Gunnar Bårdsen and Ragnar Nymoen 865Union utility depends on the consumer real wage of an unemployed worker
and the aggregate rate of unemployment, thusV
(W
P,U,Aν)
wherePdenotes the
consumer price index.^7 The partial derivative with respect to wages is positive,
and negative with respect to unemployment (VW′ >0 andV′U≤0).Aνrepresents
other factors in union preferences. The fallback or reference utility of the union
depends on the overall real wage level and the rate of unemployment, henceV 0 =
V 0
(W ̄
P,U)
whereW ̄ is the average level of nominal wages, which is one of the
factors determining the size of strike funds. If the aggregate rate of unemployment
is high, strike funds may run low, in which case the partial derivative ofV 0 with
respect toUis negative (V 0 ′U < 0 ). However, there are other factors working in
the other direction, for example that the probability of entering a labor market
program, which gives laid-off workers higher utility than open unemployment, is
positively related toU.
With these specifications of utility and break-points, the Nash product, denoted
N, can be written as:
N={
V(
W
P
,U,Aν)
−V 0(
W ̄
P
,U)}{(
1 −W( 1 +T 1 )
Q1
Z)
Y} 1 −
,or:
N={
V(
RW
Pq( 1 +T 1 ),U,Aν)
−V 0(
W ̄
P,U)}{(
1 −RW
1
Z)
Y} 1 −
,whereRW=W( 1 +T 1 )/Qis the producer real wage, andPq( 1 +T 1 )=P( 1 +T 1 )/Q
is thewedgebetween the consumer and producer real wage.
Note also that, unlike many expositions of the so-called “bargaining approach”
to wage modeling (e.g., Layard, Nickell and Jackman, 1991, Ch. 7), there is no
aggregate labor demand function – employment as a function of the real wage –
subsumed in the Nash product. In this we follow Hahn and Solow (1997, Ch.
5.3), who point out that bargaining is usually over the nominal wage and not over
employment.
The first-order condition for a maximum is given byNRW=0, or:
V′W(
RW
Pq( 1 +T 1 ),U,Aν)V(
RW
Pq( 1 +T 1 ),U,Aν)
−V 0( ̄
W
P,U)=( 1 −)1
( Z
1 −RW^1 Z). (17.17)In a symmetric equilibrium,W=W ̄, leading toPq(RW 1 +T 1 )=
W ̄
P in equation (17.17).
The aggregate bargained real wageRWb is defined implicitly as:
RWb=F(Pq( 1 +T 1 ),Z,,U), (17.18)or, using the definition:
RWb≡Wb( 1 +T 1 )/Q,