsusing Monte Carlo simulation. You might simulate 10.000 se-quences of length 1000. Remember to reset the the state of therandom number generator for each simulation (not each sequenceof each simulation) – e.g. randn(’state’,0).

(c) Update the aggregate capital stock and factor prices.(d) Find the equilibrium aggregate capital stock using bisection method.

4. Plot the decision rules, value functions, and distributions. The assetdistribution might look almost degenerate. Give a very brief economicexplanation for why it looks the way it does.

Hint: Start by modifying the code for the Aiyagari model.

2