THE SUPPLY OF LABOR AND HOUSEHOLD PRODUCTION

Transcription

1 THE SUPPLY OF LABOR AND HOUSEHOLD PRODUCTION João Rogério Sanson* Abstract Labor suly is seen as an outut from household roduction. Given by the hysical effort of a erson, working in the market also requires secific inuts. This rocess may be described with the hel of a general technology that comrises oint roduction. At least one of the oututs is labor suly. With the hel of a simlified version of the model, initially the choice among different tyes of market work is discussed. Within this discussion, it is shown how different estimates of the oortunity cost of time naturally aear, all in standard microeconomic results. Then, the definition of net result of the worker is related to economic rent due to the fact that the consumer-roducer can not alter the time endowment. JEL codes: J2, D. Key words: labor suly, household roduction.. Introduction Labor suly, in standard models, is given by the residual time of the economic agent after the quantity of leisure is chosen. This is generalized in the household roduction models by defining consumtion activities that use, as inuts, urchased goods and own time. Even then, labor suly is modeled in the same way: from total available time in a given eriod, time in household activities is subtracted so as to get the time sold in the market. An alternative modeling is to have leisure time as a residual. Another alternative is to have both labor and leisure in the reference function, as in Johnson (966), Georgescu-Roegen (968, ), and Sanson (987 and 99). Johnson (966) introduced a model with leisure and labor in the reference function and at least one inut for the roduction of labor. The aer was highly influential in the literature on urban transortation demand, although gradually suerseded by Beckerian models. This was true esecially of DeSera (97). He included all forms of time use together with the Beckerian commodities in the reference function. Gronau (986 and 987) also includes the labor activity in the reference function in a Beckerian model. * Deartment of Economics, Federal University of Santa Catarina (UFSC), Florianoolis (SC), Brazil. rsanson@mbox.ufsc.br. The author thanks Idaleto M.Aued, José Rubens D.Garli, Pedro H.V.Mendes, Jean-Luc Rosinger, Luciana T.Sanson, and Fernando Seabra for suggestions during the research. He also thanks the National Council for Scientific and Technological Develoment (CNPq), from Brazil, for a grant that also included research assistance from Michael Ax Wilhelm and Antonio M.Fontoura.

2 When labor suly is defined as a time residual, a roduction function for labor is thus exlicitly left out. In such a situation, labor time does not aear in the reference function, at least in Walrasian models, one of which is the household roduction model of Becker (965). Besides this, Beckerian models define roduction functions only for the activities that are listed in the reference function. It is true that the household roduction model can be interreted as allowing for a linear roduction function for the labor activity in which time is the only inut. As usual, the level of this activity is measured by the amount of time used in it. However, many tyes of labor may be considered in which this equivalence is not valid. One examle would be labor erformed and sold by tasks instead of by the hour. Also, there are secific inuts in the roductive rocess of human work. Therefore, a roduction function that has only time as inut does not cover the general case. It amounts to ignoring the roductive rocess for labor. Many insights on the roduction of human work exist in the literature, although in scattered form. Leontief's closed model, which is insired in the Tableau Économique of the hysiocrats, treats consumers as a sector that secializes in sulying labor. Due to the unrealistic assumtion of fixed coefficients for the consumtion of final goods, the more oular model in which consumtion and labor are exogenous has suerseded it. As in any household roductive activity, labor roduction requires several inuts, some of which have been studied. Singh, Squire, and Strauss (986), Suen and Mo (994), and Caillavet, Guyomard, and Lifran (994) resent models of roductive consumtion that include, for instance, slee, nutrition and health exenditures. An earlier suggestion for treating labor as a household outut, as an equal to commodities, aeared in DeSera (97,.829n.), although the analysis in the aer was made with fixed suly of labor. Gronau (986 and 987) treats labor as an activity with its own inuts. However, the analysis is very brief and the results are somewhat different from this aer. In human caital theory, exenditures that increase future caacity of getting income are considered as inut or, better, as investment. However, such an analysis is done intertemorally and seems not to include an exlicit roduction function for labor. The literature on collective household roduction seems to follow Becker's treatment as far as it refers to the roduction of labor. 2 Therefore, it aears that there is some sace for a See Dorfman, Samuelson, and Solow (958, ch. 0) and Pasinetti (977, ch. 4). 2 For a recent survey of collective household models, with emhasis on their econometric imlementation, see Vermeulen (2002). 2

3 rediscussion of the model in which labor suly is treated as any outut of the household roductive rocess. In this aer, we summarize and extend research resented in Sanson (997 and 2002). The resent article is restricted to a static situation. Besides, slee and leisure activities are treated as inuts that are not secific to labor roduction since they affect the whole roductive rocess of the household. They are among the activities directly included in the reference function. The aer attemts the design of a model of labor suly that should fit in the standard consumer and firm theories. Ideally, the model should also fit in the general equilibrium framework. For this, the endowment treatment is fundamental. We will see that the endowment of work caacity leads naturally for an agent that looks for a maximum rent for the human caital. Currently, the household roduction theory, desite its growing sohistication, still stays only in the dark alleys of the more oular microeconomics texts, intermediary and advanced. That is not consistent with the imortance of the theme in emirical alications of economics. The main obective of the article is to formulate a model of household roduction in which labor is one of the roductive activities whose outut may be sold in the market. This will be done in the context of Walrasian models, in which the time allocated to work in the market is given as a residual. Section 2 covers the notion of time endowment, with emhasis on its interretation as a eriod in which the human organism is available for working. This highlights the fact that, in contrast to other tyes of Walrasian endowments, the time endowment can not be altered. Section rooses a Beckerian household roduction model that has, as a distinguishing characteristic, roduction functions for different tyes of labor. Section 4 illustrates the general model with the secial case of two commodities, where one is a good that does not require time and the other only requires time, being a Beckerian version of the income-leisure model of textbooks. However, the roduction functions for labor are ket. This model allows for a new discussion of the shadow rice of time. Finally, section 5 adats the notion of rofit or net result, used in models of household roduction from agricultural economics, to the roduction of labor.

4 2. The endowment of labor Before the resentation of the model of labor suly, it will be necessary to define labor as a result of a roductive rocess. A related question is the definition of the endowment of time for each economic agent. Its interretation is deendent on the definition of labor, and it imlies a modification in the concet of oortunity cost of time. The concet of human work resuoses an activity by a erson during a time interval. A erson, in fact, is a kind of hysical caital that renders services during a given eriod. 4 Thus this erson has an occuation that may include different kinds of work, being infinite the number of ossible tasks. What counts is the combination of ersonal abilities to be used in each task. The human caital is comlemented by different inuts to render different goods and services, not necessarily tied to a aid ob. As it is taught by the theory of household roduction, work is then one of the many ossible oututs from a roductive rocess centered on a erson or a family. The organism itself is the basic caital of the erson, and being a worker is to secialize in utting for rent the own work caacity. The market structure in which this work caacity is rented varies with the tye of work. It goes from quasi-monooly markets in which the labor activity is deendent on scarce abilities, to the cometitive markets of nonqualified labor. The second question related to labor roduction is the nature of the endowment of labor force for each erson. This endowment is given by Nature, as far as the time interval considered. Thus, for a eriod of a month the erson has a maximum of man-hours that the organism can be at work, not given by the organism itself but by the time interval. The intensity of work, however, is variable and is deendent on ersonal limits and the environment in which the organism oerates. Similarly, there are many tyes of activities that a given erson may exercise during the eriod. Because of this, it is ossible to think of a roduction function for labor in which one of See Debreu (959,.0-) and Arrow e Hahn (97, ch.4). Whinston (992,.5-6, 6-64) emhasizes that many textbooks consider time itself as the inut. In fact, time only delimits the duration and the direction of the labor rocess. A favorable interretation to the textbooks is that an imlicit assumtion is made. Time is only a short for a number of man-hours of homogeneous human work er eriod. If every worker delivers the same number of manhours, then it is enough to count the number of workers. In models of individual suly, the reference to time endowment must have a number of man-hours. Another semantic question is the use of the words labor and work. In order to avoid a discussion that would easily get into hysics, these words are synonymous here. 4 Léon Walras refers to consumable services that are obtainable from ersonal abilities. See Walras (98, lessons 5 to 8). 4

5 the inuts, the essential one, is the man-hours the erson allocates to a given task. 5 A simle examle is a gardener who charges for the service by the area covered and then allocates hours of work to the contracted task. From the current viewoint, the gardener uses, besides the services of gardening hysical equiment, the services of the organism itself, treated as caital. The human organism may be able to carry on more than one activity at the same time or at least during the same eriod. One examle is to work on a lato during an air flight, where both activities are related to an occuation. 6 Another examle is the leisure of the theory class during academic meetings. This ossibility is not considered in the standard models, due to the determination of labor suly as a residual. Therefore, selling labor caacity and using it in unaid activities are mutually exclusive. Searating the endowment of work caacity from the different tyes of work that are roduced allows for a treatment of this caacity as an inut that can be ointly used in the many ossible activities of a household. In terms of the Walrasian treatment of endowments, only the availability of the human caital, measured in man-hours, as it is done for any caital good, would be treated as any other good. 7 The labor outut, of course, would deend on effort and on the technology imlicit in its roduction function. A oint that the Walrasian endowment aroach brings in the discussion is its fixity. Endowments of goods may be altered, and this is the essence of models of excess demand and reservation rices in general equilibrium analysis. In the case of labor caacity, this is not ossible. A erson may hire external labor in order to substitute for own labor in a given activity, but this does not alter own labor caacity. It ust means an alternative use for the saved labor caacity. 5 This availability of the worker for the execution of alternative tasks may be seen, desite the differences in theoretical aradigms, as the equivalent to the Marxian concet of "abstract human labor". The conversion of abstract labor into different tyes of labor could be made within the resent roduction function aroach. The usual conversion with fixed coefficients would be a secial case. An irresistible question is would such a solution avoid the aggregation roblems that created insurmountable difficulties for the labor theory of value? 6 In this examle, the service of the organism enters simultaneously in the labor activity, which is an outut, and in the transortation activity, which can be a case of an inut activity to the work activity being sold. 7 See Arrow and Hahn (97,.75-76, 65-66), in which an integration of Becker's model of time allocation is roosed for Walrasian endowments. In that solution, there are several tyes of ossible endowments of leisure time, as roosed in Arrow and Debreu (954). There is corresonding set of activities or tasks that require time with fixed coefficients. The sum of the excess demands for different tyes of time use can not be higher than the total endowment of time of the erson. There is no restriction on the sign of each excess demand. In the resent solution, in contrast, there is only one endowment, labor caacity, which may be used in any activity, including the several tyes of leisure and labor. Both solutions have in common the limit on time use given by Nature. 5

6 In the context of household roduction theory, the endowment of labor caacity may be interreted as a fixed factor for a firm. Deending on the technology of a firm, a fixed factor has ossible uses for the roduction of different tyes of goods, and may be used ointly. In household roduction theory, the decisions about consumtion and roduction are not searable, as it is suosed in the theory of the firm, in which there is no interaction between the consumtion decisions of the owners of firm and their, or of their emloyees, decisions on roduction. Thus, the caacity of labor, as a fixed factor in the household roduction rocess, affects simultaneously the consumtion and roduction decisions, including labor. The main difference is that this fixed factor is not alterable in the long run. Investments in education, formal or from exerience, are equivalent to technological rogress. This Walrasian endowment aroach in the household roduction theory results in modifications in the interretation of the oortunity cost of the labor caacity. In the basic income-leisure model, if a erson gives u selling art of the labor caacity in the market to consume it, the best wage given u is the oortunity cost. In these models, it is also suosed that there is oint use of the human caital in several activities. This is so because oint use destroys the one-to-one tradeoff between labor and leisure. With the roduction function aroach for labor suly, a man-hour can have many alternative uses, which include consumtion activities or many tyes of marketable labor. In terms of sacrificed income, now the oortunity cost is given by the best occuation available to the erson. However, given the labor suly, there is also an oortunity cost involved in choosing consumtion activities. In both situations, it will be seen that the marginal rate of transformation between a air of activities, associated to a man-hour, as a measure of labor caacity, is now art of the definition of oortunity cost.. A general household roduction model Take a Beckerian model with a given list of commodities with no erfect substitutes in the market. 8 Suose that any good bought in the market requires some ositive amount of manhours to be consumed, which turns it into a commodity. The erson maximizes the reference function 8 Had the commodities erfect substitutes, it would be necessary to distinguish, for each of them, between what is consumed domestically and what is sold, as it is done in the international trade theory. See Strauss (986) and Sanson (997). 6

7 where z is a vector that reresents the quantities of commodities. Technology is shown by the transformation function 9 u = u(z), () F ( z, y, t) = 0, (2) where y is a vector of tradable goods. Inuts are reresented by nonositive elements of the vector. One examle of outut is labor, in fact the tasks erformed in a given occuation. Vector t reresents the tyes of labor that the erson erforms domestically and in the market through different occuations. If the number of commodities and tradable goods is greater than the number of tyes of work, then there are work activities that are ointly used in several activities of the household. This ossibility is covered by (2), as a general technology of roduction. The budget restriction is given by T y = 0, () where is a rice vector with the same number of comonents as y. The only source of income is the sale of oututs from the roductive rocess. 0 As the inuts are negative variables, their costs are being covered by (). In contrast with usual resentations of household roduction models, this version treats the goods bought in the market as inuts, which, similarly to the advanced theory of the firm, are treated as negative variables. labor: The labor restriction is simly the sum of all the ossible uses of the total available J T t =, (4) where J is the summation vector and the endowment of labor is normalized to unity. From the Lagrangean function T L(z,y,t,λ) = u( z) + λ F( z, y, t) + λ ( J t + λ 2 ) the following first order conditions for a maximum are obtained: 2 T y 9 A variant of this generalized roduction function is used by Pollak and Wachter (975,.26). 0 Income from other sources is suosed to be equal to zero, i.e., m = 0. Calling the usual time restriction as labor restriction is consistent with the argument that a erson really has an endowment of labor caacity that the organism may erform during the eriod under consideration. Walras himself 7

8 u + λ = 0 z z T, (5) y T T λ + λ = 0, (6) T T λ λ2j = 0. (7) The restrictions (2), (), and (4) should also be listed as art of this matrix equation system. λ i reresent the three Lagrangean multiliers. The analytical solution of this model is clearly difficult. But some insights can be gained by oening u some of the equations. From the elements of (5), after eliminating the Lagrangean multilier, it follows that: u z u z i zi =. z (8) This means that the marginal rate of substitution between two commodities is equal to the corresonding marginal rate of transformation. This corresonds to an efficiency condition in the household roduction rocess. From (6) and (7), by also eliminating the multiliers, it follows that t i y = k t i y k (9) Consider, for instance, y and y k as tyes of traded labor. Then it is ossible to interret (9) as saying that alternative uses of the labor caacity result in the equality of their values of marginal roducts. An otimal condition for a air of goods bought from the market follows from (6): made the suggestion that the labor caacity were called leisure to differentiate it from market labor, desite the fact that only a fraction of this activity is truly leisure. 2 The null vectors may have different dimensions in each equation. Paretian efficiency conditions have been exlicitly incororated in modeling collective decisions on consumtion and the suly of labor in collective models of household roduction. See As and Rees (997) and Chiaori (997). 8

9 y y k l k = (0) l Now the marginal rate of technical substitution equals the corresonding inut rice ratio. This condition is associated to cost minimization. However, let y l be interreted as a given tye of traded labor and k as an inut for this kind of labor. If (0) were then rewritten as l y y k l = k () the result would be the equality between the value of marginal roduct and the resective rice of inut y k that is used in the roduction of labor y l. Conditions of tye (0) or () are familiar from the theory of the firm. However, they can not, at least for a general model such as this, be solved indeendently from the other equations. The roduction decisions and the consumtion decisions are simultaneously taken A simle case with occuational choice A secial case of the above model can be cast after the textbook case of income and leisure. In the model, as a simlification, there is only one commodity. The reference function is described by u x, t ), where x reresents a commodity and t reresents leisure. 5 For ( simlification, each unit of this commodity requires one unit of a secific good and no man-hours for its consumtion. In this case, the convention of treating all variables as nonnegative is followed. Therefore, the vector of goods is given by y = x, x, x, h, ). The variable x ( 2 h2 only aears in vector z = x, t ). The roduction functions of the commodity and of leisure are ( 4 For an early discussion of the cases in which recursiveness of these roblems is ossible, see Singh, Squire, and Strauss (986). 5 In u ( x, t ), the arguments of the function include two commodities that have secialized inuts. Also, the consumer directly values only this use of time, while the other uses of time, in the roduction of work, will be determined as a residual from the endowment of labor ower. As far as the oortunity cost of time is concerned, the individual will only consider the oortunity cost of sacrificing leisure time. DeSera (97) and Pollak and Wachter (975,.27) introduce the time inuts for commodities in the utility function as a way to consider references on time use itself. This might be double counting, since a Beckerian commodity is a ackage that includes the use of own time in its roduction and consumtion. A much older tradition in the literature is to consider the labor suly itself in the reference function, a rocedure more related to the resent model. On the treatment of labor suly in the reference function, see footnote n., above. 9

10 simly z = x and z 2 = t. This simlified roduction function allows for the model to focus on the roduction and suly of labor. The erson has a tye of human caital that makes it viable to suly two kinds of market labor. The roduction function for each labor tye is given by h = f ( x, t ), where x and t reresent secific inuts for the labor activity h. 6 The variable t shows the duration of the availability of the erson's own organism for the labor activity h. This roduction function reflects the convention that eole carry out tasks that require hysical effort. With variable coefficients of roduction, the substitution between bought inuts and the use of this human caital is ossible. The income restriction of this worker is given by x = wh x + w2h2 2 x2 where i is the rice of goods bought in the market and w is the unit wage. This budget restriction may be interreted as the equality between the exenditure on one commodity and the net income from each tye of labor. 7 given by The labor caacity restriction, usually referred to as time restriction, is t + t + t, 2 = where t and t 2 refer to the use of labor caacity in two tyes of occuations or tasks. In order to otimize, it is convenient to substitute the roduction functions in the budget restriction and write the following Lagrangean function: L = u x, t ) + λ [ w f ( x, t ) + w f ( x, t ) x x x ] + λ ( t t ) ( t The first order conditions in relation to x 's and t 's are: u x λ = 0 (2) u λ 2 = 0 () 6 Notice that f (.) is being used as a generic symbol for a roduction function. It should not be taken as a artial derivative. 7 Johnson (966,.42-4) has the tri to work as a time use that should be added to the work time. Also, the cost of the tri should be subtracted from the wage obtained in each tri, although this result is deendent on the transort inut being given by a fixed coefficient. The resent model could describe this articular case by using a Leontief roduction function for the labor activity. This roduction function, in fact, is the one used for each commodity in the first Beckerian models. 0

11 λ w = 0, {,2 }, x (4) λ w λ 2 0, {,2 }. = (5) Eliminating the Lagrangean multiliers and omitting the restrictions, the first order conditions may be rewritten as: u u x = w, (6) w = w 2 2 2, (7) w = x, =, 2. (8) At least theoretically, these four equations lus the budget restriction, the work restriction, and the two roduction functions allow for the determination of the value of the following variables: x, x 2, x, t, t 2, t, h, and h 2. Equation (6) is analogous to (8) in the general model. However, the roduction function of the commodity is extremely simlified, as already noted, since z = x. Thus, instead of a marginal rate of transformation between a air of commodities, (6) shows the ratio between the oortunity cost of labor caacity used in a leisure activity, w h, and the rice of the good used as inut for commodity z. Equation (6) may be rewritten as w t u = u x and it may be interreted as the equality between the value of the marginal roduct of the labor caacity used in the first occuation and the oortunity cost of leisure time in terms of the commodity. It is no surrise that the equilibrium conditions of household roduction recall the conditions for the firm, since consumtion and roduction decisions are interrelated. It also

12 rovides, at least in the one commodity model, an alternative estimate for the oortunity cost of leisure. Equation (7) is equivalent to (9) and shows that the value of marginal roductivity of labor caacity for each occuation is equal in equilibrium. They are also equal to the oortunity cost of leisure as ust seen above. This benefit of each unit of work refers only to market revenue. It is a consequence of not considering the references of the consumer-roducer for the different tyes of work. Introducing h in u(.) could do this. It would extend the labor-leisure models that were referred to in the Introduction. Finally, equations in (8), similar to (), say that the value of the marginal roduct of each inut bought in the market is equal to its marginal cost. If the unit wage for each occuation were isolated in each equation, it would be equal to the corresonding marginal cost. This result is here indeendent from references, as it did in the general model. It is also ossible to examine the marginal rates of technical substitution in the roduction of each tye of work. They are given by x = w, =, 2 (9) As exected, the marginal rates of technical substitution between the two inuts for each occuation are equal to a rice ratio of inuts. Yet, the labor caacity rice is an oortunity cost, given by the revenue that could be obtained by selling that work unit in the market. Equations in (9) are also variants of (9), now defined for airs of labor inuts. These results rovide different estimates for the oortunity cost of time, as summarized in the following set of equalities: w = x u =, =, 2 u x The first set of equalities says that the values of the marginal roduct of working in different occuations are the same. They are also equal to the roduct between the rice of an inut x and the marginal rate of technical substitution of t for x. This exression can be rewritten in terms of the marginal cost of roducing h and the marginal roductivity of working in occuation. The marginal cost is given by the ratio of the rice of an inut and its marginal roductivity: 2

13 x x Finally, the oortunity cost is equal to the roduct between the rice of commodity x and the marginal rate of substitution of commodity x for leisure, t. There are thus three different ways for estimating the oortunity cost of time: two of them from the roductive rocess of tradable work and one from consumtion. 8 In the maximizing osition of the consumer-roducer, they are all equal. Of course, out of equilibrium they can be different and serve as a guide in decisions on where to better allocate time. In more general models, with roduction functions also for the Beckerian goods, there will be a greater number of alternative estimates of the oortunity cost of time. The relationshi of this model with others can be made in several ways. First, the erson may secialize in one occuation. This may occur due to references, technology, relative rices or social rules. Then the model could be secified in nonlinear rogramming terms, with the resence of nonnegativity conditions for each variable and inequalities for some of the restrictions. Second, to obtain the usual income-leisure model, it would be sufficient to suose a linear roduction function for labor, with labor suly given by the number of man-hours sold in the market, t. The use of other secific inuts in the roduction of labor would also be ignored. In short, the income-leisure model ust resented is general enough to also include roduction functions for labor. It seems to be useful for illustrating the mix of reference and roduction decisions in the suly of labor. It also seems to give some alternative insights in the discussion of the oortunity cost of time. As a final characterization of the labor suly, it is useful to discuss a few results in the comarative statics of the secialized model. The study of the comensated demand faces two difficulties. To begin with, the equivalent to the budget restriction is now a nonlinear function, when the roduction functions of labor are substituted in the income restriction. Then, as oosed 8 These exressions could serve as a basis for a grahical illustration of the simlified model. With the three alternative uses of the endowment of work caacity, three different individual demand curves could be drawn. The first of them would reflect the demand for leisure and would give the own demand for work caacity. With it, it would be ossible to comute a reservation rice that could be zero. The other two demand curves would reflect the demand for work caacity as inuts to the roduction of the two tyes of market work. The horizontal addition of the

14 to the traditional income-leisure model, the elimination of the labor caacity restriction is not simle, so it is ket searate here. The definition of the substitution matrix is now deendent on which exogenous variable and restriction is used for defining the comensated functions. As usual, the nonlabor income from the income restriction will be chosen. In building the Slutsky equations, whether decomosing for commodities, for household oututs or for inuts it makes a difference. Only commodities enter the utility function, therefore it is natural to interret a comensated demand as a movement along an indifference curve. Comensated demands for inuts imly movements along isoquants, and the decomosition could be made in terms of substitution and outut effects. As for the outut, it is not usual to decomose the rice effect. Instead, the short-run and long-run suly functions are studied. Along these lines, the Slutsky decomosition will only be discussed for the commodities. In the income restriction of the simlified model, the value of nonlabor income is given by m = x + 2x2 + x wh w2h2 Minimizing this value, subect to a given utility level and the remaining restrictions, results in the comensated, or Hicksian, demand functions: x H (, 2,, w, w, u) and t H (, 2,, w, w, u). Establishing the identity between the Marshallian and Hicksian demand functions and alying the Enveloe Theorem yield the following Slutsky equations for the commodity x : H x x x = x i, i =, 2, m i i H x x x = + h, =, 2 w w m The only rice that directly acts uon the quantity demanded of this commodity is, even then only if the budget restriction is linearized around the equilibrium osition. 9 All the other rices, including wages, act through the nonlinear budget, as usual in Beckerian models in which the three individual demand curves would cross the vertical line at the endowment oint that would reresent the suly of work caacity, and, there, it would determine the oortunity cost of time for the consumer-roducer. 9 Theorem 6 from Blomquist (989) considers nonlinear budget restrictions in which a term involving a market rice and the corresonding good can be additively searated and how some substitution effects even so become redictable. With this theorem, it seems ossible to redict that the substitution term x H is nonositive. For the other terms in the Slutsky decomositions, it would be necessary to deal with a linearized exression of the budget restriction, based on shadow rices for the commodities and the alternative uses of time. For these rices the standard results in comarative statics are valid. But such rices would be functions of the market rices and these indirect 4

15 roduction functions do not have the roerty of constant returns to scale. The commodity t, being time used outside the market, does not have a rice. Here, wage can not be used as usual. The Slutsky decomosition can then be made only with resect to rices of inuts or wages. The exressions are similar to the ones above, excet for the variables in the numerator of each derivative. All the effects work through the nonlinear budget function. 5. The net result from labor suly Provided secific inuts to the roductive rocess of labor are identifiable, it is ossible to define gross and net revenue from market work. This recalls farm models in which art of the total income is given by the net result, also called rofit, of the goods sold in the market. 20 However, given the discussion of the endowment of work, given by Nature, this net result has the nature of an economic rent. The concet of economic or ure rofit in the theory of the firm is clearly defined by the net results obtained over the oortunity cost of own caital alied in the firm. Caital is defined in terms of the market value of the roerty rights on hysical and financial caital. When this caital is fixed in the long run, the ure rofit is called economic rent, a concet that follows from a null oortunity cost. However, this absence of oortunity cost is only valid when the owner has no alternative ersonal use for the resource; said differently, only when the reservation rice is zero. Making an analogy among wages, rofits and rent might seem odd, almost a rovocation, given the characteristics of the corresonding factor markets. Even neoclassical economists have ket the analysis of each factor searate, in light of the sociological characteristics of each factor market. So, the urose of the following observations is to find alternative interretations for the costs of sulying labor. This is, in fact, a ractical question in income taxation, where endless discussions occur on what kinds of deductions should be allowed. The net result for each occuation is equal to w h x in the model of the revious section. Notice that this is not the rofit concet, as ust defined, since it does not consider the effects should be treated searately. General results connecting the rices of goods to commodity demand are unlikely. 20 See Squire, and Strauss (986,.8, 7-72). There, the definition is used for a farm outut and referred to as rofit or net result. 5

16 oortunity cost of using this time in nonmarket activities or erhas in a different occuation with a better remuneration. Instead, the net result is equivalent to the concet of quasi-rent, as used in the theory of the firm, being given by the difference between total revenue and variable costs. The exressions of the net results for the two occuations of the income-leisure model are obtainable from the budget and time restrictions. Initially, multily both sides of the time restriction by the shadow rice of a man-hour, then add the result to the right-hand side of the budget restriction, and finally write: 2 + = + x w t w wh w t x = (20) The left-hand side of (20) shows the exenditures on inuts for the commodities, including the oortunity cost of time dedicated to one of them, leisure. The right-hand side shows the redefined Beckerian full-income, also in terms of the shadow wage. It is given by the value of the endowment of labor caacity lus the economic rofit attained in each occuation. However, this discussion is made in relation to secific inuts for labor. In the case of a firm, the roduction of traded goods is suosedly searate from the other activities of the erson. It is exactly when consumtion and roduction activities are searable that the idea of firms as a means for the social division of labor can be soken of. It is as a consequence of this ossibility that the concet of circular flows between firms and consumers may be used. But it is an age-old wisdom that the suly of labor is different from the suly of other tradable goods and services, esecially in the way the erson is involved. Suose a erson owns a caital good, such as a lawn mower. It can be used to sell services of gardening or it can be ut to rent. 2 That is not the case with labor. When a erson is hired for work, the organism itself is the source of the service. Thus to searate decisions is imossible, since all consumtion that is not secific to work is also consumtion by the same organism. The rocedure of suosing a secific roduction function for labor can only cature art of the inuts. It leaves out most of the 2 Renting a caital good involves a social arrangement with recognized roerty rights. Even then, there is the work of administering these roerty rights or at least of checking the services of those hired to manage these roerty rights. But the amount of work and the degree of effort required are certainly smaller than the oeration of most caital goods, although the stress involved in the uncertainties of returns on financial caital might be high, esecially for risk-averse ersons. 6

17 inuts that are necessary for keeing alive and in the good shae the human organism. It seems that the most that can be done by the analyst, in comuting the rofit of an occuation, is to find the contribution of this outut to total income. Note, however, that the contribution of the net revenue of market work to total income is defined after the equilibrium of the consumer-roducer is found. It has the nature of an economic rent, and it defines the net roduct or le roduit net of the erson, as the hysiocrats would say. Perhas, this can be illustrated by rewriting the above secial model in terms of cost functions. Take t as given and define the restricted or short-run cost functions, defined without the oortunity costs of time. 22 Minimize x, subect to ( ) work. This will result in the restricted cost functions c c ( h, t, ) f x, t = h for each kind of market =. Now, the budget restriction becomes x 2 = = ( w h c ( h, t ), Thus by subtracting variable costs from the total revenue of a given tye of market work, the consumer-roducer gets the economic rent for a given level of work caacity (time) allocated to that activity. Given (t, t 2 ), the value of the goods used as inuts for commodities has the nature of an economic rent. This is consistent with the Walrasian aroach of considering a erson as a kind of caital good, as discussed above. A consumer-roducer, secialized in selling work and with no income from other roerty rights, might have a very low level of exenditures on consumtion of goods beyond the necessary for working. This would occur with different abilities and for a marginal worker. However, the real income of a erson also involves leisure or, as Becker would ut it, commodities that are intensive in the use of time. As seen above, this consumtion of leisure is the origin of the oortunity cost of time when this is used as inut in order to sell work. This oortunity cost is then the equivalent to a marginal cost of labor. The revenue from work above these variable costs of labor is then the equivalent to an economic net result for the worker and it would be the equivalent to the economic rofit for caital goods. Peole can not be sold in an economy where slavery is banned. Anyway, it there were slavery a erson would be the 22 It would be a straightforward extension to use net revenue functions. Nevertheless, cost functions are better suited for the analysis of the returns to human caital. 7

18 equivalent to a caital good. Thus, for a free erson, the resent value of the net result that might be obtained along the working life is in fact the value of freedom. 7. Conclusion Time itself is not a roductive inut in household roduction. Its role is simly to delimit the interval of time in which roduction and consumtion occur. The true inut is the work done during such a time interval or eriod. However, as a simlification, it is customary to describe labor inut by the number of hours during which a erson exerts such activity. Besides that the amount of work deends on the level of effort of the erson during a given eriod. Thus, the roduction of work by a erson seems fit for a descrition by a roduction function, in which the idea of maximum outut for a given set of secific inuts is essential. This aer initially resented a general Beckerian model with a technology that includes the ossibility of oint roduction and roductive consumtion. At least one of the oututs of this roductive rocess is labor, meant to be sold in the market. Imlicit in the model is fact that the effort level deends on attaining the roduction frontier. With this secification the household roduction model, including labor suly, might be more amenable for its integration into general equilibrium theory. The model is illustrated by the secial case of income-leisure, adated for the inclusion of roduction functions for two tyes of labor. With this model, it is ossible to obtain obective estimates for the shadow rice of the labor caacity from relations that involve marginal roductivities of the inuts in labor outut. Finally, the notion of net result is adated for the roduction of labor. Because of the unalterable endowment of labor, the net result is related to economic rent. However, the own use of this endowment creates the equivalent to quasi-rent or economic rofit, although this exression is inadequate for a kind of caital that has no market in a modern society, excet for its services. From the viewoint of the consumer-roducer, a way to increase the net result is by reducing costs of roducing labor. Investments in technological rogress at the domestic factory may reduce these costs. Another form, erhas with better gains in the long run, is to invest in differentiating the kind of market labor to be offered. It is as if the erson tried to become a different tye of erson as far as the suly of work is concerned. Instead of economies of scale, 8

19 which is not ossible to exlore, given the fixity of the time endowment, the consumer-roducer creates new oututs, by differentiating the labor suly along Chamberlinian lines. There are various extensions of the model that could be made. First, it is the comarative statics, esecially with view to econometric work, but also for the urose of integration of the model in general equilibrium theory. Second, some of the results from the literature on collective household models might be changed. References APPS, Patricia F.; REES, Ray. Collective labor suly and household roduction. Journal of Political Economy, vol.05, n.,.78-90, Jan.997. ARROW, Kenneth J.; DEBREU, Gérard. Existence of equilibrium in a cometitive economy. Econometrica, vol.22, , 954. ARROW, Kenneth J.; HAHN, Frank H. General Cometitive Analysis. Amsterdam: North- Holland, 97. BECKER, Gary S. A theory of allocation of time. Economic Journal, Set.965. v.75,.49-57, BLOMQUIST, N.Sören. Comarative statics for utility maximization models with nonlinear budget constraints. International Economic Review, v.0, n.2, , May 989. CAILLAVET, France; GUYOMARD, Hervé; LIFRAN, Robert. (Orgs.) Agricultural Household Modelling and Family Economics. Amsterdam: Elsevier, 994. CHIAPPORI, Pierre-André. Introducing household roduction in collective models of labor suly. Journal of Political Economy, v. 05, n.,.9-209, Jan DEBREU, Gérard. Theory of Value. New Haven: Yale University Press, 959. DeSERPA, Allan C. A theory of the economics of time. Economic Journal, v.8, , Dec.97. DORFMAN, Robert; SAMUELSON, Paul A.; SOLOW, Robert. Linear Programming and Economic Analysis. New York: McGraw-Hill, 958. GEORGESCU-ROEGEN, Nicholas. Utility. In: International Encycloedia of the Social Sciences. New York : Macmillan, 968. Vol.6, GRONAU, Reuben. Home roduction: A survey. In: ASHENFELTER, O.; LAYARD, R. Handbook of Labor Economics. Amsterdam: Elsevier, 986. Vol.I, GRONAU, Reuben. Value of time. In: EATWELL, J.; MILGATE, M.; NEWMAN, P. The New Palgrave: A Dictionary of Economics. London: Macmillan, 987. Vol.4, JOHNSON, M.Bruce. Travel time and the rice of leisure. Western Economic Journal, v.4,.5-45, Sring

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