Report Final

Empirical methods project (Econ 597)
Oleksii Khvastunov
Abstract
My empirical methods project is based on Per Krusell and Anthony
Smith paper ”Income and Wealth Heterogeneity in the Macroeconomy”
(JPE, 1998).The paper presents a simple benchmark incomplete market
heterogeneous model with two types of risk: idiosyncratic (unemploy-
ment) and aggregate (productivity). In my project I am solving bench-
mark model following the steps presented in the paper.
1

Contents
1 Introduction 3
2 Model 4
3 Algorithm 7
4 My results 9
5 Appendix 12
5.1 Short description of the functions . . . . . . . . . . . . . . . . . . 12
5.2 Main function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 Function that ﬁnds value functions for given aggregate capital
law of motion parameters . . . . . . . . . . . . . . . . . . . . . . 16
5.4 Function that performs value function itteration step . . . . . . . 19
5.5 Simulation routine . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.6 Auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2

1 Introduction
A large number of dynamic general equilibrium macroeconomic models rely
on the assumption of representative agent. In some cases it could be a rea-
sonable assumption like in the case of complete markets. However the models
that want to have strong microfoundation require agents heterogeneity. This set
up makes models much more complicated and most of them can not be solved
analytically.
Numerical solution of such models is not an easy task as well. Rational
agents in this type of models make decisions based on the entire distribution of
individuals state variables. In general case this object is infinitely dimensional
that makes problem extremely complicated. The paper by Per Krusell and An-
thony Smith (henceforth KS) presents methodology of numerical approximation
of equilibrium in heterogeneous agent models.
The key assumption that is used to calculate the approximate equilibrium
is the following: agents have a limited ability to predict the evolution of the
distribution of individual state variables. It is shown in the paper that this
bound does not significantly restrict agents. As a result, the law of motion that
agents perceive could be described using finite-dimensional vector of distribu-
tion parameters (moments could be used for this purpose).
The rest of the report is organized as follows: in Model section I briefly
explain the model, intuition behind the methodology and give model parame-
ters that are used. In Algorithm section I will explain in details what are the
steps for equilibrium computation and how I implemented them. In the results
section I present my results and discuss them.
3

2 Model
The benchmark model consists of continuum (measure one) ex-ante iden-
tical consumers who maximize expected discounted utility subject to budget
constraints in every period:
max E0Σ∞
t=0βt
U(ct) (1)
subject to
c + k − (1 − δ)k = y (2)
where y is individual’s income that consists of two parts: return on capital that
individual owns and wage that individual gets if he is employed. U(c) - CRRA
function. Market for capital and labor is competitive so individual gets return
and wage equal to marginal product.
Production function is Cobb-Douglas:
F(k, l) = z¯kα¯l1−α
(3)
where z - aggregate productivity shock, ¯k - aggregate capital in the economy, ¯l
- total amount of labor supplied.
There are two types of uncertainty in the model: aggregate (productivity
shock - z) and idiosyncratic (individual unemployment shock - ). Both shocks
follow first order Markov process with two possible states: zg and zb for aggre-
gate productivity shock and 0 and 1 for idiosyncratic shock (0 means that the
agent is unemployed). Parameters of the model specify the average duration of
boom and recession in the economy. The authors are trying to match durations
to the ones observed in the data (8 quaters for boom and recession). As a result,
the transition matrix for aggregate shock is:
Bad Good
Bad 0.875 0.125
Good 0.125 0.875
The shock are dependent in the following sense: aggregate shock affects indi-
vidial, but individual does not affect aggregate. The intuition behind this fact is
the following: aggregate shock determines the state of the economy, in particular
employment (unemployment in boom ug and recession ub are parameters of the
model) that affects the evolution of the individual shock. The authors impose
additional conditions for the dependence of aggregate and individual shock in
order to identify the joint transition matrix. I wrote the function ”emprpr” to
find this matrix by solving the system of linear equations.As a result, the joint
transion matrix has the following form:
Bad0 Bad1 Good0 Good1
Bad0 0.5250 0.35 0.0312 0.0938
Bad1 0.0389 0.8361 0.0021 0.1229
Good0 0.0937 0.0313 0.2917 0.5833
Good1 0.0091 0.1159 0.0243 0.8507
4

Elements of this matrix satisfy system of the linear equations as I mentioned
above. Four of these equations make sure that unemployment shares remains
consistent with the model parameters ug and ub.
Individual’s Bellman equation has the following form:
v(k, ; Γ, z) = max
c,k
{U(c) + βE[v(k , ; Γ , z )|z, ]} (4)
subject to
c + k = r(¯k, ¯l, z)k + w(¯k, ¯l, z)˜l + (1 − δ)k,
Γ = H(Γ, z, z ),
k ≥ 0
where k - individual capital in current period, - current employment, Γ -
current distribution of capital, z - current aggregate shock.
Notice that when agent is employed he supplies exogenously given amount
of labor ˜l and gets wage. Otherwise an agent does not get a wage and return
on capital is the only source of his income.
The key assumption that allows compute equilibrium numerically is that
agents are boundedly rational in their perception of law of motion. Therefore,
current and future prices are assumed to be dependent only on the current
moments of capital distribution.
As a result, agent’s problem becomes:
v(k, ; ¯k, z) = max
c,k
{U(c) + βE[v(k , ; ¯k , z )|z, ]} (5)
subject to
c + k = r(¯k, ¯l, z)k + w(¯k, ¯l, z)˜l + (1 − δ)k,
log(¯k ) = a0 + a1log(¯k) if z = zg
log(¯k ) = b0 + b1log(¯k) if z = zb
k ≥ 0
where w(¯k, ¯l, z) = (1 − α)z(¯k/¯l)α
and r(¯k, ¯l, z) = αz(¯k/¯l)α−1
5

For model simulations I have used the following parameters:
Parameter Value Meaning
β 0.99 Discount factor
δ 0.025 Capital depreciation
σ 1 Utility risk-aversion parameter
α 0.36 Capital share in the production
function
zg 1.01 Productivity in a good state
zb 0.99 Productivity in a bad state
ug 0.04 Unemployment in a good state
ub 0.1 Unemployment in a bad state
˜l 0.3271 Individual labor supply
I want to point out that for the draft version I used the value of parameter
˜l = 1.11 that is not the one used in the paper. The reason was that the value
of this parameter is not given in the published version of the paper. Therefore
I looked in the other papers that are based on KS. There ˜l is normalized such
that labor supply in a bad state is equal to one (˜l = 1
1−ub
= 1
0.9 ≈ 1.11). Having
obtained results that are far from the ones in the paper I looked at the working
paper version of KS, where side note 9 on the page 10 is the following:
For the ﬁnal version of the Empirical Methods paper I am using the value
of exogenous labor supply ˜l = 0.3271 that is the one exploited in KS.
6

3 Algorithm
In this section I discuss equilibrium computation algorithm and it’s imple-
mentation.
Step 1: Compute transition probabilities. There are 4 possible states for in-
dividual in each period: (z, ) ∈ {(b, 0), (b, 1), (g, 0), (g, 1)} and paper contains
16 linear equations that determine these probabilities. As a mentioned in a
Model section evolution of (z, ) follows first order Markov process.
The function ”emprpr” contains these 16 equation and when we apply fsolve
to this function we obtain transition probabilities. These equations reflect the
following facts: individual employment can not affect aggregate economy state,
unemployment for each aggregate state should be consistent with model param-
eters ub and ug, etc.
Step 2: Choose parameters (a0, a1, b0, b1) for the law of motion of aggregate
capital. Chose grid for k and ¯k. KS suggest to have 70-130 points in k dimen-
sion with more points close to 0; and 4-6 points in ¯k dimension. Then starting
with zero initial value function do the following: do the value function iteration
step for the points on the grid, interpolate obtained result and continue these
two procedures until policy functions on a grid for two consecutive iterations
will be close enough. To be more precise the value function iteration step in
this set up is the following (denote f ≡ r(¯k, ¯l, z)k + w(¯k, ¯l, z)˜l + (1 − δ)k):
vt+1(k, ; ¯k, z) = max
k ∈[0,f]
{log(k − k ) + β
z ,
π(z , |z, )vt(k , ; ¯k , z )} (6)
Notice that vt(k , ; ¯k , z ) is defined for all k ∈ [0, f], so when we are trying
to find vt+1(k, ; ¯k, z) for the points on the grid we solve continuous optimiza-
tion problem. Having obtained vt+1(k, ; ¯k, z) for the points on the grid we
interpolate this function so that it can be used in the next iteration. We start
with v0(k, ; ¯k, z) = 0 for all k and k’. This procedure allows me to find policy
functions. In fact I need to find four policy functions because there are four
combinations of aggregate and idiosyncratic shocks.
The function ”emprrhs” computes next iteration interpolated value function
given as inputs interpolated value functions from the previous iteration. I did
not maximize value functions in each point of the grid separately. I maximize
sum of value function that should produce the same result given that value
functions at each point of the grid have different arguments. To be more precise
if you denote mij(k ) ≡ log(kj − k ) + β z , π(z , |z, )vt(k , ; ¯ki , z ) (it is
a function that you need to optimize in order to get value function in a grid
point (kj, ¯ki)), then you can find kij ∈ argmax{mij(k )} either by optimizing all
mij(k ) separately or sum them up and maximize jointly. These two procedures
should lead to the same result because all optimizations problems are indepen-
dent, however I joint optimization allows to reduce computation time (it is the
case because I can supply gradient for optimization routine). For interpolation
7

purpose I used not spline routine but pchip (Piecewise Cubic Hermite Inter-
polating Polynomials) because it allows to produces functions without wiggles
even for relatively small number of grid points (it means that value functions
will be increasing and concave).
Step 3: Given policy functions simulate 5000 agents and 11000 periods. This
procedure includes simulation of aggregate and idiosyncratic shocks according
to their conditional distribution. Policy functions give us individual capital ev-
ery period, so we can calculate actual aggregate capital. Then we can regress
log of actual aggregate capital on log of previous period actual aggregate capital
in good and bad times. This procedure gives us new parameters (a0, a1, b0, b1)
of the aggregate law of motion. Then we can go back to the Step 2. We repeat
Steps 2 and 3 until R2
for the regression of the log of actual capital is high
enough or until parameters (a0, a1, b0, b1) don’t change too much. It means
|ai
0 − ai+1
0 | + |ai
1 − ai+1
1 | + |bi
0 − bi+1
0 | + |bi
1 − bi+1
1 | < eps.
The function ”emprsim3” does simulation given policy functions and returns
implied sequence of aggregate capital and shocks. The diﬀerence with the func-
tion ”emprsim” that I used in the draft version is the following: I managed to
vectorize computations in the agent’s dimension that reduced required for sim-
ulations time from 3 hours to 1 minute (these numbers reﬂect computational
time on ”hammer” machine).
8

4 My results
For the results part I started with the following: I took the true values of
the parameters of the law of motion and tried to reproduce Figure 1 and Figure
2 from the paper.
log(¯k ) = 0.095 + 0.962log(¯k) if z = zg
log(¯k ) = 0.085 + 0.965log(¯k) if z = zb
In order to solve agent’s Bellman equation I used the following grid:
k = [0.00001,0.00004,0.00008,0.0001,0.001:0.013:0.04,0.05:0.15:1,1:1:35]; (50 points)
¯k = [7.7 : 4 : 19.7]; (4 points)
I am using less grid points in both dimensions than in the draft version
because Piecewise Cubic Hermite Interpolating Polynomials that I am using
allows me to do that. These polynomials don’t produce wiggly function that
permits me to use less grid points without substantial lose of interpolating preci-
sion. Less grid point allows me to reduce time of one iteration for value functions
iterations. So within the same time period I can perform more iterations and
solve policy function more precisely on a grid. The stopping criteria was that
average absolute change of the policy functions for two consecutive iterations
(across all grid points for all value functions) should be less than 0.00005. In the
draft version it was 0.005. I use average absolute change not maximum absolute
change, but it does not make a significant difference here because convergence is
moreless uniform on a grid points. I have got the following figure that contains
policy function for agent in a good state.
Analog of Figure 2: Tomorrow’s Individual Capital as a function of Today’s
Individual Capital
9

Then I simulate economy with 5000 individuals (who follow obtained policy
functions) for 11000 periods. I started with the good aggregate state where all
individuals hold 10 units of capital. I discarded first 1000 periods and tried to
reproduce Figure 1 in the paper.
Analog of Figure 1: Tomorrow’s Aggregate Capital as a function of Today’s
Aggregate Capital
As we can see this figure is slightly different from the one in the paper
(see Figure 1 from the paper on the next page). When I run regression of the
next period aggregate capital on today’s period aggregate capital I have got the
following results:
In good times:
log(¯k ) = 0.0993 + 0.9601log(¯k); R2
= 0.999941 ˆσ = 0.0186%
In bad times:
log(¯k ) = 0.0877 + 0.9624log(¯k); R2
= 0.999851 ˆσ = 0.0291%
10

Figure 1 from the paper: Tomorrow’s Aggregate Capital as a function of
Today’s Aggregate Capital
As we can see, obtained coefficients are close but not exactly the ones we
started from. In fact the sum of absolute coefficients change is of magnitude
0.01 that can not be considered as very precise.
Then I run the entire procedure - I started with the following guesses for the
aggregate capital law of motion:
log(¯k ) = 0 + log(¯k) if z = zg
log(¯k ) = 0 + log(¯k) if z = zb
The results of the first four iterations are presented in the table below:
The smallest change in coefficients is observed in the step from third to forth
iteration. I did not incude more iterations in the table because the procedure
diverges. The procedure does not converge because at some point iterations
produce range for the aggregate capital that fall outside the range where I have
grid points for the aggregate capital.
11

5 Appendix
5.1 Short description of the functions
Section 5.2:
emprmainbchm - main function that performs iterations for the aggregate cap-
ital law of motion coefficients.
Section 5.3:
emprmain1 - function that finds value functions for given aggregate capital law
of motion coefficients.
Section 5.4:
emprrhs - function that performs value iteration step for given aggregate capital
law of motion coefficients.
Section 5.5:
emprsim3 - function that simulates time series for the aggregate capital given
agents’ policy functions
Section 5.6:
emprcon2 - function that constructs linear constraints for the optimization re-
quired for value function iterations. This constraint insures that we will have
increasing in individual capital value function. This constraint was needed when
I used spline interpolation. For Hermite interpolation this constraint does play
a role.
emprder - function that computes coefficients of the piecewise polynomials
derivative, this function is used to supply optimization routine in the value
iteration step a gradient.
emprderu - function that returns the derivative of the utility function
emprf00, emprf01, emprf10, emprf11 - functions that are optimized in order
to perform value function iteration step.
emprfun - function that computes next period aggregate capital for today’s
corresponding grid points for aggregate capital
emprpr - function that returns vector of zeros when all conditions for tran-
sition probabilitis is satisfied. Applying fsolve to this function allows to get 4x4
transition probabilities matrix.
emprr - function that computes returns on capital.
empru - function that returns value of the utility function.
12

emprvalev - function that computes value function in all grid points.
emprw - function that computes wage.
13

5.2 Main function
clear;
diary(’output6.txt’);
diary on;
global a0 a1 b0 b1;
a0=0;
a1=1;
b0=0;
b1=1;
ccoef=zeros(50,8);
eps=1;
cccount=1;
lll=0.9;
while ((eps>0.01)&&(cccount<50))
display(’Start of outer iteration’);
cccount
a0
a1
b0
b1
emprmain1;
[res,res2]=emprsim3();
stats=regstats(log(res(find(ismember(res2(1:end-1),1))+1)),...
log(res(find(ismember(res2(1:end-1),1)))),’linear’);
stats2=regstats(log(res(find(ismember(res2(1:end-1),0))+1)),...
log(res(find(ismember(res2(1:end-1),0)))),’linear’);
display(’Good state’);
aa0=stats.beta(1,1)
aa1=stats.beta(2,1)
stats.rsquare
(stats.mse)^0.5
ccoef(cccount,1)=aa0;
ccoef(cccount,2)=aa1;
ccoef(cccount,3)=stats.rsquare;
ccoef(cccount,4)=(stats.mse)^0.5;
display(’Bad state’);
bb0=stats2.beta(1,1)
bb1=stats2.beta(2,1)
stats2.rsquare
(stats2.mse)^0.5
14

ccoef(cccount,5)=bb0;
ccoef(cccount,6)=bb1;
ccoef(cccount,7)=stats2.rsquare;
ccoef(cccount,8)=(stats2.mse)^0.5;
clear stats;
clear stats2;
eps=abs(a0-aa0)+abs(b0-bb0)+abs(a1-aa1)+abs(b1-bb1)
a0=lll*aa0+(1-lll)*a0;
b0=lll*bb0+(1-lll)*b0;
a1=lll*aa1+(1-lll)*a1;
b1=lll*bb1+(1-lll)*b1;
display(’End of outer itteration’);
cccount
cccount=cccount+1;
end;
diary off;
15

5.3 Function that ﬁnds value functions for given aggregate
capital law of motion parameters
tic
global bbeta delta sig alpha zg zb ug ub prag durub durug mkk mkkb trpr kk
kkb a0 a1 b0 b1 in00 in01 in10 in11 plcov fnorm mc00 mc01 mc10 mc11 limp
con ltild mouse;
% prag(i,j) - probabilities of transition for the aggregate state from i to
% j where 1 - bad, 2 - good
% mkk - matrix that contains grid points for k, each row is a grid for k,
% number of rows is equal to the number of grid points for kbar, all rows
% are the same
% mkkb - matrix that contains grid points for kbar, each column is a grid
% for kbar, number of colums is equal to the number of grid points for k,
% all columns are the same
% trpr - transition probabilities of moving from different conbinations of
% aggregate and idiosyncratic shocks
% orger of rows and columns is the following
% 1 - b0
% 2 - b1
% 3 - g0
% 4 - g1
% kk - grid for k
% kkb - grid for kbar
% a0, a1, b0, b1 - coefficients for the aggregate capital law of motion
% in00, in01, in10, in11 - initial value for the policy function (it is
% needed for maximization routine). The results from the previous step of
% the value function itteration is used here.
% plcov - (policy convergence) this value contains the distance between
% policy functions for the consecutive itterations.
% If it is small then value function iterations converged
plcov=2;
% average duration of bad and good times
durb=8;
durg=8;
prag=[1-1/durb, 1/durb; 1/durg, 1-1/durg];
% average duration of unemployment in good and bad times
durub=2.5;
16

durug=1.5;
ub=0.1;
ug=0.04;
zb=0.99;
zg=1.01;
alpha=0.36;
% see workpaper version
ltild=0.3271;
sig=1;
bbeta=0.99;
delta=0.025;
kk=[0.00001,0.00004,0.00008,0.0001,0.001:0.013:0.04,0.05:0.15:1,1:1:35];
kkb=[7.7:4:19.7];
% mkk(i,j) - amount of capital if kbar corresponds to i-s point on the grid
% and kk corresponds to j-s point on the grid
mkk=repmat(kk,size(kkb,2),1);
mkkb=repmat(kkb’,1,size(kk,2));
in00=zeros(size(mkk));
% calculation of transition probabilities
f=@(p) emprpr(p);
[trpr,fval,exitflag,output] = fsolve(f,ones(4,4));
mc00=emprr(0).*mkk+(1-delta).*mkk;
% 1 - aggregate state is good
% 0 - person is unemployed
mc10=emprr(1).*mkk+(1-delta).*mkk;
mc01=emprr(0).*mkk+emprw(0)*ltild+(1-delta).*mkk;
mc11=emprr(1).*mkk+emprw(1)*ltild+(1-delta).*mkk;
%Zero initial value function
y00=zeros(size(mkk));
v00=pchip(kk,y00);
v01=pchip(kk,y01);
v10=pchip(kk,y10);
17

v11=pchip(kk,y11);
fnorm=10;
con=emprcon2();
limp=1;
ccount=0;
while (((plcov>5*(1e-5)*4*size(mkk,1)*size(mkk,2))&&(ccount<370))
||(ccount==1)||(ccount==0))
[res00,res10,res01,res11]=emprrhs(v00,v10,v01,v11);
ccount=ccount+1;
ccount
v00=res00;
v01=res01;
v10=res10;
v11=res11;
plcov
end;
toc
18

5.4 Function that performs value function itteration step
function [res00,res10,res01,res11]=emprrhs(v00,v10,v01,v11)
% function that computes rhs for the value function iteration
% v00 - array that contains: v00(i) - function of kprime given i-s point on
% the grid for kbar and shocks 0-unemployed and 0-bad times
global kk kkb mkk delta trpr bbeta in00 in01 in10 in11 plcov fnorm
mc00 mc01 mc10 mc11 vp00 vp01 vp10 vp11 limp con mouse;
kkpg=emprfun(1);
kkpb=emprfun(0);
help00=(ppval(v00,kk));
h00=pchip(kkb,help00’);
he00=(ppval(h00,kkpb));
he10=(ppval(h10,kkpg));
he01=(ppval(h01,kkpb));
he11=(ppval(h11,kkpg));
% functions of kprime for every kbarprime that correspond to kbar
vp00=pchip(kk,he00’);
% derivatives of these functions
vp00d=fnder(vp00,1);
vp00d.coefs=emprder(vp00.coefs);
nvar=4*size(mkk,1)*size(mkk,2);
epss=1e-8;
19

options=optimset(’Display’,’off’,’Algorithm’,’interior-point’,’TolFun’,
1e-8,’TolX’,1e-8,’MaxFunEvals’,2000000,’TolCon’,1e-8,’MaxIter’,2000);
emprf00(in00)
ini00=max(min(mc00-epss,in00),epss*ones(size(mc00)));
hf=-epss*ones(size(mkk,1)*(size(mkk,2)-1),1);
[pl00,fval,exitflag,output]=ktrlink(@emprf00,ini00,con,hf,[],[],
epss*ones(size(mkk)),mc00-epss,[],options);
pl00
pl01
pl10
pl11
plcov=(sum(sum(abs(pl00-ini00)))+sum(sum(abs(pl01-ini01)))+
sum(sum(abs(pl10-ini10)))+sum(sum(abs(pl11-ini11))));
in00=pl00;
in01=pl01;
in10=pl10;
in11=pl11;
fv00=(limp*empru(mc00-pl00)+bbeta*(trpr(1,1)*emprvalev(vp00,pl00)+ ...
+trpr(1,2)*emprvalev(vp01,pl00)+trpr(1,3)*emprvalev(vp10,pl00)+...
+trpr(1,4)*emprvalev(vp11,pl00)));
20

res00=pchip([kk],[fv00]);
21

5.5 Simulation routine
function [res,res2]=emprsim3()
% function that simulates nn consumers for tt time periods and we burn
% first burn periods, because aggregate capital there is affected by
% initial wealth distribution
global ub ug kkb in00 in01 in10 in11 trpr prag kk;
nn=5000;
tt=11000;
burn=1000;
% suppose initially everybody has the same level of capital ink and
% aggregate state is good
ink=10;
kap0=ink*ones(nn,1);
kap=zeros(nn,1);
agst=zeros(1,tt);
empl=(rand(nn,1)>ug);
agkap=zeros(1,tt);
agkap(1)=ink;
kap=kap0;
kap2=zeros(nn,1);
empl2=zeros(nn,1);
agst(1,1)=1;
% first index is aggreagate state, second - employment
ii00=pchip(kkb,in00’);
help=1; % good initial state
for i=2:1:tt
i
he00=(ppval(ii00,agkap(i-1)));
i00=pchip(kk,he00’);
help2=rand(1,1);
22

if (help==0)
if (help2<prag(1,1))
help3=0;
else
help3=1;
end;
end;
if (help==1)
if (help2<prag(2,1))
help3=0;
else
help3=1;
end;
end;
agst(1,i)=help3;
help4=rand(nn,1);
if ((help==0) && (help3==0))
thresh=(1-empl)*trpr(1,1)/prag(1,1)+empl*trpr(2,1)/prag(1,1);
empl2=(help4>thresh);
hh=sum(ismember(empl2,1)); % number of employed guys
% adjustment of employment in order to have exact unemployment
% rates
if (hh>nn*(1-ub))
hhh=find(ismember(empl2,1));
emppl2(hhh(1:(hh-nn*(1-ub)),1),1)=0;
end;
if (hh<nn*(1-ub))
emppl2(hhh(1:(hh-nn*(1-ub)),1),1)=1;
end;
kap2(find(ismember(empl,1)))=ppval(i01,kap(find(ismember(empl,1))));
end;
if ((help==1) && (help3==0))
23

% rates
if (hh>nn*(1-ub))
emppl2(hhh(1:(hh-nn*(1-ub)),1),1)=0;
end;
if (hh<nn*(1-ub))
emppl2(hhh(1:(hh-nn*(1-ub)),1),1)=1;
end;
end;
if ((help==0) && (help3==1))
% rates
if (hh>nn*(1-ug))
emppl2(hhh(1:(hh-nn*(1-ug)),1),1)=0;
end;
if (hh<nn*(1-ug))
emppl2(hhh(1:(hh-nn*(1-ug)),1),1)=1;
end;
end;
if ((help==1) && (help3==1))
24

% rates
if (hh>nn*(1-ug))
emppl2(hhh(1:(hh-nn*(1-ug)),1),1)=0;
end;
if (hh<nn*(1-ug))
emppl2(hhh(1:(hh-nn*(1-ug)),1),1)=1;
end;
end;
help=help3;
agkap(1,i)=sum(kap2)/nn;
agkap(1,i)
kap=kap2;
empl=empl2;
end;
res=agkap(1,burn+1:end);
res2=agst(1,burn+1:end);
25

5.6 Auxiliary functions
function res=emprcon2()
global mkk;
h1=ones(size(mkk,1)*(size(mkk,2)-1),1);
h2=eye(size(mkk,1)*size(mkk,2))+diag(-h1,size(mkk,1));
res=h2(1:end-size(mkk,1),:);
*****************************************************************************
function res=emprder(coef)
% coef - each row contains coefficients of piecewise polynomials at some
% interval, for instance [1,0,2] means x^2+2
col=size(coef,2);
row=size(coef,1);
help=repmat(col-1:-1:1,row,1);
res=coef(:,1:end-1).*help;
*****************************************************************************
function res=emprderu(c)
global sig;
if (sig==1)
res=1./c;
else
res=(c.^(-sig));
end;
*****************************************************************************
function [res,res2]=emprf00(kp)
global mc00 trpr vp00 vp01 vp10 vp11 bbeta limp vp00d vp01d vp10d vp11d;
% res - value of the function
%kp=reshape(kp,size(mc00,1),size(mc00,2));
res= -1*(sum(sum(limp*empru(mc00-kp)+bbeta*(trpr(1,1)*emprvalev(vp00,kp)+ ...
+trpr(1,2)*emprvalev(vp01,kp)+trpr(1,3)*emprvalev(vp10,kp)+...+
+trpr(1,4)*emprvalev(vp11,kp)))));
% res2 - drivative (since function has matrix argument derivative is matrix as well)
res2= -1*(-1*limp*emprderu(mc00-kp)+bbeta*(trpr(1,1)*emprvalev(vp00d,kp)+ ...
+trpr(1,2)*emprvalev(vp01d,kp)+trpr(1,3)*emprvalev(vp10d,kp)+...
+trpr(1,4)*emprvalev(vp11d,kp)));
26

+trpr(2,2)*emprvalev(vp01,kp)+trpr(2,3)*emprvalev(vp10,kp)+...
res2=-1*(-1*limp*emprderu(mc01-kp)+bbeta*(trpr(2,1)*emprvalev(vp00d,kp)+ ...
*****************************************************************************
res=-1*(sum(sum(limp*empru(mc10-kp)+bbeta*(trpr(3,1)*emprvalev(vp00,kp)+ ...
res2=-1*(-1*limp*emprderu(mc10-kp)+bbeta*(trpr(3,1)*emprvalev(vp00d,kp)+ ...
*****************************************************************************
res2= -1*(-1*limp*emprderu(mc11-kp)+bbeta*(trpr(4,1)*emprvalev(vp00d,kp)+ ...
27

function res=emprfun(flag)
% function that computes next period aggregate capital for today’s
% correponding grid points for aggregate capital
% kkb - grid for aggregate capital
% flag=0 - bad state
% flag=1 - good state
global a0 a1 b0 b1 kkb;
if (flag==0)
res=exp(b0+b1*log(kkb));
elseif (flag==1)
res=exp(a0+a1*log(kkb));
end;
*****************************************************************************
function res=emprpr(pprob)
% function that returns vector of zeros when all conditions for transition
% probabilitis is satisfied
% pprob is 4x4
% pprob(i,j) is probability of transition from state i to j
% where i,j belongs to 1 - b0, 2 - b1, 3 - g0, 4 - g1
global ug ub prag durub durug;
res=zeros(16,1);
res(1,1)=1-1/durub-pprob(1,1)/prag(1,1);
res(2,1)=1-1/durug-pprob(3,3)/prag(2,2);
res(3,1)=pprob(1,1)+pprob(1,2)-prag(1,1);
res(11,1)=pprob(3,1)*prag(1,1)-1.25*pprob(1,1)*prag(2,1);
res(12,1)=pprob(1,3)*prag(2,2)-0.75*pprob(3,3)*prag(1,2);
res(13,1)=ub*pprob(1,1)+(1-ub)*pprob(2,1)-ub*prag(1,1);
res(14,1)=ub*pprob(1,3)+(1-ub)*pprob(2,3)-ug*prag(1,2);
res(15,1)=ug*pprob(3,1)+(1-ug)*pprob(4,1)-ub*prag(2,1);
res(16,1)=ug*pprob(3,3)+(1-ug)*pprob(4,3)-ug*prag(2,2);
28

function res=emprr(flag)
% function computes return on capital
% flag=0 - bad aggregate state
% flag=1 - good aggregate state
global ub ug alpha zb zg mkkb ltild;
if (flag==0)
res=alpha*zb*(mkkb.^(alpha-1))/((ltild*(1-ub))^(alpha-1));
elseif (flag==1)
res=alpha*zg*(mkkb.^(alpha-1))/((ltild*(1-ug))^(alpha-1));
end;
*****************************************************************************
function res=empru(c)
global sig;
if (sig==1)
res=log(c);
else
res=(c.^(1-sig))/(1-sig);
end;
*****************************************************************************
function res=emprvalev(cs,kpr)
% function that computes values for the value function
% res(i,j) = cs(i)(kpr(i,j))
help=ppval(cs,kpr);
help2=repmat(eye(size(kpr,1),size(kpr,1)),[1,1,size(kpr,2)]);
help3=help.*help2;
help4=sum(help3);
res=squeeze(help4);
*****************************************************************************
function res=emprw(flag)
% function computes wages
% flag=0 - bad aggregate state
% flag=1 - good aggregate state
global ub ug alpha zb zg mkkb ltild;
if (flag==0)
res=(1-alpha)*zb*(mkkb.^(alpha))/((ltild*(1-ub))^(alpha));
elseif (flag==1)
res=(1-alpha)*zg*(mkkb.^(alpha))/((ltild*(1-ug))^(alpha));
end;
29

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