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Bi-criteria utility functions: Analytical considerations and implications in the short-run labour market
European Journal of Operational Research 122 (2000) 91±100
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Theory and Methodology
Bi-criteria utility functions: Analytical considerations and implications in the short-run labour market Carlos Romero
*
Departamento de Economõa y Gesti on, E.T.S. Ingenieros de Montes, Universidad Polit ecnica de Madrid, Avenida Complutense s/n, Madrid 28040, Spain Received 14 March 1998; accepted 11 January 1999
Abstract An analytical framework to address the optimisation of bi-criteria utility functions u S; B de®ned in the sales revenue±pro®t space is presented. Some of the macroeconomics implications underlying these kinds of utility functions are elicited. Thus, the short-run aggregate demand for labour in a context where ®rms maximise a utility function u S; B is obtained and compared with the classic demand for labour predictions derived from a pro®t maximisation behaviour. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Economics; Multiple criteria decision making; Compromise programming
1. Introduction Several economists have postulated for a long time that the utility function of many kinds of ®rms may have several arguments and not a single one (pro®ts) as it is traditionally assumed. One of the pioneers of this type of argument is Baumol (1967) who proposed in his classic book Business Behavior, Value and Growth his pro®t constrainedrevenue maximisation hypothesis, which postulates that the behaviour of certain big ®rms is explained by a sales revenue objective function subject to a minimum pro®t restraint. Although
*
Tel.: +34 91 3366393; fax: +34 91 5439557. E-mail address: [email protected] (C. Romero).
Baumol placed his hypothesis within an oligopolistic context, it can be extended to other market structures, whenever the size of the ®rm is big. Fisher (1960) in his review of Baumol's book suggested a symmetrical hypothesis, that is, big ®rms seek a maximum pro®t subject to a constraint on sales revenue. Later Osborne (1964) found that both hypotheses are similar in many aspects. Hall (1966) provided more insights on the relationships between the two hypotheses. However, Rosenberg (1971) pointed out that a particular form of the objective function underlying this kind of hypothesis is very questionable if not untenable for the following reasons. To conjecture that the behaviour of certain big ®rms is explained by a sales revenue objective
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
function subject to a minimum pro®t restraint implies accepting a lexicographic ordering of preferences between both attributes ± pro®ts and sales revenue. Thus, only when minimum pro®t has been achieved, the other attribute (sales revenue) is taken into consideration. In consequence, the marginal rate of substitution of sales revenue for pro®t is in®nite when pro®t surpasses the minimum bound established and is equal to zero when pro®t is below this minimum bound. In other words, it is not possible to trade o sales for pro®ts or viceversa, which seems counter-intuitive in business scenarios. In fact, it is sensible to assume that there will always be a potential increment in sales revenue (pro®ts) which compensates a potential decrease in pro®ts (sales revenue). Rosenberg's criticism to Baumol's hypothesis can be extended to Fisher's hypothesis. Despite these criticisms, the essence of Baumol's seminal idea is quite valuable: big ®rms are not solely motivated by achieving large pro®ts but also by achieving other objectives like large sales revenue. The consideration of another criteria besides pro®ts in the utility function of big ®rms is also theoretically justi®ed from the perspective of the behavioural theory of the ®rm. Thus, the founders of this theory (Simon, 1961; Cyert and March, 1963) sustain that ®rms ± chie¯y big ®rms ± are organisations designed to achieve the objectives of several groups of economic agents such as stockholders, management, employees, etc., and therefore, a ®rm has several objectives since each group aims at his own goals. In a related sense, Williamson (1964) postulated that managers seek to maximise a utility functions with the following arguments: sta employed, emoluments of the executives and discretionary pro®ts. In connection with the above theoretical proposals, several empirical tests were developed in the late seventies. These tests are based on the assumption that executives are compensated in accordance with the ®rm's objectives. Hence, if the executive remuneration is explained by a function of sales revenue and pro®ts then the evidence gives support to the existence of a bi-criteria utility function. On the contrary, if the executive remuneration is explained only by pro®ts or only by sales revenue then the empirical evidence only
supports the corresponding single criterion hypothesis. The most conclusive research according to the above ideas is perhaps the work undertaken by Smith et al. (1975, Chaps. 8 and 9). These researchers found that for 557 large US ®rms sales revenue and pro®ts were objectives followed by the ®rms. They even quantify the trade-o between both objectives. They also found that for 49 regulated ®rms the evidence supports sales revenue maximisation rather than pro®t maximisation. Other authors have resorted to the formulation of econometric models using balance sheets and income account data to test the hypothesis of multi-criteria ®rms utility functions. Thus, Beedles (1977) using time series data for the 1929±1973 time period for three big ®rms found clear evidence supporting the argument that the three ®rms had pursued three objectives: sales revenue, pro®ts and stock price. Despite the above theoretical and empirical arguments the interest for these kinds of managerial theories of the ®rm has decreased in the last few years. One possible explanation for this loss of interest lies in the diculties associated with the building of multi-criteria utility functions which underlie all managerial theories of the ®rm. Although in some particular cases it is possible to obtain a reliable representation of a multi-criteria utility function (see Keeney and Raia, 1976) in general this problem is almost unsolvable. This paper attempts to take a further step in the direction of considering bi-criteria ®rm utility functions with two arguments: sales revenue and pro®ts. In a previous paper (Ballestero and Romero, 1994a) an analytical framework to tackle the optimisation of these kinds of bi-criteria utility functions was established. This framework provides a good surrogate which allows the utility optimum to be approximated even when the utility function is virtually unknown. In this research, these ideas will be developed with the main purpose of providing further analytical insights as well as deriving some macroeconomics implications implied from this kind of bi-criteria utility functions. The importance of connecting Operational Research (OR) methods with economic analysis
C. Romero / European Journal of Operational Research 122 (2000) 91±100
has been well established. This paper attempts to take a further step in this direction, showing how this kind of connection can open new perspectives to researchers in economics and in OR. 2. Sales revenue±pro®t utility functions and its in¯uence on demand for labour 2.1. Derivation of the aggregate frontier In this section, we will study how the aggregate demand for labour, which is usually derived by accepting an assumption of pro®ts maximisation, can be deduced in a context where the ®rms maximise a bi-criteria utility function de®ned in the sales revenue±pro®t space. Thus we will focus in the short-run competitive aggregate labour market with real wages perfectly ¯exible in an upward as well as downward direction and a demand for labour which is a function of these real wages. Part of this basic analytical procedure is applicable to other labour market structures although the simple case considered might be a good initial referential scenario. As readers know, demand for labour N d predicted by the `classic' theory in a competitive market derives from 0
h N w=P :
1
The above function implies an aggregated short-run production function Q h N which maps output level Q on input labour N (both aggregated); h0 N is the marginal productivity of labour and w=P the real wages. In a monopoly market, the `classic' prediction turns into 1 1=e h0 N w=P ;
2
where e < 0 is the elasticity of the monopolist's demand curve. Hence, considering ®nite values of elasticity e, the demand for labour at the same level of real wages is less in a monopolist industry than in a competitive industry. These common results in classic macroeconomics raise some question: (a) a smaller demand for labour in monopolist markets than in competitive markets seems counter-intuitive and not
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empirically corroborated, (b) the maximum pro®t assumption underlying the `classic' analysis is questionable for the reasons pointed out in Section 1, (c) how could a sales revenue±pro®t utility function be integrated within a macroeconomics model? In what follows these questions will be analysed with the help of conventional and nonconventional microeconomics tools. The ®rst step in our analysis will consist in determining the aggregate demand for labour in a economy where the behaviour of the ®rms is explained by the maximisation of a utility function with two arguments, sales revenue and pro®ts. Let us start by establishing the frontier (or transformation curve) sales revenue±pro®t within a `classic' macroeconomics model context. From the aggregate short-run production function Q h N the equations measuring pro®t B and sales revenue S (both aggregate) can be straightforwardly written as: B Ph N ÿ wN ; S Ph N :
3 4
By substituting Eq. (4) into Eq. (3) and taking out N we get N S ÿ B=w:
5
From Eq. (4) we have Q h N S=P :
6
From Eqs. (5) and (6), the following aggregate frontier sales revenue±pro®t is obtained: h S ÿ B=w ÿ S=P 0:
7
Frontier Eq. (7) represents the set of combinations pro®ts±sales that can be obtained by varying the amount produced (and consequently the labour input). Let us assume a quadratic technology such as h N b1 N ÿ b2 N 2 :
8
Thus, the aggregate frontier (7) turns into b2 S ÿ B=w2 ÿ b1 S ÿ B=w S=P 0:
9
In order to determine the equilibrium point and to derive from it the demand for labour the desirable approach will now be to determine the
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
point of tangency between the family of iso-utility curves u S; B k and the aggregate frontier (see Fig. 1). This function is not a `collective utility function' of all ®rms in the economy, but the utility function of a `representative ®rm'. Because of the enormous diculties to obtain a reliable representation of function u S; B we are going to seek for a surrogate to the above economic optimum. 2.2. Surrogate of the optimum equilibrium As a ®rst step in our search of a surrogate of the optimum equilibrium let us introduce two impor-
tant points in our space of reference sales revenue± pro®ts. These points are termed in decision theory as ideal and anti-ideal (or nadir) points, respectively. The ideal values S and B represent the maximum values for sales revenue and pro®ts over the frontier, while the anti-ideal or nadir values S and B represent the value of each attribute when its antagonist has achieved its maximum value (see Fig. 1). By elementary calculus, these ideal and anti-ideal values are obtained. Table 1 shows their corresponding values. Now let us introduce as an approximated solution of the economic optimum the point of the aggregate frontier that is nearest to the ideal pointZeleny's axiom of choice (see Zeleny, 1973, 1982).
Fig. 1. Equilibrium in the sales revenue±pro®ts space.
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Table 1 Pay-o matrix for sales revenue and pro®ts Sales revenue Sales revenue
S
Pro®ts
S
Pro®ts
0:25b21 P b2
0:25b21 P 2 ÿ 0:25w2 b2 P
Several solutions appear for dierent metrics p by solving the following optimisation problem: p p 1=p 10 Min Lp ap1 S ÿ S ap1 B ÿ B subject to Eq. (9), where a1 and a2 play two dierent roles: (a) that of normalising in order to make the comparison of the S and B attributes possible, and (b) that of measuring the decision maker's preferences for each objective. Yu (1973) proved that for bi-criteria cases metrics p 1 and p 1 de®ne a subset on the ecient frontier called the compromise set, where the other best-compromise solutions fall. Therefore, by solving Eq. (10) for metrics p 1 and p 1, the portion of the frontier nearest to the ideal is obtained for every metric (see Fig. 1). Let us determine L1 and L1 bounds for our aggregate frontier T S; B. It has been proved elsewhere (Ballestero and Romero, 1991) that the L1 bound of the compromise set is the point where a2 T1 ÿ a1 T2 0 ± called the L1 path ± intercepts the frontier, being T1 and T2 the partial derivatives of the frontier with respect to the two attributes under consideration (i.e. S and B for our particular case). For frontier (9) the L1 path is given by S ÿ B b1 1 b1 SÿB ÿ 2b2 ÿ a1 a2 2b2 ÿ w p w w2 w2 0:
11
To determine the L1 bound, we resort to a lemma which suitably adapted to our context reads as follows: the L1 bound of the compromise set is the point where the straight line a1 S ÿ S a2 B ÿ B ± called the L1 path ± intercepts the frontier (Ballestero and Romero, 1991). Taking into account the ideal values S and B (see Table 1) the L1 path is given by
a1
B
0:25b21 P ÿ 0:5b1 w b2
B
0:25b21 P 2 ÿ 0:5b1 wP 0:25w2 b2 P
0:25 b21 p2 ÿS b2 p
a2
0:25 b21 p2 ÿ 0:5 b1 wp 0:25 w2 ÿB : b2 p 12
To calculate the L1 and L1 paths, the problem of interpreting weights a1 and a2 has to be addressed. Of the two roles played by the weights (normalises and indicators of the entrepreneur's preferences), a1 and a2 should only play the role of normalises since the entrepreneur's preferences are actually given by the iso-utility curves de®ned in the S ÿ B space and perceived by the own entrepreneur. In this situation, taking a1 and a2 as a preference measure would be super¯uous and even contradictory. A sensible and commonly used normalising system (e.g. Romero (1991, Chap. 2) and Ballestero and Romero (1993) for a justi®cation from a shadow price perspective) is a1
1 ; S ÿ S
a2
1 ; B ÿ B
13
that is, weights inversely proportional to the ranges of both criteria. According to Table 1 we have the following variables: b2 p b2 p ; a2 : 14 0:25 w2 0:25 w2 Therefore, for the quadratic technology chosen (see expression (8)) both weights coincide. It is also easy to check that for this weighting, both paths L1 and L1 (see expressions (11) and (12)) coincide in the following equation: ÿ 15 S ÿ B 0:5 wb1 p ÿ 0:25 w2 = b2 p:
a1
From Eq. (15), the demand for labour can be straightforwardly deduced as will be shown below.
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
Let us now justify Eq. (15) as a good surrogate of the unknown utility optimum. For this purpose, we resort to a Theorem proved elsewhere (Ballestero and Romero, 1991) which suitably adapted to our S±B space reads as follows. With any utility function u S; B involving a non-satiation scenario, the condition under which the maximum of u always belongs to the compromise set on every given transformation curve is Marginal rate of substitution MRS S; B u1 =u2 a1 =a2 on the L1 path a1 S ÿ S
3. On demand for labour predictions: A comparison
a2 B ÿ B: The condition underlying the theorem seems sensible since it simply implies a behaviour coherent with the diminishing MRS law. In fact, the Theorem underlies the following DM's behaviour: if I already have a large amount of sales, I can exchange a signi®cant amount of sales for a marginal unit of pro®t without losing utility (and vice versa). On the contrary, when my sales±pro®t is already balanced (i.e. belongs to the L1 path) there is no convincing reason for sacri®cing large sales in favour of small additional pro®ts (or vice versa); that is, my MRS 1 when the sales±pro®t allocation is balanced (i.e. belongs to the L1 path). When the DM thinks this way, the traditional equilibrium point (max u S; B subject to the frontier restraint) lies on the compromise set which in our case with a quadratic technology reduces to a single point. More details about the economic plausibility of the Theorem in a general context can be seen in Ballestero and Romero (1994b). In Mor on et al. (1996) is proved the existence of a large family of utility functions holding the condition supporting the theorem. The microfoundations of the theorem in the bi-criteria space sales revenue±pro®ts can be seen in Ballestero and Romero (1994a). From Eq. (15) we can straightforwardly derive the demand for labour. In fact, taking into account that S ÿ B WN , the following demand for labour is obtained: d 0:5 NSB
b1 0:25 w ÿ : b2 p b2
d In other words, NSB represents an aggregate demand for labour in a market where the entrepreneur's behaviour is coherent with the maximisation of a utility function u S; B and they face a quadratic technology. It should be pointed out that for another technologies the bounds L1 and L1 do not coincide and therefore the demand for labour function is not unique. However, it is possible to make relatively accurate predictions on the demand for labour (see Section 4).
16
d Let us now compare the demand for labour NSB with the demands for labour which corresponds to: (a) a purely sales revenue maximise industry NSd , (b) a purely pro®t maximise industry NBd and (c) a purely pro®t maximise `a la Cournot' monopoly NCd . After simple algebraic manipulations the following expressions are obtained:
NSd 0:5
b1 ; b2
17
NBd 0:5
b1 0:5 w ÿ ; b2 b2 p
18
NCd 0:5
b1 0:5 e w : ÿ b2 b2 1 e p
19
The demand for labour Eqs. (16)±(19) is depicted in Fig. 2. The slopes of the above functions represent the marginal increase in employment generated by a marginal decrease in real wages. Thus, for a given decrease in real wages, the maximum increase in employment is obtained with a demand for labour `a la Cournot' (19), afterwards with a traditional demand function (18) and ®nally with a demand function derived from a bicriteria utility function (16) just half of the increase with respect to the traditional demand function. On the other hand, the bi-criteria demand function (16) predicts an amount of labour larger than the traditional demand function for the same level of real wages. Obviously, the maximum demand prediction is obtained when entrepreneurs in the industry maximise their sales (see function NSd in Fig. 2).
C. Romero / European Journal of Operational Research 122 (2000) 91±100
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Fig. 2. Aggregate demand for labor predictions.
These results let us explain the questions raised at the beginning of Section 2.1. Thus, it has been shown how a bi-criteria utility function de®ned in the space sales revenue±pro®ts can be fruitfully incorporated into a macroeconomics model. Moreover, the results obtained in this section reconcile the theoretical prediction on monopolist ®rms demand for labour (less labour than the competitive ®rms) with the dubious empirical corroboration of this phenomenon. In what follows, this matter will be clari®ed.
In fact, it can be conjectured that in some markets ± chie¯y oligopolistic or monopolistic markets ± there are two opposite eects. One re¯ects the in¯uence on ®rm's decisions of the sales revenue objective through the utility function u S; B. This eect ± which will be termed Baumol eect ± represents something like a `rising pull' which shifts the classic demand for labour NBd upwards. For a quadratic technology the demand functions are unique and linear and the Baumol eect multiplies by 0.5 the slope of the classic de-
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
mand for labour. The other eect re¯ects the in¯uence on ®rm's decisions of monopolistic trends. This eect ± which will be termed Cournot eect ± represents something like a `downward pull' which shifts the classic demand for labour NBd downwards. With a quadratic technology the Cournot eect multiplies by e= e 1 the slope of the classic demand for labour. In a few words, the large size of a ®rm implies an in¯uence of sales revenue on ®rm's decisions which increases the demand for labour as well as an increase in the concentration (monopolistic) trends which decreases the demand for labour. The joint action of both eects can shift the classic demand for labour upwards or downwards. Therefore, our articulation of a utility function u S; B seems to justify a demand curve for labour in a monopolist market which is lower, coincidental or higher than the classic curve. Thus, the demand for labour in a monopoly market will coincide with the demand NCd when Baumol eect is nil. Depending upon the relative value of Baumol eect with respect to the value of elasticity e then, the demand for labour in a monopolistic market can be situated lower or over the classic competitive curve NBd . Let us specify this conclusion by considering the following scenarios within a monopoly market: (a) Baumol eect (`rising pull') is exactly compensated by Cournot eect (`downward pull'). d ÿ NBd Mathematically this situation implies NSB d d NB ÿ NC . When elasticity e takes the value ÿ3 then this condition holds. In other words, when the elasticity of the monopolist's demand curve is equal to ÿ3 the amount of labour demanded coincides with the classic prediction for competitive ®rms. (b) Baumol eect (`rising pull') is larger than Cournot eect (`downward pull'). Mathematically d ÿ NBd > NBd ÿ NCd . Again this situation implies NSB it is easy to check that this situation arises when the elasticity takes a value lower than ÿ3. In this case, the amount of labour demanded is larger than the classic prediction for competitive ®rms. (c) Baumol eect (`rising pull') is less than Cournot eect (`downward pull'). Mathematically d ÿ NBd < NBd ÿ NCd . Again this situation implies NSB it is easy to check that this situation arises when
the elasticity takes a value larger than ÿ3. In this case, the amount of labour demanded is less than the classic prediction for competitive ®rms. Therefore in a monopoly market where the monopolist has a bi-criteria utility function u S; B and faces a very elastic demand curve (i.e. elasticity e larger than 3 in absolute value) then the amount of labour demanded will be higher than that of a competitive producer (assuming that both have identical quadratic production functions) for a given real wage. In the next section, these ideas will be illustrated with the help of some numerical examples. 4. Some numerical illustrations In order to illustrate the theory developed above, let us assume the following data for a case of quadratic technology: b1 24;
b2 1;
W 20;
P 5:
For these data and according to Eq. (9) the frontier is
SÿB 20
2
ÿ 24
SÿB 20
0:2 S 0:
The corresponding labour demand functions according to Eqs. (16)±(19) are NSd 12;
w ; p w NBd 12 ÿ 0:50 ; p e w d : NC 12 ÿ 0:25 1e p d 12 ÿ 0:25 NSB
For the real wage given w=p 4 the following labour demand predictions are obtained: NSd 12;
d NSB 11; e : NCd 12 ÿ 2 1e
NBd 10;
Therefore, the `rising pull' caused by the Baumol eect is equal to 11 ÿ 10 1 while the `downward pull' caused by the Cournot eect de-
C. Romero / European Journal of Operational Research 122 (2000) 91±100
99
pends upon the value of the elasticity e. It is easy to check that if e ÿ3 both eects coincide. However, if e takes a value less than ÿ3 then Baumol eect is larger than Cournot eect. Thus, for e ÿ4, we have NCd 9:33, that is the `rising pull' surpasses in 0.66 units the `downward pull'. On the contrary, if e takes a value larger than ÿ3 the ®nal result is the opposite. Thus, for e ÿ2, we have NCd 8, that is the `downward pull' surpasses in one unit the corresponding `rising pull'. The robustness of our conclusion relies heavily upon the assumption of quadratic technology. Let us now attempt an extension of the results to other technologies by means of an example. A more general aggregate production function such as
are compensated (i.e. elasticity e equal to ÿ3). The latter is equal to 178:5 ÿ 143 35:5 while the former is equal to 220:75 ÿ 178:5 42:5 for metric p 1 and equal to 215:25 ÿ 178:5 36:75 for metric p 1. Although both eects do not exactly coincide as happens when the production function is quadratic the similarity among them is strong. This similarity is specially remarkable for the metric p 1 (a Baumol eect of 36.75 versus a Cournot eect of 35.5). This numerical example although tentative seems to keep the theoretical argument in an essential way.
h N b1 N ÿ b2 N h ;
Baumol's hypothesis on the behaviour of oligopolistic ®rms (`maximisation of sales revenue subject to a pro®t constraint') although is questionable from a logical point of view is quite valuable because of its seminal idea: big ®rms are not solely motivated by achieving large pro®ts but also by achieving other objectives like large sales revenue. This kind of entrepreneurial motivation implies the consideration of bi-criteria utility functions u S; B as it has been postulated for a long time for many economists. However they have not oered a sound and operational framework to tackle their optimisation over the attainable frontier. In this paper, an analytical framework for addressing ± in a convincing way ± the optimisation of these kinds of bi-criteria utility functions is developed. Furthermore, the aggregate demand for labour has been derived within a context where ®rms maximise a utility function u S; B instead of pro®ts. Thus, a S; B bi-criteria behaviour leads to demand for labour predictions which dier from the traditional scheme, where predictions are built according to the pro®t maximisation hypothesis. In fact, when entrepreneurs follow a S; B bi-criteria policy, the demand for labour is higher than when they maximise pro®ts. This result has an obvious background (`sales objectives, bigger ®rms'). Moreover, with a S; B bi-criteria behaviour, the predicted marginal rate of decreasing demand for labour when real wages grow is higher than within a pro®t maximisation context.
h > 1;
leads to the following aggregate frontier: h
b2 S ÿ B=w ÿ b1 S ÿ B=w S=P 0:
For the above data the frontier becomes 1:5 SÿB SÿB 0:2 S 0: ÿ 24 20 20
Ideal and nadir values, as well as the bounds of the compromise set are S 10240; B 5993; L1 ! S 10091; L1 ! S 10040;
S 9562; B 5120; B 5676; B 5735:
In order to compare demands for labour (speci®ed in Sections 2 and 3) we have computed their dierent amounts: NSd 526; NBd 178:5; NCd (with e ÿ3 143; d NSB (on the L1 bound 220:75; d (on the L1 bound) 215:25: NSB
For non-quadratic technologies there is not a d function but two predictions according single NSB to the metric used. Let us now calculate Baumol and Cournot eect for the case when both eects
5. Concluding remarks
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
The explicit recognition of a S; B bi-criteria behaviour reconciles the theoretical prediction on monopolist ®rms demand for labour with the factual character of that phenomenon. In fact, a monopolist ®rm can follow a S; B policy which would justify a demand for labour situated over the classic competitive curve. This situation is specially likely when the monopolist's demand curve is very elastic (i.e., jej > 3). Finally, it should be noted that the paper concentrates on the `supply side' of an economy. A possible extension of this research will consist in considering also the `demand side', through an aggregate demand function. In this way, it should be possible to study the consequences in the macroeconomic equilibrium when a S; B bi-criteria behaviour is considered. Acknowledgements English language was checked by Christine Mendez. This research was supported by `Comisi on Interministerial de Ciencia y Tecnologõa (CICYT)' and `Consejerõa de Educaci on y Cultura' de la Comunidad de Madrid. Comments and technical help by Dr. Dõaz-Balteiro are highly appreciated. Thanks are also given to one reviewer for his helpful suggestions that have greatly improved the economic accuracy of the paper. References Ballestero, E., Romero, C., 1991. A theorem connecting utility function optimization and compromise programming. Operations Research Letters 10, 421±427. Ballestero, E., Romero, C., 1993. Weighting in compromise programming: A theorem on shadow prices. Operations Research Letters 13, 325±329.
Ballestero, E., Romero, C., 1994a. Multiple criteria decision making: Some connections with the economic analysis. In: S. Rios (Ed.), Decision Theory and Decision Analysis: Trends and Challenges. Kluwer, Boston, pp. 223±242. Ballestero, E., Romero, C., 1994b. Utility optimization when the utility function is virtually unknown. Theory and Decision 37, 233±243. Baumol, W.J., 1967. Business Behavior, Value and Growth, rev. ed. Harcourt Brace and World, New York. Beedles, W.L., 1977. A micro-econometric investigation of multi-objective ®rms. Journal of Finance 32, 1217±1233. Cyert, R.M., March, J.G., 1963. A Behavioural Theory of the Firm. Prentice-Hall, Englewood Clis, NJ. Fisher, F.M., 1960. Review of Business Behaviour, Value and Growth. Journal of Political Economy 68, 314±315. Hall, M., 1966. On the goals of the ®rm: Comment. Quarterly Journal of Economics 80, 154±158. Keeney, R.L., Raia, H., 1976. Decision with Multiple Objectives: Preferences and Value Tradeos. John Wiley, New York. Mor on, M.A., Romero, C., Ruiz del Portal, F.R., 1996. Generating well-behaved utility functions for compromise programming. Journal of Optimization Theory and Applications 91, 643±649. Osborne, D.K., 1964. On the goals of the ®rm. Quarterly Journal of Economics 78, 592±603. Romero, C., 1991. Handbook of Critical Issues in Goal Programming. Pergamon Press, Oxford. Rosenberg, R., 1971. Pro®t constrained revenue maximization: Note. American Economic Review 61, 208±209. Simon, H., 1961. Administrative Behavior. Macmillan, New York. Smith, D.J., Boyes, W.J., Peseau, D.E., 1975. Size, Gowth, Pro®ts and Executive Compensation in the Large Corporation. Macmillan, London. Williamson, O.E., 1964. The Economics of Discretionary Behaviour, Marginal Objectives in a Theory of Firm. Prentice-Hall, Englewood Clis, NJ. Yu, P.L., 1973. A class of solutions for group decision problems. Management Science 19, 936±946. Zeleny, M., 1973, Compromise programming. In: Cochrane, J.L., Zeleny. M. (Eds.), Multiple Criteria Decision Making. University of South Carolina Press, Columbia, pp. 262±301. Zeleny, M., 1982. Multiple Criteria Decision Making. McGraw Hill, New York.
www.elsevier.com/locate/orms
Theory and Methodology
Bi-criteria utility functions: Analytical considerations and implications in the short-run labour market Carlos Romero
*
Departamento de Economõa y Gesti on, E.T.S. Ingenieros de Montes, Universidad Polit ecnica de Madrid, Avenida Complutense s/n, Madrid 28040, Spain Received 14 March 1998; accepted 11 January 1999
Abstract An analytical framework to address the optimisation of bi-criteria utility functions u S; B de®ned in the sales revenue±pro®t space is presented. Some of the macroeconomics implications underlying these kinds of utility functions are elicited. Thus, the short-run aggregate demand for labour in a context where ®rms maximise a utility function u S; B is obtained and compared with the classic demand for labour predictions derived from a pro®t maximisation behaviour. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Economics; Multiple criteria decision making; Compromise programming
1. Introduction Several economists have postulated for a long time that the utility function of many kinds of ®rms may have several arguments and not a single one (pro®ts) as it is traditionally assumed. One of the pioneers of this type of argument is Baumol (1967) who proposed in his classic book Business Behavior, Value and Growth his pro®t constrainedrevenue maximisation hypothesis, which postulates that the behaviour of certain big ®rms is explained by a sales revenue objective function subject to a minimum pro®t restraint. Although
*
Tel.: +34 91 3366393; fax: +34 91 5439557. E-mail address: [email protected] (C. Romero).
Baumol placed his hypothesis within an oligopolistic context, it can be extended to other market structures, whenever the size of the ®rm is big. Fisher (1960) in his review of Baumol's book suggested a symmetrical hypothesis, that is, big ®rms seek a maximum pro®t subject to a constraint on sales revenue. Later Osborne (1964) found that both hypotheses are similar in many aspects. Hall (1966) provided more insights on the relationships between the two hypotheses. However, Rosenberg (1971) pointed out that a particular form of the objective function underlying this kind of hypothesis is very questionable if not untenable for the following reasons. To conjecture that the behaviour of certain big ®rms is explained by a sales revenue objective
0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 0 7 4 - 0
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
function subject to a minimum pro®t restraint implies accepting a lexicographic ordering of preferences between both attributes ± pro®ts and sales revenue. Thus, only when minimum pro®t has been achieved, the other attribute (sales revenue) is taken into consideration. In consequence, the marginal rate of substitution of sales revenue for pro®t is in®nite when pro®t surpasses the minimum bound established and is equal to zero when pro®t is below this minimum bound. In other words, it is not possible to trade o sales for pro®ts or viceversa, which seems counter-intuitive in business scenarios. In fact, it is sensible to assume that there will always be a potential increment in sales revenue (pro®ts) which compensates a potential decrease in pro®ts (sales revenue). Rosenberg's criticism to Baumol's hypothesis can be extended to Fisher's hypothesis. Despite these criticisms, the essence of Baumol's seminal idea is quite valuable: big ®rms are not solely motivated by achieving large pro®ts but also by achieving other objectives like large sales revenue. The consideration of another criteria besides pro®ts in the utility function of big ®rms is also theoretically justi®ed from the perspective of the behavioural theory of the ®rm. Thus, the founders of this theory (Simon, 1961; Cyert and March, 1963) sustain that ®rms ± chie¯y big ®rms ± are organisations designed to achieve the objectives of several groups of economic agents such as stockholders, management, employees, etc., and therefore, a ®rm has several objectives since each group aims at his own goals. In a related sense, Williamson (1964) postulated that managers seek to maximise a utility functions with the following arguments: sta employed, emoluments of the executives and discretionary pro®ts. In connection with the above theoretical proposals, several empirical tests were developed in the late seventies. These tests are based on the assumption that executives are compensated in accordance with the ®rm's objectives. Hence, if the executive remuneration is explained by a function of sales revenue and pro®ts then the evidence gives support to the existence of a bi-criteria utility function. On the contrary, if the executive remuneration is explained only by pro®ts or only by sales revenue then the empirical evidence only
supports the corresponding single criterion hypothesis. The most conclusive research according to the above ideas is perhaps the work undertaken by Smith et al. (1975, Chaps. 8 and 9). These researchers found that for 557 large US ®rms sales revenue and pro®ts were objectives followed by the ®rms. They even quantify the trade-o between both objectives. They also found that for 49 regulated ®rms the evidence supports sales revenue maximisation rather than pro®t maximisation. Other authors have resorted to the formulation of econometric models using balance sheets and income account data to test the hypothesis of multi-criteria ®rms utility functions. Thus, Beedles (1977) using time series data for the 1929±1973 time period for three big ®rms found clear evidence supporting the argument that the three ®rms had pursued three objectives: sales revenue, pro®ts and stock price. Despite the above theoretical and empirical arguments the interest for these kinds of managerial theories of the ®rm has decreased in the last few years. One possible explanation for this loss of interest lies in the diculties associated with the building of multi-criteria utility functions which underlie all managerial theories of the ®rm. Although in some particular cases it is possible to obtain a reliable representation of a multi-criteria utility function (see Keeney and Raia, 1976) in general this problem is almost unsolvable. This paper attempts to take a further step in the direction of considering bi-criteria ®rm utility functions with two arguments: sales revenue and pro®ts. In a previous paper (Ballestero and Romero, 1994a) an analytical framework to tackle the optimisation of these kinds of bi-criteria utility functions was established. This framework provides a good surrogate which allows the utility optimum to be approximated even when the utility function is virtually unknown. In this research, these ideas will be developed with the main purpose of providing further analytical insights as well as deriving some macroeconomics implications implied from this kind of bi-criteria utility functions. The importance of connecting Operational Research (OR) methods with economic analysis
C. Romero / European Journal of Operational Research 122 (2000) 91±100
has been well established. This paper attempts to take a further step in this direction, showing how this kind of connection can open new perspectives to researchers in economics and in OR. 2. Sales revenue±pro®t utility functions and its in¯uence on demand for labour 2.1. Derivation of the aggregate frontier In this section, we will study how the aggregate demand for labour, which is usually derived by accepting an assumption of pro®ts maximisation, can be deduced in a context where the ®rms maximise a bi-criteria utility function de®ned in the sales revenue±pro®t space. Thus we will focus in the short-run competitive aggregate labour market with real wages perfectly ¯exible in an upward as well as downward direction and a demand for labour which is a function of these real wages. Part of this basic analytical procedure is applicable to other labour market structures although the simple case considered might be a good initial referential scenario. As readers know, demand for labour N d predicted by the `classic' theory in a competitive market derives from 0
h N w=P :
1
The above function implies an aggregated short-run production function Q h N which maps output level Q on input labour N (both aggregated); h0 N is the marginal productivity of labour and w=P the real wages. In a monopoly market, the `classic' prediction turns into 1 1=e h0 N w=P ;
2
where e < 0 is the elasticity of the monopolist's demand curve. Hence, considering ®nite values of elasticity e, the demand for labour at the same level of real wages is less in a monopolist industry than in a competitive industry. These common results in classic macroeconomics raise some question: (a) a smaller demand for labour in monopolist markets than in competitive markets seems counter-intuitive and not
93
empirically corroborated, (b) the maximum pro®t assumption underlying the `classic' analysis is questionable for the reasons pointed out in Section 1, (c) how could a sales revenue±pro®t utility function be integrated within a macroeconomics model? In what follows these questions will be analysed with the help of conventional and nonconventional microeconomics tools. The ®rst step in our analysis will consist in determining the aggregate demand for labour in a economy where the behaviour of the ®rms is explained by the maximisation of a utility function with two arguments, sales revenue and pro®ts. Let us start by establishing the frontier (or transformation curve) sales revenue±pro®t within a `classic' macroeconomics model context. From the aggregate short-run production function Q h N the equations measuring pro®t B and sales revenue S (both aggregate) can be straightforwardly written as: B Ph N ÿ wN ; S Ph N :
3 4
By substituting Eq. (4) into Eq. (3) and taking out N we get N S ÿ B=w:
5
From Eq. (4) we have Q h N S=P :
6
From Eqs. (5) and (6), the following aggregate frontier sales revenue±pro®t is obtained: h S ÿ B=w ÿ S=P 0:
7
Frontier Eq. (7) represents the set of combinations pro®ts±sales that can be obtained by varying the amount produced (and consequently the labour input). Let us assume a quadratic technology such as h N b1 N ÿ b2 N 2 :
8
Thus, the aggregate frontier (7) turns into b2 S ÿ B=w2 ÿ b1 S ÿ B=w S=P 0:
9
In order to determine the equilibrium point and to derive from it the demand for labour the desirable approach will now be to determine the
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
point of tangency between the family of iso-utility curves u S; B k and the aggregate frontier (see Fig. 1). This function is not a `collective utility function' of all ®rms in the economy, but the utility function of a `representative ®rm'. Because of the enormous diculties to obtain a reliable representation of function u S; B we are going to seek for a surrogate to the above economic optimum. 2.2. Surrogate of the optimum equilibrium As a ®rst step in our search of a surrogate of the optimum equilibrium let us introduce two impor-
tant points in our space of reference sales revenue± pro®ts. These points are termed in decision theory as ideal and anti-ideal (or nadir) points, respectively. The ideal values S and B represent the maximum values for sales revenue and pro®ts over the frontier, while the anti-ideal or nadir values S and B represent the value of each attribute when its antagonist has achieved its maximum value (see Fig. 1). By elementary calculus, these ideal and anti-ideal values are obtained. Table 1 shows their corresponding values. Now let us introduce as an approximated solution of the economic optimum the point of the aggregate frontier that is nearest to the ideal pointZeleny's axiom of choice (see Zeleny, 1973, 1982).
Fig. 1. Equilibrium in the sales revenue±pro®ts space.
C. Romero / European Journal of Operational Research 122 (2000) 91±100
95
Table 1 Pay-o matrix for sales revenue and pro®ts Sales revenue Sales revenue
S
Pro®ts
S
Pro®ts
0:25b21 P b2
0:25b21 P 2 ÿ 0:25w2 b2 P
Several solutions appear for dierent metrics p by solving the following optimisation problem: p p 1=p 10 Min Lp ap1 S ÿ S ap1 B ÿ B subject to Eq. (9), where a1 and a2 play two dierent roles: (a) that of normalising in order to make the comparison of the S and B attributes possible, and (b) that of measuring the decision maker's preferences for each objective. Yu (1973) proved that for bi-criteria cases metrics p 1 and p 1 de®ne a subset on the ecient frontier called the compromise set, where the other best-compromise solutions fall. Therefore, by solving Eq. (10) for metrics p 1 and p 1, the portion of the frontier nearest to the ideal is obtained for every metric (see Fig. 1). Let us determine L1 and L1 bounds for our aggregate frontier T S; B. It has been proved elsewhere (Ballestero and Romero, 1991) that the L1 bound of the compromise set is the point where a2 T1 ÿ a1 T2 0 ± called the L1 path ± intercepts the frontier, being T1 and T2 the partial derivatives of the frontier with respect to the two attributes under consideration (i.e. S and B for our particular case). For frontier (9) the L1 path is given by S ÿ B b1 1 b1 SÿB ÿ 2b2 ÿ a1 a2 2b2 ÿ w p w w2 w2 0:
11
To determine the L1 bound, we resort to a lemma which suitably adapted to our context reads as follows: the L1 bound of the compromise set is the point where the straight line a1 S ÿ S a2 B ÿ B ± called the L1 path ± intercepts the frontier (Ballestero and Romero, 1991). Taking into account the ideal values S and B (see Table 1) the L1 path is given by
a1
B
0:25b21 P ÿ 0:5b1 w b2
B
0:25b21 P 2 ÿ 0:5b1 wP 0:25w2 b2 P
0:25 b21 p2 ÿS b2 p
a2
0:25 b21 p2 ÿ 0:5 b1 wp 0:25 w2 ÿB : b2 p 12
To calculate the L1 and L1 paths, the problem of interpreting weights a1 and a2 has to be addressed. Of the two roles played by the weights (normalises and indicators of the entrepreneur's preferences), a1 and a2 should only play the role of normalises since the entrepreneur's preferences are actually given by the iso-utility curves de®ned in the S ÿ B space and perceived by the own entrepreneur. In this situation, taking a1 and a2 as a preference measure would be super¯uous and even contradictory. A sensible and commonly used normalising system (e.g. Romero (1991, Chap. 2) and Ballestero and Romero (1993) for a justi®cation from a shadow price perspective) is a1
1 ; S ÿ S
a2
1 ; B ÿ B
13
that is, weights inversely proportional to the ranges of both criteria. According to Table 1 we have the following variables: b2 p b2 p ; a2 : 14 0:25 w2 0:25 w2 Therefore, for the quadratic technology chosen (see expression (8)) both weights coincide. It is also easy to check that for this weighting, both paths L1 and L1 (see expressions (11) and (12)) coincide in the following equation: ÿ 15 S ÿ B 0:5 wb1 p ÿ 0:25 w2 = b2 p:
a1
From Eq. (15), the demand for labour can be straightforwardly deduced as will be shown below.
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
Let us now justify Eq. (15) as a good surrogate of the unknown utility optimum. For this purpose, we resort to a Theorem proved elsewhere (Ballestero and Romero, 1991) which suitably adapted to our S±B space reads as follows. With any utility function u S; B involving a non-satiation scenario, the condition under which the maximum of u always belongs to the compromise set on every given transformation curve is Marginal rate of substitution MRS S; B u1 =u2 a1 =a2 on the L1 path a1 S ÿ S
3. On demand for labour predictions: A comparison
a2 B ÿ B: The condition underlying the theorem seems sensible since it simply implies a behaviour coherent with the diminishing MRS law. In fact, the Theorem underlies the following DM's behaviour: if I already have a large amount of sales, I can exchange a signi®cant amount of sales for a marginal unit of pro®t without losing utility (and vice versa). On the contrary, when my sales±pro®t is already balanced (i.e. belongs to the L1 path) there is no convincing reason for sacri®cing large sales in favour of small additional pro®ts (or vice versa); that is, my MRS 1 when the sales±pro®t allocation is balanced (i.e. belongs to the L1 path). When the DM thinks this way, the traditional equilibrium point (max u S; B subject to the frontier restraint) lies on the compromise set which in our case with a quadratic technology reduces to a single point. More details about the economic plausibility of the Theorem in a general context can be seen in Ballestero and Romero (1994b). In Mor on et al. (1996) is proved the existence of a large family of utility functions holding the condition supporting the theorem. The microfoundations of the theorem in the bi-criteria space sales revenue±pro®ts can be seen in Ballestero and Romero (1994a). From Eq. (15) we can straightforwardly derive the demand for labour. In fact, taking into account that S ÿ B WN , the following demand for labour is obtained: d 0:5 NSB
b1 0:25 w ÿ : b2 p b2
d In other words, NSB represents an aggregate demand for labour in a market where the entrepreneur's behaviour is coherent with the maximisation of a utility function u S; B and they face a quadratic technology. It should be pointed out that for another technologies the bounds L1 and L1 do not coincide and therefore the demand for labour function is not unique. However, it is possible to make relatively accurate predictions on the demand for labour (see Section 4).
16
d Let us now compare the demand for labour NSB with the demands for labour which corresponds to: (a) a purely sales revenue maximise industry NSd , (b) a purely pro®t maximise industry NBd and (c) a purely pro®t maximise `a la Cournot' monopoly NCd . After simple algebraic manipulations the following expressions are obtained:
NSd 0:5
b1 ; b2
17
NBd 0:5
b1 0:5 w ÿ ; b2 b2 p
18
NCd 0:5
b1 0:5 e w : ÿ b2 b2 1 e p
19
The demand for labour Eqs. (16)±(19) is depicted in Fig. 2. The slopes of the above functions represent the marginal increase in employment generated by a marginal decrease in real wages. Thus, for a given decrease in real wages, the maximum increase in employment is obtained with a demand for labour `a la Cournot' (19), afterwards with a traditional demand function (18) and ®nally with a demand function derived from a bicriteria utility function (16) just half of the increase with respect to the traditional demand function. On the other hand, the bi-criteria demand function (16) predicts an amount of labour larger than the traditional demand function for the same level of real wages. Obviously, the maximum demand prediction is obtained when entrepreneurs in the industry maximise their sales (see function NSd in Fig. 2).
C. Romero / European Journal of Operational Research 122 (2000) 91±100
97
Fig. 2. Aggregate demand for labor predictions.
These results let us explain the questions raised at the beginning of Section 2.1. Thus, it has been shown how a bi-criteria utility function de®ned in the space sales revenue±pro®ts can be fruitfully incorporated into a macroeconomics model. Moreover, the results obtained in this section reconcile the theoretical prediction on monopolist ®rms demand for labour (less labour than the competitive ®rms) with the dubious empirical corroboration of this phenomenon. In what follows, this matter will be clari®ed.
In fact, it can be conjectured that in some markets ± chie¯y oligopolistic or monopolistic markets ± there are two opposite eects. One re¯ects the in¯uence on ®rm's decisions of the sales revenue objective through the utility function u S; B. This eect ± which will be termed Baumol eect ± represents something like a `rising pull' which shifts the classic demand for labour NBd upwards. For a quadratic technology the demand functions are unique and linear and the Baumol eect multiplies by 0.5 the slope of the classic de-
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C. Romero / European Journal of Operational Research 122 (2000) 91±100
mand for labour. The other eect re¯ects the in¯uence on ®rm's decisions of monopolistic trends. This eect ± which will be termed Cournot eect ± represents something like a `downward pull' which shifts the classic demand for labour NBd downwards. With a quadratic technology the Cournot eect multiplies by e= e 1 the slope of the classic demand for labour. In a few words, the large size of a ®rm implies an in¯uence of sales revenue on ®rm's decisions which increases the demand for labour as well as an increase in the concentration (monopolistic) trends which decreases the demand for labour. The joint action of both eects can shift the classic demand for labour upwards or downwards. Therefore, our articulation of a utility function u S; B seems to justify a demand curve for labour in a monopolist market which is lower, coincidental or higher than the classic curve. Thus, the demand for labour in a monopoly market will coincide with the demand NCd when Baumol eect is nil. Depending upon the relative value of Baumol eect with respect to the value of elasticity e then, the demand for labour in a monopolistic market can be situated lower or over the classic competitive curve NBd . Let us specify this conclusion by considering the following scenarios within a monopoly market: (a) Baumol eect (`rising pull') is exactly compensated by Cournot eect (`downward pull'). d ÿ NBd Mathematically this situation implies NSB d d NB ÿ NC . When elasticity e takes the value ÿ3 then this condition holds. In other words, when the elasticity of the monopolist's demand curve is equal to ÿ3 the amount of labour demanded coincides with the classic prediction for competitive ®rms. (b) Baumol eect (`rising pull') is larger than Cournot eect (`downward pull'). Mathematically d ÿ NBd > NBd ÿ NCd . Again this situation implies NSB it is easy to check that this situation arises when the elasticity takes a value lower than ÿ3. In this case, the amount of labour demanded is larger than the classic prediction for competitive ®rms. (c) Baumol eect (`rising pull') is less than Cournot eect (`downward pull'). Mathematically d ÿ NBd < NBd ÿ NCd . Again this situation implies NSB it is easy to check that this situation arises when
the elasticity takes a value larger than ÿ3. In this case, the amount of labour demanded is less than the classic prediction for competitive ®rms. Therefore in a monopoly market where the monopolist has a bi-criteria utility function u S; B and faces a very elastic demand curve (i.e. elasticity e larger than 3 in absolute value) then the amount of labour demanded will be higher than that of a competitive producer (assuming that both have identical quadratic production functions) for a given real wage. In the next section, these ideas will be illustrated with the help of some numerical examples. 4. Some numerical illustrations In order to illustrate the theory developed above, let us assume the following data for a case of quadratic technology: b1 24;
b2 1;
W 20;
P 5:
For these data and according to Eq. (9) the frontier is
SÿB 20
2
ÿ 24
SÿB 20
0:2 S 0:
The corresponding labour demand functions according to Eqs. (16)±(19) are NSd 12;
w ; p w NBd 12 ÿ 0:50 ; p e w d : NC 12 ÿ 0:25 1e p d 12 ÿ 0:25 NSB
For the real wage given w=p 4 the following labour demand predictions are obtained: NSd 12;
d NSB 11; e : NCd 12 ÿ 2 1e
NBd 10;
Therefore, the `rising pull' caused by the Baumol eect is equal to 11 ÿ 10 1 while the `downward pull' caused by the Cournot eect de-
C. Romero / European Journal of Operational Research 122 (2000) 91±100
99
pends upon the value of the elasticity e. It is easy to check that if e ÿ3 both eects coincide. However, if e takes a value less than ÿ3 then Baumol eect is larger than Cournot eect. Thus, for e ÿ4, we have NCd 9:33, that is the `rising pull' surpasses in 0.66 units the `downward pull'. On the contrary, if e takes a value larger than ÿ3 the ®nal result is the opposite. Thus, for e ÿ2, we have NCd 8, that is the `downward pull' surpasses in one unit the corresponding `rising pull'. The robustness of our conclusion relies heavily upon the assumption of quadratic technology. Let us now attempt an extension of the results to other technologies by means of an example. A more general aggregate production function such as
are compensated (i.e. elasticity e equal to ÿ3). The latter is equal to 178:5 ÿ 143 35:5 while the former is equal to 220:75 ÿ 178:5 42:5 for metric p 1 and equal to 215:25 ÿ 178:5 36:75 for metric p 1. Although both eects do not exactly coincide as happens when the production function is quadratic the similarity among them is strong. This similarity is specially remarkable for the metric p 1 (a Baumol eect of 36.75 versus a Cournot eect of 35.5). This numerical example although tentative seems to keep the theoretical argument in an essential way.
h N b1 N ÿ b2 N h ;
Baumol's hypothesis on the behaviour of oligopolistic ®rms (`maximisation of sales revenue subject to a pro®t constraint') although is questionable from a logical point of view is quite valuable because of its seminal idea: big ®rms are not solely motivated by achieving large pro®ts but also by achieving other objectives like large sales revenue. This kind of entrepreneurial motivation implies the consideration of bi-criteria utility functions u S; B as it has been postulated for a long time for many economists. However they have not oered a sound and operational framework to tackle their optimisation over the attainable frontier. In this paper, an analytical framework for addressing ± in a convincing way ± the optimisation of these kinds of bi-criteria utility functions is developed. Furthermore, the aggregate demand for labour has been derived within a context where ®rms maximise a utility function u S; B instead of pro®ts. Thus, a S; B bi-criteria behaviour leads to demand for labour predictions which dier from the traditional scheme, where predictions are built according to the pro®t maximisation hypothesis. In fact, when entrepreneurs follow a S; B bi-criteria policy, the demand for labour is higher than when they maximise pro®ts. This result has an obvious background (`sales objectives, bigger ®rms'). Moreover, with a S; B bi-criteria behaviour, the predicted marginal rate of decreasing demand for labour when real wages grow is higher than within a pro®t maximisation context.
h > 1;
leads to the following aggregate frontier: h
b2 S ÿ B=w ÿ b1 S ÿ B=w S=P 0:
For the above data the frontier becomes 1:5 SÿB SÿB 0:2 S 0: ÿ 24 20 20
Ideal and nadir values, as well as the bounds of the compromise set are S 10240; B 5993; L1 ! S 10091; L1 ! S 10040;
S 9562; B 5120; B 5676; B 5735:
In order to compare demands for labour (speci®ed in Sections 2 and 3) we have computed their dierent amounts: NSd 526; NBd 178:5; NCd (with e ÿ3 143; d NSB (on the L1 bound 220:75; d (on the L1 bound) 215:25: NSB
For non-quadratic technologies there is not a d function but two predictions according single NSB to the metric used. Let us now calculate Baumol and Cournot eect for the case when both eects
5. Concluding remarks
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The explicit recognition of a S; B bi-criteria behaviour reconciles the theoretical prediction on monopolist ®rms demand for labour with the factual character of that phenomenon. In fact, a monopolist ®rm can follow a S; B policy which would justify a demand for labour situated over the classic competitive curve. This situation is specially likely when the monopolist's demand curve is very elastic (i.e., jej > 3). Finally, it should be noted that the paper concentrates on the `supply side' of an economy. A possible extension of this research will consist in considering also the `demand side', through an aggregate demand function. In this way, it should be possible to study the consequences in the macroeconomic equilibrium when a S; B bi-criteria behaviour is considered. Acknowledgements English language was checked by Christine Mendez. This research was supported by `Comisi on Interministerial de Ciencia y Tecnologõa (CICYT)' and `Consejerõa de Educaci on y Cultura' de la Comunidad de Madrid. Comments and technical help by Dr. Dõaz-Balteiro are highly appreciated. Thanks are also given to one reviewer for his helpful suggestions that have greatly improved the economic accuracy of the paper. References Ballestero, E., Romero, C., 1991. A theorem connecting utility function optimization and compromise programming. Operations Research Letters 10, 421±427. Ballestero, E., Romero, C., 1993. Weighting in compromise programming: A theorem on shadow prices. Operations Research Letters 13, 325±329.
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