Weighted choice probability models

Example: Labor supply

, ,U C h z

,C f hw I

, , , ( )U C h z v C h z

, , , ,h W I v f hW I h

Number of jobs

• Let m(h) be the number of jobs in the choice set B(h). Although m(h) may also depend on the wage rate, we suppress this in the notation here. The non-market choice consists of one alternative so that m(0)=1.

0( )

hm h

( ) ( )g h m h

( ) ( )m h g h

Max utility under constraints

• The assumptions made above allow us to derive the probability that an agent will choose a job with hours of work h within the choice set B(h). By well-known results, it follows that if the agent is a utility maximizer, then the probability that job z within the choice set B(h) is chosen, is given by

( )

( )

, , , ,, , ( ) max max , , ( )

, , ( ) , ,

x k B x

x D x Dk B x

h W I h W IP h W I z x W I k

x W I m x x W I

Choice probability

• The choice of any job:

; ,h W I

( )( )

( )0,

0,

; , , , , ( ) max max , , ( )

, , ( ) , ,

( ) ; , 0,0, ( ) , ,

( ) , ,.

0,0, ( ) , ,

x k B xz B h

z B hx D x x D

x x D

h W I P H hW I P h W I z x W I k

h W I m h h W I

m x x W I I m x x W I

g h h W I

I g x x W I