Labor Economics

Winter 2014/2015

Problem set 11

December 8, 2014

Surplus in the basic search and matching model

Question 1

Value functions

W = w + β(1− ρ)(W− U) + βU

U = b+ βF (W− U) + βU

J = p− w + β(1− ρ)(J− V) + βV

V = −κ+ βQ(J− V) + βV

Under free entry V = 0, which leads to

κ/Q = βJ

J = p− w + β(1− ρ)J

Question 2

η(W− U)η−1J1−η =(1− η)(W− U)ηJ−η (1)

ηJ =(1− η)(W− U) (2)

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Question 3

S =w + β(1− ρ)(W− U)− b− βF (W− U) + p− w + β(1− ρ)J (3)

=p− b+ β(1− ρ)S− βFη/(1− η)J (4)

=p− b− η/(1− η)κθ + β(1− ρ)S (5)

Multiply the above by 1− η to get

(1− η)S = (1− η)(p− b)− ηκθ + β(1− ρ)(1− η)S

When (1 − η)S = J, this is equivalent to the expression for J in which you substitute

in the Nash wage.

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