Solving Stochastic Infinite Horizon Model Using Log-Linearization (1)
Tanapong Potipiti
Infinite Period Cake-Eating
Solving c
1
0
1 1
1
1
( 1 )
t
1
t
t
c c k
k
c
Time-Invariant Relationship: c
t= f(k
t)
Notice that
We can show that for all t
Choice and State Variables
• The above equation shows the time-invariant relationship between
– choice variable at time t: ct and kt+1
– state variable at time t: kt
• Choice variables = what the agent chooses.
• State variables contains all relevant information for making decision.
A Simple Stochastic Linear Economy
3 key equations
(The general form of TVC for infinite horizon model)
Solving c
tTaking limit and expectation of the resource constraint
Solving c
tSolving c
tSimulation Code
k(1) = 0; B = 0.9; r = 1/B-1;
y = grand(1000, 1, 'uin', 0,1);
//grand generates random output for t=1:500
c(t) = (1-B)/B*k(t) + (1-B)*y(t) + B/2;
k(t+1) = (1+r)*k(t) + y(t) - c(t);
end
plot(c);
Random-Walk Consumption Path
0 50 100 150 200 250 300 350 400 450 500 -1.0
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
Why Linear Model is Easy to Solve
• We can solve the previous model directly and analytically because the first order + TVC
conditions are linear.
In general, linear models are easy to solve because
– We can apply Matrix algebra.
– With linear function f we can deal with expectation easily: for a linear function f
E(f(x)) = f(E(x)).
Linear Approximation (Linearization)
Linear approximation of y = f(x) around point (x0,y0)
Log-Linear Approximation
(Log Linearization)
Basic Rules for Log-linearization
Basic Rules for Log-linearization
• In case that you can apply more than one rule, the basic rules have to be applied in the following order
1) Addition rule 2) Product rule 3) Power rule
4) Anything in blanket
