Production and Distribution in Political-Economic Systems: a Non-Atomic Game" (2017)

University of Massachusetts Amherst ScholarWorks@UMass Amherst Economics Department Working Paper Series Economics 2017 Production and Distribution in Political-Economic Systems: a Non- Atomic Game Alex Coram University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/econ_workingpaper Part of the Economics Commons Recommended Citation Coram, Alex, "Production and Distribution in Political-Economic Systems: a Non-Atomic Game" (2017). UMass Economics Working Papers. 236. https://doi.org/10.7275/10943155 This Article is brought to you for free and open access by the Economics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Economics Department Working Paper Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Production and distribution in political-economic systems: a non-atomic game [Alex Coram. Economics. University of Massachusetts. Draft 3. September o17] Abstract Societies are often compared in terms of the relative shares of the domestic product controlled by government but we don’t have much by way of analytical tools to allow us to think about the ways in which changes in these shares affect the characteristics of production and distribution. This paper treats production and distribution as a cooperative game with a continuum of players. Outcomes depend on politics and economics. It has a number of unexpected results. Among these are that relative shares do not depend on the majority rule the production function and that the characteristics of inegalitarian and egalitarian systems may converge. 1 Introduction Societies are often compared in terms of the relative shares of the total domestic product transferred through government action and there is extensive discussion of the effects of differences in these shares on things like output, economic participation and distribution. Obvious examples are debates about welfare payments and a universal income. So far this discussion has mostly been historical and empirical. Despite the progress made with these approaches it would also be useful if it were possible to construct some sort of analytical framework that would help understand the factors influencing these differences and their affects at a more abstract level. This might help serve as a reference point for discussion in the same way as abstract models of, say, economic growth help us talk about particular economies. For example we might want to ask: What would be the properties of an allocation if every agent had a share based on political power and productive capacities? How does participation in production change as the share of distribution controlled by the political system changes? What happens as political power and the allocation of resources changes? The purpose of this paper is to attempt to contribute to constructing at least one version of such a framework. Rather than starting with micro-foundations it looks at a distribution across large systems of political and economic interactions. I also undertake some comparative static analysis of systems with different levels of transfer. 1 The approach taken here is to treat the production and distribution problem as a cooperative game with an arbitrarily large number of agents thought of as a continuum. In contrast, most of the current theoretical literature on allocation through the political process concentrates on competitions between candidates or parties that offer resources to voters. Individual voters respond by trying to maximize their payoffs. This is usually done by making standard assumptions about utility functions and information and restricting the number of coalitions that form.1 These restrictions strengthen the analysis in some directions by allowing the tools of optimization theory to be used, but they weaken it in others. In particular they leave out the macro-effects produced by the possibility of shifting sets of possible coalitions. For example with 30 voters and a majoritarian voting rule there are about 150 million possibilities. It might be objected that the decision to ignore micro-foundations and use a model based on the value of the game is inconsistent with much contemporary practice in economics. This misses the point that the questions I am asking are different from micro-foundations questions. It also assumes that micro-foundations models are on a sound basis, or at least on a better basis than the approach used here.2 Many of the theoretical foundations for this paper have been developed by Aumann and Shapley [1], Aumann and Kurz [3], Shapiro and Shapley [17] and Milnor and Shapely [10]. In particular the second and third sections of the paper relies heavily on results in [1]. In the conclusion I summarise the characteristics of the systems studied and provide some answers to the questions in the first paragraph. Some results are reasonably obvious after the event. Others are less intu- itive. One example is that the proportion of votes required to make a winning coalition and the nature of the production function does not effect the relative distribution. Another is that participation in production is similar in systems with equal distribution of endowments and systems with unequal distribution of political power. I do not claim that these results map into the real world. On the other hand some of the results on the characteristics of systems with different shares of transfer are striking and, perhaps, should not be ignored in attempting to understand more specific cases. I set out the paper as follows. In §.2 the problem and the model are presented. §.3 gives the main theorem. This is analyzed in §.4. 2 The model 2.1. Statement of the model Suppose we have a social system made up of an economy and a political process. The problem presented 1See Roemer [14] for an example of a two party model and good recent discussion of the literature on vote competition. Osborne and Slivinski [12] give a model where the number of candidates is not restricted but voter preferences are linear. 2There is a large literature which questions whether voters and economic agents behave in the way required for a micro- foundations approach. See for example Shapiro and Green [18]. 2 in the introduction might be restated as that of determining how the total product would be distributed across all the members of a given system when everyone gets a share that is related to its political and economic capacities. The economy is described in terms of a production function, endowments of capital and labour capacity, and the political process represented by a voting mechanism. The total value for an individual is the sum of returns from participating in the political and production processes. The economy might be thought of as either capitalist in which each individual has an initial endowment of resources and skills, or a system in which all resources are collectively owned and workers have different skills. Although the model is written in terms of capitalism it holds for both cases. It is assumed that individuals do not extracts rent and everyone has to work with their endowments. The system might be thought of as some sort of lemonade stand capitalism. There is full employment and everyone who wants to work has a job. The collective choice rule says that any coalition with a weight greater than half is winning. A winning coalition can distribute some portion of the produced goods to its members. This portion is fixed in the short run. In many cases discussions of the effect of transfers from government on participation in the economy assume that there is a loss of utility associated with work. A similar assumption is made in this paper. The size of a set, or coalition, of agents is defined by the Lebesque measure µ on the interval I = [0, 1]. This means µ(I)= inf i νi = i µi = 1 where vi is a sequence of intervals covering I. Let S be a coalition of agents. The measureP of S is µP(S)= µ(S ∩ I). The majority rule is given by the number θ. A coalition is 1 winning when µ(S) ≥ θ ≥ 2 . Each individual i is represented by an infinitesimally small sub-interval written dt or µ(i) or µ depending on the point to be made. To begin let µ(i)= µ(j) for all i,j. I will change this assumption later in the paper. There is a single all purpose produced good and utilities in this good are linear. Utility of work is expressed in the same units as utilities of the produced good. Individuals have different productive capacities depending on their endowments of resources and skills. Without worrying too much about measurement it is assumed that these endowments are substitutable. They can be thought of either as a vector or as a simple function g that takes on a finite number of values (g1,...,gk) on the interval I. Write the set of individuals with endowments gr as Gr and think of grdt as the endowments of an individual in Gr. Since g is Lebesque integrable define a vector γ(S):=( g1dt, g2dt,..., gkdt) ZG1∩S ZG2∩S ZGq ∩S r r with γ (S) := Gr ∩S g dt. It will be useful to express pendowments as a measure by defining R r r γ (S) w (S) := j j γ (I) P where j = (1,...,k). This means that wr(S) is the endowments of type r in S expressed as a proportion of the total. It follows that 3 wr(I)=1= µ(I) (1) Xr It is assumed, for the time being, that w is a non-trivial partition of I. In other words there is no r : wr(I)=1 and wj (I)=0 for j 6= r.

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