Bellman equation
V15 max ulǜ BE 以川了
K⽐⻔
s ⼆Aflk 11 S k Clanofmotion
where A is TFPshockwhichis id
The economy it Production and consumption of asinglegood
At beginningofpernod stocks of the good
is available
Resource chant
5 4k
t One continuous state variable SE lo
One continuous action variable KETOD
collocation Strategy
us Us ⼆点gg is
⻘cjdjcsi max ul Sik 焦Ěwzgg 的 k城北⻔KEN s
constraint㓤neverbind
We approximate me winefunction using Chebyshev polynomial
and nodegauduediutize.me TFPShockusing
Gaussian Quadrature The utility function is givenly
Ir
灬 9等 州⼆
HadA is iid lognormally distributed
Maximization problem We can use a brute force approach
to solvefor k for any given State Si Foreach state
we construct mepayof function
Ik Is ucsi kytfjjwzgdjlctbktAz.tk'D
Pickme k which yields the largest value f J
hodel Simion
tho a SequenceofproductivityShocks
Use the optimalpolicy to calculate the
capital stock and tu stock of thegood
Dtus repeatedly formany randomly
generated stews of productivity Shock
Idea i Suppose we have kiglSD.li and we here
a sequenceof productivityshocks A Apj
initial resources are given by Si
Next periodstockof megood isgivenby
SP tsjglsiytA.fi以
The next step is givenby
si 1Dgcsi ⼗ 时gcsi and soforth
Note the function g is onlyknown on megudpoint
IS 521 sn了 but the simulationtakeson valuesof
the grid so that we needto interpolate funding
fromGaussianquadrature
To drewvmdomsqunasoffoduetiutgshochs.ve
make useof me discrete rv ZEE Zug with
prob diminution W Wig Wedraw uniformlydistributed
GQ.ws and use it to determine me realization of me
probability shock The CDF of thediscrete N is the
cumulative sum of the pwb.IN With ⻔
if the uniform w fins forexample betweenuand w.tw then we assign Zz and so forth
ūauate a function of optimalchoices
findthe maximization point
name distance