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Workers' investment funds and the dynamic inefficiency of capitalism
Journal
of Public
Economics
20 (1983) 271-279.
WORKERS’ INVESTMENT INEFFICIENCY Matti
North-Holland
March
Company
FUNDS AND THE DYNAMIC OF CAPITALISM POHJOLA*
University of Helsinki, SF-00100 Received
Publishing
Helsinki 10, Finland
1982, revised version
received
July 1982
This paper examines the effects of the partial transfer of control over the investment decision from capitalists to workers, which is an essential feature of plans to establish workers’ investment funds, in a two-class model of capitalism formulated as a differential game. It is shown that the institutional change which provides the workers with means to invest their savings directly, instead of having to use the capitalists as agents for accumulation, improves economic efliciency. As a consequence, the economy saves, invests, grows and consumes more than the pure capitalist society, which is characterized by the Keynesian separation of saving and investment decisions.
1. Introduction The past 15 years have seen a considerable growth of interest in Western Europe in workers’ investment funds. As for example, Atkinson (1974, ch. 11) points out in his survey, the proposals which have been put forward to establish such funds share the common objective of a more equal distribution of both wealth and control in capitalist society, and their origin is in the apparent failure of the trade union movement to achieve this objective through collective wage bargaining. In this paper we shall dwell neither on the details of these proposals nor on the results of previous studies [see, for example, Atkinson (1972) Brems (1975a, 1975b), Maital (1978) Ndslund and Sellstedt (1978, ch. S)]. Instead, we shall concentrate on the particular function of investment funds in providing workers with the means to control the use of their savings, i.e. the accumulation of their wealth. The economic effects of this transfer of control from capitalists to workers will be examined in the two-person differential game formulation of capitalism developed by Lancaster (1973). We give three reasons for this choice of the object of our examination. First, one of the principal arguments raised against workers’ investment funds, especially against profit-sharing schemes, is that they would discourage investment and lead to a capital flight [see, for example, Atkinson (1974, ch. 1 l)]. *Financial support from gratefully acknowledged. 0047-2727/83/$03.00
the
Kymenlaakso
Fund
of the
Finnish
Cultural
Foundation
is
212
M. Pohjola,
Workers’ investmentfunds
It might be of interest to find out in the differential game formulation of capitalism whether this belief is correct. Secondly, it is known from Lancaster’s (1973) analysis that the Keynesian separation of saving and investment decisions leads to a dynamic welfare loss. Workers’ investment funds might then reduce the inefficiency of capitalism by providing workers with some control over how their savings will be used. Thirdly, the fact that the trade union movement in a number of countries is looking for an alternative to conventional incomes policies reveals workers’ dissatisfaction with the current practice. Tarling and Wilkinson (1977) bring this out clearly in their account of British post-war incomes policies. Their explanation for the failure of these policies, especially in 1975-1977, is that British governments have not been able to achieve a substantial improvement in economic performance in return for the sacrifice in current consumption which the trade union movement has been willing to accept. Studying schemes which allow workers to retain the rights over the fruits of their labour is thus not only of theoretical interest but also of practical relevance, We shall set up our model in the next section. The effects of workers’ control over their savings on investment, growth and welfare will be examined in section 3. In concluding the paper we shall also discuss the restrictive assumptions of our analysis.
2. The model and its solution In his analysis of capitalism Lancaster (1973) considers an economy consisting of two homogeneous social groups the workers and the capitalists. The former may consume or save, the latter consume or invest. Both groups are assumed to be interested in maximizing their own undiscounted consumption over a fixed time horizon which is the same for the two classes. The workers can control the share of their consumption in the total output. Their decision to forgo present consumption in exchange for future consumption means essentially that the workers hand over their savings to the capitalists who, unlike the workers, have full control over the investment decision. The capitalists will invest only if it is in their own interest to do so. By formulating the accumulation problem as a noncooperative differential game Lancaster (1973) and Hoe1 (1975, 1978) have demonstrated how the disjunction between saving and investment decisions leads to a welfare loss and how society would benefit from a coordination of these decisions. We now assume that this social organization is changed in such a way that the workers can decide whether their savings will be invested or not. This means that they no longer need the capitalists as agents for accumulation, which in our opinion is an essential feature of workers’ investment funds. All the other assumptions made by Lancaster (1973) are
M. Pohjola, Workers’ investment funds
273
retained. Thus, we consider a one-sector single technique economy whose output X(t) at any given time t may be consumed or added to the existing capital stock K(t) and where labour is never limiting and capital lasts for ever. It is assumed that the workers can control the shares of their consumption and investment in the total output, ur(t) = C,(t)/X(t) and vi(t) = Zr(t)/X(t), respectively. We further suppose that u,(t)zc, u,(t)20 and ui(t)+u,(t)5 b for all t, c and b being constants such that O
w,=~C,(t)dt=;~K(t)[b-q(t),dt 0
subject
0
to
x(t) = II(t) + I&) = aK(t)[vl(t) + (1 - b)u,(t)]
(2)
and K(0) =K, for the anticipated choice of u2: [0, T] -CO, l] by the capitalists. The crucial assumption here is that the workers can carry out their investments as efficiently as the capitalists. As long as the workers are certain that their savings will be invested they could hand them over to the capitalists with the same results. Without any loss of generality we can, however, assume that also in this case they proceed directly instead of using the capitalists as their agents. Denoting capitalists’ consumption by C,(t), we can express their problem as that of choosing a feasible piecewise continuous
M. Pohjola, Workers’ investment funds
214
function
u2 so as to maximize w,= jC,(t)dt= 0
jrrK(t)(l
-b)[l
-u,(t)]dt
0
subject to (2) and the anticipated choice of v1 by the workers. The noncooperative open-loop (Nash) equilibrium of this game, (ti1,ti2), can be characterized as follows (see the Appendix for details): b-c,
C,(t)=
for
tE[O,fl,
for
tE(cn;
1,
for
tE[O,CJ,
i 0,
for
te(t:T],
where t^= t”- In [b(l - c)/(l - b)(2b - c)]/a(b -c) and t”= T - l/ah. To simplify matters T is assumed large enough for both t^ and t” to be positive. Observe that t”> 6 since b(1 -c)/(l - b)(2b -c) > 1. The optimal solution is then seen to consist of three phases. In the first phase both social groups consume minimally and invest maximally. In the second the capitalists have stopped investing and now consume at the maximum rate while the workers still accumulate. In the final phase both classes consume maximally and so accumulation has ceased. It is interesting to observe that the capitalists stop investing earlier than the workers. This is explained as follows. In devising their optimal policies both classes form their valuation, in terms of their own consumption over the time horizon, of a marginal increase in capital stock at each time instant. Accumulation ceases when the value of a unit of output invested falls below the value of a unit of output consumed. The workers and the capitalists stop investing at different points in time simply because the total output is not divided equally between the two groups. We made the realistic assumption that the workers can get more than half of it (b> l/2). It is this fact that makes them save and invest over a longer period than the capita1ists.l The assumption that the workers can invest is critical for this result. If the social organization is such that they have no control over how their savings will be used and they have to use the capitalists as agents for accumulation, it is not optimal for them to save when the capitalists have stopped investing. The workers cannot force the capitalists to invest against the latter’s best interest. Accumulation ceases when the value of investment to the capitalists falls below the value of consumption, even if the workers and society value ‘If b = l/2, i= f and both groups stop investing accumulating earlier than the capitalists.
at the same instant.
If b < l/2, the workers
stop
M. Pohjola, Workers’ investment finds
275
investment higher than consumption. This was found by Lancaster (1973). We can derive the solution for this case by restricting the workers’ investment to zero for the planning horizon [O, r] and by treating ur(t) as their control variable. The equilibrium solution (U;, V;) consists of two phases. In the first phase both classes consume minimally, i.e. zZr(t)=c, v;(t)= 1 for t E [O, ij, where t= T - l/a(l -b), and in the second they consume maximally, i.e. zil(t) = b, CJt)=O for t E(< T] [see Lancaster (1973) for details]. This solution is game-stable. Provided that both groups understand the game situation they are in, the one that deviates from equilibrium will be exploited in the sense that its anticipation will not be realized. As a consequence, the strategies converge to equilibrium. Against this background we can then understand workers’ reluctance [see Tarling and Wilkinson (1977)] to plans which require them to save but do not offer any guarantee that these savings will increase their future consumption. The welfare effects of an institutional change which provides the workers with such guarantees can be ascertained by comparing the two programs developed here.
3. The implications
of workers’ investment
We have already seen that under a scheme which provides the workers with the means to control how their savings will be used they save and invest for a longer period than the capitalists. From the results of the previous section we also obtain that ?>< i.e. that the workers now switch later to consume maximally than in Lancaster’s (1973) capitalist society. What is more surprising, however, is that also the capitalists invest for a longer time, i.e. that i> E This can be proved by denoting z = b( 1 - c)/( 1 - b)(2b - c) and by utilizing the results of section 2 to show that a(b- c)(?-- t) =(b -c)(2b - l)/b(l -b)-lnz>z-l-lnz>O, since z>l. Recognition of the fact that the workers save and invest the fraction b-c of the total output until t= t”>? induces the capitalists to invest maximally right up to t =F> z Setting up workers’ investment funds then increases investment rather than resulting in a capital flight, which could here be described as the capitalists switching to consume maximally earlier than in the pure capitalist society. The conclusion we can draw is that an economy where workers can invest their savings directly saves, invests and grows more (i.e. obtains a higher end-of-horizon capital stock) than one where they have to use the capitalists as agents for accumulation. Next we investigate who benefits from the institutional change which allows the workers to invest directly. This can be done by comparing the welfare of each social class under the two programs. To do this let us first characterize how the economy grows under these programs. Denote by Z?: [0, T]-+R the trajectory of the capital stock generated by the solution (Or,CJ of the differential game where the workers invest their own savings.
276
M. Pohjola,
Workers’ investment funds
From the characteristics of the solution and from the dynamics of the capital stock (2) we can deduce that for t~[O,fi, for te[t?FJ, for tE[tlT]. The corresponding trajectory Lancaster’s (1973) economy is Koe”(l -c)t
K: [IO,T]+R,
generated
by
(U1,uz), in
for te[O,Fj,
R(t) =
!-- W),
’
for t E [( T].
A priori, one would expect the workers to gain from the transfer of control in their favour. This can be confirmed by comparing their welfare (1) under the two schemes:
W,(ti,,i&)- W,(U,,V,)= ;&(t)[b-c’,(t)]dt=K(t){[c/(l
-c)+(b-c+bc)/(l
j&(t)C,(t)dt 0
0
-b)(2b--~)]e~(~-~)(~-~)
-c/(1-c)-b/(1-b)}>O, which can, with more patience than difficulty, be shown to hold by utilizing the fact that e”” -c)(i-i) > 1+ a( 1 - c)(F- 0 > 1 + (1 - c)(2b - l)(b - c)/b( 1 - b)(2b -c). Besides the workers, the capitalists also gain from this institutional change: W,(B,,L:,)-W,(u,,v,)=ja~(r)(l-h)[l-d,(t)]dt 0
+I@)[1
-uI(t)][l
-v;(t)]dt=R(+R(t)>O,
since t^>E We may then conclude that the social organization where the workers can invest their savings directly is Pareto-superior to the system where they have to use the capitalists as agents for accumulation. The conclusion that the capitalists have no reason to resist plans to establish workers’ investment funds is, however, conditional on the assumption that their welfare only depends on their consumption and not, for example, on
277
M. Pohjola, Workers’ investment funds
capital ownership. important variable sake of increasing capitalist society.
It may well be that consumption is not the most in their criterion function and that they invest for the their control, per se, of the economic institutions of
4. Concluding remarks We have reached the conclusion that, as far as the partial transfer of control over the investment decision from capitalists to workers is an essential feature of workers’ investment funds, the institutional change establishing such funds improves economic efficiency in capitalist society. As a consequence, the economy saves, invests, grows and consumes more. For analytical simplicity we have here assumed that the workers can directly invest all their savings. The qualitative results obtained would not, however, change if they were allowed to control the use of only a fraction of their savings, which is closer to the idea of plans for workers’ investment funds. The limitations of our analysis are rather obvious. We have, for example, divided the population into only two social classes, treated each class as a single decision unit, and studied the equilibrium solution of the resulting non-cooperative differential game. Within this specification, however, the simple model constructed could be generalized in a number of respects [see Lancaster (1973) and Hoe1 (1975, 1978)]. The results obtained in this paper would not be affected provided that in the generalized model the value of investment is higher to the workers than to the capitalists, as is the case in Lancaster’s original model. If, for example, we extend the basic model by introducing the discounting of future consumption, the conclusions obtained hold as long as the workers are not essentially less patient than the capitalists.’ In the real world they may well be, as is sometimes argued in the debate about Yugoslavian worker-managed firms. The concern of the trade union movement for workers’ investment funds, however, is an indication to the contrary.
Appendix: The solution of the differential
game
Let (i?i,z?J be an open-loop equilibrium of the game, defined in (l)-(3), generating the trajectory R: [O, 7J+R. Then we know that there exist two piecewise differentiable functions yi: [0, T] -+R, i = 1,2, such that the following
2Denoting the condition
the workers’ discount rate by p1 and assuming the capitalists’ p,
discount
rate zero,
218
M. Pohjola, Workers’ investment funds
hold almost
everywhere
on [0, T]:
(A.11 YiCT)=O,
64.2)
for all v,~[O,b-c],
(A.3)
for all u2 E [0,11,
(A.4)
where
=
From
~W)[lb- ~I(01+ Y,ww)lI~,(t)
+(I -bMt)l,
(A.3) and (A.4) we can now easily deduce
b-c,
if yl(t)>
1,
that
1,
if Y,(t)> 1,
0,
if y2(t)< 1.
&(t) =
C,(t) = 0,
if yl(t) < 1;
The interpretation is that both social classes invest maximally until the value of a marginal increase in the capital stock, yi(t), falls below unity, the value of a unit of output consumed. As we have just seen, there are only four possible combinations of the values of control variables which can appear in the solution of the game. From (A.2) we obtain that Cl(t)=G,(t) =0 in the interval (6 T], where t”is yet to be determined. It can be found by applying these values of the control variables to the differential equation (A.l), which can be solved to yield
Yz(4 =Yk2)-41
-M-b)>
M. Pohjola, Workers’ investment funds
279
where t, and t, are time instants at which yl(t)= 1 and y*(t)= 1, respectively. Using the transversality condition (A.21, we obtain t, = rl‘- l/ah and t, = T - l/a(l -b). Now t, > t, since b > l/2. Therefore, t”= t,. We know that y,(g= 1 and y,(g= l/b- 1 < 1. It can then be deduced from the differential equation (A.l) that y,(t)> 1 and yZ(t)< 1 in the immediate time interval preceding < Therefore, r?,(f)= b-c and Cz(t)=O in this interval which we denote by (f ?I. r^can be found from (A.l) when i=2. It is the time instant at which the capitalists start consuming maximally, i.e. at which y2(t) = 1. Applying the solution and the boundary condition y&j= l/b- 1, the differential equation can be solved to yield y2(t) = [( 1 - b)/b+ (1 - b)/(b-c)] e -“(b-cw -‘) - (I-
b)/(b- c),
from which t can be solved by using the fact that y,(t^,= 1. This gives F=F - In [b( I- c)/(I- b)(2b - c)]/a(b -c). We can again use (A.l) to infer that both yl(t)> 1 and y,(t)> 1 in the immediate time interval preceding L Therefore, Gl(t)= b-c and v^Jt)= 1 for t E [0, ?J. Consequently, the solution of the differential game has the form described in section 2.
References Atkinson, A.B., 1972, Capital-growth-sharing schemes and the behaviour of the firm, Economica 39, 237-249. Atkinson, A.B., 1974, Unequal shares, rev. edn. (Penguin Books, Harmondsworth). Brems, Hans, 1975a, An investment wage and a wage earners’ investment fund under siead!state growth, Swedish Journal of Economics 77, 13-30. Brems, Hans, 1975b, Profit sharing and a wage earners’ investment fund under steady-state growth, Kyklos 28, 94-l 16. Hoel, Michael, 1975, Aspects of distribution and growth in a capitalist economy, Memorandum from Institute of Economics, University of Oslo. Hoel, Michael, 1978, Distribution and growth as a different game between workers and capitalists, international Economic Review 19, 335-350. Lancaster, Kelvin, 1973, The dynamic inefficiency of capitalism, Journal of Political Economy 81, 1092-l 109. Maital, Shlomo, 1978, Linking wages to prices without causing inflation: Deferred income in theory and practice, Cambridge Journal of Economics 2, 83-98. Nislund, Bertil and Bo Sellstedt, 1978, Neo-Ricardian theory, Lecture Notes in Economics and Mathematical Systems 156 (Springer-Verlag, New York). Tarling, Roger and Frank Wilkinson, 1977, The social contract: Post-war incomes policies and their inflationary impact, Cambridge Journal of Economics 1, 395414.
of Public
Economics
20 (1983) 271-279.
WORKERS’ INVESTMENT INEFFICIENCY Matti
North-Holland
March
Company
FUNDS AND THE DYNAMIC OF CAPITALISM POHJOLA*
University of Helsinki, SF-00100 Received
Publishing
Helsinki 10, Finland
1982, revised version
received
July 1982
This paper examines the effects of the partial transfer of control over the investment decision from capitalists to workers, which is an essential feature of plans to establish workers’ investment funds, in a two-class model of capitalism formulated as a differential game. It is shown that the institutional change which provides the workers with means to invest their savings directly, instead of having to use the capitalists as agents for accumulation, improves economic efliciency. As a consequence, the economy saves, invests, grows and consumes more than the pure capitalist society, which is characterized by the Keynesian separation of saving and investment decisions.
1. Introduction The past 15 years have seen a considerable growth of interest in Western Europe in workers’ investment funds. As for example, Atkinson (1974, ch. 11) points out in his survey, the proposals which have been put forward to establish such funds share the common objective of a more equal distribution of both wealth and control in capitalist society, and their origin is in the apparent failure of the trade union movement to achieve this objective through collective wage bargaining. In this paper we shall dwell neither on the details of these proposals nor on the results of previous studies [see, for example, Atkinson (1972) Brems (1975a, 1975b), Maital (1978) Ndslund and Sellstedt (1978, ch. S)]. Instead, we shall concentrate on the particular function of investment funds in providing workers with the means to control the use of their savings, i.e. the accumulation of their wealth. The economic effects of this transfer of control from capitalists to workers will be examined in the two-person differential game formulation of capitalism developed by Lancaster (1973). We give three reasons for this choice of the object of our examination. First, one of the principal arguments raised against workers’ investment funds, especially against profit-sharing schemes, is that they would discourage investment and lead to a capital flight [see, for example, Atkinson (1974, ch. 1 l)]. *Financial support from gratefully acknowledged. 0047-2727/83/$03.00
the
Kymenlaakso
Fund
of the
Finnish
Cultural
Foundation
is
212
M. Pohjola,
Workers’ investmentfunds
It might be of interest to find out in the differential game formulation of capitalism whether this belief is correct. Secondly, it is known from Lancaster’s (1973) analysis that the Keynesian separation of saving and investment decisions leads to a dynamic welfare loss. Workers’ investment funds might then reduce the inefficiency of capitalism by providing workers with some control over how their savings will be used. Thirdly, the fact that the trade union movement in a number of countries is looking for an alternative to conventional incomes policies reveals workers’ dissatisfaction with the current practice. Tarling and Wilkinson (1977) bring this out clearly in their account of British post-war incomes policies. Their explanation for the failure of these policies, especially in 1975-1977, is that British governments have not been able to achieve a substantial improvement in economic performance in return for the sacrifice in current consumption which the trade union movement has been willing to accept. Studying schemes which allow workers to retain the rights over the fruits of their labour is thus not only of theoretical interest but also of practical relevance, We shall set up our model in the next section. The effects of workers’ control over their savings on investment, growth and welfare will be examined in section 3. In concluding the paper we shall also discuss the restrictive assumptions of our analysis.
2. The model and its solution In his analysis of capitalism Lancaster (1973) considers an economy consisting of two homogeneous social groups the workers and the capitalists. The former may consume or save, the latter consume or invest. Both groups are assumed to be interested in maximizing their own undiscounted consumption over a fixed time horizon which is the same for the two classes. The workers can control the share of their consumption in the total output. Their decision to forgo present consumption in exchange for future consumption means essentially that the workers hand over their savings to the capitalists who, unlike the workers, have full control over the investment decision. The capitalists will invest only if it is in their own interest to do so. By formulating the accumulation problem as a noncooperative differential game Lancaster (1973) and Hoe1 (1975, 1978) have demonstrated how the disjunction between saving and investment decisions leads to a welfare loss and how society would benefit from a coordination of these decisions. We now assume that this social organization is changed in such a way that the workers can decide whether their savings will be invested or not. This means that they no longer need the capitalists as agents for accumulation, which in our opinion is an essential feature of workers’ investment funds. All the other assumptions made by Lancaster (1973) are
M. Pohjola, Workers’ investment funds
273
retained. Thus, we consider a one-sector single technique economy whose output X(t) at any given time t may be consumed or added to the existing capital stock K(t) and where labour is never limiting and capital lasts for ever. It is assumed that the workers can control the shares of their consumption and investment in the total output, ur(t) = C,(t)/X(t) and vi(t) = Zr(t)/X(t), respectively. We further suppose that u,(t)zc, u,(t)20 and ui(t)+u,(t)5 b for all t, c and b being constants such that O
w,=~C,(t)dt=;~K(t)[b-q(t),dt 0
subject
0
to
x(t) = II(t) + I&) = aK(t)[vl(t) + (1 - b)u,(t)]
(2)
and K(0) =K, for the anticipated choice of u2: [0, T] -CO, l] by the capitalists. The crucial assumption here is that the workers can carry out their investments as efficiently as the capitalists. As long as the workers are certain that their savings will be invested they could hand them over to the capitalists with the same results. Without any loss of generality we can, however, assume that also in this case they proceed directly instead of using the capitalists as their agents. Denoting capitalists’ consumption by C,(t), we can express their problem as that of choosing a feasible piecewise continuous
M. Pohjola, Workers’ investment funds
214
function
u2 so as to maximize w,= jC,(t)dt= 0
jrrK(t)(l
-b)[l
-u,(t)]dt
0
subject to (2) and the anticipated choice of v1 by the workers. The noncooperative open-loop (Nash) equilibrium of this game, (ti1,ti2), can be characterized as follows (see the Appendix for details): b-c,
C,(t)=
for
tE[O,fl,
for
tE(cn;
1,
for
tE[O,CJ,
i 0,
for
te(t:T],
where t^= t”- In [b(l - c)/(l - b)(2b - c)]/a(b -c) and t”= T - l/ah. To simplify matters T is assumed large enough for both t^ and t” to be positive. Observe that t”> 6 since b(1 -c)/(l - b)(2b -c) > 1. The optimal solution is then seen to consist of three phases. In the first phase both social groups consume minimally and invest maximally. In the second the capitalists have stopped investing and now consume at the maximum rate while the workers still accumulate. In the final phase both classes consume maximally and so accumulation has ceased. It is interesting to observe that the capitalists stop investing earlier than the workers. This is explained as follows. In devising their optimal policies both classes form their valuation, in terms of their own consumption over the time horizon, of a marginal increase in capital stock at each time instant. Accumulation ceases when the value of a unit of output invested falls below the value of a unit of output consumed. The workers and the capitalists stop investing at different points in time simply because the total output is not divided equally between the two groups. We made the realistic assumption that the workers can get more than half of it (b> l/2). It is this fact that makes them save and invest over a longer period than the capita1ists.l The assumption that the workers can invest is critical for this result. If the social organization is such that they have no control over how their savings will be used and they have to use the capitalists as agents for accumulation, it is not optimal for them to save when the capitalists have stopped investing. The workers cannot force the capitalists to invest against the latter’s best interest. Accumulation ceases when the value of investment to the capitalists falls below the value of consumption, even if the workers and society value ‘If b = l/2, i= f and both groups stop investing accumulating earlier than the capitalists.
at the same instant.
If b < l/2, the workers
stop
M. Pohjola, Workers’ investment finds
275
investment higher than consumption. This was found by Lancaster (1973). We can derive the solution for this case by restricting the workers’ investment to zero for the planning horizon [O, r] and by treating ur(t) as their control variable. The equilibrium solution (U;, V;) consists of two phases. In the first phase both classes consume minimally, i.e. zZr(t)=c, v;(t)= 1 for t E [O, ij, where t= T - l/a(l -b), and in the second they consume maximally, i.e. zil(t) = b, CJt)=O for t E(< T] [see Lancaster (1973) for details]. This solution is game-stable. Provided that both groups understand the game situation they are in, the one that deviates from equilibrium will be exploited in the sense that its anticipation will not be realized. As a consequence, the strategies converge to equilibrium. Against this background we can then understand workers’ reluctance [see Tarling and Wilkinson (1977)] to plans which require them to save but do not offer any guarantee that these savings will increase their future consumption. The welfare effects of an institutional change which provides the workers with such guarantees can be ascertained by comparing the two programs developed here.
3. The implications
of workers’ investment
We have already seen that under a scheme which provides the workers with the means to control how their savings will be used they save and invest for a longer period than the capitalists. From the results of the previous section we also obtain that ?>< i.e. that the workers now switch later to consume maximally than in Lancaster’s (1973) capitalist society. What is more surprising, however, is that also the capitalists invest for a longer time, i.e. that i> E This can be proved by denoting z = b( 1 - c)/( 1 - b)(2b - c) and by utilizing the results of section 2 to show that a(b- c)(?-- t) =(b -c)(2b - l)/b(l -b)-lnz>z-l-lnz>O, since z>l. Recognition of the fact that the workers save and invest the fraction b-c of the total output until t= t”>? induces the capitalists to invest maximally right up to t =F> z Setting up workers’ investment funds then increases investment rather than resulting in a capital flight, which could here be described as the capitalists switching to consume maximally earlier than in the pure capitalist society. The conclusion we can draw is that an economy where workers can invest their savings directly saves, invests and grows more (i.e. obtains a higher end-of-horizon capital stock) than one where they have to use the capitalists as agents for accumulation. Next we investigate who benefits from the institutional change which allows the workers to invest directly. This can be done by comparing the welfare of each social class under the two programs. To do this let us first characterize how the economy grows under these programs. Denote by Z?: [0, T]-+R the trajectory of the capital stock generated by the solution (Or,CJ of the differential game where the workers invest their own savings.
276
M. Pohjola,
Workers’ investment funds
From the characteristics of the solution and from the dynamics of the capital stock (2) we can deduce that for t~[O,fi, for te[t?FJ, for tE[tlT]. The corresponding trajectory Lancaster’s (1973) economy is Koe”(l -c)t
K: [IO,T]+R,
generated
by
(U1,uz), in
for te[O,Fj,
R(t) =
!-- W),
’
for t E [( T].
A priori, one would expect the workers to gain from the transfer of control in their favour. This can be confirmed by comparing their welfare (1) under the two schemes:
W,(ti,,i&)- W,(U,,V,)= ;&(t)[b-c’,(t)]dt=K(t){[c/(l
-c)+(b-c+bc)/(l
j&(t)C,(t)dt 0
0
-b)(2b--~)]e~(~-~)(~-~)
-c/(1-c)-b/(1-b)}>O, which can, with more patience than difficulty, be shown to hold by utilizing the fact that e”” -c)(i-i) > 1+ a( 1 - c)(F- 0 > 1 + (1 - c)(2b - l)(b - c)/b( 1 - b)(2b -c). Besides the workers, the capitalists also gain from this institutional change: W,(B,,L:,)-W,(u,,v,)=ja~(r)(l-h)[l-d,(t)]dt 0
+I@)[1
-uI(t)][l
-v;(t)]dt=R(+R(t)>O,
since t^>E We may then conclude that the social organization where the workers can invest their savings directly is Pareto-superior to the system where they have to use the capitalists as agents for accumulation. The conclusion that the capitalists have no reason to resist plans to establish workers’ investment funds is, however, conditional on the assumption that their welfare only depends on their consumption and not, for example, on
277
M. Pohjola, Workers’ investment funds
capital ownership. important variable sake of increasing capitalist society.
It may well be that consumption is not the most in their criterion function and that they invest for the their control, per se, of the economic institutions of
4. Concluding remarks We have reached the conclusion that, as far as the partial transfer of control over the investment decision from capitalists to workers is an essential feature of workers’ investment funds, the institutional change establishing such funds improves economic efficiency in capitalist society. As a consequence, the economy saves, invests, grows and consumes more. For analytical simplicity we have here assumed that the workers can directly invest all their savings. The qualitative results obtained would not, however, change if they were allowed to control the use of only a fraction of their savings, which is closer to the idea of plans for workers’ investment funds. The limitations of our analysis are rather obvious. We have, for example, divided the population into only two social classes, treated each class as a single decision unit, and studied the equilibrium solution of the resulting non-cooperative differential game. Within this specification, however, the simple model constructed could be generalized in a number of respects [see Lancaster (1973) and Hoe1 (1975, 1978)]. The results obtained in this paper would not be affected provided that in the generalized model the value of investment is higher to the workers than to the capitalists, as is the case in Lancaster’s original model. If, for example, we extend the basic model by introducing the discounting of future consumption, the conclusions obtained hold as long as the workers are not essentially less patient than the capitalists.’ In the real world they may well be, as is sometimes argued in the debate about Yugoslavian worker-managed firms. The concern of the trade union movement for workers’ investment funds, however, is an indication to the contrary.
Appendix: The solution of the differential
game
Let (i?i,z?J be an open-loop equilibrium of the game, defined in (l)-(3), generating the trajectory R: [O, 7J+R. Then we know that there exist two piecewise differentiable functions yi: [0, T] -+R, i = 1,2, such that the following
2Denoting the condition
the workers’ discount rate by p1 and assuming the capitalists’ p,
discount
rate zero,
218
M. Pohjola, Workers’ investment funds
hold almost
everywhere
on [0, T]:
(A.11 YiCT)=O,
64.2)
for all v,~[O,b-c],
(A.3)
for all u2 E [0,11,
(A.4)
where
=
From
~W)[lb- ~I(01+ Y,ww)lI~,(t)
+(I -bMt)l,
(A.3) and (A.4) we can now easily deduce
b-c,
if yl(t)>
1,
that
1,
if Y,(t)> 1,
0,
if y2(t)< 1.
&(t) =
C,(t) = 0,
if yl(t) < 1;
The interpretation is that both social classes invest maximally until the value of a marginal increase in the capital stock, yi(t), falls below unity, the value of a unit of output consumed. As we have just seen, there are only four possible combinations of the values of control variables which can appear in the solution of the game. From (A.2) we obtain that Cl(t)=G,(t) =0 in the interval (6 T], where t”is yet to be determined. It can be found by applying these values of the control variables to the differential equation (A.l), which can be solved to yield
Yz(4 =Yk2)-41
-M-b)>
M. Pohjola, Workers’ investment funds
279
where t, and t, are time instants at which yl(t)= 1 and y*(t)= 1, respectively. Using the transversality condition (A.21, we obtain t, = rl‘- l/ah and t, = T - l/a(l -b). Now t, > t, since b > l/2. Therefore, t”= t,. We know that y,(g= 1 and y,(g= l/b- 1 < 1. It can then be deduced from the differential equation (A.l) that y,(t)> 1 and yZ(t)< 1 in the immediate time interval preceding < Therefore, r?,(f)= b-c and Cz(t)=O in this interval which we denote by (f ?I. r^can be found from (A.l) when i=2. It is the time instant at which the capitalists start consuming maximally, i.e. at which y2(t) = 1. Applying the solution and the boundary condition y&j= l/b- 1, the differential equation can be solved to yield y2(t) = [( 1 - b)/b+ (1 - b)/(b-c)] e -“(b-cw -‘) - (I-
b)/(b- c),
from which t can be solved by using the fact that y,(t^,= 1. This gives F=F - In [b( I- c)/(I- b)(2b - c)]/a(b -c). We can again use (A.l) to infer that both yl(t)> 1 and y,(t)> 1 in the immediate time interval preceding L Therefore, Gl(t)= b-c and v^Jt)= 1 for t E [0, ?J. Consequently, the solution of the differential game has the form described in section 2.
References Atkinson, A.B., 1972, Capital-growth-sharing schemes and the behaviour of the firm, Economica 39, 237-249. Atkinson, A.B., 1974, Unequal shares, rev. edn. (Penguin Books, Harmondsworth). Brems, Hans, 1975a, An investment wage and a wage earners’ investment fund under siead!state growth, Swedish Journal of Economics 77, 13-30. Brems, Hans, 1975b, Profit sharing and a wage earners’ investment fund under steady-state growth, Kyklos 28, 94-l 16. Hoel, Michael, 1975, Aspects of distribution and growth in a capitalist economy, Memorandum from Institute of Economics, University of Oslo. Hoel, Michael, 1978, Distribution and growth as a different game between workers and capitalists, international Economic Review 19, 335-350. Lancaster, Kelvin, 1973, The dynamic inefficiency of capitalism, Journal of Political Economy 81, 1092-l 109. Maital, Shlomo, 1978, Linking wages to prices without causing inflation: Deferred income in theory and practice, Cambridge Journal of Economics 2, 83-98. Nislund, Bertil and Bo Sellstedt, 1978, Neo-Ricardian theory, Lecture Notes in Economics and Mathematical Systems 156 (Springer-Verlag, New York). Tarling, Roger and Frank Wilkinson, 1977, The social contract: Post-war incomes policies and their inflationary impact, Cambridge Journal of Economics 1, 395414.