Notes on McCall s Model of Job Seach Timothy J Kehoe Mach Fv ( ) pob( v), [, ] Choice: accept age offe o eceive b and seach again next peiod An unemployed oke solves hee max E t t y t y t if job offe has been accepted b if seaching ellman s equation fo an unemployed oke: V( ) max, b EV( ') V( ) max, b V( ') df( ') Suppose that e have solved this poblem and found V( ) Then V b V( ') df( ') is just a constant Let be such that V b V( ') df( ') Then e can gaph the value function and chaacteize the optimal decision by the unemployed oke as tun don age offes and accept age offes If e can find that satisfies the above elationship, e have found V( )
V( ) V ' b VdF( ') df( ') ' b df( ') df( ') Since df ( ') df ( ') df ( ') e can ite ' df( ') df( ') b df( ') df( ') ( ) ' df( ') b df( ') df( ') df ( ') b ( ' df ) ( ') Adding ( ) df( ') df( ') to both sides of this equation, e obtain
b ( ' ) df( ') b is the cost of tuning don a age offe to continue seaching, and ( ' ) ( ') df is the expected discounted benefit of tuning don a age offe to continue seaching Let Then Notice that h ( ) ( ' df ) ( ') b h( ) E h() h ( ) h'( ) F( ) h ''( ) '( ) F Note: To find the expession fo h'( ), e apply Leibnitz s ule fo diffeentiating functions ith integals, to h: ( ) b( x) gx ( ) f ( xydy, ) a( x) b( x) f( x, y) g'( x) f( x, a( x)) a'( x) f( x, b( x)) b'( x) dy a( x) x h ( ) ( ' df ) ( ') h'( ) ( ) df( ') 3
h'( ) F( ) Given these chaacteistics of h, ( ) e can da a gaph illustate ho is detemined: E b ( ' ) ( ') df b Incease in unemployment benefits Compaative statics: An incease in b leads to an incease in We can see this by shifting the line b in the gaph donad We can also do this algebaically by applying the implicit function theoem: b ( ' ) df( ') b ( ) b ( ' b ( )) df( ') b ( ) b ( ) b hb ( ( )) '( b) h'( ( b)) '( b) h'( ( b)) '( b) '( b) h'( ( b)) 4
Example Unifom distibutions on [, ] ith / if F (, ) if if Fo, h ( ) ' ( ') ( ') df df ' h( ) df( ') df( ') h ( ) ( ) h ( ) ( ) Fo, Fo, h ( ) ( ' df ) ( ') ( ') h ( ) ' ( )( ) ( ) h ( ) ( ) ( )( ) h ( ) ( ) ( ) ( )( ) ( ) h ( ) ( )( ) h ( ) 5
Consequently, ( ) if ( ) ( ) h ( ) if ( )( ) if ( ) b ( ) ( )( ) b Check popeties of h: ( ) h() ( ) h ( ) h ( ) if ( ) ( ) h'( ) if ( )( ) if if h''( ) if ( )( ) if 6
Notice that h ( ) and h'( ) but not h''( ) ae continuous at and at Digession on mean peseving speads Anothe expession fo the mean of a andom vaiable: E df( ) Integation by pats: b b b a a a vdu uv udv ( F ( )) d ( F ( )) df ( ) df( ) ( F( )) d E F( ) d Class of distibutions that depend on a paamete : E F(, ) d Requiement that F (, ) and F (, have the same mean: F F d (, ) (, () Single cossing popety: Let W be the subset of [, ] that is the union of the suppots of F (, ) and F (, Then F (, ) and F (, have the single cossing popety if thee exists such that ( F (, F (, ) if and W F (, ) F (, ) if and W Example The class of unifom distibutions on [, ] ith / 7
These distibutions have the same mean, E / They also satisfy the single cossing popety Let us compae F (,) ith F, (, ) / Hee and in tems of the equal mean popety () and the single cossing popety ( Density function f (, ) df(, ) : f( p, ) f (, ) f(,) Distibution function F: (, ) F (, ) F (, ) F (,) 8
Mean peseving speads F (, ) F (, ) F (, Popeties () and ( imply popeties () and v (3) F (, F (, ) d fo all v, v W Popeties () and (3) do not imply popety () and (, hoeve That is, hen combined ith the equal mean popety (), popety (3) is eake than the single cossing popety ( Rothschild and Stiglitz (97) say that, if F (, ) and F (, satisfy popeties () and (3), then F (, is a mean peseving spead of F (, ) Equivalently, e can also say that F (, ) is a mean peseving contaction of F (, Rothschild and Stiglitz (97) use popeties () and (3) athe than the stonge popeties () and ( because popeties () and (3) geneate a tansitive patial odeing of distibutions F: (, ) If F (, is a mean peseving spead of F (, ) and F (, 3) is a mean peseving spead of F (,, then F (, 3) is a mean peseving spead of F (, ) This is not tue if e use popeties () and ( (The odeing is patial because, fo to abitay distibutions F (, is not necessaily eithe a mean peseving spead o a mean peseving contaction of F (, ) ) 9
To see that () and ( do not define a tansitive patial ode, e da a gaph in hich F (, 3) cosses F (, once, and F (, cosses F (, ) once, but F (, 3) cosses F (, ) moe than once F (, ) F (, ) F (, 3) F (, The concept of mean peseving spead is attactive because, as Rothschild and Stiglitz (97) sho, the folloing thee definitions of an incease in iskiness of a da ae equivalent: Any isk avese oke pefes to take a da fom F (, ) to taking a da fom F (, F (, assigns moe pobability to its tails than does F (, ) 3 F (, equals F (, ) plus noise If e ee to define F (, as iskie than F (, ) if F (, had the same mean as F (, ) but a highe vaiance, then e ould have a tansitive patial odeing, but it ould not have these thee attactive popeties Incease in isk in the labo maket Reite b ( ' ) df( ')
as b ( ' ) df( ') ( ' ) df( ') ( ' ) df( ') b ( ' ) df( ') ( ' ) df( ') E b ( ' ) df ( ') Integation by pats: ( ) be ( ' ) df( ') Theefoe b b b a a a vdu uv udv ( ' ) df ( ') ( ' ) F ( ') F ( ') d ' ( ' ) df( ') F( ') d' ( ) be F( ') d' b( Eb) F( ') d' Let g ( ) Fd ( ') ' Then b ( Eb) g( ) Notice that g() g ( ) g'( ) F( ) g''( ) F'( ) Fo distibution functions in the class F, (, )
g (, ) F ( ', d ) ' Notice that, if F (, is a mean peseving spead of F (, ), then To detemine e solve F ( ', ) F ( ', ) d ' F ( ', ) d' F ( ', d ) ' g (, ) g (, ) fo all W b( Eb) F( ', ) d' b ( Eb) g(, ) b ( E b) ( E b) g(, ) b Compaative statics: An incease in isk leads to an incease in We can see this by shifting the cuve ( E b) g(, ) in the gaph upad Suppose that a decease in is a mean peseving spead The diffeential vesions of popeties () and (3) ae (4) (5) F (, ) d v F (, ) d fo all v W
e caeful: We ae folloing the opposite convention as Ljungqvist and Sagent hee an incease in is a mean peseving spead We ae doing so because it matches ou example ith unifom distibutions We can also do this algebaically by applying the implicit function theoem: b( Eb) F( ', ) d' b ( ) () b ( Eb) F( ',) d' ( ) F ( ', ) '( ) F( ( ), ) '( ) d' ( ) F ( ', ) F ( ( ), ) '( ) d' Notice that F ( ( ), ), hich implies that ( ) F ( ', ) d ' '( ) F ( ( ), ) An incease in isk inceases the esevation age and inceases the expected utility of an unemployed oke Example Unifom distibutions on [, ] ith / Fo, Fo, g ( ) ' g ( ) d' ( ') g ( ) ( ) g ( ) ( ) 3
Fo, g ( ) ' g( ) d' d' ( ) g ( ) ( ) ( ) 4 6 ( ) g ( ) Consequently, if g ( ) if ( ) if b b b g (, ) b Check popeties of g: ( ) g() g ( ) fo all [, ] 4
if g'( ) if if if g''( ) if if Notice that g ( ) and g'( ) but not g''( ) ae continuous at and at Mean aiting time Let n be the aveage numbe of peiods that a oke is unemployed Since F is constant and the policy ule that the unemployed oke follos is stationay, n is the expected aiting time fo the oke to find a job no matte ho may peiods the oke has aleady been seaching Let df ( ) be the pobability that a job offe is ejected Then n ( ) ( n) That is, in peiod t the expected aiting time n can be decomposed into the pobability ( ) of accepting the job offe in peiod t, in hich case the aiting time ill have been, and the pobability of ejecting the job offe in peiod t, in hich case the oke ill have aited one peiod and no again have the expected aiting time n n Notice that the highe is, the highe is and the highe is n 5
Quits Suppose a oke has a job that pays and quits The next peiod, he expected utility is V b V( ') df( ') That is, quitting loes utility Fies Suppose that once a oke has accepted a job offe, she faces a constant pobability,, of being fied ellman s equation fo an unemployed oke: V( ) max ( ) V( ) b V( ') df( '), b V( ') df( ') Let C([, ]) be the set of continuous bounded functions on [, ] Define the opeato T : C([, ]) C([, ]) by the ule T( V)( ) max ( ) V( ) b V( ') df( '), b V( ') df( ') Notice that, ( ) V( ) b V( ') df( ') is inceasing in if ( ) V is inceasing in, hile b V( ') df( ') does not depend on Consequently, T maps inceasing functions into inceasing functions This implies that the optimal V( ) is inceasing Once again, suppose that e have solved fo V and let V be the constant value of being unemployed: V b V( ') df( ') Since V( ) is inceasing, e can chaacteize the value function as befoe: 6
Hee b V( ') df( ') if V( ) ( ) b V( ') df( ') if V( ) V V ( ) Let V V ( ) V ( ) V V Once again, the optimal decision of the unemployed oke is to tun don age offes and to accept age offes Folloing the same steps as in the model ith, e find ' b df( ') df( ') ( ) 7
' df ( ') df ( ') b df ( ') df ( ') ( ) ' df ( ') b df ( ') ( ) ( ) ' df ( ') df ( ') b df ( ') ( ) ( ) b ( ' ) df( ') ( ) b is the cost of tuning don a age offe to continue seaching, and ( ' ) ( ') ( ) df is the expected discounted benefit of tuning don a age offe to continue seaching E ( ) b ( ' ) ( ') ( ) df b We can analyze the impact of an incease in unemployment benefits b o of a mean peseving spead in F as in the model ith and obtain simila esults Compaative statics: An incease in leads to an decease in We can see this by shifting the cuve /( ( )) ( ' ) df ( ') in the gaph donad 8
Algebaically, e can apply the implicit function theoem: b ( ' ) df( ') ( ) ( ) b ( ' ( )) df( ') ( ) ( ) '( ) ( ' ( )) df( ') F( ( ) '( ) ( ( )) ( ) ( ) F ( ( ) '( ) ( ' ( )) df ( ') ( ) ( ( )) ( ) ( ' ( )) df( ') ( ) '( ) ( ( )) ( ( )) F ( ( ) An incease in deceases the expected utility of both unemployed okes and employed okes Unemployment ate Suppose that thee is a continuum of ex ante identical okes ho move beteen peiods of employment and unemployment: In the stationay solution u t ( ut) F( ) ut u u u uˆ ( uˆ) F( ) uˆ uˆ F ( ) ˆ t t We have aleady calculated the mean duation of unemployment as n u F ( ) The mean duation of employment can be found by solving n ( )( n ) e e 9
n e Notice that flos in and out of employment ae govened by a stationay Makov chain: employment F ( ) unemployment The invaiant distibution is F ( ) F ( ) F ( ) uˆ F ( ) ˆ uˆ F ( ) Refeences Las Ljungqvist and Thomas J Sagent (4), Recusive Macoeconomic Theoy Second edition MIT Pess, Chapte 6 John J McCall (97), Economics of Infomation and Job Seach, Quately Jounal of Economics, 84, 3 6 Michael Rothschild and Joseph Stiglitz (97), Inceasing Risk I: A Definition, Jounal of Economic Theoy,, 5 43 Michael Rothschild and Joseph Stiglitz (97), Inceasing Risk II: Its Economic Consequences, Jounal of Economic Theoy, 3, 66 84