Abstract
Labour supply is seen as an output from household production. Given the physical effort of a person, working in the market also requires specific inputs. This process may be described with the help of a joint-production technology, where at least one of the outputs is labour supply. With the help of a simplified version of the model, the choice among different types of market work is initially discussed. Within this discussion, it is shown how different estimates of the opportunity cost of time naturally appear. Then, the definition of net result of the worker is related to economic rent due to the fact that the consumer–producer cannot alter the time endowment. As a result, the household production model, including labour supply, might be more amenable to integration into general equilibrium theory and microeconomic theory in general.
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Notes
- 1.
- 2.
One recent example of this literature is Steger (2002).
- 3.
For a recent survey of collective household models with emphasis on their econometric implementation, see Vermeulen (2002).
- 4.
See Debreu (1959, pp. 30–31) and Arrow and Hahn (1971, chap. 4). Whinston (1982, pp. 15–16, 163–164) argues that many textbooks consider time itself as the input. In fact, time only delimits the duration and the direction of the labour process. A favourable interpretation for the textbooks is that an implicit assumption is made. Time is only a short for a number of man-hours of homogeneous human work per period. If every worker delivers the same number of man-hours, then it is enough to count the number of workers. In models of individual supply, the reference to time endowment must be to a number of man-hours. Another semantic question is the use of the words labour and work. In order to avoid a discussion that would easily get into physics, these words are synonymous here.
- 5.
Walras refers to “… personal capital or persons, capable of yielding personal incomes or services of persons, which we shall also call labour …” See Walras (1977, p. 215, italics in the original).
- 6.
This availability of the worker for the execution of alternative tasks may be seen, despite the differences in theoretical paradigms, as the equivalent to the Marxian concept of “abstract human labour”. The conversion of abstract labour into different types of labour could be made within the present production function approach. The usual conversion with fixed coefficients would be a special case. An irresistible question is: Would such a solution avoid the aggregation problems that created insurmountable difficulties for the labour theory of value?
- 7.
In this example, the service of the organism enters simultaneously in the labour activity, which is an output, and in the transportation activity, which can be a case of an input activity to the work activity being sold.
- 8.
- 9.
This covers the treatment of productive consumption. Suen and Mo (1994) follow a tradition of having a continuum of types of work, given by wage as a function of productive consumption. The present model implies a discrete number of work types. This is more akin to the usual textbook general equilibrium models. However, the present treatment has in common with that followed by Arrow and Hahn (1971) only the limit on total time use, given by Nature. There is no need to define endowments for each type of leisure. Each Beckerian commodity might require work time that will come out from the total work capacity of the person.
- 10.
Income from other sources is supposed to be equal to zero, i.e. m = 0. If positive, it would be added to the left-hand side of (4.3).
- 11.
Calling the usual time restriction a labour restriction is consistent with the argument that a person really has an endowment of labour capacity that the organism may perform during the period under consideration. Walras himself made the suggestion that the labour capacity should be called leisure to differentiate it from market labour, despite the fact that only a fraction of this activity is truly leisure. For simplicity, the inequality conditions on time use are omitted and only interior solutions are considered.
- 12.
The null vectors may have different dimensions in each equation.
- 13.
- 14.
Since these are implicit derivatives, this interpretation would best be made with negative signs in both sides of (4.9).
- 15.
For an early discussion of cases in which recursion of these problems is possible, see Singh et al. (1986).
- 16.
The reason for the non-coincidence of the indices of commodities and inputs will soon be clear. It has to do with the treatment of the inputs of the occupations.
- 17.
In u (x 3, t 3), the consumer directly values only leisure time, while the other uses of time, in the production of work, will be determined as a residual from the endowment of labour power. As far as the opportunity cost of time is concerned, the individual will only consider the opportunity cost of sacrificing leisure time. DeSerpa (1971) and Pollak and Wachter (1975, p. 271) introduce the time inputs for commodities in the utility function as a way to consider preferences on time use itself. This might be double counting, since a Beckerian commodity is a package that includes the use of own time in its production and consumption. A much older tradition in the literature is to consider the labour supply itself in the preference function, a procedure that seems more related to the present model.
- 18.
Notice that f j (.) is being used as a generic symbol for a production function. It should not be taken as a partial derivative.
- 19.
Johnson (1966, pp. 142–143) has the trip to work as a time use that should be added to the work time. Also, the cost of the trip should be subtracted from the wage obtained in each trip, although this result is dependent on the transport input being given by a fixed coefficient. The present model could describe this particular case by using a Leontief production function for the labour activity. This production function, in fact, is the one used for each commodity in the first Beckerian models.
- 20.
This benefit of each unit of work refers only to market revenue. It is a consequence of not considering the preferences of the consumer–producer for the different types of work. Introducing h j in u(.) could do this. It would extend the labour–leisure models that were referred to in the Introduction.
- 21.
These expressions could serve as a basis for a graphical illustration of the simplified model. With the three alternative uses of the endowment of work capacity, 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 capacity. With it, it would be possible to compute a reservation price that could be zero. The other two demand curves would reflect the demand for work capacity as inputs to the production of the two types of market work. The horizontal addition of the three individual demand curves would cross the vertical line at the endowment point that would represent the supply of work capacity, and, there, it would determine the opportunity cost of time for the consumer–producer. Besides, with simple algebraic operations, it can be shown that these expressions for the opportunity cost of time are equivalent to the ratio λ 1 /λ 2, widely used in the economics of the allocation of time (Gronau 1986).
- 22.
Theorem 6 from Blomquist (1989) considers non-linear budget restrictions in which a term involving a market price and the corresponding good can be additively separated. He then shows how some substitution effects, even so, become predictable. With this theorem, it seems possibleto predict that the substitution term \(\partial {z_1}^H/\partial {p_3}\) is non-positive. For the other terms in the Slutsky decompositions, it would be necessary to deal with a linearized expression of the budget restriction, based on shadow prices for the commodities and the alternative uses of time. For these prices the standard results in comparative statics are valid. But such prices would be functions of the market prices and these indirect effects should be treated separately. General results connecting the prices of goods to commodity demand are unlikely.
- 23.
See Singh et al. (1986, pp. 18, 71–72). There, the definition is used for farm output and referred to as profit or net result.
- 24.
Renting a capital good involves a social arrangement with accepted property rights. Even then, there is the work of administrating these property rights or at least of checking the services of those hired to manage these property rights. But the amount of work and the degree of effort required are certainly smaller than the operation of most capital goods, although the stress involved in the uncertainties of returns on financial capital might be high, especially for risk-averse persons.
- 25.
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 capital.
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Acknowledgments
The author thanks Idaleto M. Aued, José Rubens D. Garlipp, Pedro H.V. Mendes, Jean-Luc Rosinger, Joseph Lacey, Luciana T.Sanson, and Fernando Seabra for suggestions during the research. He thanks the National Council for Scientific and Technological Development (CNPq), from Brazil, for a grant that also included research assistance from Michael Ax Wilhelm and Antonio M. Fontoura.
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Sanson, J. (2009). The Supply of Labour and Household Production. In: Musella, M., Destefanis, S. (eds) Paid and Unpaid Labour in the Social Economy. AIEL Series in Labour Economics. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2137-6_5
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