Toward the Constructive Theory of Human Social Behavior. V. Labor Behavior under Piecework Pay YURI B. CHERNYAK? ARKADII I. LEONOV,b AND ALEXANDR Y. LERNERC

aDivision of Health Sciences and Technology Harvard University-Massachusetts Institute of Technology Cambridge, Massachusetts 02139 hDepartment of Polymer Engineering Center for Polymer Engineering University ofAkron Akron, Ohio 44325 Weizmann Institute of Science Rehovot 76100. Israel

INTRODUCTION This paper is the fifth in our series (see references 1-4) aimed at describing the functioning of a human community (for example, a firm, a region, o r a state) on the basis of a “microscopic” picture that represents individual deeds (behavior, in fact). General principles are outlined in reference 1, where we formulate an extremal principle that governs the behavior of individs and, thereby, the life of whole community. The description is, generally, closed, by which it is implied that the goods and services consumed by each individ result mainly from the labor of other members of the society (see also references 5 and 6). References 2-4, being concerned with applications of our general approach to the consumer and labor behavior of a separated individ, disconnect the closed scheme for this purpose. Recent quantization of the notion of quality (see, for example, references 7 and 8) has permitted us to formalize the notion of qualification as a parameter Q (0 I Q I 1) of the individ. With this aim in view we proposed a test experiment during which the individ is working with maximum effort and thoroughness. The production ratey is related in this case only to the product quality q (0 I q 5 l), and the relation chosen is linear y

=

(1 + c)Q - cq

(c = const

> 0)

(1)

where y is presumed to be normalized to the ultimate production rate achievable under the given technology at q = 1, and where constant c is determined primarily by the technology. Equation (l),which describes on the ( q , y ) plane a curve referred to below as a qualification curve, shows that the higher the production rate y (at q = const), the higher the qualification Q of the individ under test, and that the higher the qualityq (aty = const), the higher, yet again, the qualification Q. It is also 324

CHERNYAK et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

325

seen from (1) that the individ with Q = 1, the "genius of the job," provides efficiency y = 1 at the highest qualityq = 1. Under ordinary working conditions the individ would never work so efficiently and thoroughly as under the test. This fact can be takcn into account by the introduction of the two behavior variables I and z such that by choosing their values the individ controls the values ofy and q. According to reference 3 these variables are introduced by the equation

y = (1

+ c)QI - cq/z

(4

where the values of the behavior variables I and z (0 I I I 1; 0 < z 5 1) are chosen by the individ in compliance with his/her preferences. To describe these we will use quite a traditional approach and introduce a function of all the behavior variables, comfort function S, which is maximized by the individ when implementing his/her mode of conduct. In accordance with our general scheme,' the total comfort function breaks into the sum of quadratic-linear partial comfort functions, each of a particular behavior variable. In this paper we will study the activity of an individ who works implementing the optimal values 1" and zo of behavior variables I and z, respectively, and also consumes choosing the optimal valuexO of the generalized variablex representing the total amount of the goods and services c o n s ~ m e dNote . ~ that if one is given the value ofx and the list of prices of all the goods on the market, one can, if one is interested, split the x into the familiar consumption vector giving a comprehensive description of the consumption. This can be done with the aid of our consumer behavior theory* on the basis of the same general scheme.' Thus, the total comfort function of the individ under study is of the form S(x, I , 2 ) = P,x(2xm - x)

+ P,I(21m - I ) + P,z(2zm - 2 )

(3)

where the positive constants P,, PI, and P, have the meaning of the amplitudes of the corresponding partial comfort functions, while x,,,, I,,,, and z, indicate the positions of their absolute maximums (in fact, saturation points). It is clear that maximization of comfort function (3) is equivalent to minimization of the following discomfort function:

D(x, I , 2 ) = S(x,, I,,

2,)

- S(x, I, 2 ) = P,(x, - x)*

+ P,(Im - I )2 + Pz(zm- z)2

(4)

This discomfort function represents a dissatisfaction index for the individ. The surface S(x, I , z ) = const (or D(x, I, z ) = const) will be referred to as an isocomfort surface or an isocomf. The family of isocomfs for a given individ comprises a family of similar ellipsoids centered at the point (x,, Im, z m ) . Note that the variables and parameters entering into equations (3) and (4) satisfy the inequalities O ~ x < m ,

0 5 1 ~ 1 , O 0. Because of these features, the individs of the second group may be referred to as normal employeeseager consumers-strivers for quantity.

FIGURE 4. Domains R, ( i = 1,2,3) corresponding to different types of tangency of the 2 & 3 shown at the section z = 1. When 0, E R I , the isocomf to the MCS from FIGURES tangency occurs on edge BD (in FIGS.2 & 3); when 0, E 0 2 , the tangency is smooth and occurs at an interior point; when 0, E R3,the tangency takes place at vertex C. The straight line BE (or B) can be obtained from the condition that the tangency at point B (or C) is smooth.

The third group consists of the individs who have exhausted all their possibilities (I = 1,z = 1) to increase their income and thereby their consumption. These individs can be referred to as hard workers-unsatisfied eager consumers. Thus, under the piecework pay there appear four types of behavior among those individs who are capable of the given job. Three of these types are different from those arising under the fixed wage.4 Details of normal employee behavior naturally depend on the values of amplitudesp,, P,, and P, in the total comfort function (3).We will not devote any space here to the cumbersome expressions for coordinates of the extremum point, because the details are not significant for the purpose of the present eIt should be recalled that according to reference 1 we consider developed communities where each individ is supplied with a minimum of goods to survive. The value x of consumed goods is counted off from this minimum. fIn contrast to those who “strive for quality” (see the next section).

CHERNYAK et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

333

paper. Let us now proceed to the analysis of the individ's behavior under piecework pay again, but in a more realistic formulation.

Limitations on Quality and Production Rate In the previous subsection, we considered the situation in which there was no limit on how small the production rate could be, in which it could even be equal to zero. Such a situation cannot be tolerated by the employer. To eliminate the complete dependence of the employer on the degree of hidher employees' wish to earn money and consume, the employer usually sets a minimal admissible production ratey * > 0 such that ify < y , the wage (or salary) is unpaid and the corresponding employee is advised to start another working activity (for example, a managerial one). So the labor results (q,y) must obey the inequalities q * I q I l ,

y * I y I l

(26)

Equation (11) for the individ's labor remuneration I , can now be placed by

It is clear that the value of y * cannot be too high and that the production rate y * must be achievable at least by the individ having qualification Q = 1 and working with maximum effort and thoroughness (I = 1, z = 1). This yields

Besides, under (q *, y * ) given, there exists one and only one qualification Q = Q +. such that the corresponding qualification curve (1) passes through the point (q *, y * ) (compare FIG.l), which gives

Q*=

Y* + c q * l+c

(29)

Note that Q * + Qo* wheny * + 0. The individs with Q < Q * can in no way provide the production rate and the quality required. We shall call them low-qualification employees and shall not consider them here. Let us analyze the behavior of an individ with qualification Q E [Ql * ] under the remuneration method defined by relation (27). In accordance with basic equation (2) and conditions (26),we have 1

1

On the other hand, the limitation (15) imposed on the value of consumption, consistently with equations (7), yields

which also takes account of the condition q

2

q*.

334

ANNALS NEW YORK ACADEMY OF SCIENCES

Inequalities (30) and (31) must be fulfilled simultaneously; that is, the value of 1 must not be less than the greatest of the right-hand sides of these relations. Therefore, by setting l*(x,z) = -

cq,lz,x I By,lP,

+ cq ,/ z , x 2 By ,IP,

one finds that the admissible domain fix& of variables (x, I, z ) for the individ with given qualification Q is determined now by relations (5) and the inequality 12 l,(x,z)

(33)

The domain fix,z corresponds now to the remuneration (27), and the surface 1 = l*(x,z)

(34)

confining it consists of two pieces of different hyperbolic cylinders (FIG.5). Since the obtained in upper line in equality (32) coincides with the expression for I,@) reference 4, for the case of fixed wage (while the lower-with (17)) the piece ABDF of surface (34) is similar to that considered in reference 4, and the surface piece DBC is a part of the surface ABC shown in FIGURE 2. Thus, the problem of the individ’s behavior under pay scheme (27), in fact, reduces to one of the two problems considered in reference 4 and in the previous subsection. In much the same way as before, we come to the following conclusions: 1) If an individ’s qualification satisfies the inequality

FIGURE 5. Domain slxb (ABCDEF) and the MCS (ABCDF) for the piecework pay scheme (26). On the piece ABDF, y = y * and q = q * ; on the piece BCD, q = q * and y > y * . Here X* =~*Blpr.

CHERNYAK et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

335

FIGURE 6. Domains 0,corresponding to different types of tangency of the isocomf to the MCS from FIGURE 5 (at the section z = 1). The individs with 0, € 0 1 are normal employeesmoderate consumers; those with 0, € 0, are lazy workers-listless consumers; those with 0, € 0 3 are strivers for quantity-eager consumers; those with 0, € 0 4 are hard workers-unsatisfied eager consumers.

such that the individ reaches the the isocomf center 0, = (x,,,, l,, z,) E absolute maximum of hislher comfort function in the point (XI],lo,L O ) = 0, where D(xo,lo,zo) = 0. 2 ) If the strict inequality takes place in (35) there remains a degree of freedom for the corresponding individ to choose the working point in the ( q , y ) plane. As this takes place, quantities q and y are in general higher than the minimal admissible ones, and the excess of labor income over the need results in positive savings. Thus, the individs with Q 2 Q ,are in our classification the creative employees-moderate consumers. The situation in which

corresponds to the case of a normal employee-moderate consumer (the same group appeared under the constant wage scheme in reference 4. The center of the isocomfs is situated between the piece A B D F of the hyperbolic cylinder and axis x and also between plane x = 0 and x = y ,BIP, (plane BDG in FIG.5). The extremum point (xo, lo,zo) is the point of tangency of the isocomf to surface ABDF (FIG.5). The labor results of such an individ are the minimal admissible ones (q = q ,,y = y ,), while the value of the discomfort function is conditioned only by the labor variables ( I o > I,,,, zo > 2,). SincexO = x, < By ,/P,, an individ from the group considered has positive savings. Finally, if

the individs break down into three subgroups similar to those considered in the previous subsection. FIGURE 6 illustrates in two dimensions the situations that arise. The individs that satisfy (36) and have 0, in domain 0, are normal employeesmoderate consumers and behave as described above. The normal employees with 0, in domain n2from FIGURE 6 have their working point on edge BD (FIG.5); they may

336

ANNALS NEW YORK ACADEMY OF SCIENCES

be called lazy workers-listless consumers because they prefer not to overburden themselves with their jobs even if their needs grow. The behavior of an individ with 0, in domain il3 from FIGURE6 is typical for the normal employee-eager consumer-striver for quantity. The extremum point is here the point of tangency of the isocomfort ellipsoid to an interior point of either piece BCD o r edges BC or D C (FIG. 5). The quality of the labor results is minimal ( q = q * ), but the production rate is higher thany * and grows as needs (x,) increase. Finally, if 0,of an individ belongs to domain 0 4 shown in FIGURE6, that is, if h i d h e r needs again satisfy inequality (25), then vertex c becomes the tangency point of the isocomf to the MCS. The individ’s behavior in this case is typical of the hard worker-unsatisfied eager consumer described in the previous subsection. Note that the discomfort of the individs from the last three groups is conditioned by all the behavior variables; that is, relations (22) and (24) are valid again. To conclude this section, it may be observed that if constant B in relations (11) and (27) is the same, the domain ABCDFE in FIGURE 5 is a part of the domain ABCD in FIGURE 2. Thus, introducing the limitation y 2 y * with the labor price B kept constant, one diminishes the domain of maximum comfort, and consequently some of the creative employees are transferred into the category of normal employees.

BEHAVIOR OF AN I N D M D UNDER PIECEWORK PAY WITH A PREMIUM BONUS FOR QUALITY

No Limitation on Production Rate Let us consider an individ’s behavior under the remuneration method that generalizes expression (11) as follows:

where constants B > 0, b > 0, and q * > 0 are set by the terms of the hiring. A new element in expression (38) is the term b (q - q * )0(q - q * ), a specific premium for the quality of labor results. How this parameter affects the individ’s behavior is now to be studied. We will again assume that condition (12) is fulfilled, that is, that the individ’s total income coincides with his wage (salary) I,. In this case limitation (7) o n consumption takes the form XP, I I ,

(39)

Since, according to (38) the conditions of hiring stipulate only a limitation on quality q of the labor results, the qualification Q of the individs who are capable of the given job must obey inequality (14) with the lower bound given by (13). As before, the solution of problem (10) for a given individ consists primarily in constructing the domain Rxl, of admissible values of behavior variables I, z, a n d x that are consistent with all the problem limitations. For this purpose we will find the MCS

CHERNYAK er al.: CONSTRUCTIVE THEORY PIECEWORK PAY

337

that is the greatest lower bound of behavior variables 1 and z and the least upper bound of variable x . Using the basic equation (2) and relations (38) and (39) we obtain

P;x I B [ ( c + 1)Ql - c ~ / z +] b ( q - q * ) where q

2

q * . Let us introduce the designations

p = b/B,

z*

= clp

(411

where p characterizes a relationship between the premium for quality and the remuneration for the production rate. Using inequality (40) one can write the expression for the minimal admissible value I * of variable 1 for an individ with a given qualification Q in the form

with the lower bound calculated under conditions y20,

q * 1 q 1 1

(43)

It is clear that the MCS equation takes the form

I

= l*(x,z)

(44)

and that the shape of this surface depends significantly on the values of parameters p andz, =z,(p) from (41). If p is sufficiently small (p I c or z , 2 l ) , the term pq(z* - z ) / z in expression (42) is nonnegative (because z E (0, I]) and the greatest lower bound is reached at q = q *, which yields

This expression coincides with the right-hand side of relation (17), which corrcsponds to the case p = 0. Therefore, the domain Clr,z and the corresponding MCS coincide with those defined in the previous section for the method of remuneration (11) where no additional pay for quality was suggested (FIG.2). This fact, in turn, means that the individs break down into the same groups under 0 < p I c as under p = 0. Since for normal employees q = q *, they will receive no premium for quality, as immediately follows from (38). A creative employee’s behavior also remains the same as under p = 0, but only the amount of savings can alter the remaining “instability” of quality mentioned in the previous section: if a creative employee wants to increase his pecuniary savings, he/she is working with a dcgradation of quality. Thus, a minor (p I c) additional premium for quality docs not affect the individ’s behavior. Let us consider now the case p > c o r z * < 1. If 0 < z < z *, the greatest lower

ANNALS NEW YORK ACADEMY OF SCIENCES

338

bound in expression (42) is reached at q = q * . The relation (42) can therefore be rewritten as

It is clear that the minimum in this expression is obviously reached at a certain greatest value of q = qmm,which, in general, depends on x and z (qmm= qmax(x,z ) ) , and is consistent with conditions (43) and (39). Using relations (38) and (39), and taking into account the fact that the surface sought, 1 = 1 , (x, z ) , is the surface of complete consumption, where inequality (39) reduces to the equality, we obtain Pr Pq=gx+Pq*

~ * ( x , z ) = '~ z * ( x J ) ;

z* < z I 1 ,

-Y

(47)

x~x*=(l-q*)BIP,

(49)

q=1)

(SO)

with the function qm(x) and the bound x * defined in (48). Formulas (SO) explicitly express that 1 ) the part of the surface defined by 1 = 11 * (x, z ) is characterized by the minimal admissible quality q = q *, the production ratey increasing in proportion to x; 2 ) the part of the surface defined by 1 = 12* (x, z ) is determined by the conditions y = 0 and q = q m ( x ) ;and 3) the part of the surface defined by 1 = l3 * (x, z ) and where the production rate y increases as (x - x , ) is where the highest quality q = 1 is reached.

CHERNYAK el al.: CONSTRUCTIVE THEORY PIECEWORK PAY

339

To clear up the mutual arrangement of the different pieces of the surface I I * (x, z ) , we consider the differences

=

and

b\-

One can infer from formulas (51) and (50) that the first and the second pieces of the surface 1 = I , ( x , z ) meet in the plane x = 0 along a curve that may be

1- -

\I

I

I I 01

---

1 I

I

I I

21

2.

2

FIGURE 7. The sections of the surface I = I , (x, z ) (49)by planes x = const showing how the pieces I = I I * ( x , z ) and I = Iz,(x,z) of the MCS join. Curve 1 corresponds to x = 0 where II * (0,z ) = 12, (0, z) = Il2(2). Curves 2 and 3 arise at the sectionsx = x2 andx = x3 underxz < xj. The straight line z = zI is determined by the condition /I~(ZI) = 1. If z , < 21, the surface piece I , ,= I , * (x, z ) lies beyond the main parallelepiped ( 5 ) .

conveniently considered as the line of intersection between this plane and the hyperbolic cylinder

Comparison between formulas (50) and (53) shows that in the regionx > 0 both the surfaces II +. (x, z ) and 12* (x, z ) are situated higher than the cylinder (53). It can also be inferred from relations (50) and (51) that the surfaces 11 + (x, z ) and l2 * (x, z ) are joined nonsmoothly in the plane z = z , with the first surface arranged under the second one at z < z *, and in the opposite way at z > z * (FIG.7).

ANNALS NEW YORK ACADEMY OF SCIENCES

340

Similarly, one can infer from formulas (50) and (52) that the surfaces 12 * (x, z ) and l3* (x, z ) intersect in the plane x = x * along a hyperbole that also arises from cutting the hyperbolic cylinder

by the plane x = x * . In the region of x > x * the surface 1 = l3* (x, z) is arranged higher than the surface 1 = l2 * (x, z ) and also higher than cylinder (54). For constructing the domain flh of the admissible values of (x, 1,z) it thus remains to find the intersection of the domain 1 2 1 * (x, z) with the basic parallelepiped (5). To this end we observe that the scale along the axis 1 is fixed, and the multiplierP,/B in formulas (50) acts only as a scale factor with respect to axisx. The surface 1 = 1* (x, z ) can therefore be considered as a family of surfaces depending on two parameters Q and P whose values determine the position and the general shape of the domain flxb sought. The position of the surface 1 = 1 * (x, z ) is determined 1/Q), and, if Q < Qo * = cq * /(c + l), the whole mainly by the qualification Q(l* surface under consideration lies above the basic parallelepiped (I * < 1 at 0 < z Il), so the set fldz is empty (see (14)). Furthermore, since the piece 1 = l3* (x, z) atx > x * is located above the cylinder 1 = 123(z),the specified piece of the surface 1 , (x, z ) is arranged beyond the basic parallelepiped when lZ3(l)> 1; that is, Q > c/(c + 1). Thus, there appears a domain of the medium qualifications Q

-

such that the curvilinear surface (MCS) that confines consists of no more than two pieces, 1 = l1* (x, z ) and 1 = lz* (x, z ) (compare FIGS.2 & 8a). If relation (55) is fulfilled andz, 2 1 or

pIpl=c

(56)

the surface 1 = lz* (XJ) is sited beyond the basic parallelepiped (FIG. 7), and the MCS consists of a single piece 1 = lI * (x, I). This may be immediately seen from 7 that if z , < formula (49) and has already been discussed. One can see from FIGURE z * I 1 (where z1 is determined by the condition 1 1 2 ( 2 1 ) = l), that is, if the values of P satisfy the conditions

PI < P < PZ

(C

+ l)Q/q*

(57)

the MCS consists of two pieces, lI * (x, z) and 12* (x, z ) (FIG. 8a). Next, for the medium qualifications (55) and small values of z * (FIG.7), that is, for z * < z1 or

the surface lI * (x, z ) falls out of the main parallelepiped and the MCS is given by the equation 1 = 12 * (x, I) (FIG.8b). It is important to note that parameter P characterizes according to (41) the additional remuneration for quality (because B = const), and that as a result the expansion of domain flXlz with P increasing has the natural interpretation: the individ’s comfort (function) can only increase as the pay in-

CHERNYAK et af.: CONSTRUCTIVE THEORY PIECEWORK PAY

341

a

b

FIGURE 8. Domain 4 1 , and the MCS for individs with medium qualification (Q * o < Q I QI * ) under remuneration (38). (a) The case of intermediate premium for quality, < p 5 p2: on A E F I = I, *, q = q * a n d y > 0; on EFBC: I = /2* , y = 0, and q > q * . (b) The case of highest premium for quality, p 2 pz:on ABC (corresponding to p = pz) and on ABC (corresponding to p > p l ) , y = 0 a n d q > q * .The surface ABC corresponding to small p I (shown in FIG.2) is also given for comparison.

creases. Let us now direct our attention to the case of individs with the highest qualification from the interval

In this case domain is wider than that for individs with the medium qualification (55). If the value of p is small (p _< PI) the MCS consists of a single piece I =

ANNALS NEW YORK ACADEMY OF SCIENCES

342 a

b

I

1

and the MCS for the individs with high qualification Q > Ql* under FIGURE 9. Domain remuneration (38). (a) The case < p < p2: on A1FlEl:y > 0, q = q * ;on HGK:y > 0, q = . ; on HGBIFIEl:y = 0, q > q * . (b) The case p > p2: on A I B I G ~ H = I :0,~ q > q * ; on HLGIK~: y > 0 , q = 1.

Il * (x, z) as above, but its surface AIBICl (FIG.2) lies below the similar surface ABC corresponding t o the medium qualifications (55) and small p (56). If the values of p are intermediate and fall within interval (57), the MCS consists of all three pieces given by (49) (FIG.9a). Finally, if f3 > p2, or, which is the same, z * < zl, the piece I = I I * (x, 2) lies beyond the basic parallelepiped and the MCS consists of the two pieces 9b. shown in FIGURE Concluding the analysis of the geometrical situation, one can state that there exist two qualification regions, the medium (55) and the highest (59), and two critical

C H E W A K et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

343

values and p2 of parameter p that divide the interval 0 I p < m into three regions: (56), (57),and (58). The results are summed up in FIGURE 10. It only remains to describe the behavior that results from different combinations of the parameters Q and p (these combinations are responsible for the shape of the MCS) and from different combinations of the individ's parameters entering into the comfort function (these combinations are responsible for the position of the extremum point on the MCS or inside Clx12). Just as in the previous section of this paper, and in the previous paper in this series: one can separate here the set of creative employees-moderate consumers for l,, 2 , ) belongs to and whose behavior is fully similar to whom the point Om(xm, that considered there. We should thus discuss only the normal employees with OmC nxb, and with the extremum point that lies on the MCS and is determined by the tangency of the minimum ellipsoid-isocomf to the MCS. If the premium for quality is small, p I PI= c , the MCS takes the shape shown in FIGURE2, regardless of the individ's qualification, and the individ's behavior is, therefore, exactly the same as in the case discussed in great detail in the previous section, when there was no additional remuneration for quality (q = q * ). Let the premium for quality have an intermediate value from region (57).If the individ possesses the medium qualification (55), there arise four groups of normal employees corresponding to the four different positions of the tangency point on the MCS shown in FIGURE 8a. The first group consists of the lazy workers-moderate consumers with the tangency point on edge AFB. They produce nothing ( y = 0) and earn and consume

Fig. 8b

t'= L, ( A q ,9= W),J=O

Flg. 9b

& (AiB&H,), $=R[z),r=0

i-c IH 0). When the tangency point lies in the interior of EFBCl ( I = l2+. (x, z)) the individ can be ascribed to the third group of strivers for quantity-eager consumers because on this piece of the M C S y = 0 while q = qm(x) and grows when needs increase. Although such a behavior looks rather ridiculous, it does not, alas, seem to b e unrealistic. The fourth group of normal employees comprises the individs with the working point in vertex C1.They can be described as hard but uneffective workers-unsatisfied eager consumers (Y = 0, q > 4 * ). If the premium for quality is intermediate, p, < p < p2, as above, but the 9a, HGK, is individ’s qualification is high (see (59)),the third piece shown in FIGURE added to the MCS and in this way gives rise to the three new types of behavior (the fifth, sixth, and seventh groups). The fifth group consists of the individs whose working point belongs to curvilinear edge HG where q = 1 but y = 0. The individs of this group are reluctant to increase their production rate when their needs grow. The sixth group is formed by the individs with the comfort maximum reached at an interior point of piece HGK, while the seventh group corresponds to the tangency in 9a). The last two groups are most interesting for the employer vertex K (FIGURE because they are really the best workers: the quality of their labor results is maximal, q = 1, and the production ratey is positive. This situation occurs because the normal employee has exhausted all the chances to increase his income by upgradingq, so the only possibility that remains is to increasey at q = 1. These two types of behavior can take place only if the individ also has sufficiently great values of l,, ,z, andx,. Finally, if the premium for quality is high, p > p2, the behavior of the “lazy workers” remains unaltered, but the majority of normal employees turn into the “strivers for quality” (FIGS.8b & 9b). All the individs with the medium qualification (55) will prefer to work with the zero production rate, however, and their working points (xo,lo,zo) will belong to the piece 1 = l2 * (x, z) of the MCS wherey = 0 and q = q m ( x ) . Only those employees who have the highest qualification (Q > Q , , ) and values of 1, and z, sufficiently close to unity will provide the positive production rate that will save the employer from ruination. It is rather obvious that all normal employees under the remuneration method (38) have positive values of partial discomfort functions with respect to all variables (x, I , 2); however, the value of D , that is, the dissatisfaction index, declines as p increases. Thus, from the viewpoint of an employer the linear remuneration method (38) is acceptable only for quite a diligent and thorough normal employee with high qualification and who is also an eager consumer. Although creative employees may evidently provide y > 0 as well as sufficiently high quality, their behavior, again, as shown in the previous section, is unstable with respect to their intentions to

CHERNYAK et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

345

accumulate money, which can make their working point jump on the MCS piece l2 * (x, z) where y = 0. Let us discuss, in conclusion, a rather interesting point that could not appear in the previous section because the MCSs there were convex surfaces. Now, for p1 < p < p2,the MCS represents a nonconvex surface and the simultaneous tangency of the isocomf to the MCS in two points (near segment FE in FIG. 8a or in FIG. 9a) becomes possible. It is clear that for each ensemble of individs with the same parameters Q , P,, Pz, and Px, there exists such a separating surface (separatrix) that the indicated ambiguity takes place when the point O,(x,, l,, I,) falls on it. The separatrix subdivides the whole domain of the behavior variables of normal employees into two subdomains that are similar to the attractors" in the phase space of a dynamic system.8 A transition from one to another subdomain through the separatrix will correspond to the jump in the individual behavior, that is, to the transformation of the individ from being a striver for quantity to being a striver for quality and vice versa. The case of 0, lying on the separatrix is not interesting, however, because the familiar roughness conditions"J* are not satisfied (the case is not generic).

Minimums Prescribed for Quality and Production Rate It has been ascertained in the previous subsection that the pay scheme (38) is often unsatisfactory from the employer's point of view, because some of the employees transfer to the mode with the production rate vanishing ( y = 0). To eliminate this problem one can introduce a nonlinear dependency of the remuneration I , on y and q such as I , y (see reference 6) or put administrative limitations on the production rate. W e will consider briefly in this subsection the last case, in which the labor remuneration can be represented in the form generalizing relation (38) as

-

)I%

Y -Y * ) (60) where y * is a minimal admissible production rate ( y * > 0). We will assume again 1, = [BY + b ( q - 4 *

-4* M

that there is no independent income I,, so formula (60) determines the total income of the individ under study. The solution of the extremal problem (10) that now describes the individ's behavior under remuneration (60) can be analyzed in a way perfectly analogous to that demonstrated in the previous subsection, so we will just list the main results. As above, the individs with the qualification sufficient to ensure the requirements y 2 y , and q 2 q * fall into two groups: first, the employees with a medium qualification Q from the region

and, second, the employees with the highest qualification such that

gNote that there is no problem in writing formally the differential equations describing the time evolution of a dynamic system with the phase space ( x , I , z ) subdivided into these attractors.

ANNALS NEW YORK ACADEMY OF SCIENCES

346

The latter group exists if the minimal admissible production rate y * is properly choosen ( y * < l), which will be assumed further on. As above, the region [0, m), to which the parameter P belongs, is broken down into three subdomains by two critical values, PI and P2, with P1 determined by relation (56)as before, and P2 given now by

Note that as a consequence of the first inequality (61) we have P 2 > PI. The domain [0, m) of x is now broken down, generally speaking, into three subdomains by the following characteristic points:

=y*B/f+,

X*

= [ Y * +P(l-q*)IB/Pr

Let us consider the situations that may occur. If the premium or bonus for quality is small, f3 I equation, regardless of qualification, takes the form

P1

(64)

(or z * I l), the MCS

‘lo* (x, z), x I XI * l=l*(x,z)= 4*(x,z), o < z < z * , ,12*(x,z)

I* I 2

X>XI*

5 1, x

(68)

>XI*

FIGUREl l a shows the MCS corresponding to formula (68). As compared to FIGURE 8a, there appears a new piece of the cylindrical surface PRB2A2,which is joined to the part of the surface AFBCIE shown in FIG.8a. On this new surface 1 = lo* (x,z), y = y , , q = q * , and x = x,,, I x1*; that is, a normal employee with the tangency point sited on the surface PRB2A2 (FIG.l l a ) provides only the minimal admissible norms, which suffices to achieve the maximum desirable consumption

CHERNYAK et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

347

(moderate consumer) (the situation is similar to that under the fixed age^,^). The surface piece A2F2Eis described by the equation I = II * (x, z ) where q = q * a n d y = xPJB > y * (x * < x < x,,, ); that is, the corresponding normal employee is the striver for quantity-eager consumer. FIGURE 1l a shows, in particular, EF2BzCl, the piece of a

b

FIGURE 11. Domain ClI,2 and the MCS resulting from remuneration (60) with the medium premium for quality, < p < p2: (a) Q * < Q < QI *; (b) Q l * 5 Q s 1.

the MCS that is described by the equation I = l2 * (x, z ) and corresponds to the strivers for quality-eager consumers, because in this case y = y *, but q > q * and grows withx increasing (see (69)). It is quite clear that among the individs corresponding to the MCS shown in FIGURE l l a , there are two large groups, one operating on

ANNALS NEW YORK ACADEMY OF SCIENCES

348

edge AFB2 (lazy workers-eager consumers) and the other with the working point in vertex C1 (hard workers-unsatisfied eager consumers). If the quality premium or bonus is intermediate again, that is, as in (66), but the individ has the highest qualification QI* < Q I 1

(70)

then for smallx (x 5 x1* ) the MCS is determined by the first line from (68), while for x > x1* the MCS equation takes the following form:

I = /3*(x,z)

=

1 ~

(c

+ 1)Q

[XPJB - P(1 - q * )

+ c/z]

FIGURE l l b shows the MCS corresponding to conditions (66) and (70). It consists of four pieces: the first three pieces are similar to the corresponding pieces shown in FIGUREl l a , while the fourth piece (KHG in FIG.l l b ) is determined by equation (71) and is characterized by the highest quality (q = 1) and by production ratey = y * (x - x * )PJB > y ,, which grows as the consumer’s demand increases. So these normal employees can be identified as thorough workers-strivers for quality-eager consumers. There is also an intermediate case corresponding to the tangency on edge HG in FIGURE 1lb. Here the individ manifests a kind of apathy toward increasing his production rate. Next, let us consider the case of the extremely high quality premium or bonus

+

P>

P2

For the group of individs with medium qualifications (67) the MCS piece determined by the equation I = lo* (x, z ) disappears now, and the MCS consists of two pieces (FIG.12a). In this case, among the normal employees there are four groups. The first one operates on AzB2RP according to minimal norms, y = y *, q = q * , and x = x, I x, (moderate consumers). The second group, with the working point on edge AzB2 (FIG.12a) wherey = y *, q = q *, andx = x * < x,, comprises the lazy workers-eager consumers. The third group of employees comprises the strivers for quality-eager consumers who operate on the interior points of AzBzC~ where y = y * ,q = qm(x) > q *, and x * < x < x,. The fourth group comprises the unsatisfied consumers-hard workers whose working point is in vertex Cz (FIG.12a). Finally, when condition (72) holds and the individ possesses a high qualification (Q > Ql * ), the MCS is formed by the three pieces shown in FIGURE 12b. This MCS is similar to the surface shown in FIGURE l l b but differs in the absence of piece A3F3E1,the points of which correspond to the strivers for quantity. All the geometrical situations that arise under remuneration (60) are summed up in FIGURE13. Note that the considered subdivision of the behavior space into the different domains can again be interpreted in terms of the subdivision of the phase space into the attractors.

GENERALIZATIONS AND RELATED PROBLEMS

It is of interest to discuss first of all what novel features of the labor and consumer behavior can arise when the individ is allowed not to work and may also possess a

CHERNYAK el al.: CONSTRUCTIVE THEORY PIECEWORK PAY

349

source of an independent income 1, > 0 (that is, when legislation toward working and income is permissive). Note that there can be two types of the independent income, unconditional income, which is obtained independent of whether the individ is working or not, and conditional income, which is available under the precondition that the individ gives up working. The effect of unconditional income has been

b

FIGURE 12. Domain ClX,: and the MCS resulting from remuneration (60) with the high premium for quality, p 2 p2: (a) Q * < Q < Ql* ; (b) &I * 5 Q I 1.

considered in references 4 and 6 for the case of fixed wages, so we shall dwell here only briefly on the effect of conditional income. Since the restriction that the individ is obliged to work has been omitted, there arises a new opportunity of the idle life. In terms of the admissible values of the behavior variables it implies that a new onc-dimensional domain X = (x, 1,z:

ANNALS NEW YORK ACADEMY OF SCIENCES

350

1 = 0, z = 0,O I x I Io/Pr)is added to domain Clxb to form the following doubly connected domain

a,

=

x u adz

(73)

If the minimum isocomf is tangent to the piece Cl,, the individ works as before; if it is tangent to the piece O w , the individ is idle. The choice between these alternatives is equivalent to the assignment of a value to a new binary behavior variable o,which should be introduced into the comfort (discomfort) function to describe the new degree of freedom (to work or not to work). When o takes on one of the two possible values, the individ works, when

FIGURE 13. The geometrical situations arising for normal employees under the remuneration determined by (60).

another, he/she does not. It is convenient to mark the comfort (discomfort) function corresponding to the former value of o with a superscript plus (+), and to the latter with a superscript minus (-). When the individ is idle one should set 1 = 0 and z = 0, and the discomfort function becomes

D -(x, 1, z)

= Px(xm- x)2

+ P,l$ + P,Zi

(74)

Thus, the labor variables give a constant contribution into the discomfort, which now varies only with the consumption x . If the needs of the individ are sufficiently low as compared to the independent income Zo, that is, if

xmPr I I ,

(75)

CHERNYAK el nl.: CONSTRUCTIVE THEORY: PIECEWORK PAY

35 1

the individ's optimum consumption in the idle state will be equal tox,, and hence the value of the corresponding discomfort function becomes

Therefore, the discomfort stems only from the lack of a job. Otherwise, when

the discomfort is given by the expression

so the discomfort in this case is caused by both labor and consumption variables. When considering the behavior of a particular individ, one should first find out which of the groups (75) or (77) the individ belongs to, and then compare the value of Dj- (i = 0, 1) with the value of the discomfort D +(xn, I ",z o ) that corresponds to the working state, and which is calculated on the solution (x", I", zo) of the extremal problem discussed above. It is evident that when

Di- < D +(xo,lo,z o )

(79)

the individ from group i prefers the idle life to that complicated by the job. In the opposite case, when

Di- > D+(xn,I o , z O )

(80)

the individ prefers to work so as to gain a higher level of consumption. Note, that for the creative employees-moderate consumers we always have D + ( x o ,l o , z n ) = 0. However great their independent income may be, they never give up working. The boundary case

D,- = D +(xo, I", z") corresponds to ambiguous behavior (the individ is not sure whether to work or not). Condition (81) constitutes an equation of the separatrix in the space of the personality parameters, the separatrix dividing the set of individs into subsets of working and idle individs. Case (81) in itself is not generic, and therefore is of no particular interest? The next problem to be studied within the above framework is the problem of the individ's choice from among n different available jobs. When there is such a choice one can find the set of admissible domains each corresponding t o job i (i = 1, 2, . . . n ) and solve n extremal problems

D(')(x,I , z ) = min,

(x, I, z ) E

(82)

hThe ambiguity disappears when a more fine and detailed study is made. The behavior becomes more definite when additional preferences and restrictions (such as prestige and time expenditures) are taken into consideration. The account of such factors increases the dimensionality of the parameter space, so the "relative probability" of having the individ fall within the separatrix "decreases infinitely" with each new dimensionality added.

ANNALS NEW YORK ACADEMY OF SCIENCES

352

taking into account the fact that the qualification and other labor parameters related to different jobs are generally different. The choice of the job reduces now to minimization of [ D ( f from ) ] (82) with respect to integer variable i (1 I i I n), which is a mere generalization of the “to work or not to work” problem. A purely academic study of the job choice problem is not very interesting because there are too many parameters involved. But the study of this problem in connection with a real, particular situation, when some of the parameters are fixed, is fairly interesting. The next problem to be mentioned is the effect of an insufficient supply of goods on the market. Such a situation may be caused either by legislation or by natural development of the society. Let us assume that the limitation x 11

(83)

is imposed on the consumer activity of the working individ. It is clear that the admissible domain s2,,transfers (83) into the new one =

fl ( x , l , z : x I i ,l E R+,z E R , )

(84)

which is, generally, smaller than the original domain 0 d z . In geometrical terms domain hdzresults from cutting off the region of the highest production rate and/or the highest quality from original domain sZxbby the planex = i .As has been shown in previous sections, the domain sl,, expands with the qualification growing, so one comes to the conclusion that the higher the individ’s qualification the greater the negative impact of limitation (83) on the individ’s labor results (quality and/or production rate). A limitation such as (83) is also disadvantageous from the managerial viewpoint. Due to the limitation the individ becomes less controllable by the management. For example, additional remuneration, suggested earlier, would have no effect, whereas additional remuneration would result in a positive effect in the absence of restriction (83). Note that a limitation such as (83) can also arise from the time expenditures’** devoted to the purchase and utilization of goods. Therefore, queues and poor service may give birth to a heavy restriction of this kind. Within the framework of our general scheme covering the whole society,’ goods consumed are produced by other individs who themselves should obey restriction (83). This suggests that there arises a positive feedback loop around productivity and living standards. That is, a cycle develops in which restriction (83) causes developments that in turn give the restriction more force, and so on. As a concluding remark, we should stress that the present study, considered as an application of our general scheme’ to the simplified problem of individual labor behavior, has revealed a number of interesting and realistic features that are rather characteristic of actual human behavior. And this has proved possible even within the framework of a very unsophisticated quadratic-linear comfort function. This fact strongly supports our hope that the general scheme can be successfully applied to more general problems concerning the functioning of the whole society.

CHERNYAK et al.: CONSTRUCTIVE THEORY PIECEWORK PAY

353

REFERENCES

Y. B., A. I. LEONOV& A. Y. LERNER.1992. Toward the constructive theory of 1. CHERNYAK, human social behavior. I. Fundamental principles. Ann. N.Y. Acad. Sci. This volume. 2. CHERNYAK, & A. Y. LERNER. 1992. Toward the constructive theory of Y. B., A. 1. LEONOV human social behavior. 11. Consumer Behavior. Ann. N.Y. Acad. Sci. This volume. Y. B., A. I. LEONOV& A. Y. LERNER.1992. Toward the constructive theory of 3. CHERNYAK, human social behavior. 111. Labor behavior with due regard for consumption. Ann. N.Y. Acad. Sci. This volume. 4. CHERNYAK, Y. B., A. 1. LEONOV& A. Y. LERNER.1992. Toward the constructive theory of human social behavior. IV. Labor behavior under fixed remuneration. Ann. N.Y. Acad. Sci. This volume. & A. Y. LERNER.1985. On the theory of mass behavior in Y. B., A. 1. LEONOV 5. CHERNYAK, developed human communities. Ann. N.Y. Acad. Sci. 452 44-53. 6. CHERNYAK, & A. Y. LERNER. 1990. On a theory of developed human Y. B., A. I. LEONOV communities: Individual behavior in case of performing labor activity and consumption. I n Collection of Works Dedicated to Alexander Y. Lerner on the Occasion of His Seventieth Birthday. J. Acad. Proc. Sov. Jewry (H. S. Publications) 2 37-103. G. P. & E. P. RAIKHMAN. 1973. On Qualimetry (in Russian). Izdatelstvo 7. AZDALDOV, Standartov. Moscow. T. V., V. A. KUKHADZE & BOIKOVSKII. 1974. Quantitative Methods to Evaluate 8. KASHUBA, Quality and Technical Level of Industrial Output (in Russian). TzNIITEI. Obzornaya Informatziya. Moscow. 1968. Nonlinear Programming. John Wiley & Sons. 9. FIACCO,A. V. & G. P. MCCORMICK. New York, NY. E. 1971. Lecons De Theorie Microeconomique. Dunod. (Russian transla10. MALINVAUD, tion: 1985.) Nauka. Moscow. 1983. Regular and Stochastic Motion. SpringerA. J. & M. A. LIEBERMAN. 11. LICHTENBERG, Verlag. New York, NY. A. A., A. A. VITT & S. E. KHAIKIN.1981. Theory of Oscillations (in Russian). 12. ANDRONOV, Nauka. Moscow.

Toward the constructive theory of human social behavior. V. Labor behavior under piecework pay.

Toward the Constructive Theory of Human Social Behavior. V. Labor Behavior under Piecework Pay YURI B. CHERNYAK? ARKADII I. LEONOV,b AND ALEXANDR Y. L...
1MB Sizes 0 Downloads 0 Views