- Email: [email protected]
Fixprice analysis of labor-managed economies
JOURNAL
OF COMPARATIVE
ECONOMICS
13,227-253
(1989)
Fixprice Analysis of Labor-Managed Economies’ FERNANDO B. SALDANHA World Bank, Washington, DC 20433 Received July 15, 1987; revised October 5, 1988
Saldanha,FernandoB.-Fixprice
Analysis of Labor-Managed Economies
The macroeconomics of labor-managed economies ( LMEs) are investigated in the context of fixprice theory. Two models are constructed. In a traditional Ward LME, firms maximize profit per worker, and workers supply labor taking the wage rate as given. In a 8en LME, workers decide jointly on employment and leisure. In both cases fixprice equilibria exist and are unique, under reasonable assumptions. The three traditional regimes (Classical, Keynesian, and Inflationary) of traditional fixprice theory are possible in the Ward LME. In the simpler Sen LME the only possibilities arc excess supply or demand for goods. A conjecture of Vanek and Meade is shown to be false in the context of either model: monetary and fiscal policies have the usual qualitative effects in the LME and in the profit-managed economies. J. Comp. Econom., June 1989, 13(2), pp. 227-253. World Bank, Washington, DC 20433 o 1989 Academic press, Inc.
Journal of Economic Literature Classification Numbers 022,023, 052.
1. INTRODUCTION A labor-managed economy (LME) is distinguished by three operating rules: ( 1) a firm is managed by its members, who also comprise its labor force; (2) a firm’s members are residual income claimants, thus their collective income is the firm’s profit; and (3) firms and consumers operate autonomously, with interactions among agents occurring through free markets. The original analysis of the labor-managed firm (LMF) is due to Ward ( 1958 ) . He assumed that a LMF is required to evenly distribute profits among all its members, making the LMF a perfectly egalitarian cooperative, and proved that if the firm maximizes per-member profits, or wages, then it has ’ I am grateful to Jean-Pascal Bcnassy, David Conn, Deborah Hartz, and two anonymous referees for useful comments. The College of Business and Public Administration, University of Arizona, has generously provided financial assistance for this project. Of course, any errors are mine. 227
0147-5967189 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
228
FERNANDO
B. SALDANHA
a negatively sloped output supply function. This result is guaranteed if labor is the only variable input for the LMF and is likely when there are several variable inputs. Since the publication of Ward’s seminal work the literature on the LMF and the LME has expanded rapidly. The hypothesis of the LMF’s perverse supply curve was extensively discussed ( Meade, 1972; Vanek, 1970; Steinherr and Thisse, 1979), and another branch of the literature focused on the efficiency of general equilibrium (D&e, 1976; Ichiishi, 1977, 198 1) . In contrast, almost no attention was given to the macroeconomics of LMEs. Some remarks on the problems of unemployment and inflation are scattered in the literature (Vanek, 1970; Meade, 1972; Tyson, 1980)) but there was no attempt to elaborate a specific macrotheory of the LME. We begin to fill this gap by constructing a disequilibrium macromodel of the LME. Our analysis is developed in the theoretical framework of “disequilibrium” or “fixprice” theory (Barr0 and Grossman, 197 1; Malinvaud, 1977; Benassy, 1982). The book by Benassy ( 1982) is a good reference for the reader unfamiliar with fixprice theory. Both Vanek ( 1970) and Meade ( 1972) conjectured that the perversity of Ward’s LMF supply curve might lead to macroeconomic instability. Meade claimed that “to rely on Keynesian policies to expand effective demand in times of unemployment would be at best ineffective, and at worst might lead to a reduction in output and employment.” In Section 2 we analyze this question in the context of Ward’s model and find out that the conjecture is false in the short run. The Ward economy may operate under the same three regimes, Classical, Keynesian, and Inflationary, as the PME, and all shortrun multipliers have the same signs that they would have in a capitalist economy. Ward’s assumption that the LMF maximizes per-member wages has been criticized by several authors ( Steinherr and Thisse, 1979; Miyazaki and Neat-y, 1983) on several grounds. Most relevant in our context is the fact that in Ward’s model what essentially constitutes a single decision of workers about employment and labor supply is artificially split, so that the firm maximizes wages per worker, and workers, taking that wage as given, choose their labor supplies.* More realistically, the firm as an independent decision unit should be abolished from the analysis, and workers should choose the employment level so as to maximize their utilities. Sen ( 1966) was the first to introduce a model of this type. Unfortunately, realistically modeling the Sen LME can involve considerable complication. In the models of Steinherr and Thisse ’ Here we follow D&x ( 1976), who assumes that the worker treats the wage rate as given when choosing hi labor supply, even though it depends on the amount of labor employed. Alternatively, one can interpret the worker’s behavior in Fellner’s ( 1947) bilateral monopoly framework. Law (1977) shows that ifthe worker’s (or the union’s) indil%rencz curves are horizontal the firm displays the behavior postulated by Ward.
FIXPRICE
LABOR-MANAGED
229
ECONOMIES
( 1979) and Miyazaki and Neary ( 1983), for example, workers have to cope with uncertainty, since layoffs are random. We adopt instead a simple model in which each firm has a single member and analyze its macroeconomic properties in the context of fixprice theory. The worker never considers “quitting,” that is, dissolving the firm, and of course cannot be laid off. He faces only a labor-leisure choice. Although our model is quite specific, one conclusion generalizes to the whole class of models: there are only two possible regimes for the economy, one with excess demand for the good (Inflationary) and the other with excess supply (Keynesian). This is because there is no supply or demand for labor, but only a labor-leisure decision. It also turns out that Meade’s conjecture is false in our specific version of the Sen economy. The structure of the paper is as follows. In Section 2 we analyze the macroeconomics of the Ward economy. This section is subdivided into subsections dealing with definitions and setting up of the model, existence and uniqueness of equilibrium, and comparative statics. Section 3 contains a similar analysis of the Sen economy. Finally, Section 4 contains our conclusions and some directions for further research. 2. FIXPRICE
ANALYSIS
OF THE WARD
ECONOMY
In this section we develop a fixprice model of a Ward economy. Firms are modeled as independent agents that maximize wages per worker. Workers take wages as given and behave as their counterparts in the PME.
2.1. The Economy We consider an economy with three commodities: a consumption good, labor, and money. No future contracts are allowed. The price p of the consumption good is exogenously fixed, and the good market clears by quantity rationing on the long side. There is no labor market in the usual sense. Firms are labor-managed, and maximize per worker revenue net of nonlabor costs, as in Ward ( 1958). This average net revenue w is henceforth called the wage rate, since it plays a role similar to that of the wage rate in the PME. Both the demand for labor and the desired (by firms) wage rate are functions of the good price. This means labor demand is not a function of the wage rate. Labor supply is, as in a capitalist economy, a function of the wage rate. One can still speak of excess demand or supply of labor, but such conditions do not affect the wage rate according to the Law of Supply and Demand.3 Money is the only store of value. Agents are one representative consumer-worker, one representative firm, and the govemment.4 3 This means a situation of excessdemand (supply) for labor does not necessarily imply that the wage rate must rise (fall). ’ All of the analysis that follows could have been developed in an economy with many identical firms and consumers. For simplicity, we assume that there is only one agent of each type.
230.
FERNANDO
B. SALDANHA
In a Ward economy, the employment and labor supply decisions are artificially split. The firm maximizes wages per worker, and the worker takes this behavior of the lirrn as given and maximizes preferences given prices, the wage rate, and endowments. 2.2. Notional and Efective Demands The distinction between notional and eflective demands and supplies is fundamental in fixprice analysis. Notional demands and supplies depend only on prices and kndowments, these are the standard demands and supplies of microeconomic theory. Effective demands and supplies also depend on quantity signals, usually in the form of quantity constraints. Several alternative formulations of effective demand have been proposed in the lixprice literature.5 The one we adopt is based in the dual-decision hypothesis. We assume that agents face quantity constraints on their purchases and sales and that an agent’s demand or supply for a commodity is determined by maximizing the agent’s objective function given all quantity constraints except for the one referent to that commodity. We return to this point below when we define the effective demands and supplies of the firm and the worker. 2.3. The Firm The labor-managed firm has a production function f: I?+ + R,, twice continuously differentiable, monotonic, and strictly concave. There is also a fixed (in nominal terms) cost K. Let L denote the membership of the firm. Then the wage rate is given by *(L
9
p)=pf(L)-K
L
*
(1)
The notional labor demand Lc (p, K) is given by arg max w( L, p) .6 IA y:(p), K) be the notional supply of the consumption good. Of course, yc(p, K) = f( L:(p, K)). It is well known that 5 Svensson ( 1980) discusses several concepts of effective demand. 6 In this section we adopt the following conventions on notation. (a) Subscript capital letters denote different types of agents: W stands for the worker, F for the firm. (b) There are three types of demand and supply functions: notional, effective, and LM-effxtive. Notional demands and supplies am denoted by a superscript *. Worker’s LM-effbctive demand and supply functions, de&xi in Section 2.4, are topped by a caret -. Traditional effective functions have no superscripts or carets. (c) A tilde - over a variable indicates a fixprice equilibrium value of that variable.
FIXPRICE
LABOR-MANAGED
MP, m < o ap
ECONOMIES
231
(2)
(Ward’s Theorem ) (3)
Now, the firm can be constrained in the labor or good markets, whence the need for effective demand and supply schedules.7 For example, if the firm cannot sell all it would desire to sell at a given price, then its demand for labor is correspondingly affected: it demands just enough labor to produce the quantity demanded. Similarly, a constraint on the available manpower determines a reduction of output supply. The effective demand and supply schedules of the firm must then have quantities as arguments: good demand for the effective demand for labor function and labor supply for the effective output supply function. Let &(p, jj) and ~+(p, L) be respectively the effective demand for labor and the effective supply of goods. Our definition of effective demands and supplies, which is based on the dual-decision hypothesis, implies that &(p, J) is the solution of
subject to
L 2 0.
Note that the firm’s effective demand for labor depends only on the quantity constraint on the goods market. The firm cannot sell more than $, and its demand for labor is correspondingly affected. The quantity constraint on the labor market itself does not appear in the optimization problem above. This is the meaning of the dual-decision hypothesis. The firm’s effective supply of goods y~(p, L) is given as the solution of max
PY
L
subject to Y =f(L) OGLGL. ’ It turns out that the firm cannot be constrained simultaneously in both markets. It follows that a disequilibrium regime with excess supply of goods and excess demand for labor is not possible in the context of the present model.
232
FERNANDO
B. SALDANHA
Again the dual-decision hypothesis implies that the constraint in the good market is ignored, and only the constraint in the labor market is considered. The firm can hire at most L of labor, and this affects its supply of the good. Wage maximization implies Y>- d(P) YG A(P)
(4)
LkL,*(p) L G L F*(p). The equations (4) can be interpreted as follows. If the good demand is enough to absorb the firm’s notional good supply (~7 2 J$ ) then effective and notional labor demand coincide. Essentially, the firm is unconstrained in the good market, so the ineffective constraint does not affect its labor demand. But if jj < J$ then the firm just demands enough labor to produce the amount of output it can sell. Similarly, if labor supply is enough to produce the notional supply of the good (12 L: = f - ’ ( J$ ) ) , then effective and notional good supplies coincide. The firm is essentially unconstrained in the labor market, and the ineffective constraint L does not affect its supply of the good. On the other hand, if L < Lz then output must be correspondingly reduced. 2.4. The Worker Given that we are dealing with a Ward economy, in which employment and labor supply decisions are separately made, the notional and effective demand and supply functions of the worker are the ones used in the standard fixprice theory of the profit-managed economy (PME) . Let x be the consumption of the private sector, in real terms. We denote by L$(p, w, m) and by x$(p, w, m) the worker’s notional labor supply and good demand functions. Now, the worker can be rationed in one or both of the labor and good markets. This may affect its demand for the good or supply of labor. If the worker cannot sell his labor time, then he has to reduce his purchases of the good. Similarly, a constraint on the availability of the good is likely to adversely affect the labor supply: there is no incentive to work if there are no goods to be bought. The dual decision hypothesis implies that the worker’s effective demand for goods xw(p, w, m, L) is given by the solution to mm Vx, L, II) subject to px+pLm+wL 0 < x,
O
FIXPRICE
LABOR-MANAGED
ECONOMIES
233
Here U is of course the worker’s utility function, which depends on current consumption x, on current labor supply L, and on money balances CLcarried to future periods. Le is the worker’s endowment of labor. Money enters indirectly in the utility function. That is, it is not necessary to assume that the worker can derive direct satisfaction from the possession of money to write U in the form above. One can see the worker as solving a multiperiod optimization problem, where future utility depends on the amount of money carried from period 1 to period 2. Grandmont ( 1974) establishes the conditions under which a utility function of the form above is well defined and has the usual properties of monotonicity, continuity, and quasi-concavity. Similarly, the worker’s effective supply of labor LW ( p, w, m, X) is given by the solution to max W,
L P)
subject to px+p
0 =s L < Le.
Of course, if the quantity constraint faced by the worker in a market is greater than his notional demand or supply in that market, then that constraint remains inactive. This means that the functions xw and Lw must also satisfy XW(P,
W, m,
G
=
x~%P,
W, m)
if
L > LG(p, w, m)
Lw@,
w,
2)
=
G-b,
w,
if
2 2 x$(p,
m,
m)
w, m).
In our model, the wage rate w is a function of the employment level, unlike the wage rate that is taken as fixed in fixprice models of PMEs. It would be a mistake to treat w as an exogenously fixed variable, as is usual in fixprice analyses of the PME. The functions xw and LW are then only indirectly useful in the analysis of the LME: defining them is the first step in the construction of LM-effective demand and supply functions that take into account the dependence of w on the output and employment levels. Suppose that the worker perceives a quantity constraint in the labor market at E, and E is also the employment level. It follows that the wage rate is w( E, p) . Accordingly, we define the worker’s LM-effective demand as m, L) = xw(p, w(E,p>,
m, t).
(5) Similarly, if there is a constraint 2 in the good market, g is real government expenditure, and the output level equals X + g, then the worker is paid at the rate w(f-’ (X + g) , p), and his LM-effective labor supply is Zw(p,
~w(P,
m,
Xl
E Lw(P,
W-‘(X+
g)),
m,
Xl.
(6)
In order to prove the existence of equilibrium and perform comparative statics analysis one has to make specific assumptions about the signs of the
234
FERNANDO
derivatives of the LM-effective sumptions is*
B. SALDANHA
functions of workers. A set of reasonable as-
a.fw,O
aLwGo
2%”
am
azw d o aK azwao Tic-
aLw ~ o ’
aK
’
aiw,o --z-”
(WLM)
The inequalities in (WLM) are assumed to hold strictly, except when this is not possible due to a nonnegativity condition, or when a quantity constraint is not binding. 2.5. Existence of Equilibrium The existence of fixprice equilibria for capitalist economies has been established (cf. Bohm, 1978 ) by means of an essentially graphic argument. Here we use an alternative line of reasoning, based on Brouwer’s fixed point theorem. A notion of equilibrium for a model like the one described above has to differ from the usual concept of a Walrasian equilibrium, in which notional supplies equal notional demands. The essence of ajixprice equilibrium is that it is a fixed point in the space of quantity constraints. That is, if agents expect to face a given vector of quantity constraints, then their behavior is such that they actually face those constraints. DEFINITION 1. We say that (z, 2) is a fixprice equilibrium for the economy defined by (p, m, g, K) if f(z) = 2 + g and there are (E, 2) such that
Z = min { L&p, X + g), Lw(p, m, a)}
(7)
2
(8)
=
min(fi(P,
0
-g,
2~0,
m,
L)},
where E=L
if
L = Lw(p,
m, 2)
JC=oO
otherwise
(10)
x=x”
if
(11)
X=cO
otherwise.
i = ~?~(p, m, L)
(9)
(12)
Definition 1 can be interpreted as follows. First, workers face quantity constraints that are given by the actual output and employment levels C?+ g and * See the Appendix for a discussion of the plausibility of ( WLM ) .
FIXPRICE LABOR-MANAGED ECONOMIES
235
L’. Second, the same is true for firms in a market (I= L” or X = x”) if workers areontheshortsideofthatmarket(~=~w(p,m,x”)or~==w(p,m,L1)), but otherwise firms are unconstrained (E = cc or X = CO). Third, exchange is voluntary; the left-hand sides of (7) and (8) cannot be strictly greater than the respective right-hand sides. Finally, there cannot be unsatisfied agents in both sides of a market; the left-hand sides of (7) and (8) cannot be strictly smaller than the respective right-hand sides. The fact that constraints are equal to actual quantities embodies the idea that a fixprice equilibrium is a fixed point in the space of constraints. Given Definition 1, we can prove THEOREM 1. Given (p, m, g, K) such that m > 0, suppose that the set E consisting of all (L, x) E l!?$ such that
&(~,m,L)~f(L)-g Gv(P,
m,
xl
2
f-lb
+ g)
is nonempty. Then aJixprice equilibrium for the economy defined by (p, m, g, K) exists. Proof The proof depends on Theorem 2 below. See the Appendix. We now introduce an alternative but equivalent definition of fixprice equilibrium as a fixed point of a mapping. DEFINITION 2. Let G be defined by
G=
{(L,x)ER:Ix+g=f(L)<
y;(p)}
and let Q = G fl E, where E is defined as in the statement of Theorem 1. Define \k: !J --* R by
WL,x) =(&f(X)-g), where X = mini
Lb,
m,
x)X’(&(P,
m,
L)
+ g),
G(P)}.
Here G is the set of all (L , x) that can be voluntarily produced by the firm at the given prices, and E is the set of all (L , x) that can be voluntarily traded by the worker. Clearly Q = G rl E is compact, and \k is well defined and continuous. For any r = (L, x) E a, \k( {) is the largest (further away from the origin) allocation in Q such that the quantity constraints on households determined by { are satisfied. That is, if the initial allocation r is such that the corresponding quantity constraints on households are satisfied, then P expands output and employment as much as possible given that those same constraints remain satisfied. And if ris not compatible with voluntary exchange on the part of the households, the q contracts output and employment so that the quantity constraints associated to 5 are just satisfied.
FJ%NANDO B. SALDANHA
236
THEOREM 2. Let (p, m, g, K) bejixed. Then (L, x”) is afixprice equilibrium for the economy defined by (p, m, g, K) if and only if(L, x”) E Q and q(L, 2) = (L, 2). ProoJ See the Appendix.
There are several possible types of fixprice equilibria, which are also called regimes of the economy.
DEFINITION 3. Let (z, 2) be an equilibrium
for the economy defined by
(p, m, g, K). We say that (2, 2) is:
(i) A Classical equilibrium if L” < t&p, m, x”) and 2 < a,(~, m, z) (Excess supply of labor, excess demand for the good.) (ii) A Keynesian equilibrium if L” < &(p, m, 2) and 2 = y~(p, w, E) - g. (Supply exceeds demand in all markets. ) (iii) An Inflationary equilibrium if L’ = LF(p, w, X + g) and x” < a,(~, m, L”). (Demand exceeds supply in all markets.) (iv) AWalrasianequilibriumifL”=I&(p,m,Z)= LF*(p)andZ=&(p, m, L”) = yz (p) - g. (Supply equals demand in all markets, firms are unconstrained.) This classification of equilibria is not exhaustive, but will serve for our purposes. Each of these types of equilibria corresponds to a region in (p, m , g, K) space. Other types of equilibria could be defined, corresponding to the boundaries between these regions. Figures 1-4 depict the different types of equilibria in (L , x) space. In these figures three types of curves are depicted. The curve Lw (x) gives the worker’s LM-effective labor supply as a function of his constraint in the good market. The curve Zw ( L) gives the worker’s LM-effective good demand as a function of his constraint in the labor market. The curve f( L) - g shows output net of government demand, the output available to the worker. Inflationary and Keynesian equilibria are given by the intersection off ( L) - g with one of the two other curves. When the firm’s notional point (Lz , yr - g) lies in the interior of the region delimited by the two curves -Ilw ( L) and &v ( x) that point is a Classical equilibrium. Finally, if (Lc , yr? - g) coincides with the point of intersection of 2w ( L) and tw (x) then this triple intersection point is a Walrasian equilibrium. The reader should check that these are the only points where Definition 1 is satisfied and that there is a correspondence between Figs. l-4 and the different types of equilibria of Definition 3. 2.6. Uniqueness of Equilibrium
In order to prove the uniqueness of fixprice equilibrium, Bohm ( 1978) essentially assumes that the slope of the net production function f( * ) - g is
L
L FIG.
FIG.
1. Classical equilibrium.
2. Keynesian equilibrium. 231
238
FERNANDO
FIG.
FIG.
B. SALDANHA
3. Inflationary equilibrium.
4. Walrasian equilibrium.
239
FIXPRICE LABOR-MANAGED ECONOMIES
always greater than the slope of the effective demand function x( * ) and smaller than the slope of the inverse labor supply function L$ ( * ) . He justifies this assumption relating it to the well-known condition that the marginal propensity to consume be less than one. More generally, a relaxation of a constraint on sales (purchases) should not generate increased purchases (sales) of greater value.’ This condition was also used by Schulz ( 1983) to prove the uniqueness of fixprice equilibrium for a capitalist exchange economy and in our notation may be formalized by (13) aLw ax
P
(14)
In the context of our model it is sensible to replace ( 13 ) and ( 14) by
agw aoa, 00 pTZ< aE
(15) W)
a(w(p,E)L) ai:, -1 aL cp a2 *
L-1
(16)
The first inequality in (U) states that if the worker’s quantity constraint in the labor market is relaxed, so that he can sell additional labor, then not all of the additional revenue is spent on the good, but part of it is saved in the form of money. Similarly, ( 16) states that if the worker’s quantity constraint in the good market is relaxed, so that he can purchase additional units of the good, then part of those additional purchases will be financed out of savings. THEOREM 3. Suppose that the economy defined by (p, m, g, K) satisfies the conditions of Theorem 1 and ( U) is satisfied. Then there exists a unique jixprice equilibrium.
Proof: See the Appendix. 2.7. Representation
of Equilibria
in Parameter Space
In this section we let g and K be fixed and study the representation of equilibria in (m , p) space. The following proposition establishes conditions under which there is at most one Walrasian equilibrium. ’ Note however that these conditions are not completely analogous to the condition that the marginal propensity to consume is less than one. This is because the relaxation of a constraint affects by itself the demand or supply in the other market, independent of its effect on the worker’s income.
240 PROPOSITION
FERNANDO
B. SALDANHA
1. Suppose that the functions L$ and x$ are homogeneous
of degree one in their respective arguments, and as.wme’O
aG,, ap MP,
'
-aL;: G 0, am 0, ml
= 0,
aL;t aw
-20 V(P,
m)
ww
Then there exists at most one pair (p*, m* ) such that there exists a Walrasian equilibrium for the economy defined by (p*, m*, g, K). Proof: See the Appendix. If the conditions of Proposition 1 are satisfied the whole (m, p) space can be divided into four connected regions. Three of these are the usual Classical, Keynesian, and Inflationary regions, while the fourth corresponds to those values of m and p such that no equilibrium exists. Assuming ( WLM), it is easy to verify the following assertions: (i) The Classical region lies to the right of the Keynesian region.” (ii) The Inflationary region lies to the right of both the Keynesian and the Classical regions. Figure 5 depicts the typical partition of (m, p) space into regions corresponding to the different regimes of the economy. 2.8. Comparative Statics Under mild assumptions, we can perform simple comparative statics for the LME. It turns out that Meade’s ( 1972) conjecture that expansionist fiscal policies never have a positive effect on output level is incorrect when shortrun effects are considered. In fact, changes in g and m have the same shortrun qualitative effects as those in a PME. The government may also wish to use K as a policy instrument. For example, if the fixed costs denoted by K are partially due to taxation, the government can vary the tax rate. Given ( WLM ) , comparative statics analysis can be done in two alternative ways. First, one can implicitly differentiate the equations that define Classical, Inflationary, and Keynesian equilibria. These equations are ” The inequalities in (WW) are assumed to hold strictly unless a nonnegativity constraint prevents this. These properties could be derived, in the context of an economy with many identical consumers, from properties of the utility function of these consumers, as done by Van den Heuvel (1983). We prefer, however, to assume them directly. ” By this we mean that along the boundary between these two regions an increase in m determines a movement into the Classical region.
FIXPRICE
LABOR-MANAGED
241
ECONOMIES
L m
0 I FIGURES
(Classical)
w(L, PI = Pf’(L) x + g = ~I~w(P, f(L)=.f~(P,m,L)+g
m, x))
(Inflationary) (Keynesian ) .
Another way to perform comparative statics is to use Figs. l-3. An increase in government expenditure g shifts down the curvef( L) - g, an increase in the stock of money m stimulates demand for goods and leisure, and thus shifts upward the curve 2w ( L) and leftward the curve & ( x). An increase in Kreduces the worker’s wage, and thus shifts 2~ ( L) downward and &., ( x) leftward. Since Ward established that increases in K stimulate the LMF’s notional supply, the point (Lz , J$ - g) moves further up along the curve f( L) - g when K increases. Considering these effects of changes in g, m , and K, one can use Figs. l-3 to perform comparative statics analysis, Figure 6 shows the effect of an increase in g in the case of a Keynesian equilibrium. The qualitative effects of variations in g, m, and K on output and employment are given in Table 1. As stated above, changes in g and m have the same qualitative effects in both the LME and the PME. It may seem surprising that an increase in K has an adverse effect on employment in the Keynesian and Inflationary regions, since in the Ward economy notional supply varies directly with K.
242
FERNANDO
B. SALDANHA
FIGURE
6
Note however that it is only in conditions of Classical unemployment that notional supply determines output. In an Inflationary equilibrium, employment is determined by labor supply, which decreases with K. And increases in K reduce demand, and thus employment in the Keynesian region. 3. Fixprice analysis of the Sen Economy In this section we analyze an economy that is basically similar to that of Section 2, in the sense that the same commodities are traded, workers have the same utility functions, and the firm has the same technology. However, we drop the assumption that the firm is an independent decision unit and have workers simply choosing their labor-leisure position given the prices of TABLE 1 Region variable
Classical
Keynesian
Inflationary
m g K
0 0 +
+ +
-
FIXPRICE
LABOR-MANAGED
ECONOMIES
243
goods and the technology of the firm. Thus, employment and labor supply decisions are one and the same. The economy can operate under two alternative regimes, one of excess demand for the good (Inflationary), and the other of excess supply (Keynesian). There is no market for labor, even in the restricted sense of Section 2. That is, in the model of Section 2 there are buyers (firms) and sellers (workers) of labor, but the wage rate is not determined by a market process. In the present model there are no buyers or sellers of labor. In order to keep the model simple we assume that each firm has only one member. In the context of our specific model, Meade’s conjecture is again false: changes in g and m have the same qualitative effects on employment in the PME, the Ward, and the Sen LMEs. 3.1. Workers To simplify matters we assume that each firm has only one member. If this member is unconstrained, he solves the following maximization program: max W.7 L CL) subject to w=pf(L)-K
Let x$(p, m) and L$(p, m) be respectively the notional demand for the good and labor-leisure decision of the worker.‘* At this point it is convenient to drop the assumption of a single good and assume that there are many identical firms producing different goods that must be consumed in fixed proportions. Also, government consumption follows the same pattern as that of private expenditure, that is, g is collinear with x. This allows us to write our equations in terms of aggregates, since the ratios in which goods are produced and consumed remain constant. Without this multiple goods assumption a firm that could not sell its product in the market would still produce for the consumption of its own workers. Given that goods must be consumed in fixed proportions, this cannot happen. Allowing firms to produce for self-consumption would complicate matters considerably, especially with regards to comparative statics analysis. To simplify matters, we also assume that there is a one to one correspondence between workers and firms. Under these assumptions, we let x be the magnitude of the consumption vector and p be the magnitude of the fixed price vector. Ifthemarketdemandisy=x+g
244
FERNANDO
B. SALDANHA
for goods. The representative worker’s effective demand for goods xw ( p , m , ~7)is then given by the solution to mm W,
L P)
subject to w=pf(L)-K px+p
0 c L < Le.
Note that the restriction on the sales of the firm is actually a restriction on hours worked, and thus determines the worker’s effective demand for goods. The worker can also be restricted as a consumer in the goods markets. A worker that cannot satisfy all of his notional demand for goods actually falls short by the same proportion in every good. His labor-leisure decision and therefore his firm’s supply are correspondingly affected. The worker’s effective labor-leisure decision LW ( p, m, 2) is given by the solution to max U-T L P> subject to w=pj-(L)-K px+pGm+w OGXCX,
0 < L < Le.
The firm’s effective supply is then YW(P,
m, 2) =~(Lw(P,
my 2)).
3.2. Equilibrium
Given that we have defined aggregate effective demand and supply functions, we can also define fixprice equilibria. This is done with the help of diagrams that are somewhat similar to the traditional Keynesian 45’ diagram. In Figs. 7 and 8 we plot the functions xw (~7) + g and yw (f - g), along with a 45” line. The condition of voluntariness of exchange means that in equilibrium no agent can trade more than his desired amount. Then no equilibrium can lie above either of the curves xw (~3 + g and yw (jj - g) . Also, if in equilibrium there cannot be unsatisfied agents in both sides of a market, no equilibrium can lie (strictly) below both of these curves. A concept of equilibrium emerges naturally. We say that y” is ahprice equilibrium for the Sen economy if
* Y
y” FIG. 7. Keynesian equilibrium.
FIG.
8. Inflationary equilibrium. 245
246
FERNANDO
B. SALDANHA
If y” is a fixprice equilibrium, one of the curves xw (7) + g and J+,,(9 - g) crosses the 45” line at j?, and the other one is above the 45” line at p. There are three obvious types of equilibrium: (i) 9 is a Keynesian equilibrium (excess supply of goods) if y” = xw (9) f g* (ii) y” is an If lationary equilibrium (excess demand for goods) if y”= yw ( y” - g). (iii) j7 is a Walrasian equilibrium (equilibrium in the goods markets) if y” =xw(Y)+g=yw(9-g). Figures 7 and 8 show respectively a Keynesian and an Inflationary equilibrium. A Walrasian equilibrium (not shown) occurs when both xw (9) + g and yw (jJ - g) cross the 45” line at the same point. The existence and uniqueness of equilibria for this model can be proved under mild assumptions. 3.3. Comparative Statics
As in the previous section, specific assumptions on the signs of derivatives of effective demand and supply functions are needed to perform comparative statics analysis. We assume I3
-axw ama 0, -axwG0,
(S)
aK
The inequalities in (S) are assumed to hold strictly, except when this is prevented by active nonnegativity constraints. Consider first a Keynesian equilibrium. Increases in g and m stimulate demand, and therefore shift up the xw($) + g curve. Hence, employment increases. An increase in K shifts that same curve down, and is therefore contractionary. Now let the equilibrium be Inflationary. Increases in g and m reduce the effective willingness to work, in the first case because there are less goods to be bought, and in the second case because the demand for leisure is stimulated. Hence in both cases the yw (~7 - g) curve shifts down, and employment decreases. An increase in K has the opposite effect, since it is equivalent to a decrease in m. Figure 9 illustrates the expansionary effect of an increase in government expenditures in a Keynesian equilibrium. Except for one case, the qualitative effects of changes in g, m, and K on employment are the same as those in the Ward economy under the similarly named regime. The exception is that, under an Inflationary regime, an increase I3 See the Appendix
for a discussion
of (S) .
FIXPRICE
LABOR-MANAGED
ECONOMIES
247
Y t
0
.
y”
Y
? FIGURE
9
in K reduces output and employment in the Ward economy, if the substitution effect dominates the income effect, and has the opposite effect in the Sen economy. I4 Once more, Meade’s conjecture that Keynesian policies would be at best ineffective in reducing unemployment is not vindicated. 3.4. Eficiency and the Price Level In the Sen economy the equilibrium allocation is completely determined as a solution of the worker’s optimization problem. So one is tempted to believe that equilibria are necessarily Pareto efficient. However, this is not the case, as is clear from the possibility of unemployment. The reason why inefficient outcomes are possible is that K and m are fixed in nominal terms. An increase in p implies a smaller real value of K, which is a gain for the worker. If savings are positive (p > m), the worker benefits from a higher p in another way: it is less costly in real terms to achieve a given level of savings.‘5 Only when dissavings are exactly equal to K (i.e., when m - p = K) the loss I4 The difference between the signs of $+,/aK in (S) and of the corresponding &&,/aK in ( WLM) is due to the absence of a substitution effect in the first case. In the Sen economy a change in K does not affect the worker’s perceived marginal trade-off between leisure and consumption. Is Assuming that the elasticity of price expectations with respect to present prices is zero.
248
FERNANDO
B. SALDANHA
due to the reduced real value of these dissavings is exactly sufficient to compensate for the gain due to a reduced real value of K, so that a marginal increase in p does not affect the worker’s welfare. This is precisely the condition for Pareto optima&y. These statements can be justified more formally. It follows easily from the Envelope theorem (cf. Varian, 1984) that, for either one of the worker’s optimization problems in Section 3.1,
-au* = X(y- x), ap where U* is the optimal value of the corresponding problem, and X is the Lagrange multiplier associated with the budget constraint. Also, from wL=py-K=px+p-m it follows that p(y-x)=K+(p-m). We conclude that dU*/dp
= 0 is equivalent to K = m - P.
4. Conclusions We develop two fixprice models for labor-managed economies under alternative assumptions about decisions on employment and labor supply. In both cases the qualitative effects of variations in government expenditures and the money supply are the same as those that apply in a PME. That is, increases in these variables am expansionary in situations of generalized excess supply and contractionary in conditions of generalized excess demand. In no case do our results support Meade’s conjecture that Keynesian policies are ineffective or even counterproductive in the LME. Of course, Meade’s conjecture is based on the well-known perversity of the LMFs supply curve in a Ward economy. In our model prices are fixed, so this perversity could not play any significant role. Naturally one is then led to the question of the validity of Meade’s conjecture in the long run, with flexible prices. The answer may well be different for each of the two types of models we consider. These issues constitute important topics for further research. APPENDIX Discussion of ( WLMj . The first-order conditions that determine ?w (p , w , m, L) and & ( p, w, m, 2) in interior solutions where the quantity constraints are binding respectively reduce to
ux u,=”
and
UL -=-w. U#
RXPRICE
Differentiating
* dK
LABOR-MANAGED
249
ECONOMIES
these conditions we obtain
= L aw a3W
ai., -=
dK am ’
at, i& -3 am
aK
1
if!!
A2 aK
a.cw a(d) azw u,, - pu, -=v-aL
a& az
-=-p=+
aL
a&
am
A1
aw -1 --aw al, az aK
[1 E
u, + WV,, A2
The terms A1 and A2 are negative, from the second-order conditions. It follows that if
u,,>,oa ULr,
(A.11
the first three inequalities in (WLM) hold. (A.1 ) is satisfied if the utility function is separable or if it has the Cobb-Douglas form. The effect of a change in K on labor supply can be divided into a substitution and an income effect. Given (A. 1) , these effects work in opposite directions. If the substitution effect is the stronger of the two, which means the effective labor supply function is upward sloping (aLW/dw > 0)) the fourth inequality in (WLM) is also satisfied. When z is increased, there are two effects on the demand for the good. First, there is an income effect, which, given (A. I), stimulates consumption. Second, there is a shift in the relative desirabilities of consumption and money holdings. If consumption and leisure are strongly complementary, this effect can dominate the 6rst effect and determine a net fall in consumption. The fifth inequality in (WLM) assumes away this possibility. Finally, the effect of an increase in 2 can be divided into three parts. First, the income available for the purchase of leisure decreases. Second, the wage rate increases, which in itself has a substitution and an income effect. And third, there is a shift in the relative desirabilities of leisure and money holdings. If (A. 1) holds and the effective labor supply function is upward sloping, the first two effects work in the same direction, and the last inequality in (WLM) also holds, unless there is strong complementarity between consumption and leisure. Discussion of (22). The first-order conditions that determine xw ( p, m , L) and Lw ( p, m, 2) in interior solutions where the quantity constraints are binding respectively reduce to
UX -= UP p
and
UL - = -pf’(L). u,
250
Differentiating
FERNANDO
B. SALDANHA
these conditions, we find ax, ---=- ax” -$i? dK
Kit - PUW A3
afi = --=-ati
ULF
-
dm
+
dK
Pf ‘(L) 4
u,, ’
where the terms A3 and 4 are negative if the second-order conditions are satisfied. It follows that (A. 1) is a sufficient condition for (S). Proof of Theorem 1. The hypothesis of the theorem (E # a) implies that Q # 0, where Q is as in Definition 3. Given the last two inequalities in ( WLM), it is easy to see that \k maps each of the connected components of Q into itself. The theorem now follows from Theorem 2 and the Brouwer fixed point theorem. Proof of Theorem 2. (Necessity). Suppose that (z, 2) is a fixprice equilib rium and
WQ)
= (kf(X)-g),
where X # L. Clearly, from (7), (8), and the definition of 9, i < A, and hence l< X. Then ( 10) and ( 12) imply that e = X = cc and (7) implies that L = LF(P, x + g).
(A.2)
From ff = co it follows that LF(P,
if
+ g)
=
LF*
(A.3)
(P)-
Putting (A.2) and (A.3) together we find a contradiction of L < X. (Sufficiency). Suppose that (l, 2) is a fixed point of 9. Define L=L
if
It = &(p,
E=oo jf=f
otherwise if
x=a3
otherwise.
x” = &(p,
m, 2)
m, i)
We must show that ( 7) and ( 8) are satisfied for this choice of (E, X). There are three possible cases: ( 1) L’ = i&p,
m, 2’) 6 L:(p).
Suppose that (7) does not hold. Then
L&p, 2 + g) < x G L:(p), and hence X < cc. It follows that X = 2 = iw(p,
m, L),
(A.41
FIXPRICE
LABOR-MANAGED
251
ECONOMIES
and therefore LF(p, 2 + g) = LF(p, x” + g) = L&f(i)). Then, from (4), either LF(P, 2 + g) = L’ or LF(P, 2 + g) = L:(p), and in both cases we have a contradiction of (A.4). Now suppose that (8) does not hold. Then there are two possibilities: (i) J&P, L) - g -C R. A reasoning similar to the above then obtains a contradiction. (ii) 2 < min ( yF(p, E) - g, &( p, m, L)} . Then, using the definition of \k, we find that z = L” = iw(p, m, x”), hence J&PI a - g
- g = x”,
and again we have a contradiction. (2) L’ =f-‘(&(p, m, z) + g) < L:(p). A reasoning analogous to that developed for case 1) shows that (7) and (8 ) are satisfied. (3) L = L:(p) < &(p, m, x”) and 2 = y:(p) - g G &(p, m, L). It follows that I:
=
L:(P)
=
LF(P,
d(P))
=
LF(P,
2 +
g)-
(A.5)
Now, 2 can take two possible values, 2 or cc, and it follows from (A.5) that in both of these cases (7) is satisfied. A similar reasoning shows that (8) holds. Since (7) and (8) are satisfied in all three cases, the theorem now follows from the observation that (9)-( 12) hold by construction. Proof of Theorem 3. It is easy to see that d(W(P, Jw) aJ5
and hence (U) implies that a.&
x
= pf’(Jc).
af,-1. [ 1
< ajz
(A-6)
It follows immediately from (A.6) that the set G of Definition 3 can intersect the boundary dE of E at most once. If G fl aE = @ then there exists a unique fixprice equilibrium, which must be of the Classical type. If G Cl aE # 0 then the point of intersection corresponds to a unique fixprice equilibrium, which may be of either the Keynesian or the Inflationary types. Proof of Proposition 1. Homogeneity implies that LG and x$ can be written as functions of real (as opposed to nominal ) quantities. From ( WW) , the Implicit Function theorem can be used to define two functions m’(p) and m*(p) as
FERNANDO
252
B. SALDANHA
where w*(p) is the optimal value of the firm’s maximization problem at price p. Since L$ and xz are monotonic functions of m, the functions m ’ and m* cannot have more than one branch. Also, it is easy to compute that
aLF* -____
am’ -=-+p
m
ap
P
(( ap
m
ad ---
P
(( ap
am* -=-+p ap
a&$ a(w*jp) ww
. )/&)
ap
a&
a(w*ip)
~(WIP)
)/$$-y).
at,
(A-7)
(‘4.8)
Now, a quick calculation shows that
aw*m = K ap P2LF*(P)
>o .
Substituting in (A.7) and (A.8) and taking the difference, it follows immediately from homogeneity, ( WW), and the perversity of the LMF’s supply curve that am2 --ap
am1 < 0, ap
and the curves m ’ and m* cross at most once. The proposition now follows from the observation that m* = m’(p*) = m’(p*) if and only if there is a Walrasian equilibrium (L*, x* ) such that L* = LF(p*)
-W*(P)
= L; [
P*
x*=x;t1 -1 w*(p*) P*
-m* ’ P*
1
m*
’ P*
=YF*(P*)--g
associated to (p*, m*, g, K). REFERENCES Bar~o, Robert J., and Grossman, Herschel I., “A General Disequilibrium Model of Income and Employment.*’ Amer. Econ. Rev. 61, 1:82-93, Mar. 1971. Benassy, Jean-Pascal. The Economics of Market Disequilibrium. New York: Academic Press, 1982. B6hm, Vblker, “Disequilibrium Dynamics in a Simple Macroeconomic Model.” J. &on. Theory 17,2:179-199, Apr. 1978.
FIXPRICE
LABOR-MANAGED
ECONOMIES
253
Bonin, John P., “The Theory of the Labor-Managed Firm from the Membership’s Perspective with Implications for Marshallian Industry Supply.” .I Camp. Econom. 5, 4:337-351, Dec. 1981. D&e, Jacques, “Some Theory of Labor-Management and Participation.” Econometrica 44,6: 1125-l 139, Jun. 1976. Fellner, William, “Prices and Wages under Bilateral Monopoly.” Quart. .I. Econ. 61, 503-532, Aug. 1947. Grandmont, Jean-Michel, “On the Short-Run Equilibrium in a Monetary Economy.” In Jacques D&e, Ed., Allocation under Uncertainty: Equilibrium and Optimality, pp. 2 13-228. London: Macmillan & Co., 1974. Ichiishi, Tatsuro, “Coalition Structure in a Labor-Managed Economy.” Econometrica 45,2:341360, Mar. 1977. Ichiishi, Tatsuro, “A Social Coalition Equilibrium Existence Lemma.” Econometrica 49,2:369378, Mar. 1981. Law, Peter J., “The Illyrian Firm and Fellner’s Union-Management Model.” J. Econ. Stud. 4, 1:29-37, May 1977. Malinvaud, Edmond, The Theory of Unemployment Reconsidered. Oxford: Blackwell, 1977. Meade, James E., “The Theory of Labour-Managed Firms and of Profit Sharing.” Econ. .I. 82, 325s:402428, Mar. 1972. Miyazaki, Hajime, and Neary, Hugh M., “The Illyrian Firm Revisited.” Bell. .I. Econ. 14, 1:259270, Spring 1983. Schulz, Norbert, “On the Global Uniqueness of Fix-Price Equilibria.” Econometrica 51, 1:4768, Jan. 1983. Sen, Amartya K., “Labour Allocation in a Cooperative Enterprise.” Rev. Econ. Stud. 33,4:361371, Oct. 1966. Steinherr, A., and J-F. Thisse, “Are Labor-Managers Really Perverse?” Econ. Lett. 2: 137-142, 1979. Svensson, Iars E. O., “Effective Demand and Stochastic Rationing.” Rev. Econ. Stud. 41,2:339356, Jan. 1980. Tyson, Laura D., The Yugoslav Economic System and Its Pe$ormance in the 1970s. Institute of International Studies, University of California, Berkeley Research Series, 44, 1980. Vanek, Jamslav, The General Theory of Labor-Managed Market Economies. Ithaca, NY/London: Cornell Univ. Press, 1970. Varian, Hal R., Microeconomic Analysis. New York: Norton, 1984. Ward, Benjamin, “The Firm in Illyria: Market Syndicalism,” Amer. Econ. Rev. 48, 4:566-589, Sept. 1958.
OF COMPARATIVE
ECONOMICS
13,227-253
(1989)
Fixprice Analysis of Labor-Managed Economies’ FERNANDO B. SALDANHA World Bank, Washington, DC 20433 Received July 15, 1987; revised October 5, 1988
Saldanha,FernandoB.-Fixprice
Analysis of Labor-Managed Economies
The macroeconomics of labor-managed economies ( LMEs) are investigated in the context of fixprice theory. Two models are constructed. In a traditional Ward LME, firms maximize profit per worker, and workers supply labor taking the wage rate as given. In a 8en LME, workers decide jointly on employment and leisure. In both cases fixprice equilibria exist and are unique, under reasonable assumptions. The three traditional regimes (Classical, Keynesian, and Inflationary) of traditional fixprice theory are possible in the Ward LME. In the simpler Sen LME the only possibilities arc excess supply or demand for goods. A conjecture of Vanek and Meade is shown to be false in the context of either model: monetary and fiscal policies have the usual qualitative effects in the LME and in the profit-managed economies. J. Comp. Econom., June 1989, 13(2), pp. 227-253. World Bank, Washington, DC 20433 o 1989 Academic press, Inc.
Journal of Economic Literature Classification Numbers 022,023, 052.
1. INTRODUCTION A labor-managed economy (LME) is distinguished by three operating rules: ( 1) a firm is managed by its members, who also comprise its labor force; (2) a firm’s members are residual income claimants, thus their collective income is the firm’s profit; and (3) firms and consumers operate autonomously, with interactions among agents occurring through free markets. The original analysis of the labor-managed firm (LMF) is due to Ward ( 1958 ) . He assumed that a LMF is required to evenly distribute profits among all its members, making the LMF a perfectly egalitarian cooperative, and proved that if the firm maximizes per-member profits, or wages, then it has ’ I am grateful to Jean-Pascal Bcnassy, David Conn, Deborah Hartz, and two anonymous referees for useful comments. The College of Business and Public Administration, University of Arizona, has generously provided financial assistance for this project. Of course, any errors are mine. 227
0147-5967189 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
228
FERNANDO
B. SALDANHA
a negatively sloped output supply function. This result is guaranteed if labor is the only variable input for the LMF and is likely when there are several variable inputs. Since the publication of Ward’s seminal work the literature on the LMF and the LME has expanded rapidly. The hypothesis of the LMF’s perverse supply curve was extensively discussed ( Meade, 1972; Vanek, 1970; Steinherr and Thisse, 1979), and another branch of the literature focused on the efficiency of general equilibrium (D&e, 1976; Ichiishi, 1977, 198 1) . In contrast, almost no attention was given to the macroeconomics of LMEs. Some remarks on the problems of unemployment and inflation are scattered in the literature (Vanek, 1970; Meade, 1972; Tyson, 1980)) but there was no attempt to elaborate a specific macrotheory of the LME. We begin to fill this gap by constructing a disequilibrium macromodel of the LME. Our analysis is developed in the theoretical framework of “disequilibrium” or “fixprice” theory (Barr0 and Grossman, 197 1; Malinvaud, 1977; Benassy, 1982). The book by Benassy ( 1982) is a good reference for the reader unfamiliar with fixprice theory. Both Vanek ( 1970) and Meade ( 1972) conjectured that the perversity of Ward’s LMF supply curve might lead to macroeconomic instability. Meade claimed that “to rely on Keynesian policies to expand effective demand in times of unemployment would be at best ineffective, and at worst might lead to a reduction in output and employment.” In Section 2 we analyze this question in the context of Ward’s model and find out that the conjecture is false in the short run. The Ward economy may operate under the same three regimes, Classical, Keynesian, and Inflationary, as the PME, and all shortrun multipliers have the same signs that they would have in a capitalist economy. Ward’s assumption that the LMF maximizes per-member wages has been criticized by several authors ( Steinherr and Thisse, 1979; Miyazaki and Neat-y, 1983) on several grounds. Most relevant in our context is the fact that in Ward’s model what essentially constitutes a single decision of workers about employment and labor supply is artificially split, so that the firm maximizes wages per worker, and workers, taking that wage as given, choose their labor supplies.* More realistically, the firm as an independent decision unit should be abolished from the analysis, and workers should choose the employment level so as to maximize their utilities. Sen ( 1966) was the first to introduce a model of this type. Unfortunately, realistically modeling the Sen LME can involve considerable complication. In the models of Steinherr and Thisse ’ Here we follow D&x ( 1976), who assumes that the worker treats the wage rate as given when choosing hi labor supply, even though it depends on the amount of labor employed. Alternatively, one can interpret the worker’s behavior in Fellner’s ( 1947) bilateral monopoly framework. Law (1977) shows that ifthe worker’s (or the union’s) indil%rencz curves are horizontal the firm displays the behavior postulated by Ward.
FIXPRICE
LABOR-MANAGED
229
ECONOMIES
( 1979) and Miyazaki and Neary ( 1983), for example, workers have to cope with uncertainty, since layoffs are random. We adopt instead a simple model in which each firm has a single member and analyze its macroeconomic properties in the context of fixprice theory. The worker never considers “quitting,” that is, dissolving the firm, and of course cannot be laid off. He faces only a labor-leisure choice. Although our model is quite specific, one conclusion generalizes to the whole class of models: there are only two possible regimes for the economy, one with excess demand for the good (Inflationary) and the other with excess supply (Keynesian). This is because there is no supply or demand for labor, but only a labor-leisure decision. It also turns out that Meade’s conjecture is false in our specific version of the Sen economy. The structure of the paper is as follows. In Section 2 we analyze the macroeconomics of the Ward economy. This section is subdivided into subsections dealing with definitions and setting up of the model, existence and uniqueness of equilibrium, and comparative statics. Section 3 contains a similar analysis of the Sen economy. Finally, Section 4 contains our conclusions and some directions for further research. 2. FIXPRICE
ANALYSIS
OF THE WARD
ECONOMY
In this section we develop a fixprice model of a Ward economy. Firms are modeled as independent agents that maximize wages per worker. Workers take wages as given and behave as their counterparts in the PME.
2.1. The Economy We consider an economy with three commodities: a consumption good, labor, and money. No future contracts are allowed. The price p of the consumption good is exogenously fixed, and the good market clears by quantity rationing on the long side. There is no labor market in the usual sense. Firms are labor-managed, and maximize per worker revenue net of nonlabor costs, as in Ward ( 1958). This average net revenue w is henceforth called the wage rate, since it plays a role similar to that of the wage rate in the PME. Both the demand for labor and the desired (by firms) wage rate are functions of the good price. This means labor demand is not a function of the wage rate. Labor supply is, as in a capitalist economy, a function of the wage rate. One can still speak of excess demand or supply of labor, but such conditions do not affect the wage rate according to the Law of Supply and Demand.3 Money is the only store of value. Agents are one representative consumer-worker, one representative firm, and the govemment.4 3 This means a situation of excessdemand (supply) for labor does not necessarily imply that the wage rate must rise (fall). ’ All of the analysis that follows could have been developed in an economy with many identical firms and consumers. For simplicity, we assume that there is only one agent of each type.
230.
FERNANDO
B. SALDANHA
In a Ward economy, the employment and labor supply decisions are artificially split. The firm maximizes wages per worker, and the worker takes this behavior of the lirrn as given and maximizes preferences given prices, the wage rate, and endowments. 2.2. Notional and Efective Demands The distinction between notional and eflective demands and supplies is fundamental in fixprice analysis. Notional demands and supplies depend only on prices and kndowments, these are the standard demands and supplies of microeconomic theory. Effective demands and supplies also depend on quantity signals, usually in the form of quantity constraints. Several alternative formulations of effective demand have been proposed in the lixprice literature.5 The one we adopt is based in the dual-decision hypothesis. We assume that agents face quantity constraints on their purchases and sales and that an agent’s demand or supply for a commodity is determined by maximizing the agent’s objective function given all quantity constraints except for the one referent to that commodity. We return to this point below when we define the effective demands and supplies of the firm and the worker. 2.3. The Firm The labor-managed firm has a production function f: I?+ + R,, twice continuously differentiable, monotonic, and strictly concave. There is also a fixed (in nominal terms) cost K. Let L denote the membership of the firm. Then the wage rate is given by *(L
9
p)=pf(L)-K
L
*
(1)
The notional labor demand Lc (p, K) is given by arg max w( L, p) .6 IA y:(p), K) be the notional supply of the consumption good. Of course, yc(p, K) = f( L:(p, K)). It is well known that 5 Svensson ( 1980) discusses several concepts of effective demand. 6 In this section we adopt the following conventions on notation. (a) Subscript capital letters denote different types of agents: W stands for the worker, F for the firm. (b) There are three types of demand and supply functions: notional, effective, and LM-effxtive. Notional demands and supplies am denoted by a superscript *. Worker’s LM-effbctive demand and supply functions, de&xi in Section 2.4, are topped by a caret -. Traditional effective functions have no superscripts or carets. (c) A tilde - over a variable indicates a fixprice equilibrium value of that variable.
FIXPRICE
LABOR-MANAGED
MP, m < o ap
ECONOMIES
231
(2)
(Ward’s Theorem ) (3)
Now, the firm can be constrained in the labor or good markets, whence the need for effective demand and supply schedules.7 For example, if the firm cannot sell all it would desire to sell at a given price, then its demand for labor is correspondingly affected: it demands just enough labor to produce the quantity demanded. Similarly, a constraint on the available manpower determines a reduction of output supply. The effective demand and supply schedules of the firm must then have quantities as arguments: good demand for the effective demand for labor function and labor supply for the effective output supply function. Let &(p, jj) and ~+(p, L) be respectively the effective demand for labor and the effective supply of goods. Our definition of effective demands and supplies, which is based on the dual-decision hypothesis, implies that &(p, J) is the solution of
subject to
L 2 0.
Note that the firm’s effective demand for labor depends only on the quantity constraint on the goods market. The firm cannot sell more than $, and its demand for labor is correspondingly affected. The quantity constraint on the labor market itself does not appear in the optimization problem above. This is the meaning of the dual-decision hypothesis. The firm’s effective supply of goods y~(p, L) is given as the solution of max
PY
L
subject to Y =f(L) OGLGL. ’ It turns out that the firm cannot be constrained simultaneously in both markets. It follows that a disequilibrium regime with excess supply of goods and excess demand for labor is not possible in the context of the present model.
232
FERNANDO
B. SALDANHA
Again the dual-decision hypothesis implies that the constraint in the good market is ignored, and only the constraint in the labor market is considered. The firm can hire at most L of labor, and this affects its supply of the good. Wage maximization implies Y>- d(P) YG A(P)
(4)
LkL,*(p) L G L F*(p). The equations (4) can be interpreted as follows. If the good demand is enough to absorb the firm’s notional good supply (~7 2 J$ ) then effective and notional labor demand coincide. Essentially, the firm is unconstrained in the good market, so the ineffective constraint does not affect its labor demand. But if jj < J$ then the firm just demands enough labor to produce the amount of output it can sell. Similarly, if labor supply is enough to produce the notional supply of the good (12 L: = f - ’ ( J$ ) ) , then effective and notional good supplies coincide. The firm is essentially unconstrained in the labor market, and the ineffective constraint L does not affect its supply of the good. On the other hand, if L < Lz then output must be correspondingly reduced. 2.4. The Worker Given that we are dealing with a Ward economy, in which employment and labor supply decisions are separately made, the notional and effective demand and supply functions of the worker are the ones used in the standard fixprice theory of the profit-managed economy (PME) . Let x be the consumption of the private sector, in real terms. We denote by L$(p, w, m) and by x$(p, w, m) the worker’s notional labor supply and good demand functions. Now, the worker can be rationed in one or both of the labor and good markets. This may affect its demand for the good or supply of labor. If the worker cannot sell his labor time, then he has to reduce his purchases of the good. Similarly, a constraint on the availability of the good is likely to adversely affect the labor supply: there is no incentive to work if there are no goods to be bought. The dual decision hypothesis implies that the worker’s effective demand for goods xw(p, w, m, L) is given by the solution to mm Vx, L, II) subject to px+pLm+wL 0 < x,
O
FIXPRICE
LABOR-MANAGED
ECONOMIES
233
Here U is of course the worker’s utility function, which depends on current consumption x, on current labor supply L, and on money balances CLcarried to future periods. Le is the worker’s endowment of labor. Money enters indirectly in the utility function. That is, it is not necessary to assume that the worker can derive direct satisfaction from the possession of money to write U in the form above. One can see the worker as solving a multiperiod optimization problem, where future utility depends on the amount of money carried from period 1 to period 2. Grandmont ( 1974) establishes the conditions under which a utility function of the form above is well defined and has the usual properties of monotonicity, continuity, and quasi-concavity. Similarly, the worker’s effective supply of labor LW ( p, w, m, X) is given by the solution to max W,
L P)
subject to px+p
0 =s L < Le.
Of course, if the quantity constraint faced by the worker in a market is greater than his notional demand or supply in that market, then that constraint remains inactive. This means that the functions xw and Lw must also satisfy XW(P,
W, m,
G
=
x~%P,
W, m)
if
L > LG(p, w, m)
Lw@,
w,
2)
=
G-b,
w,
if
2 2 x$(p,
m,
m)
w, m).
In our model, the wage rate w is a function of the employment level, unlike the wage rate that is taken as fixed in fixprice models of PMEs. It would be a mistake to treat w as an exogenously fixed variable, as is usual in fixprice analyses of the PME. The functions xw and LW are then only indirectly useful in the analysis of the LME: defining them is the first step in the construction of LM-effective demand and supply functions that take into account the dependence of w on the output and employment levels. Suppose that the worker perceives a quantity constraint in the labor market at E, and E is also the employment level. It follows that the wage rate is w( E, p) . Accordingly, we define the worker’s LM-effective demand as m, L) = xw(p, w(E,p>,
m, t).
(5) Similarly, if there is a constraint 2 in the good market, g is real government expenditure, and the output level equals X + g, then the worker is paid at the rate w(f-’ (X + g) , p), and his LM-effective labor supply is Zw(p,
~w(P,
m,
Xl
E Lw(P,
W-‘(X+
g)),
m,
Xl.
(6)
In order to prove the existence of equilibrium and perform comparative statics analysis one has to make specific assumptions about the signs of the
234
FERNANDO
derivatives of the LM-effective sumptions is*
B. SALDANHA
functions of workers. A set of reasonable as-
a.fw,O
aLwGo
2%”
am
azw d o aK azwao Tic-
aLw ~ o ’
aK
’
aiw,o --z-”
(WLM)
The inequalities in (WLM) are assumed to hold strictly, except when this is not possible due to a nonnegativity condition, or when a quantity constraint is not binding. 2.5. Existence of Equilibrium The existence of fixprice equilibria for capitalist economies has been established (cf. Bohm, 1978 ) by means of an essentially graphic argument. Here we use an alternative line of reasoning, based on Brouwer’s fixed point theorem. A notion of equilibrium for a model like the one described above has to differ from the usual concept of a Walrasian equilibrium, in which notional supplies equal notional demands. The essence of ajixprice equilibrium is that it is a fixed point in the space of quantity constraints. That is, if agents expect to face a given vector of quantity constraints, then their behavior is such that they actually face those constraints. DEFINITION 1. We say that (z, 2) is a fixprice equilibrium for the economy defined by (p, m, g, K) if f(z) = 2 + g and there are (E, 2) such that
Z = min { L&p, X + g), Lw(p, m, a)}
(7)
2
(8)
=
min(fi(P,
0
-g,
2~0,
m,
L)},
where E=L
if
L = Lw(p,
m, 2)
JC=oO
otherwise
(10)
x=x”
if
(11)
X=cO
otherwise.
i = ~?~(p, m, L)
(9)
(12)
Definition 1 can be interpreted as follows. First, workers face quantity constraints that are given by the actual output and employment levels C?+ g and * See the Appendix for a discussion of the plausibility of ( WLM ) .
FIXPRICE LABOR-MANAGED ECONOMIES
235
L’. Second, the same is true for firms in a market (I= L” or X = x”) if workers areontheshortsideofthatmarket(~=~w(p,m,x”)or~==w(p,m,L1)), but otherwise firms are unconstrained (E = cc or X = CO). Third, exchange is voluntary; the left-hand sides of (7) and (8) cannot be strictly greater than the respective right-hand sides. Finally, there cannot be unsatisfied agents in both sides of a market; the left-hand sides of (7) and (8) cannot be strictly smaller than the respective right-hand sides. The fact that constraints are equal to actual quantities embodies the idea that a fixprice equilibrium is a fixed point in the space of constraints. Given Definition 1, we can prove THEOREM 1. Given (p, m, g, K) such that m > 0, suppose that the set E consisting of all (L, x) E l!?$ such that
&(~,m,L)~f(L)-g Gv(P,
m,
xl
2
f-lb
+ g)
is nonempty. Then aJixprice equilibrium for the economy defined by (p, m, g, K) exists. Proof The proof depends on Theorem 2 below. See the Appendix. We now introduce an alternative but equivalent definition of fixprice equilibrium as a fixed point of a mapping. DEFINITION 2. Let G be defined by
G=
{(L,x)ER:Ix+g=f(L)<
y;(p)}
and let Q = G fl E, where E is defined as in the statement of Theorem 1. Define \k: !J --* R by
WL,x) =(&f(X)-g), where X = mini
Lb,
m,
x)X’(&(P,
m,
L)
+ g),
G(P)}.
Here G is the set of all (L , x) that can be voluntarily produced by the firm at the given prices, and E is the set of all (L , x) that can be voluntarily traded by the worker. Clearly Q = G rl E is compact, and \k is well defined and continuous. For any r = (L, x) E a, \k( {) is the largest (further away from the origin) allocation in Q such that the quantity constraints on households determined by { are satisfied. That is, if the initial allocation r is such that the corresponding quantity constraints on households are satisfied, then P expands output and employment as much as possible given that those same constraints remain satisfied. And if ris not compatible with voluntary exchange on the part of the households, the q contracts output and employment so that the quantity constraints associated to 5 are just satisfied.
FJ%NANDO B. SALDANHA
236
THEOREM 2. Let (p, m, g, K) bejixed. Then (L, x”) is afixprice equilibrium for the economy defined by (p, m, g, K) if and only if(L, x”) E Q and q(L, 2) = (L, 2). ProoJ See the Appendix.
There are several possible types of fixprice equilibria, which are also called regimes of the economy.
DEFINITION 3. Let (z, 2) be an equilibrium
for the economy defined by
(p, m, g, K). We say that (2, 2) is:
(i) A Classical equilibrium if L” < t&p, m, x”) and 2 < a,(~, m, z) (Excess supply of labor, excess demand for the good.) (ii) A Keynesian equilibrium if L” < &(p, m, 2) and 2 = y~(p, w, E) - g. (Supply exceeds demand in all markets. ) (iii) An Inflationary equilibrium if L’ = LF(p, w, X + g) and x” < a,(~, m, L”). (Demand exceeds supply in all markets.) (iv) AWalrasianequilibriumifL”=I&(p,m,Z)= LF*(p)andZ=&(p, m, L”) = yz (p) - g. (Supply equals demand in all markets, firms are unconstrained.) This classification of equilibria is not exhaustive, but will serve for our purposes. Each of these types of equilibria corresponds to a region in (p, m , g, K) space. Other types of equilibria could be defined, corresponding to the boundaries between these regions. Figures 1-4 depict the different types of equilibria in (L , x) space. In these figures three types of curves are depicted. The curve Lw (x) gives the worker’s LM-effective labor supply as a function of his constraint in the good market. The curve Zw ( L) gives the worker’s LM-effective good demand as a function of his constraint in the labor market. The curve f( L) - g shows output net of government demand, the output available to the worker. Inflationary and Keynesian equilibria are given by the intersection off ( L) - g with one of the two other curves. When the firm’s notional point (Lz , yr - g) lies in the interior of the region delimited by the two curves -Ilw ( L) and &v ( x) that point is a Classical equilibrium. Finally, if (Lc , yr? - g) coincides with the point of intersection of 2w ( L) and tw (x) then this triple intersection point is a Walrasian equilibrium. The reader should check that these are the only points where Definition 1 is satisfied and that there is a correspondence between Figs. l-4 and the different types of equilibria of Definition 3. 2.6. Uniqueness of Equilibrium
In order to prove the uniqueness of fixprice equilibrium, Bohm ( 1978) essentially assumes that the slope of the net production function f( * ) - g is
L
L FIG.
FIG.
1. Classical equilibrium.
2. Keynesian equilibrium. 231
238
FERNANDO
FIG.
FIG.
B. SALDANHA
3. Inflationary equilibrium.
4. Walrasian equilibrium.
239
FIXPRICE LABOR-MANAGED ECONOMIES
always greater than the slope of the effective demand function x( * ) and smaller than the slope of the inverse labor supply function L$ ( * ) . He justifies this assumption relating it to the well-known condition that the marginal propensity to consume be less than one. More generally, a relaxation of a constraint on sales (purchases) should not generate increased purchases (sales) of greater value.’ This condition was also used by Schulz ( 1983) to prove the uniqueness of fixprice equilibrium for a capitalist exchange economy and in our notation may be formalized by (13) aLw ax
P
(14)
In the context of our model it is sensible to replace ( 13 ) and ( 14) by
agw aoa, 00 pTZ< aE
(15) W)
a(w(p,E)L) ai:, -1 aL cp a2 *
L-1
(16)
The first inequality in (U) states that if the worker’s quantity constraint in the labor market is relaxed, so that he can sell additional labor, then not all of the additional revenue is spent on the good, but part of it is saved in the form of money. Similarly, ( 16) states that if the worker’s quantity constraint in the good market is relaxed, so that he can purchase additional units of the good, then part of those additional purchases will be financed out of savings. THEOREM 3. Suppose that the economy defined by (p, m, g, K) satisfies the conditions of Theorem 1 and ( U) is satisfied. Then there exists a unique jixprice equilibrium.
Proof: See the Appendix. 2.7. Representation
of Equilibria
in Parameter Space
In this section we let g and K be fixed and study the representation of equilibria in (m , p) space. The following proposition establishes conditions under which there is at most one Walrasian equilibrium. ’ Note however that these conditions are not completely analogous to the condition that the marginal propensity to consume is less than one. This is because the relaxation of a constraint affects by itself the demand or supply in the other market, independent of its effect on the worker’s income.
240 PROPOSITION
FERNANDO
B. SALDANHA
1. Suppose that the functions L$ and x$ are homogeneous
of degree one in their respective arguments, and as.wme’O
aG,, ap MP,
'
-aL;: G 0, am 0, ml
= 0,
aL;t aw
-20 V(P,
m)
ww
Then there exists at most one pair (p*, m* ) such that there exists a Walrasian equilibrium for the economy defined by (p*, m*, g, K). Proof: See the Appendix. If the conditions of Proposition 1 are satisfied the whole (m, p) space can be divided into four connected regions. Three of these are the usual Classical, Keynesian, and Inflationary regions, while the fourth corresponds to those values of m and p such that no equilibrium exists. Assuming ( WLM), it is easy to verify the following assertions: (i) The Classical region lies to the right of the Keynesian region.” (ii) The Inflationary region lies to the right of both the Keynesian and the Classical regions. Figure 5 depicts the typical partition of (m, p) space into regions corresponding to the different regimes of the economy. 2.8. Comparative Statics Under mild assumptions, we can perform simple comparative statics for the LME. It turns out that Meade’s ( 1972) conjecture that expansionist fiscal policies never have a positive effect on output level is incorrect when shortrun effects are considered. In fact, changes in g and m have the same shortrun qualitative effects as those in a PME. The government may also wish to use K as a policy instrument. For example, if the fixed costs denoted by K are partially due to taxation, the government can vary the tax rate. Given ( WLM ) , comparative statics analysis can be done in two alternative ways. First, one can implicitly differentiate the equations that define Classical, Inflationary, and Keynesian equilibria. These equations are ” The inequalities in (WW) are assumed to hold strictly unless a nonnegativity constraint prevents this. These properties could be derived, in the context of an economy with many identical consumers, from properties of the utility function of these consumers, as done by Van den Heuvel (1983). We prefer, however, to assume them directly. ” By this we mean that along the boundary between these two regions an increase in m determines a movement into the Classical region.
FIXPRICE
LABOR-MANAGED
241
ECONOMIES
L m
0 I FIGURES
(Classical)
w(L, PI = Pf’(L) x + g = ~I~w(P, f(L)=.f~(P,m,L)+g
m, x))
(Inflationary) (Keynesian ) .
Another way to perform comparative statics is to use Figs. l-3. An increase in government expenditure g shifts down the curvef( L) - g, an increase in the stock of money m stimulates demand for goods and leisure, and thus shifts upward the curve 2w ( L) and leftward the curve & ( x). An increase in Kreduces the worker’s wage, and thus shifts 2~ ( L) downward and &., ( x) leftward. Since Ward established that increases in K stimulate the LMF’s notional supply, the point (Lz , J$ - g) moves further up along the curve f( L) - g when K increases. Considering these effects of changes in g, m , and K, one can use Figs. l-3 to perform comparative statics analysis, Figure 6 shows the effect of an increase in g in the case of a Keynesian equilibrium. The qualitative effects of variations in g, m, and K on output and employment are given in Table 1. As stated above, changes in g and m have the same qualitative effects in both the LME and the PME. It may seem surprising that an increase in K has an adverse effect on employment in the Keynesian and Inflationary regions, since in the Ward economy notional supply varies directly with K.
242
FERNANDO
B. SALDANHA
FIGURE
6
Note however that it is only in conditions of Classical unemployment that notional supply determines output. In an Inflationary equilibrium, employment is determined by labor supply, which decreases with K. And increases in K reduce demand, and thus employment in the Keynesian region. 3. Fixprice analysis of the Sen Economy In this section we analyze an economy that is basically similar to that of Section 2, in the sense that the same commodities are traded, workers have the same utility functions, and the firm has the same technology. However, we drop the assumption that the firm is an independent decision unit and have workers simply choosing their labor-leisure position given the prices of TABLE 1 Region variable
Classical
Keynesian
Inflationary
m g K
0 0 +
+ +
-
FIXPRICE
LABOR-MANAGED
ECONOMIES
243
goods and the technology of the firm. Thus, employment and labor supply decisions are one and the same. The economy can operate under two alternative regimes, one of excess demand for the good (Inflationary), and the other of excess supply (Keynesian). There is no market for labor, even in the restricted sense of Section 2. That is, in the model of Section 2 there are buyers (firms) and sellers (workers) of labor, but the wage rate is not determined by a market process. In the present model there are no buyers or sellers of labor. In order to keep the model simple we assume that each firm has only one member. In the context of our specific model, Meade’s conjecture is again false: changes in g and m have the same qualitative effects on employment in the PME, the Ward, and the Sen LMEs. 3.1. Workers To simplify matters we assume that each firm has only one member. If this member is unconstrained, he solves the following maximization program: max W.7 L CL) subject to w=pf(L)-K
Let x$(p, m) and L$(p, m) be respectively the notional demand for the good and labor-leisure decision of the worker.‘* At this point it is convenient to drop the assumption of a single good and assume that there are many identical firms producing different goods that must be consumed in fixed proportions. Also, government consumption follows the same pattern as that of private expenditure, that is, g is collinear with x. This allows us to write our equations in terms of aggregates, since the ratios in which goods are produced and consumed remain constant. Without this multiple goods assumption a firm that could not sell its product in the market would still produce for the consumption of its own workers. Given that goods must be consumed in fixed proportions, this cannot happen. Allowing firms to produce for self-consumption would complicate matters considerably, especially with regards to comparative statics analysis. To simplify matters, we also assume that there is a one to one correspondence between workers and firms. Under these assumptions, we let x be the magnitude of the consumption vector and p be the magnitude of the fixed price vector. Ifthemarketdemandisy=x+g
244
FERNANDO
B. SALDANHA
for goods. The representative worker’s effective demand for goods xw ( p , m , ~7)is then given by the solution to mm W,
L P)
subject to w=pf(L)-K px+p
0 c L < Le.
Note that the restriction on the sales of the firm is actually a restriction on hours worked, and thus determines the worker’s effective demand for goods. The worker can also be restricted as a consumer in the goods markets. A worker that cannot satisfy all of his notional demand for goods actually falls short by the same proportion in every good. His labor-leisure decision and therefore his firm’s supply are correspondingly affected. The worker’s effective labor-leisure decision LW ( p, m, 2) is given by the solution to max U-T L P> subject to w=pj-(L)-K px+pGm+w OGXCX,
0 < L < Le.
The firm’s effective supply is then YW(P,
m, 2) =~(Lw(P,
my 2)).
3.2. Equilibrium
Given that we have defined aggregate effective demand and supply functions, we can also define fixprice equilibria. This is done with the help of diagrams that are somewhat similar to the traditional Keynesian 45’ diagram. In Figs. 7 and 8 we plot the functions xw (~7) + g and yw (f - g), along with a 45” line. The condition of voluntariness of exchange means that in equilibrium no agent can trade more than his desired amount. Then no equilibrium can lie above either of the curves xw (~3 + g and yw (jj - g) . Also, if in equilibrium there cannot be unsatisfied agents in both sides of a market, no equilibrium can lie (strictly) below both of these curves. A concept of equilibrium emerges naturally. We say that y” is ahprice equilibrium for the Sen economy if
* Y
y” FIG. 7. Keynesian equilibrium.
FIG.
8. Inflationary equilibrium. 245
246
FERNANDO
B. SALDANHA
If y” is a fixprice equilibrium, one of the curves xw (7) + g and J+,,(9 - g) crosses the 45” line at j?, and the other one is above the 45” line at p. There are three obvious types of equilibrium: (i) 9 is a Keynesian equilibrium (excess supply of goods) if y” = xw (9) f g* (ii) y” is an If lationary equilibrium (excess demand for goods) if y”= yw ( y” - g). (iii) j7 is a Walrasian equilibrium (equilibrium in the goods markets) if y” =xw(Y)+g=yw(9-g). Figures 7 and 8 show respectively a Keynesian and an Inflationary equilibrium. A Walrasian equilibrium (not shown) occurs when both xw (9) + g and yw (jJ - g) cross the 45” line at the same point. The existence and uniqueness of equilibria for this model can be proved under mild assumptions. 3.3. Comparative Statics
As in the previous section, specific assumptions on the signs of derivatives of effective demand and supply functions are needed to perform comparative statics analysis. We assume I3
-axw ama 0, -axwG0,
(S)
aK
The inequalities in (S) are assumed to hold strictly, except when this is prevented by active nonnegativity constraints. Consider first a Keynesian equilibrium. Increases in g and m stimulate demand, and therefore shift up the xw($) + g curve. Hence, employment increases. An increase in K shifts that same curve down, and is therefore contractionary. Now let the equilibrium be Inflationary. Increases in g and m reduce the effective willingness to work, in the first case because there are less goods to be bought, and in the second case because the demand for leisure is stimulated. Hence in both cases the yw (~7 - g) curve shifts down, and employment decreases. An increase in K has the opposite effect, since it is equivalent to a decrease in m. Figure 9 illustrates the expansionary effect of an increase in government expenditures in a Keynesian equilibrium. Except for one case, the qualitative effects of changes in g, m, and K on employment are the same as those in the Ward economy under the similarly named regime. The exception is that, under an Inflationary regime, an increase I3 See the Appendix
for a discussion
of (S) .
FIXPRICE
LABOR-MANAGED
ECONOMIES
247
Y t
0
.
y”
Y
? FIGURE
9
in K reduces output and employment in the Ward economy, if the substitution effect dominates the income effect, and has the opposite effect in the Sen economy. I4 Once more, Meade’s conjecture that Keynesian policies would be at best ineffective in reducing unemployment is not vindicated. 3.4. Eficiency and the Price Level In the Sen economy the equilibrium allocation is completely determined as a solution of the worker’s optimization problem. So one is tempted to believe that equilibria are necessarily Pareto efficient. However, this is not the case, as is clear from the possibility of unemployment. The reason why inefficient outcomes are possible is that K and m are fixed in nominal terms. An increase in p implies a smaller real value of K, which is a gain for the worker. If savings are positive (p > m), the worker benefits from a higher p in another way: it is less costly in real terms to achieve a given level of savings.‘5 Only when dissavings are exactly equal to K (i.e., when m - p = K) the loss I4 The difference between the signs of $+,/aK in (S) and of the corresponding &&,/aK in ( WLM) is due to the absence of a substitution effect in the first case. In the Sen economy a change in K does not affect the worker’s perceived marginal trade-off between leisure and consumption. Is Assuming that the elasticity of price expectations with respect to present prices is zero.
248
FERNANDO
B. SALDANHA
due to the reduced real value of these dissavings is exactly sufficient to compensate for the gain due to a reduced real value of K, so that a marginal increase in p does not affect the worker’s welfare. This is precisely the condition for Pareto optima&y. These statements can be justified more formally. It follows easily from the Envelope theorem (cf. Varian, 1984) that, for either one of the worker’s optimization problems in Section 3.1,
-au* = X(y- x), ap where U* is the optimal value of the corresponding problem, and X is the Lagrange multiplier associated with the budget constraint. Also, from wL=py-K=px+p-m it follows that p(y-x)=K+(p-m). We conclude that dU*/dp
= 0 is equivalent to K = m - P.
4. Conclusions We develop two fixprice models for labor-managed economies under alternative assumptions about decisions on employment and labor supply. In both cases the qualitative effects of variations in government expenditures and the money supply are the same as those that apply in a PME. That is, increases in these variables am expansionary in situations of generalized excess supply and contractionary in conditions of generalized excess demand. In no case do our results support Meade’s conjecture that Keynesian policies are ineffective or even counterproductive in the LME. Of course, Meade’s conjecture is based on the well-known perversity of the LMFs supply curve in a Ward economy. In our model prices are fixed, so this perversity could not play any significant role. Naturally one is then led to the question of the validity of Meade’s conjecture in the long run, with flexible prices. The answer may well be different for each of the two types of models we consider. These issues constitute important topics for further research. APPENDIX Discussion of ( WLMj . The first-order conditions that determine ?w (p , w , m, L) and & ( p, w, m, 2) in interior solutions where the quantity constraints are binding respectively reduce to
ux u,=”
and
UL -=-w. U#
RXPRICE
Differentiating
* dK
LABOR-MANAGED
249
ECONOMIES
these conditions we obtain
= L aw a3W
ai., -=
dK am ’
at, i& -3 am
aK
1
if!!
A2 aK
a.cw a(d) azw u,, - pu, -=v-aL
a& az
-=-p=+
aL
a&
am
A1
aw -1 --aw al, az aK
[1 E
u, + WV,, A2
The terms A1 and A2 are negative, from the second-order conditions. It follows that if
u,,>,oa ULr,
(A.11
the first three inequalities in (WLM) hold. (A.1 ) is satisfied if the utility function is separable or if it has the Cobb-Douglas form. The effect of a change in K on labor supply can be divided into a substitution and an income effect. Given (A. 1) , these effects work in opposite directions. If the substitution effect is the stronger of the two, which means the effective labor supply function is upward sloping (aLW/dw > 0)) the fourth inequality in (WLM) is also satisfied. When z is increased, there are two effects on the demand for the good. First, there is an income effect, which, given (A. I), stimulates consumption. Second, there is a shift in the relative desirabilities of consumption and money holdings. If consumption and leisure are strongly complementary, this effect can dominate the 6rst effect and determine a net fall in consumption. The fifth inequality in (WLM) assumes away this possibility. Finally, the effect of an increase in 2 can be divided into three parts. First, the income available for the purchase of leisure decreases. Second, the wage rate increases, which in itself has a substitution and an income effect. And third, there is a shift in the relative desirabilities of leisure and money holdings. If (A. 1) holds and the effective labor supply function is upward sloping, the first two effects work in the same direction, and the last inequality in (WLM) also holds, unless there is strong complementarity between consumption and leisure. Discussion of (22). The first-order conditions that determine xw ( p, m , L) and Lw ( p, m, 2) in interior solutions where the quantity constraints are binding respectively reduce to
UX -= UP p
and
UL - = -pf’(L). u,
250
Differentiating
FERNANDO
B. SALDANHA
these conditions, we find ax, ---=- ax” -$i? dK
Kit - PUW A3
afi = --=-ati
ULF
-
dm
+
dK
Pf ‘(L) 4
u,, ’
where the terms A3 and 4 are negative if the second-order conditions are satisfied. It follows that (A. 1) is a sufficient condition for (S). Proof of Theorem 1. The hypothesis of the theorem (E # a) implies that Q # 0, where Q is as in Definition 3. Given the last two inequalities in ( WLM), it is easy to see that \k maps each of the connected components of Q into itself. The theorem now follows from Theorem 2 and the Brouwer fixed point theorem. Proof of Theorem 2. (Necessity). Suppose that (z, 2) is a fixprice equilib rium and
WQ)
= (kf(X)-g),
where X # L. Clearly, from (7), (8), and the definition of 9, i < A, and hence l< X. Then ( 10) and ( 12) imply that e = X = cc and (7) implies that L = LF(P, x + g).
(A.2)
From ff = co it follows that LF(P,
if
+ g)
=
LF*
(A.3)
(P)-
Putting (A.2) and (A.3) together we find a contradiction of L < X. (Sufficiency). Suppose that (l, 2) is a fixed point of 9. Define L=L
if
It = &(p,
E=oo jf=f
otherwise if
x=a3
otherwise.
x” = &(p,
m, 2)
m, i)
We must show that ( 7) and ( 8) are satisfied for this choice of (E, X). There are three possible cases: ( 1) L’ = i&p,
m, 2’) 6 L:(p).
Suppose that (7) does not hold. Then
L&p, 2 + g) < x G L:(p), and hence X < cc. It follows that X = 2 = iw(p,
m, L),
(A.41
FIXPRICE
LABOR-MANAGED
251
ECONOMIES
and therefore LF(p, 2 + g) = LF(p, x” + g) = L&f(i)). Then, from (4), either LF(P, 2 + g) = L’ or LF(P, 2 + g) = L:(p), and in both cases we have a contradiction of (A.4). Now suppose that (8) does not hold. Then there are two possibilities: (i) J&P, L) - g -C R. A reasoning similar to the above then obtains a contradiction. (ii) 2 < min ( yF(p, E) - g, &( p, m, L)} . Then, using the definition of \k, we find that z = L” = iw(p, m, x”), hence J&PI a - g
- g = x”,
and again we have a contradiction. (2) L’ =f-‘(&(p, m, z) + g) < L:(p). A reasoning analogous to that developed for case 1) shows that (7) and (8 ) are satisfied. (3) L = L:(p) < &(p, m, x”) and 2 = y:(p) - g G &(p, m, L). It follows that I:
=
L:(P)
=
LF(P,
d(P))
=
LF(P,
2 +
g)-
(A.5)
Now, 2 can take two possible values, 2 or cc, and it follows from (A.5) that in both of these cases (7) is satisfied. A similar reasoning shows that (8) holds. Since (7) and (8) are satisfied in all three cases, the theorem now follows from the observation that (9)-( 12) hold by construction. Proof of Theorem 3. It is easy to see that d(W(P, Jw) aJ5
and hence (U) implies that a.&
x
= pf’(Jc).
af,-1. [ 1
< ajz
(A-6)
It follows immediately from (A.6) that the set G of Definition 3 can intersect the boundary dE of E at most once. If G fl aE = @ then there exists a unique fixprice equilibrium, which must be of the Classical type. If G Cl aE # 0 then the point of intersection corresponds to a unique fixprice equilibrium, which may be of either the Keynesian or the Inflationary types. Proof of Proposition 1. Homogeneity implies that LG and x$ can be written as functions of real (as opposed to nominal ) quantities. From ( WW) , the Implicit Function theorem can be used to define two functions m’(p) and m*(p) as
FERNANDO
252
B. SALDANHA
where w*(p) is the optimal value of the firm’s maximization problem at price p. Since L$ and xz are monotonic functions of m, the functions m ’ and m* cannot have more than one branch. Also, it is easy to compute that
aLF* -____
am’ -=-+p
m
ap
P
(( ap
m
ad ---
P
(( ap
am* -=-+p ap
a&$ a(w*jp) ww
. )/&)
ap
a&
a(w*ip)
~(WIP)
)/$$-y).
at,
(A-7)
(‘4.8)
Now, a quick calculation shows that
aw*m = K ap P2LF*(P)
>o .
Substituting in (A.7) and (A.8) and taking the difference, it follows immediately from homogeneity, ( WW), and the perversity of the LMF’s supply curve that am2 --ap
am1 < 0, ap
and the curves m ’ and m* cross at most once. The proposition now follows from the observation that m* = m’(p*) = m’(p*) if and only if there is a Walrasian equilibrium (L*, x* ) such that L* = LF(p*)
-W*(P)
= L; [
P*
x*=x;t1 -1 w*(p*) P*
-m* ’ P*
1
m*
’ P*
=YF*(P*)--g
associated to (p*, m*, g, K). REFERENCES Bar~o, Robert J., and Grossman, Herschel I., “A General Disequilibrium Model of Income and Employment.*’ Amer. Econ. Rev. 61, 1:82-93, Mar. 1971. Benassy, Jean-Pascal. The Economics of Market Disequilibrium. New York: Academic Press, 1982. B6hm, Vblker, “Disequilibrium Dynamics in a Simple Macroeconomic Model.” J. &on. Theory 17,2:179-199, Apr. 1978.
FIXPRICE
LABOR-MANAGED
ECONOMIES
253
Bonin, John P., “The Theory of the Labor-Managed Firm from the Membership’s Perspective with Implications for Marshallian Industry Supply.” .I Camp. Econom. 5, 4:337-351, Dec. 1981. D&e, Jacques, “Some Theory of Labor-Management and Participation.” Econometrica 44,6: 1125-l 139, Jun. 1976. Fellner, William, “Prices and Wages under Bilateral Monopoly.” Quart. .I. Econ. 61, 503-532, Aug. 1947. Grandmont, Jean-Michel, “On the Short-Run Equilibrium in a Monetary Economy.” In Jacques D&e, Ed., Allocation under Uncertainty: Equilibrium and Optimality, pp. 2 13-228. London: Macmillan & Co., 1974. Ichiishi, Tatsuro, “Coalition Structure in a Labor-Managed Economy.” Econometrica 45,2:341360, Mar. 1977. Ichiishi, Tatsuro, “A Social Coalition Equilibrium Existence Lemma.” Econometrica 49,2:369378, Mar. 1981. Law, Peter J., “The Illyrian Firm and Fellner’s Union-Management Model.” J. Econ. Stud. 4, 1:29-37, May 1977. Malinvaud, Edmond, The Theory of Unemployment Reconsidered. Oxford: Blackwell, 1977. Meade, James E., “The Theory of Labour-Managed Firms and of Profit Sharing.” Econ. .I. 82, 325s:402428, Mar. 1972. Miyazaki, Hajime, and Neary, Hugh M., “The Illyrian Firm Revisited.” Bell. .I. Econ. 14, 1:259270, Spring 1983. Schulz, Norbert, “On the Global Uniqueness of Fix-Price Equilibria.” Econometrica 51, 1:4768, Jan. 1983. Sen, Amartya K., “Labour Allocation in a Cooperative Enterprise.” Rev. Econ. Stud. 33,4:361371, Oct. 1966. Steinherr, A., and J-F. Thisse, “Are Labor-Managers Really Perverse?” Econ. Lett. 2: 137-142, 1979. Svensson, Iars E. O., “Effective Demand and Stochastic Rationing.” Rev. Econ. Stud. 41,2:339356, Jan. 1980. Tyson, Laura D., The Yugoslav Economic System and Its Pe$ormance in the 1970s. Institute of International Studies, University of California, Berkeley Research Series, 44, 1980. Vanek, Jamslav, The General Theory of Labor-Managed Market Economies. Ithaca, NY/London: Cornell Univ. Press, 1970. Varian, Hal R., Microeconomic Analysis. New York: Norton, 1984. Ward, Benjamin, “The Firm in Illyria: Market Syndicalism,” Amer. Econ. Rev. 48, 4:566-589, Sept. 1958.