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3.4. ON-THE-JOBSEARCH
311
end
for (i, x) in enumerate(jv.x_grid)
function w(z)
s, phi z
h(u) Vf[max(G(x, , phi), u)] .* pdf(F, u)
integral do_quad(h, nodes, weights)
pi_func(s) integral (1.0 pi_func(s)) Vf[G(x, phi)]
return (1.0 phi s) bet q
end
if brute_force
for in search_grid
for phi in search_grid
if phi <= 1.0
cur_val = -w((s, phi))
else
cur_val = -1.0
end
if cur_val max_val
max_val, max_s, max_phi cur_val, s, phi
end
end
end
else
max_s, max_phi minimize(w, guess, constraints=constraints;
disp=0, method="SLSQP")["x"]
max_val = -w((max_s, max_phi), , x, , a, b, Vf, jv)
end
if ret_policies
s_policy[i], phi_policy[i] max_s, max_phi
else
new_V[i] max_val
end
end
end
function bellman_operator(jv::JvWorker, V::Vector; ; brute_force=true,
ret_policies=false)
if ret_policies
out (similar(V), similar(V))
else
out similar(V)
end
bellman_operator!(jv, V, out, brute_force=brute_force,
ret_policies=ret_policies)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.4. ON-THE-JOBSEARCH
312
return out
end
"""
Extract the greedy policy (policy function) of the model.
##### Arguments
- cp::CareerWorkerProblem : Instance of CareerWorkerProblem
- v::Vector: Current guess for the value function
- out::Tuple(Vector, Vector) : Storage for output of policy rule
##### Returns
None, out is updated in place to hold the e policy y function
"""
function get_greedy!(jv::JvWorker, V::Vector, out::Tuple{Vector, , Vector};
brute_force=true)
bellman_operator!(jv, V, out, ret_policies=true)
end
function get_greedy(jv::JvWorker, V::Vector; brute_force=true)
bellman_operator(jv, V, ret_policies=true)
end
Thecodeiswrittentoberelativelygeneric—andhencereusable
• Forexample,weusegenericG(x,f)insteadofspecificA(xf)
a
Regardingtheimports
• fixed_quadisasimplenon-adaptiveintegrationroutine
• fmin_slsqpisaminimizationroutinethatpermitsinequalityconstraints
NextwebuildatypecalledJvWorkerthat
• packagesalltheparametersandotherbasicattributesofagivenmodel
• Implementsthemethodbellman_operatorforvaluefunctioniteration
Thebellman_operatormethodtakesacandidatevaluefunctionVandupdatesittoTVvia
TV(x)= min
s+f1
w(s,f)
where
w(s,f):=
x(s f)+b(p(s))V[G(x,f)]+bp(s)
Z
V[G(x,f)_u]F(du)
(3.34)
HereweareminimizinginsteadofmaximizingtofitwithSciPy’soptimizationroutines
WhenwerepresentV,itwillbewithaJuliaarrayVgivingvaluesongridx_grid
Buttoevaluatetheright-handsideof(3.34),weneedafunction,sowereplacethearraysVand
x_gridwithafunctionVfthatgiveslineariterpolationofVonx_grid
T
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ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.4. ON-THE-JOBSEARCH
313
Henceinthepreliminariesofbellman_operator
• fromthearrayVwedefinealinearinterpolationVfofitsvalues
– c1isusedtoimplementtheconstraints+f1
– c2isusedtoimplementse,anumericallystable
alternativetothetrueconstraints0
– c3doesthesameforf
Insidetheforloop,foreachxinthegridoverthestatespace,wesetupthefunctionw(z) =
w(s,f)definedin(3.34).
Thefunctionisminimizedoverallfeasible(s,f)pairs,eitherby
• arelativelysophisticatedsolverfromSciPycalledfmin_slsqp,or
• bruteforcesearchoveragrid
Theformerismuchfaster,butconvergencetotheglobaloptimumisnotguaranteed.Gridsearch
isasimplewaytocheckresults
SolvingforPolicies
Let’splottheoptimalpoliciesandseewhattheylooklike
Thecodeisinafilejv/jv_test.jlfromtheapplicationsrepositoryandlooksasfollows
s_policy, phi_policy bellman_operator(wp, V, return_policies=true)
# === plot policies === #
fig, ax subplots()
ax[:set_xlim](0, maximum(wp.x_grid))
ax[:set_ylim](-0.11.1)
ax[:plot](wp.x_grid, phi_policy, "b-", label="phi")
ax[:plot](wp.x_grid, s_policy, "g-", label="s")
ax[:set_xlabel]("x")
ax[:legend]()
plt.show()
Itproducesthefollowingfigure
Thehorizontalaxisisthestatex,whiletheverticalaxisgivess(x)andf(x)
Overall,thepoliciesmatchwellwithourpredictionsfromsection.
• Workerswitchesfromoneinvestmentstrategytotheotherdependingonrelativereturn
• Forlowvaluesofx,thebestoptionistosearchforanewjob
• Oncexislarger,workerdoesbetterbyinvestinginhumancapitalspecifictothecurrent
position
T
HOMAS
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ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.4. ON-THE-JOBSEARCH
314
Fig. 3.2:Optimalpolicies
Exercises
Exercise1 Let’slookatthedynamicsforthestateprocessfx
t
gassociatedwiththesepolicies.
Thedynamicsaregivenby(3.32)whenf
t
ands
t
arechosenaccordingtotheoptimalpolicies,and
Pfb
t+1
=1g=p(s
t
).
Sincethedynamicsarerandom,analysisisabitsubtle
Onewaytodoitistoplot,foreachxinarelativelyfinegridcalledplot_grid,alargenumberK
ofrealizationsofx
t+1
givenx
t
=x.Plotthiswithonedotforeachrealization,intheformofa45
degreediagram.Set:
50
plot_grid_max, plot_grid_size 1.2100
plot_grid linspace(0, plot_grid_max, plot_grid_size)
fig, ax subplots()
ax[:set_xlim](0, plot_grid_max)
ax[:set_ylim](0, plot_grid_max)
Byexaminingtheplot,arguethatundertheoptimalpolicies,thestatex
t
willconvergetoacon-
stantvalue¯xclosetounity
Arguethatatthesteadystate,s
t
0andf
t
0.6.
Exercise2 Intheprecedingexercisewefoundthats
t
convergestozeroandf
t
convergestoabout
0.6
Sincetheseresultswerecalculatedatavalueofbclosetoone, let’scomparethemtothebest
choiceforaninfinitelypatientworker.
Intuitively,aninfinitelypatientworkerwouldliketomaximizesteadystatewages,whicharea
functionofsteadystatecapital.
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.5. SEARCHWITHOFFERDISTRIBUTIONUNKNOWN
315
Youcantakeitasgiven—it’scertainlytrue—thattheinfinitelypatientworkerdoesnotsearchin
thelongrun(i.e.,s
t
=0forlarget)
Thus,givenf,steadystatecapitalisthepositivefixedpointx
(f)ofthemapx7!G(x,f).
Steadystatewagescanbewrittenasw
(
f)=x
(
f)(f)
Graphw
(
f)withrespecttof,andexaminethebestchoiceoff
Canyougivearoughinterpretationforthevaluethatyousee?
Solutions
Solutionnotebook
3.5 SearchwithOfferDistributionUnknown
Contents
• SearchwithOfferDistributionUnknown
– Overview
– Model
– Take1:SolutionbyVFI
– Take2:AMoreEfficientMethod
– Exercises
– Solutions
Overview
Inthislectureweconsider anextensionofthejobsearchmodeldevelopedbyJohnJ. McCall
[McC70]
IntheMcCallmodel,anunemployedworkerdecideswhentoacceptapermanentpositionata
specifiedwage,given
• hisorherdiscountrate
• thelevelofunemploymentcompensation
• thedistributionfromwhichwageoffersaredrawn
Intheversionconsideredbelow,thewagedistributionisunknownandmustbelearned
• Basedonthepresentationin[LS12],section6.6
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.5. SEARCHWITHOFFERDISTRIBUTIONUNKNOWN
316
Modelfeatures
• Infinitehorizondynamicprogrammingwithtwostatesandonebinarycontrol
• Bayesianupdatingtolearntheunknowndistribution
Model
Let’sfirstrecallthebasicMcCallmodel[McC70]andthenaddthevariationwewanttoconsider
TheBasicMcCallModel Consideranunemployedworkerwhoispresentedineachperiodwith
apermanentjobofferatwagew
t
Attimet,ourworkerhastwochoices
1. Accepttheofferandworkpermanentlyatconstantwagew
t
2. Rejecttheoffer,receiveunemploymentcompensationc,andreconsidernextperiod
Thewagesequencefw
t
gisiidandgeneratedfromknowndensityh
TheworkeraimstomaximizetheexpecteddiscountedsumofearningsE
å
¥
t=0
b
t
y
t
Trade-off:
• Waitingtoolongforagoodofferiscostly,sincethefutureisdiscounted
• Acceptingtooearlyiscostly,sincebetterofferswillarrivewithprobabilityone
LetV(w)denotethemaximalexpecteddiscountedsumofearningsthatcanbeobtainedbyan
unemployedworkerwhostartswithwageofferwinhand
ThefunctionVsatisfiestherecursion
V(w)=max
w
b
,c+b
Z
V(w
0
)h(w
0
)dw
0
(3.35)
wherethetwotermson ther.h.s. . aretherespectivepayoffsfromacceptingandrejectingthe
currentofferw
Theoptimalpolicyisamapfromstatesintoactions,andhenceabinaryfunctionofw
Notsurprisingly,itturnsouttohavetheform1fww¯g,where
• w¯isaconstantdependingon(b,h,c)calledthereservationwage
• 1fww¯gisanindicatorfunctionreturning1ifww¯and0otherwise
• 1indicates“accept”and0indicates“reject”
Forfurtherdetailssee[LS12],section6.3
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3.5. SEARCHWITHOFFERDISTRIBUTIONUNKNOWN
317
Offer DistributionUnknown Nowlet’sextendthemodel byconsideringthevariationpre-
sentedin[LS12],section6.6
Themodelisasabove,apartfromthefactthat
• thedensityhisunknown
• theworkerlearnsabouthbystartingwithapriorandupdatingbasedonwageoffersthat
he/sheobserves
TheworkerknowstherearetwopossibledistributionsFandG—withdensitiesfandg
Atthestartoftime,“nature”selectshtobeeither forg—thewagedistributionfromwhichthe
entiresequencefw
t
gwillbedrawn
Thischoiceisnotobservedbytheworker,whoputspriorprobabilityp
0
onfbeingchosen
Updaterule:worker’stimetestimateofthedistributionisp
t
f+(p
t
)g,wherep
t
updatesvia
p
t+1
=
p
t
f(w
t+1
)
p
t
f(w
t+1
)+(1 p
t
)g(w
t+1
)
(3.36)
ThislastexpressionfollowsfromBayes’rule,whichtellsusthat
Pfh=fjW=wg=
PfW=wjhfgPfh=fg
PfW=wg
and PfW=wg=
å
y2ff,gg
PfW=wjh=ygPfh=yg
Thefactthat(3.36)isrecursiveallowsustoprogresstoarecursivesolutionmethod
Letting
h
p
(w):=pf(w)+(1 p)g(wand q(w,p):=
pf(w)
pf(w)+(p)g(w)
wecanexpressthevaluefunctionfortheunemployedworkerrecursivelyasfollows
V(w,p)=max
w
b
,c+b
Z
V(w
0
,p
0
)h
p
(w
0
)dw
0
where p
0
=q(w
0
,p)
(3.37)
Noticethatthecurrentguesspisastatevariable,sinceitaffectstheworker’sperceptionofprob-
abilitiesforfuturerewards
Parameterization Followingsection6.6of[LS12],ourbaselineparameterizationwillbe
• =Beta(1,1)andg=Beta(3,1.2)
• b=0.95andc=0.6
Thedensitiesfandghavethefollowingshape
LookingForward Whatkindofoptimalpolicymightresultfrom(3.37)andtheparameterization
specifiedabove?
Intuitively,ifweacceptatw
a
andw
a
w
b
,then—allotherthingsbeinggiven—weshouldalso
acceptatw
b
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April20,2016
3.5. SEARCHWITHOFFERDISTRIBUTIONUNKNOWN
318
Thissuggestsapolicyofacceptingwheneverwexceedssomethresholdvaluew¯
Butw¯shoulddependonp—infactitshouldbedecreasinginpbecause
• fisalessattractiveofferdistributionthang
• largerpmeansmoreweighton fandlessong
Thuslargerpdepressestheworker’sassessmentofherfutureprospects,andrelativelylowcur-
rentoffersbecomemoreattractive
Summary:Weconjecturethattheoptimalpolicyisoftheform1fww¯(p)gforsomedecreasing
functionw¯
Take1: SolutionbyVFI
Let’ssetaboutsolvingthemodelandseehowourresultsmatchwithourintuition
Webeginbysolvingviavaluefunctioniteration(VFI),whichisnaturalbutultimatelyturnsout
tobesecondbest
VFIisimplementedinthefileodu/odu.jlcontainedintheQuantEcon.applicationsrepo
Thecodeisasfollows
#=
Solves the e "Offer Distribution Unknown" Model by value function
iteration and a second faster r method d discussed in the e corresponding
quantecon lecture.
@author : Spencer Lyon <spencer.lyon@nyu.edu>
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3.5. SEARCHWITHOFFERDISTRIBUTIONUNKNOWN
319
@date: 2014-08-14
References
----------
http://quant-econ.net/jl/odu.html
=#
using Distributions
"""
Unemployment/search problem where offer r distribution n is unknown
##### Fields
- bet::Real : Discount factor on (0, 1)
- c::Real : Unemployment compensation
- F::Distribution : Offer distribution F
- G::Distribution : Offer distribution G
- f::Function : : The e pdf of F
- g::Function : : The e pdf of G
- n_w::Int : Number of points on the grid d for r w
- w_max::Real : : Maximum m wage offer
- w_grid::AbstractVector : Grid of wage offers w
- n_pi::Int : Number of points on grid for r pi
- pi_min::Real : : Minimum m of pi grid
- pi_max::Real : : Maximum m of pi grid
- pi_grid::AbstractVector : Grid of probabilities pi
- quad_nodes::Vector : Notes for quadrature e ofer r offers
- quad_weights::Vector : Weights for quadrature ofer offers
"""
type SearchProblem
bet::Real
c::Real
F::Distribution
G::Distribution
f::Function
g::Function
n_w::Int
w_max::Real
w_grid::AbstractVector
n_pi::Int
pi_min::Real
pi_max::Real
pi_grid::AbstractVector
quad_nodes::Vector
quad_weights::Vector
end
"""
Constructor for r SearchProblem  with default values
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ARGENTAND
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TACHURSKI
April20,2016
3.5. SEARCHWITHOFFERDISTRIBUTIONUNKNOWN
320
##### Arguments
- bet::Real(0.95) : Discount factor in (0, 1)
- c::Real(0.6) : : Unemployment t compensation
- F_a::Real(1), F_b::Real(1) : Parameters of Beta  distribution n for F
- G_a::Real(3), G_b::Real(1.2) : Parameters of Beta distribution for G
- w_max::Real(2)  : : Maximum of wage offer grid
- w_grid_size::Int(40) : Number of points in n wage e offer grid
- pi_grid_size::Int(40) : Number of points in n probability y grid
##### Notes
There is also o a a version of this function n that t accepts keyword arguments for
each parameter
"""
function SearchProblem(bet=0.95, c=0.6, F_a=1, F_b=1, G_a=3, G_b=1.2,
w_max=2, w_grid_size=40, pi_grid_size=40)
Beta(F_a, F_b)
Beta(G_a, G_b)
# NOTE: the e x./w_max)./w_max x in these e functions s makes our dist t match
#
the scipy one with scale=w_max x given
f(x) pdf(F, x./w_max)./w_max
g(x) pdf(G, x./w_max)./w_max
pi_min 1e-3 # avoids instability
pi_max pi_min
w_grid linspace_range(0, w_max, w_grid_size)
pi_grid linspace_range(pi_min, pi_max, pi_grid_size)
nodes, weights qnwlege(210.0, w_max)
SearchProblem(bet, c, F, G, f, g,
w_grid_size, w_max, w_grid,
pi_grid_size, pi_min, pi_max, pi_grid, nodes, weights)
end
# make kwarg version
function SearchProblem(;bet=0.95, c=0.6, F_a=1, F_b=1, G_a=3, G_b=1.2,
w_max=2, w_grid_size=40, pi_grid_size=40)
SearchProblem(bet, c, F_a, F_b, G_a, , G_b, , w_max, w_grid_size,
pi_grid_size)
end
function q(sp::SearchProblem, w, pi_val)
new_pi 1.0 ./ (((pi_val) .* sp.g(w)) ./ (pi_val .* sp.f(w)))
# Return new_pi when in [pi_min, pi_max] and else end points
return clamp(new_pi, sp.pi_min, sp.pi_max)
end
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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