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James Devine/June 23, 2000

Technical Appendix 

to "The Positive Political Economy of Individualism and Collectivism: Hobbes, Locke, and Rousseau," Politics & Society, volume 28 number 2 (June 2000), 265-304.

These notes are labeled to correspond to the sections of the main paper (though not all sections have notes). They are not meant to be totally realistic or to prove my results but instead present an illustrative model, showing the kinds of assumptions and logic consistent with the discussion of the body of the paper. Because it represents the basis for the rest of the paper, all of the comments below refer to section 2.

2.B.1. The model starts with a large number (N) of individuals, each maximizing utility based on the (homogeneous) goods received during a period. Each individual i maximizes utility. This utility function is deliberately simplified, highlighting the role of differences in individual collectivism by abstracting from all other individual differences:

ln(Ui) = ln(Gi) + Pi ln(Ga)

[1]

The simple mathematical form of [1] was chosen because it implies diminishing utility returns to both Gi and Ga. Gi refers to the goods received by individual i and Ga is the number of goods received by the average person in society. Whereas the other components of the utility function are assumed invariant between individuals, Pi represents the degree of collectivism of individual i, the elasticity of individual utility with respect to the average person's goods, with varies between people. For the extreme self-denying altruist, Pi à   , while for the extreme slave to envy, Pi à  - .

Information. Though goods known to be received by others rather than those actually acquired should be valued by our hypothetical individuals, assume until sections E.(iii) and F.(iv) that information problems play no role. A classic case of imperfect information, the principal/agent problem mentioned in section 1.B., is thus simply treated as a case of externalities, in which the agent is able to impose costs on the principal independent of what was contracted for.

Single-Peakedness. Because the options are so simple, individuals who prefer either total equality of distribution or extreme inequality but reject intermediate cases seem unlikely to exist; most or almost all individual preferences fit the Single-Peakedness assumption. One condition allowing single-peakedness is that the P and S spectra are independent of all other issues or that people treat it as such. Even given the Simplicity abstraction, which abstracts from such issues, individual differences of perception of reality probably make it impossible to argue that this independence exists always or everywhere. Rather, simply assume that single-peakedness applies, i.e., that people perceive the world as being as simple as the model. This expedient, by the way, is common in the "Public Choice" literature.

2.B.2. Next, people live with others and their externalities, so that each has to deal with the impact of others' holdings of resources on his production. Assume that many (M) different types of resources exist. For each resource type k, assume that individual resource holdings xk, and thus total societal resource holdings, Xk, are constant in order to focus attention on how individuals deal with the impacts of these resources via societal institutions.

The Equality abstraction implies that each person's technology or "production function" is the same. In a stateless situation, the contribution of any input k to the production of the good for individual i is:

Gik = gk xik + zk s k Xk

[2]

Assume a linear model, with the technical coefficients gk, s k, and zk constant for each k; the first two are non-negative. The coefficient gk represents a standard input-output relationship, the direct effect: an individual's control of resource k raises his or her G production.

The product of zk and s k represents the external effects of Xk on Gi. The constant zk measures the kind of external impact that resource k has: for a resource with external benefits, zk = 1, so that its use helps others' production of Gi, while for a resource with external costs, zk = -1. (Other values of zk are not allowed.) Next, s k represents the salience of the impact of the use of resource k held by others on the production of goods by any individual. The higher this number, the greater the impact (positive or negative).

Unlike with equation [1], these external impacts are objective. Implicit in [1] was the assumption that there is no direct connection between other people's holdings of resources and one's utility. The impact is instead indirect, through their effects on their receipt of goods.

2.B.4. As noted in section (iii), societal organization is needed. By the Modernism assumption, this is assumed to be a central state. For simplicity, the state is assumed to tax resources producing external costs and subsidies those having external benefits, which gives us a way to quantify qualitative restrictions and encouragements. The only variable is the degree of collectivization (S): those resources with the most externalities are collectivized most while a more "collectivized" society regulates resources more.

To represent these conceptions, replace equation [2] with the following. The contribution of any resource k to the goods production of individual i is:

Gik = (1 + zk sk) gk xik + (1 + zk Î k sk) zk s k Xk

[3]

where sk is the degree of regulation of resource k, the magnitude of the tax or subsidy. If this subsidy or effluent tax variable equals 0, this is the same as equation [2]. The coefficient Î k represents the degree of effectiveness of government regulation of resource k, assumed to be less than or equal to one; if it equals one, government regulation of that resource is totally effective.

Following crude collective rationality, equation [3] is set up so that the individual with resources having beneficial externalities gets a subsidy and that with those having negative externalities pays taxes. In the former case, for a resource with z = 1, [3] first says that individual i receives a subsidy of sk for each unit of resource k he controls (so that the total effect of the subsidy equals sk gk xik). Second, the subsidy is treated as increasing the external effect of Xk: he benefits indirectly, from the subsidy of others' holdings of that resource (adding up to Î k sk s k Xk). Therefore, all else equal, individual i would favor an increase of sk. If Î k is high, encouraging external benefits, then this regulation would gain further favor.

In the case where the use of a resource pollutes, where zk = -1, the individual's direct material benefit of holding k falls with sk, as the effluent tax rises. But the external cost imposed on individual i by others holding it also falls, especially if Î k is high. In addition, a rise in effluent taxes helps pay for the subsidies discussed above and administrative overhead. So it is possible that the individual can gain from effluent tax.

Equation [3] hardly represents the whole story of how goods are produced for individual i by her resources. All M resources contribute to Gi. For simplicity, interaction between resources will be ignored so that the total contribution of these resources to the production of goods for i equals a simple sum of all of their effects. Further, a tax T must be paid by each person. So the total goods received by individual i equals:

Gi = S [(1 + zk sk) gk xik + zk (1 + zk Î k sk) s k Xk] - T

[4]

where this summation (and all others below) is over all M resources. Following the Equality abstraction, T is a head tax, i.e., the same for all individuals.

The total net revenue (taxes minus subsidies) of the government would be:

R = -S [zk sk Xk] + N T

[5]

The coefficient Î k does not show up in this equation, since it is assumed that even though government regulation is less than totally effective with Î k < 1, none of the revenue earned by taxing effluents is lost to the system (though it might be grabbed by government officials, who are part of the system) and that all of the subsidies of beneficial externalities are distributed. Instead, Î k refers to the degree of success of the tax or subsidy program in discouraging or encouraging externalities.

Also assume, in line with the Simplicity assumption (which does not allow for intertemporal borrowing), that the government balances its budget. Thus, the non-negative administrative overhead costs for monitoring and enforcement of the Property System (A) equals R. So the magnitude of the head tax equals:

T = [A + S zk sk Xk]/N

[6]

This equation says that as subsidies rise or effluent taxes fall, T rises to keep the budget balanced.

This completes the discussion of the limits on the way in which the government can behave in our imaginary society. Turn to questions of what kind of subsidies or taxes people want. Rather than have them decide on each program piece-meal, the concept of "sophisticated collective rationality" is assumed to apply. In order to create a benchmark, keep its definition independent of people's actual personalities: it is defined assuming P = 0 (so that utility comes from only the material receipt of G), under the Equality and Cohesion abstractions (so that collective rationality is defined by a representative individual's decision-making).

The individual maximizes utility by changing sk. For HE, d Ui/d sk = (Ui/Gi) d Gi/d sk (following [1]). Holding A constant and employing equations [4] and [6], the latter derivative equals:

d Gi/d sk = zk gk xki + Î k s k Xk + zk Xk/N

 

 

[7]

= [zk xka (gk - 1)] + Î k s k Xk

 

where the second formulation applies the Equality assumption to assume that the individual's holdings of xk equal that of the average (mean) individual xka = Xk/N.

The term in square brackets says that, contrary to intuition, an individual holding a polluting resource will gain directly from effluent taxes (and lose from subsidies), once the head tax and balanced-budget constraint are taken into account, if the degree of direct productivity of that asset to the individual is low (if gk < 1). However, that seems a rare special case that should be ignored: assume that gk   1 for all k.

Given the assumed constancy and positivity of gk, s k, xki, Î k, and Xk, d Gi/d sk cannot equal zero, so there is no optimum. The best we can say is that due to diminishing marginal utility, d Ui/d Gi asymptotically approaches 0 as Gi rises (and Ui/Gi falls), so that rising sk produces less and less marginal benefit to an individual. An actual optimum might occur if a different mathematical formulation of [1] were used or if we assume that Î k falls as sk rises (increasing marginal cost to regulation) or if there are political barriers to raising sk.

Even without this latter assumption, [7] indicates that when either s k or Î k rises, so does the marginal net benefits of regulation, all else constant, so that the individual is more likely to gain despite any direct suffering from effluent taxes. The connection between rising s k and the benefits of rising sk is the basis for sophisticated collective rationality.

Assuming that Î k is the same for all resources, in order to emphasize the overall effectiveness of government and to simplify the analysis, the simplest case of sophisticated collective rationality says that for each resource k,

sk = S s k

[8]

where S is non-negative.

This is only one possible definition of the variable S so central to the paper. Instead, we might rank all M resources according to their salience, so that collective rationality would mean that the ranking of the corresponding sk would be the same, with S being measured according to the degree of regulation for the median resource. However, definition [8] is used below because it is quantifiable.

2.B.5. The social choice concerns that of the general level of regulation (S), which in turn determines all of the sk. For decisions in the Assembly, utility maximization following [1] gives us:

dUi/dS = (Ui/Gi) [dGi/dS] + Pi (Ui/Ga) [dGa/dS]

[9]

This formula allows us to describe the marginal utility to the median voter of increased collectivization, dUm/dS. To represent decision-making by the median voter in the Assembly under the Equality abstraction, assume that the two terms in square brackets equal each other and dGm/dS, i.e., that the marginal material benefit to the median voter m is the same as that to the mean voter a. (This implies that the distribution of goods amongst individuals is symmetrical.) On the other hand, this voter does not see herself as being the representative agent for all of society and thus distinguishes between her own material benefit and that to others, represented by the average of society, so that [9] becomes:

dUm/dS = Ui Ga ([1/(Gm/Ga)] + Pm) dGm/dS

[10]

Even though the median voter does not perceive it this way, the Equality assumption means that goods she receives (Gm) should in general move in step with that of the mean voter (Ga), so that the term in square brackets should be roughly constant (and positive). If the last derivative is positive, the marginal benefit from increased regulation rises with Pm.

What is this last derivative (dGm/dS)? Assuming sophisticated collective rationality as represented by [8] and substituting S into equations [4] and [6] gives the following, holding A constant and holding Î k = Î for all resources k:

dGm/dS = S {[zk s k gk xmk] + [Î (s k)2 Xk] - [zk s k Xk/N]}

[11]

The first term in square brackets represents the net direct benefit to the individual from all the subsidies and effluent taxes. The second bracketed term represents the net indirect benefit from all the subsidies of and taxes on others' resource holdings, while the third represents changes in the head tax resulting from rising S.

If we assume that the median voter owns the same amount of each resource k that the average one does (m k = xkm = Xk/N), i.e., that she has a diversified portfolio and thus no vested interests, then [11] becomes:

dGm/dS = S  s k m k N {[zk (gk - 1)/N] + Î s k}

[12]

While the term in square brackets is nonpositive (since gk   1) when zk = -1, it is likely to be very small relative to the second term, since zk/N is close to zero. Further, the negative terms should correspond to positive terms of similar magnitude and thus cancel out in the summation. So dGm/dS   Î S  (s k)2 Xk and most of the time, dGm/dS is positive (especially when Î is high) and dUm/dS rises with Pm.

2.B.6. Assuming the "will of the people" is expressed by a Social Contract, Single-Peakedness implies that as the degree of collectivism of the median voter rises, more collectivization is chosen. This is summarized by the "Social Contract" or "social choice" function, sc:

Sm = sc(Pm, c); d Sm/d Pm > 0; d Sm/d c > 0

[13]

The degree of collectivization chosen by the majority Sm is a positive function of the median individual's personality Pm: individualistic (collectivist) people create an individualized (collectivized) society. (It should be upward sloping, rather than horizontal, because Î is likely to fall with S.) Sm is also a positive function of the vector c (for "communication") measuring other factors that vary to determine the society that people create, held constant here, i.e., the technologies of communications, transport, and societal aggregation of preferences.

2.B.7. Since society is more than a constitution, the free-rider problem persists. In an atomistic situation like that of a simple market, the individual maximizes utility assuming that her actions have no impact on the goods received by the average individual. So in this context, [1] is changed, rendering the individual's degree of collectivism irrelevant:

ln(Ui) = ln(Gi) + constant; d Ui/d Ga = 0

[1-i]

The individual thus acts in a way that seems as if she gets no pleasure from other's good fortune, i.e., maximizing her own Gi and not caring about Ga. (The person does care, if Pi > 0, but cannot act on these cares.) When "voting" for the degree of socialization in the atomized situation, the individual's degree of collectivism is similarly irrelevant:

dUi/dS = (Ui/Gi) [dGi/dS]

[9-i]

Worse, the market situation encourages the individual to evade effluent and head taxes and to milk any subsidies for all they are worth. This also involves breaking contracts with others, reinterpreting them in a self-serving way, embezzling, etc. In general, the individual acts as a free rider on the Property System.

2.B.8. In Prisoner's Dilemma terms, atomistic competition encourages defection. The result of defection is represented by the following:

So = sc(Po, c)

[14]

So is the society implied by the actions of the most ruthless competitor, the most extreme free-rider, Po. That collective and individual decisions impact society using the same function is a common assumption in game theoretic approaches to this paper's questions, which assume that the same game matrix includes both Hobbesian (So) and organized-society (Sm) results.

Reading Figure 1 vertically, we see a Gap between points G and G1 at Sm representing the free-rider problem: Po < Pm. Absent state restraint, opportunistic behavior dominates and defines the equilibrium society, at point E:

Seq = So

[15]

This equilibrium is stable, in that opportunistic behavior involves the dumping of costs on others and the exploitation of others' external economies while an atomistic situation encourages others to emulate this behavior and to reciprocate.

The tendency for the society to move toward this free-riding equilibrium is formalized by positing the following kind of free-rider dynamics that occur if society starts with any St to its right or left of Seq:

D S = -fr(Pt - Po); fr(0) = 0; fr' > 0

[16]

where Pt is the personality which would choose St in the Assembly (Pt =  sc-1(St), assuming that c is constant and that sc is a simple monotonic function of P) and Po is the personality produced by St following the cd function developed below (equation [17]). (In figure 1, Po is constant and given.) Since fr' > 0, a positive free-rider gap (the argument of fr) causes S to fall, toward Po.

More generally under these dynamics, i.e., for all of the graphs, if sc cuts cd "from below" then the equilibrium is stable, as at points E, H and RD in the figures. (In other words, at a stable equilibrium, d sc-1/d S > d cd/d S.) That the process of getting to these equilibrium points does not cause shifts in sc or cd is part of the Simplicity assumption.

2.C. The effects of society on the most opportunistic person are formalized as the custom or character-development function:

Po = cd(St, h); d cd/d St 0; d Po/d h > 0

[17]

Po is a nonnegative function of S due to the virtuous circle: the greater the success of collectivist norms, the more individuals go along with them, encouraging them to work more successfully. Po also can rise due to other characteristics of the culture (represented by h for homogeneity), including ethnic or linguistic homogeneity and sharing of values.

2.D. With no Social Contract, additional dynamics are added. First, there are democratic dynamics:

D St = dd(Pm - Pt, Democracy, Unity)

[18]

where dd is a positive function of all three arguments (and that D St = 0 when Pm = Pt). This assumes that the degrees of Democracy and societal Unity can be measured as scalars. This first indicates that, via a process of repeated collective agreements, society tends to become more collectivized if the median person is more collectivist than the personality corresponding to the actual society (Pt = sc-1(St)). It also indicates that the more democratic the government and the more Unified the society, the faster the democratic dynamics work.

Second, there are cynical or pessimistic dynamics, in which the direction of change in [18] is reversed:

D Pm = pd(St - Sm); pd(0) = 0; pd' > 0

[19]

where pd' > 0, indicating that the median individual changes his level of collectivism toward the level of collectivization that currently prevails. If the degrees of societal Unity and governmental Democracy are low, then the democratic dynamics are slow, allowing pd to take over and possibly to interact with and reinforce fr dynamics.

2.F. A society without trickle-down (a simple case of one without Cohesion) is described by a static model, revising equations [2] and [4]. Only two mutually-exclusive groups exist, W and C (workers and capitalists), possessing different sets of resources. W has only B (brains and brawn), while C possesses only K (physical capital). Each class has a constant number of members, Nw and Nc, where Nw + Nc = N. Assume also that each class is internally homogeneous and has single-peaked preferences. "Homogeneity" of classes means that assumptions of equality and of no vested interests applies within each class, while each capitalist c has the same Kc (= K/Nc) and each worker w has the same Bw (= B/Nw), where K and B are the total amounts of these items existing.  

If the K-holders have the upper hand over those holding B, W suffer external costs from, and give external benefits to, C (zK = -1 and zB = 1). Since Equality is gone, the two classes also have different technologies. Thus, for representative individuals w and c of the two classes in an imaginary pre-state position of inequality the production functions would be:

Gc = gK Kc + s B B

 

 

[2c]

Gw = gB Bw - s K K

 

where gj and s j are both positive constants for j = B and K. A more complete model would add floors on Gw, to allow survival as human beings, and Gc, to allow survival and finance accumulation.  

Assuming the crude collective rationality, equations [2c] become:

Gc = (1 - sK) gK Kc + (1 + Î B sB) Î B B - Tc

 

 

[4c]

Gw = (1 + sB) gB Bw - (1 - Î K sK) Î K K - Tw

 

C pays effluent taxes (sK) because of his negative impact on W but may benefit from subsidies for W (sB) such as public education, depending on the effectiveness (Î B) of these programs in promoting capitalist goals. W, on the other hand, gains directly from the subsidies, and also from sK, depending on the latter's effectiveness (Î K) at discouraging negative effects.

In Class Society, Tc and Tw would not be equal to each other, in general. Instead, it is more likely that each class pays a different head tax, with the ratio between them a nonnegative constant:

Tw = (1 - p ) Tc; p 1

[20]

With higher p , the tax system is more progressive (while for p  = 0, workers pay the same tax as capitalists). With p constant, a general rise in taxes is seen as a cost by members of both classes.

Next, the average tax rate T equals both the total necessary to balance the state budget divided by the number of people and the weighted average of the two tax rates:

T = ( A - sK K + sB B)/N = (Nw Tw + Nc Tc)/N

[6c]

so that:

Tc = (A - sK K + sB B)/N*

[21]

where the denominator is a positive constant (N* = p Nw) when p is constant. (Since N/Nw > 1 and 1 p , N/Nw > p , so that N > p Nw and N* > 0.) Then, Tw is defined by [20].

Consider the third type of conflict within Class Society, in which A is taken as given. Also assume that p is constant, so that the structure of the head-tax system is not a subject for debate. From equations [4c], [20], and [21]:

d Gc/d sB = B [(Î B)2 - (1/N*)]

[22a]

d Gc/d sK = -K [gK/Nc + 1/N*] < 0

[22b] 

d Gw/d sB = Bw [gB - (1 - p ) Nw/N*

[22c]

d Gw/d sK = K [(Î K)2 + (1 - p )/N*] > 0 

[22d]

Equations [22b] and [22d] indicate the obvious, that capitalists lose from, while workers gain from, taxes on K. Further, since gB   1 and 1   (1 - p ) Nw/N*, the term in brackets is in [22c] is nonnegative: workers almost always gain from subsidies of B, while to the extent that the tax system is progressive, the marginal benefit of these subsidies grows. Finally, [22a] says that for high Î B, and large N*, capitalists gain from subsidizing workers. (N* rises with the tax system's regressivity, the size of the population, and the relative size of the capitalist class. With a large Nc, the burden of the head tax is distributed more widely.)

More importantly, these equations show room for the social-democratic class compromise discussed in the text (and, by extension, compromises in Liberal Society despite lack of Cohesion). Consider a balanced-budget expansion where wage subsidies are paid for by taxes on capital and B dsB = K dsK. The benefit to capitalists from raising taxes on themselves (sk) equals:

dGc = [(d Gc/d sB) (K/B) + (d Gc/d sK)] dsK

[23]

So from [22a] and [22b],

dGc/dsK = K {[(Î B)2 - (1/N*)] - [gK/Nc + 1/N*]}

[24]

Therefore, the class compromise allowing greater taxation of capital, is possible if:

(Î B)2 > gK/(N - Nw) + 2/(N + p Nw) = RHS

[25]

This condition may be met: if N à and Nw/N is constant, the RHS à 0. Further, [25] indicates that compromise is more likely the more effective the government, the less productive K is to the individual capitalist (i.e., to more that their goods arise from exploitation of labor), the larger the capitalist class compared to the population, the larger the population, and the more regressive the tax system.

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