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Are labor-managers really perverse?
Economics Letters 2 (1979) 137-142 0 North-Holland Publishing Company
ARE LABOR-MANAGERS A. STEINHERR
REALLY PERVERSE?
and J.-F. THISSE *
Universitk Catholique de Louvain, Louvain, Belgium Received
February
1979
One major problem of labor-managed firms seems to be their perverse response to price incentives. We show that this perversity is only due to an incomplete specification of the objective function of these firms.
1. Introduction The objective function for the labor-managed and Vanek (1970) among others, is defined by
firm (LMF) used by Ward (1958)
(1)
max y = @F(K, L> - rK)/L , where y p F K L r
= labor’s value-added (or income) per capita, = market price of output (exogenous), = production function, = stock of capital, = employment, = rental price of one unit of capital (exogenous).
In this letter we argue that this objective is inappropriate to deal with changes in the economic environment of the LMF. We suggest some alternatives that seem to be more consistent with both logic and the spirit of labor-management. The focus of the analysis is a competitive firm perturbed by an increase of the price of output ’ while dosing in general equilibrium, which implies that initially average incomes of workers are equal across all firms. For conciseness, we limit the analysis to the shortrun where the stock of capital is fixed.
* The authors thank P. Dehez, P. Kleindorfer, P. Pestieau and M. Sertel for discussions. 1 Obviously, our results are pertinent to a reduction of the rental price of capital or to different types of technological progress,(including Harrod or Hicks neutral processes). 137
A. Steinherr and J.-F. Thisse /Are labor-managers really perverse?
138
2. The risk of being dismissed In general, the workers who make the decision to reduce employment in the LMF risk to be among those who will be selected for departure. This risk should be integrated into the objective function of the LMF, which is precisely what we are going to do. Denoting the parameters and the variables in the initial state with subscript zero, we have (PoFKo,
LO)
-
roKoYLo
= w.
(2)
With p 1 > p. , it is known that the optimal employment Y(L) =
CPIJWO,
L)
-
for maximization
of (3)
rKo)/L
denoted by L 1, is smaller than the initial employment Lo. We now assume that for selecting workers a random process is used, where each worker has the same probability of being dismissed. Besides simplicity, this system has the further advantage of corresponding to a well-established, albeit particular, notion of justice. Consequently, if L denotes the number of workers remaining in the firm, the probability for a worker to leave the LMF is given by IT= (Lo - L)fLo,
(4)
while the probability
to remain is
1 -n=L/Lo.
(5)
Under these circumstances, the alternative revenue of dismissed workers must be taken into account. This alternative income is equal to income from employment in other industries or to unemployment benefits. (The short-run horizon of our analysis precludes creation of employment through creation of new firms.) Hence, this income may be assumed not to be larger than initial income w. In the following it is taken as equal to w. This is not restrictive, however, since the conclusions we are going to derive remain unaltered if the alternative income is below w. We can now rewrite the objective function of the LMF as follows: max V(L)
w.r.t.
L E [0, -1,
where
V(L) = U(w)> = UP(L)] =
* (1 - n) + U(w) * 77
NY(L)1
U denoting the utility function
for
L = 0,
for
O
for
LofL,
assumed identical for all workers.
(6)
A. Steinherr and J.-F. Thisse /Are
139
labor-managers really perverse?
Theorem 1. Assume (6). Ifworkers are not risk-lovers, then the optimal level of employment is unique and given by Lo. Proofi Given the standard assumption of continuity of y(L) and given that y(L) tends to zero when L becomes arbitrarily large, it is easily seen that at least one maximizer of V(L) exists; it is denoted by L * . Let L(Kc) be the value of L such that the marginal productivity of labor FL (KO, L) increases on [0,Z(Kc)] and decreases on @(Kc), -1; clearly L(Ke) < L 1 . First, assume that L* E [0, z(KO)]. As V(0) < V(L,,), L* must belong to [0, E(Kc)]. In this case, we would have V(L*)=uLy(L*)]-~+u(w).~
< U(r(L,)]
+ {ub(L*>]
* 2. + U(w) - y
L*-Ll - V(w)} * ~ LO
since Lo-L1
.g-yw,.--
< ULl),
a contradiction.
Lo
Second, take L* E [z(K,), {U’b(L*)]
g=;
as
Y&l)
>Y(L*)V
L*
Lo]. Then, L* must satisfy the first-order condition
* CplF;(Ko,
L*) - y(L*)]
+ Ub(L*)]
- U(w)} = 0.
0
(7)
As L*
L*) >~oFi(Ko,
Lo).
Now, poFt (Ko, Lo) = w, so that PIF~(Ko,
-f)
> w.
(8)
Given (7) and (8) we obtain u’b(L*>]
* [w - y(L*)]
+ ub(L*)]
- V(w) < 0.
(9)
Moreover, Y(L*) >w,
(10) without which Lo would be better than L*, which is impossible. We now show that (9) and (10) are incompatible when U is concave. For that, put a = w and b = y(L*) and rewrite (9) as U’(b) - (a - b) + U(b) - U(a) < 0.
(11)
A. Steinherrand J.-F. Thisse/Are labor-managers reallyperverse?
140
As c = ((b - c)/(b - a))~ + (1 - (LJ- c)/(b - a)& and as U is concave, we obtain U(b) - U(a) > (b - a) -
U(b)- WC> b-c
(12)
’
Taking the limit for c + b in (12) yields U(b) - U(a) > (b - a) * U’(b),
(13)
which contradicts (11). Third, suppose that L* E [Lo, -1. As FL (KO, L) is decreasing, y(l) is also decreasing on [Lo, -1. Consequently, as L 1 V(L*), a contradiction. Fourth, and last, as L* exists, we must necessarily have L* = Lo and the proof is complete. Thus, taking into account the risk for workers to lose their jobs and assuming that workers are not risk-lovers implies that the level of employment remains unchanged when market conditions improve.
3. Group optimality Here we assume again an increase of the market price and ask the question: what would be the optimal employment when welfare of the group is the prime concern of workers? We consider two important schemes, suggested by welfare theory. The first one consists in maximization of the sum of the utilities of all initial workers, allowing employment to adjust optimally. This amounts to maximizing the following particular collective utility function: V(L) = W(w) * L() = Wqy(L)] * L + W(w) - (Lo -L) =
wY(L>l
for
L = 0,
for
O
for
LO
(14)
where W denotes the utility of income, assumed concave and identical for all workers. Clearly, objective function (14) is formally equivalent to (6), so that the argument of Theorem 1 applies. As a consequence, the optimal employment corresponds to the initial level of employment Lo. It must be emphasized, however, that the two objective functions are based on completely different behavioral and organizational patterns. The second form is suggested by the compensation principle: remaining workers have to compensate dismissed workers for their income losses. An obvious benchmark for the determination of a transfer would be the difference between the average income in the new state, if all workers had remained in the firm, and the alternative income w. Lower transfers would not compensate workers entirely and higher
141
A. Steinherr and J.-F. Thisse /Are labor-managers really perverse?
transfers would never be made as will become obvious below. The objective function can now be written as max V(L)
w.r.t.
L E [0, -1,
where V(L) = W b(Lo) - w]
- y
=y(L) =y(L)
for
L = 0,
for
O
for
LoGL,
(15)
and where the amount y(L,,) - w represents the transfer payable to each dismissed worker. Theorem 2.
Assume (15). The optimal level of employment is unique and given
by Lo. Boot Let L** be a maximizer of V(L). Using an argument similar to that of Theorem 1 shows that L** must belong to [L(Ke), L,]. Assume that L** E [z(KO), L,]. Then, L** must verify the first-order condition dV z=L**2-
plF;(Ko,
,**> - y(L**)
+
b(Lo)
- w]
= 0. I
As with (8), we obtain PrFt&,
,**> >w,
so that L**w - L**y(L**>
Denoting
+ Lob(Lo)
- w] < 0.
(17)
Y(L) = Ly(L) + (Lo - L) w, we may rewrite (17) as
Y(L()) - Y(L * *> < 0.
(18)
Consider now the maximizer of Y(L) on [0, Lo], say i. If i E [0, L,], we must have p1 FL @Co, i) = w. As pr FL (K,,, L) >poFL (Ko, L) for any L, we therefore obtain i > L,,, a contradiction. Consequently, i = Lo. Knowing that Y(L) is strictly concave on [L(KJ, L,] , it then follows that Y(L**) < Y(i) = Y(L,),
which contradicts
(18). This completes the proof.
Thus, when complete compensation has to be paid for income losses the LMF never reduces employment. Interesting enough, objective functions (14) and (1.5) are written without specification of any particular selection process which is left exogenous. For example,
142
A. Steinherr and J.-F. Thisse /Are
Iabor-managers really perverse?
our results apply, to a system of seniority (last-in-first-out). To be complete, let us briefly describe the case when the price of output decreases. Because of the reduction of prices, income per capita in all the firms of the industry falls below incomes elsewhere. Most probably, some workers of the LMF will decide to leave the sector with the obvious consequence that the quantity supplied by the firm will be reduced. To sum-up: labor-managers are not behaving perversely, as suggested by Ward (1958) and Vanek (1970), provided the objective function of the LMF is appropriately specified.
References Vanek, J., 1970, The general theory of labor-managed market economies (Cornell University Press, Ithaca, NY). Ward, B.N., 1958, The firm in Illyria: Market syndicalism, American Economic Review 48, 566-589.
ARE LABOR-MANAGERS A. STEINHERR
REALLY PERVERSE?
and J.-F. THISSE *
Universitk Catholique de Louvain, Louvain, Belgium Received
February
1979
One major problem of labor-managed firms seems to be their perverse response to price incentives. We show that this perversity is only due to an incomplete specification of the objective function of these firms.
1. Introduction The objective function for the labor-managed and Vanek (1970) among others, is defined by
firm (LMF) used by Ward (1958)
(1)
max y = @F(K, L> - rK)/L , where y p F K L r
= labor’s value-added (or income) per capita, = market price of output (exogenous), = production function, = stock of capital, = employment, = rental price of one unit of capital (exogenous).
In this letter we argue that this objective is inappropriate to deal with changes in the economic environment of the LMF. We suggest some alternatives that seem to be more consistent with both logic and the spirit of labor-management. The focus of the analysis is a competitive firm perturbed by an increase of the price of output ’ while dosing in general equilibrium, which implies that initially average incomes of workers are equal across all firms. For conciseness, we limit the analysis to the shortrun where the stock of capital is fixed.
* The authors thank P. Dehez, P. Kleindorfer, P. Pestieau and M. Sertel for discussions. 1 Obviously, our results are pertinent to a reduction of the rental price of capital or to different types of technological progress,(including Harrod or Hicks neutral processes). 137
A. Steinherr and J.-F. Thisse /Are labor-managers really perverse?
138
2. The risk of being dismissed In general, the workers who make the decision to reduce employment in the LMF risk to be among those who will be selected for departure. This risk should be integrated into the objective function of the LMF, which is precisely what we are going to do. Denoting the parameters and the variables in the initial state with subscript zero, we have (PoFKo,
LO)
-
roKoYLo
= w.
(2)
With p 1 > p. , it is known that the optimal employment Y(L) =
CPIJWO,
L)
-
for maximization
of (3)
rKo)/L
denoted by L 1, is smaller than the initial employment Lo. We now assume that for selecting workers a random process is used, where each worker has the same probability of being dismissed. Besides simplicity, this system has the further advantage of corresponding to a well-established, albeit particular, notion of justice. Consequently, if L denotes the number of workers remaining in the firm, the probability for a worker to leave the LMF is given by IT= (Lo - L)fLo,
(4)
while the probability
to remain is
1 -n=L/Lo.
(5)
Under these circumstances, the alternative revenue of dismissed workers must be taken into account. This alternative income is equal to income from employment in other industries or to unemployment benefits. (The short-run horizon of our analysis precludes creation of employment through creation of new firms.) Hence, this income may be assumed not to be larger than initial income w. In the following it is taken as equal to w. This is not restrictive, however, since the conclusions we are going to derive remain unaltered if the alternative income is below w. We can now rewrite the objective function of the LMF as follows: max V(L)
w.r.t.
L E [0, -1,
where
V(L) = U(w)> = UP(L)] =
* (1 - n) + U(w) * 77
NY(L)1
U denoting the utility function
for
L = 0,
for
O
for
LofL,
assumed identical for all workers.
(6)
A. Steinherr and J.-F. Thisse /Are
139
labor-managers really perverse?
Theorem 1. Assume (6). Ifworkers are not risk-lovers, then the optimal level of employment is unique and given by Lo. Proofi Given the standard assumption of continuity of y(L) and given that y(L) tends to zero when L becomes arbitrarily large, it is easily seen that at least one maximizer of V(L) exists; it is denoted by L * . Let L(Kc) be the value of L such that the marginal productivity of labor FL (KO, L) increases on [0,Z(Kc)] and decreases on @(Kc), -1; clearly L(Ke) < L 1 . First, assume that L* E [0, z(KO)]. As V(0) < V(L,,), L* must belong to [0, E(Kc)]. In this case, we would have V(L*)=uLy(L*)]-~+u(w).~
< U(r(L,)]
+ {ub(L*>]
* 2. + U(w) - y
L*-Ll - V(w)} * ~ LO
since Lo-L1
.g-yw,.--
< ULl),
a contradiction.
Lo
Second, take L* E [z(K,), {U’b(L*)]
g=;
as
Y&l)
>Y(L*)V
L*
Lo]. Then, L* must satisfy the first-order condition
* CplF;(Ko,
L*) - y(L*)]
+ Ub(L*)]
- U(w)} = 0.
0
(7)
As L*
L*) >~oFi(Ko,
Lo).
Now, poFt (Ko, Lo) = w, so that PIF~(Ko,
-f)
> w.
(8)
Given (7) and (8) we obtain u’b(L*>]
* [w - y(L*)]
+ ub(L*)]
- V(w) < 0.
(9)
Moreover, Y(L*) >w,
(10) without which Lo would be better than L*, which is impossible. We now show that (9) and (10) are incompatible when U is concave. For that, put a = w and b = y(L*) and rewrite (9) as U’(b) - (a - b) + U(b) - U(a) < 0.
(11)
A. Steinherrand J.-F. Thisse/Are labor-managers reallyperverse?
140
As c = ((b - c)/(b - a))~ + (1 - (LJ- c)/(b - a)& and as U is concave, we obtain U(b) - U(a) > (b - a) -
U(b)- WC> b-c
(12)
’
Taking the limit for c + b in (12) yields U(b) - U(a) > (b - a) * U’(b),
(13)
which contradicts (11). Third, suppose that L* E [Lo, -1. As FL (KO, L) is decreasing, y(l) is also decreasing on [Lo, -1. Consequently, as L 1
3. Group optimality Here we assume again an increase of the market price and ask the question: what would be the optimal employment when welfare of the group is the prime concern of workers? We consider two important schemes, suggested by welfare theory. The first one consists in maximization of the sum of the utilities of all initial workers, allowing employment to adjust optimally. This amounts to maximizing the following particular collective utility function: V(L) = W(w) * L() = Wqy(L)] * L + W(w) - (Lo -L) =
wY(L>l
for
L = 0,
for
O
for
LO
(14)
where W denotes the utility of income, assumed concave and identical for all workers. Clearly, objective function (14) is formally equivalent to (6), so that the argument of Theorem 1 applies. As a consequence, the optimal employment corresponds to the initial level of employment Lo. It must be emphasized, however, that the two objective functions are based on completely different behavioral and organizational patterns. The second form is suggested by the compensation principle: remaining workers have to compensate dismissed workers for their income losses. An obvious benchmark for the determination of a transfer would be the difference between the average income in the new state, if all workers had remained in the firm, and the alternative income w. Lower transfers would not compensate workers entirely and higher
141
A. Steinherr and J.-F. Thisse /Are labor-managers really perverse?
transfers would never be made as will become obvious below. The objective function can now be written as max V(L)
w.r.t.
L E [0, -1,
where V(L) = W b(Lo) - w]
- y
=y(L) =y(L)
for
L = 0,
for
O
for
LoGL,
(15)
and where the amount y(L,,) - w represents the transfer payable to each dismissed worker. Theorem 2.
Assume (15). The optimal level of employment is unique and given
by Lo. Boot Let L** be a maximizer of V(L). Using an argument similar to that of Theorem 1 shows that L** must belong to [L(Ke), L,]. Assume that L** E [z(KO), L,]. Then, L** must verify the first-order condition dV z=L**2-
plF;(Ko,
,**> - y(L**)
+
b(Lo)
- w]
= 0. I
As with (8), we obtain PrFt&,
,**> >w,
so that L**w - L**y(L**>
Denoting
+ Lob(Lo)
- w] < 0.
(17)
Y(L) = Ly(L) + (Lo - L) w, we may rewrite (17) as
Y(L()) - Y(L * *> < 0.
(18)
Consider now the maximizer of Y(L) on [0, Lo], say i. If i E [0, L,], we must have p1 FL @Co, i) = w. As pr FL (K,,, L) >poFL (Ko, L) for any L, we therefore obtain i > L,,, a contradiction. Consequently, i = Lo. Knowing that Y(L) is strictly concave on [L(KJ, L,] , it then follows that Y(L**) < Y(i) = Y(L,),
which contradicts
(18). This completes the proof.
Thus, when complete compensation has to be paid for income losses the LMF never reduces employment. Interesting enough, objective functions (14) and (1.5) are written without specification of any particular selection process which is left exogenous. For example,
142
A. Steinherr and J.-F. Thisse /Are
Iabor-managers really perverse?
our results apply, to a system of seniority (last-in-first-out). To be complete, let us briefly describe the case when the price of output decreases. Because of the reduction of prices, income per capita in all the firms of the industry falls below incomes elsewhere. Most probably, some workers of the LMF will decide to leave the sector with the obvious consequence that the quantity supplied by the firm will be reduced. To sum-up: labor-managers are not behaving perversely, as suggested by Ward (1958) and Vanek (1970), provided the objective function of the LMF is appropriately specified.
References Vanek, J., 1970, The general theory of labor-managed market economies (Cornell University Press, Ithaca, NY). Ward, B.N., 1958, The firm in Illyria: Market syndicalism, American Economic Review 48, 566-589.