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Profit sharing and employment
Profit sharing and employment John C. Eckalbar
Two disequilibrium macro models are constructed. Each has households and firms exchanging labour, money and storable output. One model has workers compensated by wages only, and the other has a profit sharing scheme. Contrary to earlier findings by Weitzman, both systems are found to be susceptible to a demand shock. Keywords: Profit sharing; Disequilibrium
In a recent book and series of articles [ 6 ] - [ 9 ] , Martin Weitzman has made some very strong claims for the macroeconomic benefits of profit sharing. '[C]onversion from a wage system to an equivalent-looking profit-sharing system yields unambiguously superior macroeconomic characteristics' [8]. 'The wage variant of capitalism is essentially a poorly designed system possessing some very undesirable tendencies toward inflation and unemployment. A profit sharing economy, on the other hand, has a natural inclination toward sustained, non-inflationary, market-oriented full employment' [8]. Because of these advantages, Weitzman feels the government should 'encourage, through exhortation and special tax privileges, the widespread use of profit sharing' 1"7]. In his Simple Macroeconomics of Profit Sharing, Weitzman conducts the following experiment. Two theoretical disequilibrium macro systems are built up. They are identical in every way except that one has a profit sharing compensation system and the other does not. Then, from an initial full employment equilibrium, a demand shock is introduced. In the conventional fixed wage economy, the short-run result is unemployment, while the profit sharing system stays at full employment after the disturbance. It is necessary to refer to Weitzman's papers themselves for a full explanation of the reasons for his results, but the following plays a key role. In a long-run equilibrium in a profit sharing economy the typical firm will, in a sense, have an excess demand for labour. The author is Professor of Economics, California State University, Chico, CA 95929, USA. Final manuscript received 5 March 1991. 104
That is, given the present values for the hourly base wage and profit sharing percentage, the firm would like to hire more labour than is available, though the firm will not want to alter the base wage or profit sharing percentage. Given this, any moderate shock to demand will still leave the firm with an excess demand for labour, and full employment will be maintained. If Weitzman is right, the importance of his finding could hardly be exaggerated. For all practical purposes, unemployment could be made a thing of the past. But how robust is his result ? The purpose of the present paper is to try that experiment again with a different disequilibrium system. The model used is very similar to one recently developed by the author to study inventory cycles (Eckalbar [2] ). The model has households and firms exchanging labour, money and storable output. Firms try to maintain inventories at a fixed ratio to expected sales, Thus, firms will always try to hire enough labour to produce the expected sales volume plus or minus any necessary inventory correction. Households offer to sell a fixed quantity of labour each period, and determine their output demand (ie sales) by maximizing utility subject to labour sales and budget constraints. Wages and prices are fixed (Eckalbar [3]). Following Weitzman, we create twin conventional wage and profit sharing versions of the model, start from a full employment equilibrium, introduce a downward demand shock, and carry out a comparative check of the short-run dynamics of each system. The result is that the profit sharing system is just as susceptible to unemployment as the conventional wage system. We defer further discussion of the result until the conclusion of this paper.
0264-9993/92/020104-07 © 1992 Butterworth-Heinemann Ltd
Profit sharino and employment: J.C. Eckalbar
output actually sold. Hence, we write
The general model The model has three goods - labour, money and output. Workers offer to supply a fixed quantity of labour, L, and they succeed in selling L = rain (L d, L ), where L a is labour demand and is discussed shortly. Workers use some of the proceeds of their labour sale and profit income to finance their purchase of goods according to Equation ( 1 ): $1 = f +
b_~WL+ p • cnt
(1)
where f, b and c are positive constants, the latter two being fractions, w is the nominal wage, p is the price level, nt is the workers' share of real profits, and St is sales to workers. Equation (1) can be derived from an explicit utility maximization problem, where, for instance, a Cobb-Douglas utility function U (M/P, St ) is maximized subject to f f l / p + w L / p + n t = M / p + S t , where ~ and M are initial and desired ending money balances respectively. Owners receive only profit income, n2, and spend it on goods as shown in Equation (2): S2 = e + cn 2
n=s---
wS
(5)
pd
Notice from Equations (3) and (5) that there is a chicken and egg problem with sales and profits. Workers and owners need to know or estimate profits before they can become buyers, and firms need to know sales before they can compute profits. To avoid any sequencing problems and/or difficulties with lagged variables, we will assume that workers and owners make perfect forecasts of profits. Thus, the value for n used in (3) to give S will be the value which solves (5) for that level of S. Substituting (5) into (3), we get S=
apd + bdwL
(6)
pd(1 - c) + cw
We assume that production is profitable in the sense that d > w / p , and that firms hire sufficient labour, if possible, to produce enough output to maintain a fixed target inventory-sales ratio. That is, firms want to maintain Equation (7):
(2) (7)
V = kS ~
Again, Equation (2) follows from maximization of a Cobb-Douglas utility function. Notice that workers and owners have the same marginal propensity to spend out of profits. The obvious intention of this assumption is to prevent the distribution of profits from having an effect on sales, as long as L is constant. The whole point of this exercise is to see if changing the compensation scheme can change output via a change in labour demand. Having distributional effects on goods demand would only complicate the comparison. Equations (1) and (2) are aggregated to give the economy's sales equation: bw
S=a+--L+cn
(3)
where V is the level of inventories, k is a positive constant, and S" is the firm's point sales expectation. These considerations give rise to the labour demand equation given in (8): L d = (l/d)[(1 + k ) S ~ -
V]
(8)
Equation (8) has firms trying to hire enough labour to provide for S e plus or minus any necessary inventory correction, k S ~ - V. But since firms cannot possibly hire more than L workers, they cannot produce more than Q = dL. Thus actual ouptut is given by Equation (9): Q = min[(1 + k ) S e - V, (~]
(9)
P Inventories change according to Equation (10): where n = nl + n2 and a = f + e. We assume that technology exhibits constant returns, so Q = dL
(4)
where Q is real output and d is a positive constant. Profits are taken to be the difference between sales and the current real cost of producing the quantity of ECONOMIC MODELLING April 1992
f'=Q-S
(10)
(see Eckalbar [ 1] and Honkapohja and Ito [5]). As an approximation to optimal least squares learning we assume that S e is adjusted adaptively according to Equation ( 11 ):
$" = s - s "
(11)
105
Profit sharing and employment: J.C. Eckalbar
Equations (1) to (11 ) define a dynamic system in Q, S ~, S, L, n and V. Substituting into (10) and (11) we get the two-dimensional system (12) and ( 13 ) : ~
apd
=
bw
+
pd(1 - c) + cw
pd(1 - c) + cw
x mini(1 + k ) S ~ - V, (~] - S ~ ['={1
bw
-pd(1-c)+cw
(12)
}min[(l+k)S~_V,(2]
(13)
apd pd(1 - c) + cw
apd
Obviously if a = fi -- Q.pd(1 - c) + cw - bw/pd, the system would have a full employment equilibrium. Hence, our method will be to set each system up with a = a and then shock it by reducing a to a' < a. We begin with the case of fixed positive wages,
F i x e d positive w a g e s The disequilibrium system (12) and (13) is, like most disequilibrium systems, subject to regime switching ( s e e [ l ] and [5]). Theline A = {(V, S~)I(1 + k ) M V = (~} divides the phase space into two areas or regions. In AI -- {(V,S~)I(1 + k ) S ~ - V < Q} the operative dynamic system is ~ = apd + bw[(1 + K ) S ~ - V] _ S~ pd(1 - c) + cw
apd pd(1 - c) + cw
106
(14)
bw } pd(1 -f-c) + cw
(is)
A1
o
Figure 1. Phase space partition.
while in A 2 - {(V,S*)I(1 + k ) S ~ - V > (5}, the appropriate system is ~ =
pd (1 - c) + cw - bw
f'=[(l+k)S*-V]{1-
0
l +-fT-g
We will investigate this system under two alternative sets of assumptions. We first suppose that w is fixed at some positive value, and we then compare our results with an alternative system with zero wages and all worker income in the form of profits. Our mode of proceeding will be to set each system up at a full employment equilibrium and then check the system's behaviour following a cut in demand. In equilibrium S ~ = S = Q and V = k S e = k Q . Substituting these values into (12) and (13 ) and using the fact that ge = 0 in equilibrium, we get the following equation for equilibrium output:
Q=
Se
apd + bwQ - S" pd(1 - c) + cw
.;¢ = Q. _
apd + bwQ pd(1 - c) + cw
(16)
(17)
This structure complicates the dynamic analysis considerably, but the problem is quite tractable if we patiently patch the systems together graphically. Referring to Figure 1 we see that the switching line A partitions the phase space as shown. Let S~o - {(V, Se)l (V, S ' ) ~ A 1, S" = 0}. Checking (14) we see that the locus S~o is given by Equation (18): Se =
apd - b w V pd(1 - c) + c w -
(18)
bw(1 + k)
The sign of the denominator of the righthand side of (18) is indeterminant. For the sake of concreteness we will assume that it is positive and leave it to the reader to check the alternate case. Given this, a graph of S~o is shown in Figure 2. The flow of S~o on either side of S~o is also shown in Figure 2. The direction of the flow is derived in the usual way from the equation for ~e in system (14) and (15). Define ['to -= {(V, S e ) I ( V , S * ) ~ A 1 , [ ' = 0}. ['1o is ECONOMIC M O D E L L I N G April 1992
Profit sharino and employment: J.C. Eckalbar
5e
Se
I o
V
~V
0
Figure 3. Flow around E in A~.
Figure 2. Movement of S* in At.
5e
given by Equation (19) : S~-
V
l+k
-1
apd
(1 + k ) [ p d ( 1 - c) + c w - bw] (19)
The locus f'l 0 has the same slope as the switching line A, and the intercept for I;'1o will be at or below that of A according to whether the system has a full or less than full employment equilibrium. (More on the latter point shortly.) Assuming for the moment that f'to lies below A, the flow of V in A t on either side of l;'to is as shown in Figure 3. In region A 2, system (12) and (13) becomes (16) and (17). The locus ~ o - {(V, Se)[(V, se)EA2, ~e = 0 } is given by Se =
qpd + bw(2 pd(1 - c) + cw
(20)
It should be obvious from (16) and (17) that the flow of S" around S[o. is as shown in Figure 4. There is no V2o locus. If the system has a full employment equilibrium, then 1;'---0 everywhere in A2; and if the equilibrium is at less than full employment, f" > 0 everywhere in A2. If we patch systems (14) and (15), (16) and (17) together on the assumption that a = ~, we end up with the flow shown in Figure 5. The system does not have ECONOMIC M O D E L L I N G April 1992
"e
520
V 0
Figure 4. Flow in
A 2.
a unique equilibrium, but rather an equilibrium locus consisting of the points on S~o (ie the segment J E ) . Anywhere on J E the system is producing and selling the full employment output. Firms would like to build up inventories, but they cannot as long as they sell everything they produce. We might regard this as a repressed inflation regime, or alternatively, imagine that temporary rationing of good sales would lead to E. 107
Profit sharin# and employment: J.C. Eckalbar Se
family of isoquantity lines parallel to A with output quantity proportional to the intercept value. To summarize : in a fixed price positive wage system, a reduction in parameter a from its full employment value leads to unemployment• This result will serve as a benchmark against which we will compare the pure profit sharing case.
Pure profit sharing To give Weitzman's hypothesis the best chance of success, we assume that w = 0 and that all employee compensation is in the form of profits. With this assumption, (12) and ( 13 ) reduce to ~ _ J,
Figure
a
S~
1-c
(22)
V
a
12= mini(1 + k ) S ~ - V, ~ ] - - 1--c
5. Flow around full employment equilibrium.
(23)
Again the phase space is partitioned by the same switching locus A into two regions AI and A2, as shown in Figure I. In A t the operative system is
5e
Vl 0
"C
,.
~, _
$20
a
Se
(24)
1-c
k.. 12=(1+k)S and in
_t
A2
~, _
~-
V -
a
1-c
_
(25)
it is
a
Se
(26)
1-c 12=(~_ a
= V Figure 6. Flow around unemployment equilibrium without profit sharing.
Now if parameter a were suddenly to drop to a' < fi, the patched up system would then be as shown in Figure 6. The equilibrium is now at E' in AI, with Q < (~. We know that equilibrium output is at less than full employment, since
The full employment value for parameter a is now ~(1 - c). If a = fi = (~(1 - c), it is easy to see that the phase portrait is now as shown in Figure 7. The loci S~o and S~o are both given by Se = Q
Se -
+
V l+k
(29)
(21)
everywhere in A 1 u A, and Equation (21) defines a 108
(28)
while I21o is given by
l+k Q=(1 +k)S ~- V
(27)
1--c
"e
510
which is the same as A. There is no 122o locus• As before, we have an equilibrium locus running E C O N O M I C M O D E L L I N G April 1992
Profit sharing and employment: J.C. Eckalbar Se A
A2
A=V10
~o
30
"e
d
520
$20
~
~
A1
-V 0
Figure 7. Flow when a ~- Q (1 - c).
8. Flow around unemployment equilibrium with profit sharing. Figure
from J to E, though in the present case E is a more probable rest point in that it resides at the terminus of a sliding line. For concreteness we again suppose that temporary rationing leads to a full employment equilibrium at E. We now shock the system by reducing a to a" < ~. The new equilibrium, E 1, is shown together with the old full employment equilibrium in Figure 8. The drop in a lowers the S]0, S[o and 121o loci. Again there is no I22o locus, and since a" < Q(1 - c), we now have 12> 0 everywhere in A2. Again the isoquantity loci are parallel to A and 12to, with output diminishing in proportion to the loci intercepts, so employment and output at E' is clearly less than at E. Thus the profit sharing compensation system does not insulate the economy from the adverse employment effects of demand shocks - even if the shock is very small. There is a difference between the values of the dQ/da multipliers for the wage system versus the pure profit sharing system. Solving both systems for their unemployment equilibrium output levels, we get
QE
da'
pd(1 - c) + cw - bw
E C O N O M I C M O D E L L I N G April 1992
a" 1 - c
and
dQE da"
-
1
(31)
1 -c
for the pure profit sharing system. Routine calculations show that the critical term deciding which multiplier is larger is (c - b). If the propensity to spend out of earned income, b, is less than the propensity to spend out of profit income, c, then the multiplier in the wage economy is greater than that for the profit sharing economy. Thus, the latter would be quantitatively less susceptible to demand disturbances. There is another difference between the economies. The unemployment equilibrium in the profit sharing economy (E' in Figure 8) is asymptotically stable, while it is at least possible that the equilibrium in the wage economy (E' in Figure 6) is unstable and the system has a stable co-limit cycle around E'.t
No intuitive, verbal explanation can adequately accoun t for the properties of a multiequation dynamic system, but we should at least try to discuss the reasons
pd (1 - c ) + cw . bw
pd
Q~ -
Why the difference?
a'pd
and dQE _
for the wage system, and
(30)
1Since the proof of this point is quite involved, and since a similar proof is given in Eckalbar 1-2], I will refer the reader there for guidance.
109
Profit sharin9 and employment: J.C. Eckalbar for the difference between Weitzman's result and ours. Recall that in Weitzman's model there is, at least in a sense, a residual excess demand for labour at the full employment equilibrium. Thus, a small demand shock will still leave the system with an excess demand for labour, and, therefore, with full employment. That is not the case in our profit sharing model. In our model, labour demand adjusts itself to sales expectations and inventory levels. In equilibrium, the firm sells what it expects to sell and ends up with inventories at the desired ratio to sales. A shock to sales in this environment will adversely affect labour demand for two reasons: first, when sales drop unexpectedly, inventories build up, and this leads firms to reduce labour demand in an effort to pull inventories back into line. It is worth pointing out that Weitzman's models do not have inventories. Second, with sales reduced, there is simply less need for production and employment. One can think of other probable scenarios where employment would drop after a shock, even if there were profit sharing. For instance, if labour were used in fixed proportion to some other input, then a sudden cut in the supply of that input would reduce labour demand. In general, a disequilibrium in the labour market could be due to a variety of factors other than too high a wage. It might be that labour and capital are complementary factors, and that capital is scarce; or that inventories are too high; or that medium of exchange constraints are binding on firms; or that sales expectations are low. Hence a programme like profit sharing, which may be a very effective tool for dealing with wage related disequilibria, may be useless in dealing with disequilibria arising from another source.
Conclusion We have not proven, or did we set out to prove, that
110
Weitzman is wrong about the benefits of profit sharing. Our analysis shows that profit sharing is unlikely to make macro performance any worse, and is quite likely to improve matters/f the disequilibrium is generated by too high a wage. That is, profit sharing is not a panacea : it does no good for non-wage related disequilibria. Weitzman has opened the door to what could and should become an important and exciting research area. Compensation systems very probably do have important macroeconomic side effects, and it is about time we looked into them.
References 1 J. Eckalbar, 'The stability of non-Walrasian processes : two examples ', Econometrica, Vo148, 1980, pp 371-386. 2 J. Eckalbar, 'Inventory fluctuations in a disequilibrium macro model', Economic Journal, Vol 95, 1985, pp 976-991. 3 J. Eckalbar, 'Inventories in a dynamic macro model with flexible prices', European Economic Review,Vo127, 1985, pp 201-219. 4 B. Friedman, 'Optimal expectations and the extreme information assumption of "rational expectations" macro models', Journal of Monetary Economics, Vol 5, 1979, pp 23-41. 5 S. Honkapohja and T. Ito, 'Stablity with regime switching', Journal of Economic Theory, Vol 29, 1983, pp 22-48. 6 M. Weitzman, 'Some macroeconomic implications of alternative compensation systems', Economic Journal, Vol 93, 1983, pp 763-783. 7 M. Weitzman, The Share Economy, Harvard University Press, New York, 1984. 8 M. Weitzman, 'Profit sharing as macroeconomic policy', paper delivered at Dallas AEA meeting September 1984. 9 M. Weitzman, 'The simple macroeconomics of profit sharing', MIT working paper, Cambridge, MA, 1984.
E C O N O M I C M O D E L L I N G April 1992
Two disequilibrium macro models are constructed. Each has households and firms exchanging labour, money and storable output. One model has workers compensated by wages only, and the other has a profit sharing scheme. Contrary to earlier findings by Weitzman, both systems are found to be susceptible to a demand shock. Keywords: Profit sharing; Disequilibrium
In a recent book and series of articles [ 6 ] - [ 9 ] , Martin Weitzman has made some very strong claims for the macroeconomic benefits of profit sharing. '[C]onversion from a wage system to an equivalent-looking profit-sharing system yields unambiguously superior macroeconomic characteristics' [8]. 'The wage variant of capitalism is essentially a poorly designed system possessing some very undesirable tendencies toward inflation and unemployment. A profit sharing economy, on the other hand, has a natural inclination toward sustained, non-inflationary, market-oriented full employment' [8]. Because of these advantages, Weitzman feels the government should 'encourage, through exhortation and special tax privileges, the widespread use of profit sharing' 1"7]. In his Simple Macroeconomics of Profit Sharing, Weitzman conducts the following experiment. Two theoretical disequilibrium macro systems are built up. They are identical in every way except that one has a profit sharing compensation system and the other does not. Then, from an initial full employment equilibrium, a demand shock is introduced. In the conventional fixed wage economy, the short-run result is unemployment, while the profit sharing system stays at full employment after the disturbance. It is necessary to refer to Weitzman's papers themselves for a full explanation of the reasons for his results, but the following plays a key role. In a long-run equilibrium in a profit sharing economy the typical firm will, in a sense, have an excess demand for labour. The author is Professor of Economics, California State University, Chico, CA 95929, USA. Final manuscript received 5 March 1991. 104
That is, given the present values for the hourly base wage and profit sharing percentage, the firm would like to hire more labour than is available, though the firm will not want to alter the base wage or profit sharing percentage. Given this, any moderate shock to demand will still leave the firm with an excess demand for labour, and full employment will be maintained. If Weitzman is right, the importance of his finding could hardly be exaggerated. For all practical purposes, unemployment could be made a thing of the past. But how robust is his result ? The purpose of the present paper is to try that experiment again with a different disequilibrium system. The model used is very similar to one recently developed by the author to study inventory cycles (Eckalbar [2] ). The model has households and firms exchanging labour, money and storable output. Firms try to maintain inventories at a fixed ratio to expected sales, Thus, firms will always try to hire enough labour to produce the expected sales volume plus or minus any necessary inventory correction. Households offer to sell a fixed quantity of labour each period, and determine their output demand (ie sales) by maximizing utility subject to labour sales and budget constraints. Wages and prices are fixed (Eckalbar [3]). Following Weitzman, we create twin conventional wage and profit sharing versions of the model, start from a full employment equilibrium, introduce a downward demand shock, and carry out a comparative check of the short-run dynamics of each system. The result is that the profit sharing system is just as susceptible to unemployment as the conventional wage system. We defer further discussion of the result until the conclusion of this paper.
0264-9993/92/020104-07 © 1992 Butterworth-Heinemann Ltd
Profit sharino and employment: J.C. Eckalbar
output actually sold. Hence, we write
The general model The model has three goods - labour, money and output. Workers offer to supply a fixed quantity of labour, L, and they succeed in selling L = rain (L d, L ), where L a is labour demand and is discussed shortly. Workers use some of the proceeds of their labour sale and profit income to finance their purchase of goods according to Equation ( 1 ): $1 = f +
b_~WL+ p • cnt
(1)
where f, b and c are positive constants, the latter two being fractions, w is the nominal wage, p is the price level, nt is the workers' share of real profits, and St is sales to workers. Equation (1) can be derived from an explicit utility maximization problem, where, for instance, a Cobb-Douglas utility function U (M/P, St ) is maximized subject to f f l / p + w L / p + n t = M / p + S t , where ~ and M are initial and desired ending money balances respectively. Owners receive only profit income, n2, and spend it on goods as shown in Equation (2): S2 = e + cn 2
n=s---
wS
(5)
pd
Notice from Equations (3) and (5) that there is a chicken and egg problem with sales and profits. Workers and owners need to know or estimate profits before they can become buyers, and firms need to know sales before they can compute profits. To avoid any sequencing problems and/or difficulties with lagged variables, we will assume that workers and owners make perfect forecasts of profits. Thus, the value for n used in (3) to give S will be the value which solves (5) for that level of S. Substituting (5) into (3), we get S=
apd + bdwL
(6)
pd(1 - c) + cw
We assume that production is profitable in the sense that d > w / p , and that firms hire sufficient labour, if possible, to produce enough output to maintain a fixed target inventory-sales ratio. That is, firms want to maintain Equation (7):
(2) (7)
V = kS ~
Again, Equation (2) follows from maximization of a Cobb-Douglas utility function. Notice that workers and owners have the same marginal propensity to spend out of profits. The obvious intention of this assumption is to prevent the distribution of profits from having an effect on sales, as long as L is constant. The whole point of this exercise is to see if changing the compensation scheme can change output via a change in labour demand. Having distributional effects on goods demand would only complicate the comparison. Equations (1) and (2) are aggregated to give the economy's sales equation: bw
S=a+--L+cn
(3)
where V is the level of inventories, k is a positive constant, and S" is the firm's point sales expectation. These considerations give rise to the labour demand equation given in (8): L d = (l/d)[(1 + k ) S ~ -
V]
(8)
Equation (8) has firms trying to hire enough labour to provide for S e plus or minus any necessary inventory correction, k S ~ - V. But since firms cannot possibly hire more than L workers, they cannot produce more than Q = dL. Thus actual ouptut is given by Equation (9): Q = min[(1 + k ) S e - V, (~]
(9)
P Inventories change according to Equation (10): where n = nl + n2 and a = f + e. We assume that technology exhibits constant returns, so Q = dL
(4)
where Q is real output and d is a positive constant. Profits are taken to be the difference between sales and the current real cost of producing the quantity of ECONOMIC MODELLING April 1992
f'=Q-S
(10)
(see Eckalbar [ 1] and Honkapohja and Ito [5]). As an approximation to optimal least squares learning we assume that S e is adjusted adaptively according to Equation ( 11 ):
$" = s - s "
(11)
105
Profit sharing and employment: J.C. Eckalbar
Equations (1) to (11 ) define a dynamic system in Q, S ~, S, L, n and V. Substituting into (10) and (11) we get the two-dimensional system (12) and ( 13 ) : ~
apd
=
bw
+
pd(1 - c) + cw
pd(1 - c) + cw
x mini(1 + k ) S ~ - V, (~] - S ~ ['={1
bw
-pd(1-c)+cw
(12)
}min[(l+k)S~_V,(2]
(13)
apd pd(1 - c) + cw
apd
Obviously if a = fi -- Q.pd(1 - c) + cw - bw/pd, the system would have a full employment equilibrium. Hence, our method will be to set each system up with a = a and then shock it by reducing a to a' < a. We begin with the case of fixed positive wages,
F i x e d positive w a g e s The disequilibrium system (12) and (13) is, like most disequilibrium systems, subject to regime switching ( s e e [ l ] and [5]). Theline A = {(V, S~)I(1 + k ) M V = (~} divides the phase space into two areas or regions. In AI -- {(V,S~)I(1 + k ) S ~ - V < Q} the operative dynamic system is ~ = apd + bw[(1 + K ) S ~ - V] _ S~ pd(1 - c) + cw
apd pd(1 - c) + cw
106
(14)
bw } pd(1 -f-c) + cw
(is)
A1
o
Figure 1. Phase space partition.
while in A 2 - {(V,S*)I(1 + k ) S ~ - V > (5}, the appropriate system is ~ =
pd (1 - c) + cw - bw
f'=[(l+k)S*-V]{1-
0
l +-fT-g
We will investigate this system under two alternative sets of assumptions. We first suppose that w is fixed at some positive value, and we then compare our results with an alternative system with zero wages and all worker income in the form of profits. Our mode of proceeding will be to set each system up at a full employment equilibrium and then check the system's behaviour following a cut in demand. In equilibrium S ~ = S = Q and V = k S e = k Q . Substituting these values into (12) and (13 ) and using the fact that ge = 0 in equilibrium, we get the following equation for equilibrium output:
Q=
Se
apd + bwQ - S" pd(1 - c) + cw
.;¢ = Q. _
apd + bwQ pd(1 - c) + cw
(16)
(17)
This structure complicates the dynamic analysis considerably, but the problem is quite tractable if we patiently patch the systems together graphically. Referring to Figure 1 we see that the switching line A partitions the phase space as shown. Let S~o - {(V, Se)l (V, S ' ) ~ A 1, S" = 0}. Checking (14) we see that the locus S~o is given by Equation (18): Se =
apd - b w V pd(1 - c) + c w -
(18)
bw(1 + k)
The sign of the denominator of the righthand side of (18) is indeterminant. For the sake of concreteness we will assume that it is positive and leave it to the reader to check the alternate case. Given this, a graph of S~o is shown in Figure 2. The flow of S~o on either side of S~o is also shown in Figure 2. The direction of the flow is derived in the usual way from the equation for ~e in system (14) and (15). Define ['to -= {(V, S e ) I ( V , S * ) ~ A 1 , [ ' = 0}. ['1o is ECONOMIC M O D E L L I N G April 1992
Profit sharino and employment: J.C. Eckalbar
5e
Se
I o
V
~V
0
Figure 3. Flow around E in A~.
Figure 2. Movement of S* in At.
5e
given by Equation (19) : S~-
V
l+k
-1
apd
(1 + k ) [ p d ( 1 - c) + c w - bw] (19)
The locus f'l 0 has the same slope as the switching line A, and the intercept for I;'1o will be at or below that of A according to whether the system has a full or less than full employment equilibrium. (More on the latter point shortly.) Assuming for the moment that f'to lies below A, the flow of V in A t on either side of l;'to is as shown in Figure 3. In region A 2, system (12) and (13) becomes (16) and (17). The locus ~ o - {(V, Se)[(V, se)EA2, ~e = 0 } is given by Se =
qpd + bw(2 pd(1 - c) + cw
(20)
It should be obvious from (16) and (17) that the flow of S" around S[o. is as shown in Figure 4. There is no V2o locus. If the system has a full employment equilibrium, then 1;'---0 everywhere in A2; and if the equilibrium is at less than full employment, f" > 0 everywhere in A2. If we patch systems (14) and (15), (16) and (17) together on the assumption that a = ~, we end up with the flow shown in Figure 5. The system does not have ECONOMIC M O D E L L I N G April 1992
"e
520
V 0
Figure 4. Flow in
A 2.
a unique equilibrium, but rather an equilibrium locus consisting of the points on S~o (ie the segment J E ) . Anywhere on J E the system is producing and selling the full employment output. Firms would like to build up inventories, but they cannot as long as they sell everything they produce. We might regard this as a repressed inflation regime, or alternatively, imagine that temporary rationing of good sales would lead to E. 107
Profit sharin# and employment: J.C. Eckalbar Se
family of isoquantity lines parallel to A with output quantity proportional to the intercept value. To summarize : in a fixed price positive wage system, a reduction in parameter a from its full employment value leads to unemployment• This result will serve as a benchmark against which we will compare the pure profit sharing case.
Pure profit sharing To give Weitzman's hypothesis the best chance of success, we assume that w = 0 and that all employee compensation is in the form of profits. With this assumption, (12) and ( 13 ) reduce to ~ _ J,
Figure
a
S~
1-c
(22)
V
a
12= mini(1 + k ) S ~ - V, ~ ] - - 1--c
5. Flow around full employment equilibrium.
(23)
Again the phase space is partitioned by the same switching locus A into two regions AI and A2, as shown in Figure I. In A t the operative system is
5e
Vl 0
"C
,.
~, _
$20
a
Se
(24)
1-c
k.. 12=(1+k)S and in
_t
A2
~, _
~-
V -
a
1-c
_
(25)
it is
a
Se
(26)
1-c 12=(~_ a
= V Figure 6. Flow around unemployment equilibrium without profit sharing.
Now if parameter a were suddenly to drop to a' < fi, the patched up system would then be as shown in Figure 6. The equilibrium is now at E' in AI, with Q < (~. We know that equilibrium output is at less than full employment, since
The full employment value for parameter a is now ~(1 - c). If a = fi = (~(1 - c), it is easy to see that the phase portrait is now as shown in Figure 7. The loci S~o and S~o are both given by Se = Q
Se -
+
V l+k
(29)
(21)
everywhere in A 1 u A, and Equation (21) defines a 108
(28)
while I21o is given by
l+k Q=(1 +k)S ~- V
(27)
1--c
"e
510
which is the same as A. There is no 122o locus• As before, we have an equilibrium locus running E C O N O M I C M O D E L L I N G April 1992
Profit sharing and employment: J.C. Eckalbar Se A
A2
A=V10
~o
30
"e
d
520
$20
~
~
A1
-V 0
Figure 7. Flow when a ~- Q (1 - c).
8. Flow around unemployment equilibrium with profit sharing. Figure
from J to E, though in the present case E is a more probable rest point in that it resides at the terminus of a sliding line. For concreteness we again suppose that temporary rationing leads to a full employment equilibrium at E. We now shock the system by reducing a to a" < ~. The new equilibrium, E 1, is shown together with the old full employment equilibrium in Figure 8. The drop in a lowers the S]0, S[o and 121o loci. Again there is no I22o locus, and since a" < Q(1 - c), we now have 12> 0 everywhere in A2. Again the isoquantity loci are parallel to A and 12to, with output diminishing in proportion to the loci intercepts, so employment and output at E' is clearly less than at E. Thus the profit sharing compensation system does not insulate the economy from the adverse employment effects of demand shocks - even if the shock is very small. There is a difference between the values of the dQ/da multipliers for the wage system versus the pure profit sharing system. Solving both systems for their unemployment equilibrium output levels, we get
QE
da'
pd(1 - c) + cw - bw
E C O N O M I C M O D E L L I N G April 1992
a" 1 - c
and
dQE da"
-
1
(31)
1 -c
for the pure profit sharing system. Routine calculations show that the critical term deciding which multiplier is larger is (c - b). If the propensity to spend out of earned income, b, is less than the propensity to spend out of profit income, c, then the multiplier in the wage economy is greater than that for the profit sharing economy. Thus, the latter would be quantitatively less susceptible to demand disturbances. There is another difference between the economies. The unemployment equilibrium in the profit sharing economy (E' in Figure 8) is asymptotically stable, while it is at least possible that the equilibrium in the wage economy (E' in Figure 6) is unstable and the system has a stable co-limit cycle around E'.t
No intuitive, verbal explanation can adequately accoun t for the properties of a multiequation dynamic system, but we should at least try to discuss the reasons
pd (1 - c ) + cw . bw
pd
Q~ -
Why the difference?
a'pd
and dQE _
for the wage system, and
(30)
1Since the proof of this point is quite involved, and since a similar proof is given in Eckalbar 1-2], I will refer the reader there for guidance.
109
Profit sharin9 and employment: J.C. Eckalbar for the difference between Weitzman's result and ours. Recall that in Weitzman's model there is, at least in a sense, a residual excess demand for labour at the full employment equilibrium. Thus, a small demand shock will still leave the system with an excess demand for labour, and, therefore, with full employment. That is not the case in our profit sharing model. In our model, labour demand adjusts itself to sales expectations and inventory levels. In equilibrium, the firm sells what it expects to sell and ends up with inventories at the desired ratio to sales. A shock to sales in this environment will adversely affect labour demand for two reasons: first, when sales drop unexpectedly, inventories build up, and this leads firms to reduce labour demand in an effort to pull inventories back into line. It is worth pointing out that Weitzman's models do not have inventories. Second, with sales reduced, there is simply less need for production and employment. One can think of other probable scenarios where employment would drop after a shock, even if there were profit sharing. For instance, if labour were used in fixed proportion to some other input, then a sudden cut in the supply of that input would reduce labour demand. In general, a disequilibrium in the labour market could be due to a variety of factors other than too high a wage. It might be that labour and capital are complementary factors, and that capital is scarce; or that inventories are too high; or that medium of exchange constraints are binding on firms; or that sales expectations are low. Hence a programme like profit sharing, which may be a very effective tool for dealing with wage related disequilibria, may be useless in dealing with disequilibria arising from another source.
Conclusion We have not proven, or did we set out to prove, that
110
Weitzman is wrong about the benefits of profit sharing. Our analysis shows that profit sharing is unlikely to make macro performance any worse, and is quite likely to improve matters/f the disequilibrium is generated by too high a wage. That is, profit sharing is not a panacea : it does no good for non-wage related disequilibria. Weitzman has opened the door to what could and should become an important and exciting research area. Compensation systems very probably do have important macroeconomic side effects, and it is about time we looked into them.
References 1 J. Eckalbar, 'The stability of non-Walrasian processes : two examples ', Econometrica, Vo148, 1980, pp 371-386. 2 J. Eckalbar, 'Inventory fluctuations in a disequilibrium macro model', Economic Journal, Vol 95, 1985, pp 976-991. 3 J. Eckalbar, 'Inventories in a dynamic macro model with flexible prices', European Economic Review,Vo127, 1985, pp 201-219. 4 B. Friedman, 'Optimal expectations and the extreme information assumption of "rational expectations" macro models', Journal of Monetary Economics, Vol 5, 1979, pp 23-41. 5 S. Honkapohja and T. Ito, 'Stablity with regime switching', Journal of Economic Theory, Vol 29, 1983, pp 22-48. 6 M. Weitzman, 'Some macroeconomic implications of alternative compensation systems', Economic Journal, Vol 93, 1983, pp 763-783. 7 M. Weitzman, The Share Economy, Harvard University Press, New York, 1984. 8 M. Weitzman, 'Profit sharing as macroeconomic policy', paper delivered at Dallas AEA meeting September 1984. 9 M. Weitzman, 'The simple macroeconomics of profit sharing', MIT working paper, Cambridge, MA, 1984.
E C O N O M I C M O D E L L I N G April 1992