Time Microeconomics : Optimization models 1

This paper provides just a generalization of Becker‟s theory of allocation of time, allowing joint production, throughout a 2-step optimization process, compatible and avoiding criticism in the sense of Pollak. The model yields some interesting results, analyzed for the static vision. After doing it, we just expand the vision to a dynamic context, where the stock of time and goods are introduced as explanatory variables. This yields optimal patterns along time for time, goods and commodities, which relates with both the endogenous change of tastes and the habit formation, by including past experiences in the determination of the utility. 1 The author would like to thank very specially to professors at Oxford University, Kimberly Fischer, Almudena Sevilla and Jonathan Gershuny, for their effort and patience by email, and very especially to Kimberly since without her intermediation this idea could have been abandoned instead of finally having had the chance to be presented at IATUR. As well, the author appreciates very much the fantastic dealing received from IATUR organizers, and the economic support from the University of Maryland for the travel support and from Oxford University for the Andrew Harvey Fellowship; without these supports, everything would have been extremely more difficult, or simply impossible. 2 I would like deeply to thank very much the crucial role played by my family and friends, and very specially the fundamental help received from very important professors for me as Julio Segura, Alfonso Novales, José Manuel Rey Simo, Geir Asheim and Knut Sydsaeter, -where the order is not important-, who have always supported me in good and bad times and still continue doing it. Thank you all for your time. 1. MOTIVATIONS “Several notable writers have recently charged that neoclassical economics is ́timeless ́. (...) The proper question to ask is not whether neoclassical economics is timeless but whether its treatment of time is adequate. (...) As Hayek recognized many years ago, how time is treated is an important aspect of any explanation of historical change.” (Boland, 1978) “Economists have always acknowledged the fact that tastes of consumers change (e.g., Marshall in his Principles). But the overwhelming majority took the attitude that it is not their business to be concerned with these changes of taste”( Von Weizsäcker, 1971) “Actions are constrained(...). Different constraints are decisive for different situations, but the most fundamental constraint is limited time.” (Becker, Nobel lecture 1992) “while the growing abundance of goods may reduce the value of additional goods, time becomes more valuable as goods become more abundant.” (Becker, Nobel lecture 1992) Looking at these citations, it is straightforward to realize about the interest and the power that time, as a decision variable, has in economical aspects, through the way in which it can determine the happiness of the individuals. As well, it is as easily seen as overwhelmingly ignored by traditional microeconomics (paraphrasing Von Weizsäcker) that changes of tastes may vary along time. Marshall was concerned from the very beginning about that: "(...)and the man is not the same at the beginning as at the end of it” (A. Marshall, “Principles of Economics,”, 1962.) The behaviour of the individuals, the way it changes along time, and overall the manner in which both make impact on the welfare of individuals and on the society itself is a very challenging aspect, usually and mostly taken as given throughout a nonvariable utility function. But, as shown above, there exist some literature by some of the main references in the economical thought, and not by many others. Even in the paper Dynamic Utility, by Ragnar Frisch (Econometrica © 1964), he worried about it. Once showed the importance and interest of this topic, it must be shown, as well, the importance of the nature of the individuals: “Divitae bonum non sunt (Material wealth is not the one good)”, “Rationale est homo (A man is rational)” and “Res severa est verum gaudium (Happiness is a serious thing)”, in De Beneficiis, by Lucio Seneca, spanish philosopher and Nero‟s counsellor during Roman Empire . 2. CONEXION WITH RELEVANT LITERATURE It could be good to start talking about the historical line that research in time issues within the field of economics has followed. The problem is that, on this way, one always finds different approaches, and not all of them have been developed very much, for several reasons, although several top researchers in economics along the time have met or crashed this topic at some point of time in their research. It is the case of, for example, Becker, Hayek, Samuelson, Marshall, Hicks, Koopmans or Debreu, for citing some ones. The diversity of point of views, and even of purposes, has made this research particularly heterogeneous, perhaps because none of them wanted explicitly to research either the time use or the change along time itself, but indeed to finish in other conclusions for which the presence of time use in their models was crucial, in one way or another. The benchmarks on these theories of allocation of time are Becker and Gronau contributions, though, following Boland, in Becker‟s “There is no reason for historical change; hence it cannot be explained” (Boland, 1978) One paper in the existing literature making a good compendium of what has been the research in time issues is Time in Economics vs Economics in Time: The 'Hayek Problem', by L. A. Boland published in the Canadian Journal of Economics / Revue canadienne d'Economique © 1978 Canadian Economics Association. We conceive time in both the time use perspective, and the time line one. Our implicit point always is the idea of time uses depending on time (the time line, i.e, time use varying along time). That is why Boland ́s paper is of interest, since it is somehow connecting both perspectives, analyzing how they have been evolving along time in the literature. In Boland ́s paper, a great analysis is made, whose summarize about the different approaches in using time in economics can be read as follows, looking at time as both the time line (decisions along time) and the time use (decisions about how to use the time):  Time and static models: even in a standard model in economics, implicitly is the idea of time, whenever, for example, we make comparative static. The problem is, as Boland argues, this always is a matter of interpretation of the model, and “Therefore, with respect to any given model, today ́ s values of the endogenous variables may be shown to be consistent with today ́ s values of the exogenous variables, but tomorrow their respective values may not be consistent. Since dynamic processes obviously refer to more than one point in time, the explanatory usefulness of a static model would seem rather limited” (Boland ́ s, pp.2)  Time-based variables: during the late 50s, Koopmans (1957) and Debreu (1959) introduced time-based variables, i.e., subscripts making reference to the point in time in which the goods are consumed, suggesting in this way that, as Boland comments, a hamburger is not the same hamburger for the consumer at time t ́ than at time t ́ ́. Although very interesting, and quite used, this approach receives a criticism in Boland ́ s, since he argues there is no dynamics in this model, since formally the model is like a static model over all the time range, which only multiplies the number of goods. Nevertheless, this idea is commonly used in economics.  Time preferences or the economics of time: This approach deals with the idea of including time as a commodity. Bohm-Bawerk (1889) and Becker (1971) are examples of it, and the impact of the Becker ́ s contribution is something to highlight. The former focuses more in production theory, while the latter develops more the consumer side, relating it to several fields. But, again, there is no dynamics in here. Time is something exogenous and static variable. “Neither Becker ́ s nor Bohm-Bawerk ́s can avoid the static approach of the givens (the constrains, the tastes, the production functions, time available, etc) [...] There is no reason for historical change; hence it cannot be explained” (Boland ́ s pp.5)  Variable givens or lagged variables: This is an alternative approach, attempting to determine the time path trajectory of the endogenous variables, and that change is suggested to be done because of a change in the parameters or in the exogenous variables along time, or both, being Hicks ́ model (1971) an example for the former, where he talk about an “autonomous invention”, and Kaldor ́ s growth model another one for the latter. Of course we are not forced to assume that the period of change for the exogenous variables has to be the same for the endogenous, and an example of this is Von Neumann balanced growth model. It is worthy to be read this section, specially the part in which he establishes a parallelism about the dealing what in economics is done to a point in time and what is (in physics) the dealing of a point in space.  Flow variables: Similarly, this approach is one such that is extending a static vision of a model to a dynamic one by inserting appropriately differential equations, and examples in the literature can be found in Barro and Grossman (1971) or Arrow (1959).  Time, logic and true statements: this part is related to the discussion about the neoclassical economics to be or not to be timeless, as many authors suggested in the 70s, since some of them tells that “all economical analysis has been merely logical derivation of solutions” as Boland writes in reference to Georgescu-Roegen (1971) and Shackle (1972) contributions. This is a controversial point in Boland ́ s, very discussed, so we do not enter in details here.  Time and knowledge: the Hayek problem: at this section, Boland insist more in the idea that in previous sections, the way in which time is included suggest that any reliance on only standard general equilibrium theory precludes and discard an explanation for historical changes, since all the causes, motivations and reason for changes are beyond explanation since they are exogenous to the models. He points as well that Hayek (1937) realized about this problem, and that this remains an essential consideration in most Austrian models, as in Hicks (1973) and in Lachmann (1976). This is what Boland understands as the “Hayek problem”, and the same Hayek in 1937 showed and recognized in his paper his incapability to solve it, but pointing in such very abstract paper a lot of insights, one of them collected by Boland, related to knowledge, what in turn relates with the next part we are going to relate with a little: habit formation. Boland, to conclude, suggests that the individual process of acquiring knowledge must be endogenous, and that the individual decision and process of learning/adapting must be taken in real time. As we can observe in Boland ́ s, his work is a good and quite complete description, introduction, motivation and connexion to any existing research related to how to include time in economic models, and the implication it implies, in different fields. Other fields this project is having touch and/or impact with, directly or indirectly, are such that habit formation, endogenous tastes or dynamic utility. Related to habit formation and endogenous tastes, there exist some very interesting work by Pollak(1968, 1970, 1973, 1976), the so called addiction perspective by Becker along his career, Hammond (1975), Weizsäcker (1970), Gorman (1967), Peston (1967) or Samuelson (1956), where none of them uses time use as an explanatory variable, for this concrete purpose. According to dynamic utility we can find literature in Frisch (1964). All of them, mostly, fit some of the Boland‟s specified categories of how time, in the broad sense, as time use and time line-, has been modelled in economics, and which we have tried to summarize above. Our project is comprising, within the Boland ́ s points summarized above, almost all points almost straightforwardly but the last two at the same time (this itself could be an interesting contribution), and in reference to the last one, the Von Hayek problem, we deem is on the line of the proposed solution Boland suggests, since the process is endogenous and in real time. However, all this discussion (which comprises as well the second last point in our Boland ́ s summarize) is mainly based on Von Hayek ́ s reflections and thoughts in what he recognizes himself is a very ambiguous paper (and quite abstract), which state what later on Boland calls the “Von Hayek problem”. Due to this ambiguity on the edge of at least Von Hayek ́ s capabilities, as he recognized, our value over our own job might be incorrect, but still might be a possible solution or at least a small contribution to the solution to the Von Hayek‟s problem applied to consumer theory with time inputs. 3. INTRODUCTION This serious task of happiness for individuals, then, should start with a very long philosophical speech (and nowadays in modern societies Séneca‟s philosophy applies very well), since, in the end, we think that any issue related with behaviour, decision and happiness of human beings is something that escapes surely any present knowledge in a formal scientific way. As we do not consider ourselves experts in that task, we simply claim to one point: What is, actually, the only thing we can, -as human beings, as individuals, or as consumers-, choose? Probably, this statement, -as simple as exigent-, if is thought carefully and with time enough, becomes to have a much easier response we thought at the beginning. Our point of view is simple: we think that, in the end, we can only choose one thing: How to distribute the time that has been given to us, for the mere fact of existing and being alive. This is simply our starting point and only main assumption. However, this might be a very stoicism perspective. Human beings do enjoy material things too, in a very hedonic way, and it must not be ignored. Up to Becker (1965) time was not formally introduced into an economic model, though previous works by Mincer pointed the importance of doing it. Hence, we ended with a model in economics that is modelling both hedonism and stoicism, relating it to economic concepts. Along this paper we just try to generalize Becker‟s contribution, and putting it into a dynamic context. Then, we build a generalized static model in a 2-step optimization process, keeping the essence in Becker, with the time use and market goods as inputs for the attitude that individuals demonstrate as producers of commodities, firstly, and, with commodities as the source of utility or happiness for them, secondly. Later on, we just make the dynamic version of the provided Generalized Becker Theory of Allocation of Time, introducing for the first time the stock of time used as an explanatory variable. By solving it, we get optimal patterns for demands of time and goods, along time, and their stock accumulated along time, which in the case of the time use, points to the habit formation, understanding habit as in the dictionary is defined: a pattern of behaviour acquired through frequent repetition. And for each period of time, then, ́optimal schedules ́ are defined, let us say, which is realistic, since we all do make plans. 4. STATIC MODEL The main idea turns around including as the main argument having impact on the utility function the concept of „commodities‟, something for which you have to use inputs in order for it to be produced. These inputs are market goods and time spent in the activities related to the commodity 3 . Hence, the problem includes time as an input, and requires and states a time constrain in addition to the usual budget constrain. The Becker‟s model looks like this:         1 1 1 x,T max , ,..., , . . m m m m T T i i w i m i w i U Z U f x T f x T