- Email: [email protected]
On the interaction between efficiency wages and union-firm bargaining models
Economics Letters 01651765/93/$06.00
41 (1993) 319-324 0 1993 Elsevier
319 Science
Publishers
B.V. All rights
reserved
On the interaction between efficiency wages and union-firm bargaining models Peter J. Sanfey* KeynesCollege, University Received Accepted
of Kent at Canterbury,
Canterbury
CT2 7NP, UK
4 March 1993 17 March 1993
Abstract Even when efficiency wage considerations are taken into account, firms may have no incentive to pay wages above the competitive minimum. The total value of output therefore may be increased if workers in the efficiency wage sector also possess bargaining power. Efficiency wage and insider-outsider models may also reinforce each other.
1. Introduction Models of efficiency wages and union-firm bargaining have achieved considerable popularity in the labour economics literature in recent years. In this paper, I outline a simple model with elements of both, and examine the following questions. First, suppose that even if workers’ effort responds positively to wage increases, firms still have no incentive to pay above the competitive, or ‘going’ rate; does the competitive solution necessarily maximize the total value of output? Second, what is the optimal level of bargaining power for workers in the efficiency wage sector? Third, do efficiency wage and insider-outsider models reinforce each other, contrary to the conclusion of Lindbeck and Snower (1991)? The model is outlined in section 2, some analytical results are derived in section 3, while section 4 concludes with a brief discussion of the implications for the current debate over ‘performance-related pay’.
2. The model There are two sectors in the economy, A and B, each with one firm. Sector A is subject to exogenous productivity shocks, and output is also a function of workers’ effort, which is variable, due perhaps to imperfect monitoring of effort, or greater morale from higher wages. Sector B, on the other hand, has no productivity shocks and constant effort level (perfect monitoring). ’ Output in both sectors is a function of labour only. Employment in both sectors lies along the * I wish to thank Alan Carruth, Andy Henley, Andrew Oswald and Jim Robinson ’ Both productivity and effort in this sector are normalized to 1.
for helpful
discussions.
320
P.J. Sanfey I Economics
Letters 41 (1993) 319-324
respective labour demand curves (the ‘right-to-manage’ assumption). Wages in sector B always adjust to clear the labour market; thus there is no unemployment. All workers are identical ex ante and, aside from the efficiency wage aspect of jobs in sector A, all jobs are identical. In sector A, the workers are unionized and wages are determined prior to employment according to a Nash-bargaining model. The specific functional forms are as follows:
QA = (eda
where
Q is output,
(1)
)
II is employment
and e is effort.
’ e is given
by
720,
e=(wlb)‘,
(3)
where w and b are the wages in sectors A and B, respectively. The interpretation of this is straightforward: workers expend effort in excess of unity (the fixed level required in sector B) if their wage exceeds the wage in sector B. ’ Clearly, T measures the elasticity of effort with respect to wages. The utility of an individual worker in either sector is given by the constant relative risk aversion form: 4 u(w) = The union
wvp.
in sector
(4)
A is utilitarian
U(w, n) = nu(w) =
and therefore
wishes
to maximize
n(wVP)
(5)
(Henceforth, II without the subscript refers to flA.) In their bargaining takes the ‘alternative’ wage, b, as given, and hence its fall-back utility
with the firm, is
the union
U* = n(b”lP) The firm in sector
(6)
A maximizes
7r = P(en)”
profits:
- wn ,
(7)
fall-back where P is the exogenous productivity shock. 5 In the event of disagreement, equal to zero. Hence, wages, employment and effort in sector A are the solution to M,“x /_Llog{n(wPIP s.t. 7rn =o,
- b’lp)}
+ (1 - /.L)( log{P(n”(wlb)“‘)
- wn}
profits
are
(8)
OS/_L’l,
wrb, ’ Clearly, the parameter LYcould take different values for each sector but nothing would be gained by assuming this below. 3 See, for example, Akerlof (1982). ‘Note that effort is not included directly in the utility function [see, for example, Layard et al. (1991, Annex 4.2)] for simplicity. Having utility functions of the form u = w - e [e.g. Shapiro and Stiglitz (1984)] or u = WI- e* (e.g. Hendricks and Kahn (1991)] may create problems when effort is a function of wages, since, on the assumption that e,, < 0. u,, > 0. ’ In order to avoid complications introduced by imperfect competition in the product market, one can think of the two sectors as operating in a world market, where world prices are taken as given.
P.J. Sanfey
where p measures following equations:
the bargaining
I Economics
power
Letters 41 (1993) 319-324
of workers
in sector
321
A. The
solution
is given
by the
(9) aPn”-‘(w/b)“’
- w = 0.
(10)
One could solve these equations, at least implicitly, for w and n as functions of (Y, p, T, P and b. However, in this two-sector model, b is determined endogenously in equilibrium by the no-employment condition. Since employment is on the labour demand curve, this implies a*B
a-l=b=a(N-n)a-l,
(11)
where N is the total labour force (n = nJ. Equations (9)-( 11) constitute three equations of output is given by V= P((wlb)‘n)”
in three
w, n and b. The total value
unknowns,
+ (N - n)” .
(12)
3. Results Equations (9)-(12) form the basis of the analysis below. Note two things: first, p = 0 implies the competitive solution i.e. the situation where workers in both sectors have no bargaining power; and second, if j.~ = 0 and r < 1, then w = b, i.e. the ‘Solow condition’ [Solow (1979)] that at a maximum, the elasticity of effort with respect to wages is equal to 1, cannot be satisfied and so wages are driven down to the competitive level. However, the key point of this paper is that even when firms have no incentive to pay wages above the minimum, the value of output to society may be increased if they can be forced or persuaded to pay higher wages in the efficiency wage sector. This is now shown in two stages: to keep things simple, P is set equal to 1. Result 1. When F = 0, dnldp Proof. From Eq. rearranging gives (w/b)
(9),
< 0.
p > 0 implies
= PI”-*‘((N
that
w > b. Putting
_ n)/n)a(‘-u)‘(‘-u’)
Eqs.
(10)
and
(11)
together
.
and
(13)
Therefore, if w > b and P = 1, (N - n) > n, and so, starting (N - n) = II, the result follows.
from
the point
where
p = 0, and
Result 2. If r > 0, the competitive solution, where p = 0, is suboptimal in this economy. Proof.
The value
of output
V= (N - n)“n’ where X = ~(1
- a)/(1
in society,
V, can be written
[using
Eqs.
(10) and (ll)]
as
+ (N - FI)~ , - W) and Y = (~(1 - r)/(l
(14) - (YT). The objective
is to show that dV/dp
> 0
322
P.J. Sanfey
I Economics
Letters 41 (1993) 319-324
when I_L= 0. Clearly, dV/dp = (aV/&z)(dnld~). shown that dV/dn
- n)x-‘ny
+ (N - n)xYny-’
dnldp
< 0 when
- a(N - n)“-’
p = 0, it remains
to be
.
(15)
p = 0 and P = 1, (N - n) = n, Eq. (15) becomes
that when
dV/dn = (Y -X)nX+Y-’ =
Since
- cxn~~l
(16)
(Y(1 - 27 + (YT) -(y n”-’ 1 - (Y7 1
1
= -(2ar(l-
(17)
- cx7))na~’ CO.
a))/(1
(18)
To recapitulate: the total value of output may be increased by paying a wage premium in the efficiency wage sector, but in this model, the only way this can be brought about some power to the workers in this sector. What is the optimal level of pu?
to workers is by giving
Result 3. The optimal level of p (when P = 1) satisfies the following equation: a7-((u - 1) 1 - (Y7
P +
Eqs. (10) and (ll),
Finding the rearrangement,
optimal value yields
-X+
in terms of n and exogenous
setting
Eq.
(15)
equal
relating
Result 4. The wage premium, Substituting
Eq.
(w’b)p =
/.A
which
establishes
variables
to zero
which,
after
some
-a{(N-n)lr~}~=O. the optimal
(21) level of n to LYand 7, and so combining
The amount of bargaining power implied by Eq. (19) could p = -1 (CRRA = 2), cy = 0.8 and r = 0.3 implies or. = 0.59. Two more results can be derived analytically:
Proof.
(19)
0.
p + a(1 - /J)(l - 7) j_l+ cr(1 - j_L)(l - r) + /.Lup((Y- 1) = O *
of n implies
Y{(N-n)ln}
This is an implicit equation and (21) gives Eq. (19).
_
=
1
Eq. (9) can be re-written
[Pn”_’ (N _ n)‘-a]p”-aT
I (a(‘-r)lD(‘-a))
41 - PL)(l_ 4
-ff [ ~++(l-j.L)(1-7)+((Y-l)/_@ Proof. Using as follows:
{(‘-m7)lP(‘-u’)
P + a(1 - P)(l - 7) /.L++(l-/_L)(l-7)+((Y-l)j_Lp
(w/b),
be considerable:
Eqs. (20)
for example,
if
is invariant to changes in P.
(13) into (20) gives El. + +
41 - P.)(l_ 7)
cw(1- /A)(1 - T) + j@((Y - 1) ’
the result,
since P drops
out of the equation.
(22)
P.J. Sanfey
I Economics
323
Letters 41 (1993) 319-324
In words, although workers in the efficiency wage sector earn a wage premium, exogenous increases in productivity are shared proportionately by both sectors, as competitive labour market theory suggests they should be. Result 5. The elasticity of wlb with respect to p, E, rises as r rises. Proof. From eq. (22), one can derive
E= /_L +
l-41-a)
(23)
(Y(1 - /J)(l - T) + /.$?((Y- 1)
Therefore
w-41- a)(1 - /J)
ae’a7 = {p+(Y(1 -
p)(l -r)
+ @((Y - l)}’ ‘O
According to this result, the elasticity of the wage premium to changes in bargaining power is greater, the more important are efficiency wage considerations. In this sense, therefore, efficiency wage and rent-sharing models complement each other, contrary to the implications of Lindbeck and Snower (1991). 6
4. Discussion Does this paper have implications for the current debate about ‘performance-related pay’? 7 Arguments in favour of this are usually based on efficiency wage ideas, and I have shown an example in this paper where the value of society’s output is increased if workers in the efficiency wage sector are paid a wage premium. However, this outcome may be achieved only if workers in this sector possess bargaining power, a conclusion which may be disconcerting for proponents of performance-related pay. Two other points arise from the model above. First, nothing in this paper vitiates the standard result that exogenous productivity gains in one sector should be shared across different sectors. In this sense, ‘going wage’ increases maintain a valid role in promoting efficiency. Second, the conclusion of Lindbeck and Snower (1991) that efficiency wage and insider-outsider models do not reinforce each other is not robust, and merits further examination.
References Akerlof, G.A., 1982, Labor contracts as partial gift exchange, Quarterly Journal of Economics 97, 543-569. Hendricks, W.E. and L.M. Kahn, 1991, Efficiency wages, monopoly unions and efficient bargaining, Economic Journal 101, 1149-1162. Layard, R., S. Nickel1 and R. Jackman, 1991, Unemployment: Macroeconomic performance and the labour market (Oxford University Press, Oxford). Lindbeck, A. and D.J. Sower, 1991, Interactions between the efficiency wage and insider-outsider theories, Economics Letters 37. 193-196.
’ One explanation for the apparent discrepancy is that their paper looked an elasticity. ’ Oswald (1991) contains a discussion of this issue in the U.K. context.
at a simple
cross-partial
derivative,
rather
than
324
P.J. Sanfey
I Economics
Letters 41 (1993) 319-324
Oswald, A.J., 1991, Pay-setting, self-employment and the unions, Oxford Review of Economic Policy 7, 31-40. Shapiro, C. and J.E. Stiglitz, 1984, Equilibrium unemployment as a worker discipline device, American Economic 74, 433-444. Solow, R.M., 1979, Another possible source of wage stickiness, Journal of Macroeconomics 1, 79-82.
Review
41 (1993) 319-324 0 1993 Elsevier
319 Science
Publishers
B.V. All rights
reserved
On the interaction between efficiency wages and union-firm bargaining models Peter J. Sanfey* KeynesCollege, University Received Accepted
of Kent at Canterbury,
Canterbury
CT2 7NP, UK
4 March 1993 17 March 1993
Abstract Even when efficiency wage considerations are taken into account, firms may have no incentive to pay wages above the competitive minimum. The total value of output therefore may be increased if workers in the efficiency wage sector also possess bargaining power. Efficiency wage and insider-outsider models may also reinforce each other.
1. Introduction Models of efficiency wages and union-firm bargaining have achieved considerable popularity in the labour economics literature in recent years. In this paper, I outline a simple model with elements of both, and examine the following questions. First, suppose that even if workers’ effort responds positively to wage increases, firms still have no incentive to pay above the competitive, or ‘going’ rate; does the competitive solution necessarily maximize the total value of output? Second, what is the optimal level of bargaining power for workers in the efficiency wage sector? Third, do efficiency wage and insider-outsider models reinforce each other, contrary to the conclusion of Lindbeck and Snower (1991)? The model is outlined in section 2, some analytical results are derived in section 3, while section 4 concludes with a brief discussion of the implications for the current debate over ‘performance-related pay’.
2. The model There are two sectors in the economy, A and B, each with one firm. Sector A is subject to exogenous productivity shocks, and output is also a function of workers’ effort, which is variable, due perhaps to imperfect monitoring of effort, or greater morale from higher wages. Sector B, on the other hand, has no productivity shocks and constant effort level (perfect monitoring). ’ Output in both sectors is a function of labour only. Employment in both sectors lies along the * I wish to thank Alan Carruth, Andy Henley, Andrew Oswald and Jim Robinson ’ Both productivity and effort in this sector are normalized to 1.
for helpful
discussions.
320
P.J. Sanfey I Economics
Letters 41 (1993) 319-324
respective labour demand curves (the ‘right-to-manage’ assumption). Wages in sector B always adjust to clear the labour market; thus there is no unemployment. All workers are identical ex ante and, aside from the efficiency wage aspect of jobs in sector A, all jobs are identical. In sector A, the workers are unionized and wages are determined prior to employment according to a Nash-bargaining model. The specific functional forms are as follows:
QA = (eda
where
Q is output,
(1)
)
II is employment
and e is effort.
’ e is given
by
720,
e=(wlb)‘,
(3)
where w and b are the wages in sectors A and B, respectively. The interpretation of this is straightforward: workers expend effort in excess of unity (the fixed level required in sector B) if their wage exceeds the wage in sector B. ’ Clearly, T measures the elasticity of effort with respect to wages. The utility of an individual worker in either sector is given by the constant relative risk aversion form: 4 u(w) = The union
wvp.
in sector
(4)
A is utilitarian
U(w, n) = nu(w) =
and therefore
wishes
to maximize
n(wVP)
(5)
(Henceforth, II without the subscript refers to flA.) In their bargaining takes the ‘alternative’ wage, b, as given, and hence its fall-back utility
with the firm, is
the union
U* = n(b”lP) The firm in sector
(6)
A maximizes
7r = P(en)”
profits:
- wn ,
(7)
fall-back where P is the exogenous productivity shock. 5 In the event of disagreement, equal to zero. Hence, wages, employment and effort in sector A are the solution to M,“x /_Llog{n(wPIP s.t. 7rn =o,
- b’lp)}
+ (1 - /.L)( log{P(n”(wlb)“‘)
- wn}
profits
are
(8)
OS/_L’l,
wrb, ’ Clearly, the parameter LYcould take different values for each sector but nothing would be gained by assuming this below. 3 See, for example, Akerlof (1982). ‘Note that effort is not included directly in the utility function [see, for example, Layard et al. (1991, Annex 4.2)] for simplicity. Having utility functions of the form u = w - e [e.g. Shapiro and Stiglitz (1984)] or u = WI- e* (e.g. Hendricks and Kahn (1991)] may create problems when effort is a function of wages, since, on the assumption that e,, < 0. u,, > 0. ’ In order to avoid complications introduced by imperfect competition in the product market, one can think of the two sectors as operating in a world market, where world prices are taken as given.
P.J. Sanfey
where p measures following equations:
the bargaining
I Economics
power
Letters 41 (1993) 319-324
of workers
in sector
321
A. The
solution
is given
by the
(9) aPn”-‘(w/b)“’
- w = 0.
(10)
One could solve these equations, at least implicitly, for w and n as functions of (Y, p, T, P and b. However, in this two-sector model, b is determined endogenously in equilibrium by the no-employment condition. Since employment is on the labour demand curve, this implies a*B
a-l=b=a(N-n)a-l,
(11)
where N is the total labour force (n = nJ. Equations (9)-( 11) constitute three equations of output is given by V= P((wlb)‘n)”
in three
w, n and b. The total value
unknowns,
+ (N - n)” .
(12)
3. Results Equations (9)-(12) form the basis of the analysis below. Note two things: first, p = 0 implies the competitive solution i.e. the situation where workers in both sectors have no bargaining power; and second, if j.~ = 0 and r < 1, then w = b, i.e. the ‘Solow condition’ [Solow (1979)] that at a maximum, the elasticity of effort with respect to wages is equal to 1, cannot be satisfied and so wages are driven down to the competitive level. However, the key point of this paper is that even when firms have no incentive to pay wages above the minimum, the value of output to society may be increased if they can be forced or persuaded to pay higher wages in the efficiency wage sector. This is now shown in two stages: to keep things simple, P is set equal to 1. Result 1. When F = 0, dnldp Proof. From Eq. rearranging gives (w/b)
(9),
< 0.
p > 0 implies
= PI”-*‘((N
that
w > b. Putting
_ n)/n)a(‘-u)‘(‘-u’)
Eqs.
(10)
and
(11)
together
.
and
(13)
Therefore, if w > b and P = 1, (N - n) > n, and so, starting (N - n) = II, the result follows.
from
the point
where
p = 0, and
Result 2. If r > 0, the competitive solution, where p = 0, is suboptimal in this economy. Proof.
The value
of output
V= (N - n)“n’ where X = ~(1
- a)/(1
in society,
V, can be written
[using
Eqs.
(10) and (ll)]
as
+ (N - FI)~ , - W) and Y = (~(1 - r)/(l
(14) - (YT). The objective
is to show that dV/dp
> 0
322
P.J. Sanfey
I Economics
Letters 41 (1993) 319-324
when I_L= 0. Clearly, dV/dp = (aV/&z)(dnld~). shown that dV/dn
- n)x-‘ny
+ (N - n)xYny-’
dnldp
< 0 when
- a(N - n)“-’
p = 0, it remains
to be
.
(15)
p = 0 and P = 1, (N - n) = n, Eq. (15) becomes
that when
dV/dn = (Y -X)nX+Y-’ =
Since
- cxn~~l
(16)
(Y(1 - 27 + (YT) -(y n”-’ 1 - (Y7 1
1
= -(2ar(l-
(17)
- cx7))na~’ CO.
a))/(1
(18)
To recapitulate: the total value of output may be increased by paying a wage premium in the efficiency wage sector, but in this model, the only way this can be brought about some power to the workers in this sector. What is the optimal level of pu?
to workers is by giving
Result 3. The optimal level of p (when P = 1) satisfies the following equation: a7-((u - 1) 1 - (Y7
P +
Eqs. (10) and (ll),
Finding the rearrangement,
optimal value yields
-X+
in terms of n and exogenous
setting
Eq.
(15)
equal
relating
Result 4. The wage premium, Substituting
Eq.
(w’b)p =
/.A
which
establishes
variables
to zero
which,
after
some
-a{(N-n)lr~}~=O. the optimal
(21) level of n to LYand 7, and so combining
The amount of bargaining power implied by Eq. (19) could p = -1 (CRRA = 2), cy = 0.8 and r = 0.3 implies or. = 0.59. Two more results can be derived analytically:
Proof.
(19)
0.
p + a(1 - /J)(l - 7) j_l+ cr(1 - j_L)(l - r) + /.Lup((Y- 1) = O *
of n implies
Y{(N-n)ln}
This is an implicit equation and (21) gives Eq. (19).
_
=
1
Eq. (9) can be re-written
[Pn”_’ (N _ n)‘-a]p”-aT
I (a(‘-r)lD(‘-a))
41 - PL)(l_ 4
-ff [ ~++(l-j.L)(1-7)+((Y-l)/_@ Proof. Using as follows:
{(‘-m7)lP(‘-u’)
P + a(1 - P)(l - 7) /.L++(l-/_L)(l-7)+((Y-l)j_Lp
(w/b),
be considerable:
Eqs. (20)
for example,
if
is invariant to changes in P.
(13) into (20) gives El. + +
41 - P.)(l_ 7)
cw(1- /A)(1 - T) + j@((Y - 1) ’
the result,
since P drops
out of the equation.
(22)
P.J. Sanfey
I Economics
323
Letters 41 (1993) 319-324
In words, although workers in the efficiency wage sector earn a wage premium, exogenous increases in productivity are shared proportionately by both sectors, as competitive labour market theory suggests they should be. Result 5. The elasticity of wlb with respect to p, E, rises as r rises. Proof. From eq. (22), one can derive
E= /_L +
l-41-a)
(23)
(Y(1 - /J)(l - T) + /.$?((Y- 1)
Therefore
w-41- a)(1 - /J)
ae’a7 = {p+(Y(1 -
p)(l -r)
+ @((Y - l)}’ ‘O
According to this result, the elasticity of the wage premium to changes in bargaining power is greater, the more important are efficiency wage considerations. In this sense, therefore, efficiency wage and rent-sharing models complement each other, contrary to the implications of Lindbeck and Snower (1991). 6
4. Discussion Does this paper have implications for the current debate about ‘performance-related pay’? 7 Arguments in favour of this are usually based on efficiency wage ideas, and I have shown an example in this paper where the value of society’s output is increased if workers in the efficiency wage sector are paid a wage premium. However, this outcome may be achieved only if workers in this sector possess bargaining power, a conclusion which may be disconcerting for proponents of performance-related pay. Two other points arise from the model above. First, nothing in this paper vitiates the standard result that exogenous productivity gains in one sector should be shared across different sectors. In this sense, ‘going wage’ increases maintain a valid role in promoting efficiency. Second, the conclusion of Lindbeck and Snower (1991) that efficiency wage and insider-outsider models do not reinforce each other is not robust, and merits further examination.
References Akerlof, G.A., 1982, Labor contracts as partial gift exchange, Quarterly Journal of Economics 97, 543-569. Hendricks, W.E. and L.M. Kahn, 1991, Efficiency wages, monopoly unions and efficient bargaining, Economic Journal 101, 1149-1162. Layard, R., S. Nickel1 and R. Jackman, 1991, Unemployment: Macroeconomic performance and the labour market (Oxford University Press, Oxford). Lindbeck, A. and D.J. Sower, 1991, Interactions between the efficiency wage and insider-outsider theories, Economics Letters 37. 193-196.
’ One explanation for the apparent discrepancy is that their paper looked an elasticity. ’ Oswald (1991) contains a discussion of this issue in the U.K. context.
at a simple
cross-partial
derivative,
rather
than
324
P.J. Sanfey
I Economics
Letters 41 (1993) 319-324
Oswald, A.J., 1991, Pay-setting, self-employment and the unions, Oxford Review of Economic Policy 7, 31-40. Shapiro, C. and J.E. Stiglitz, 1984, Equilibrium unemployment as a worker discipline device, American Economic 74, 433-444. Solow, R.M., 1979, Another possible source of wage stickiness, Journal of Macroeconomics 1, 79-82.
Review