Public Finance, econ 132, Foldvary

Holcombe, Chapter 7

A Theory of Collective Action

Note: Some of these notes are not in Holcombe and were not discussed in class.
You may skip those sections not covered in the book or class.

An economic club is a voluntary organization serving its members.

The theory of clubs has applications to government, even though government is coercive.

Collective decision requires assembling individual preferences into a group preference.

Can be by voting, demand revelation, the generation of spacial rent, surveys.

But the “group” itself does not prefer or think. Labeling it so is personification.

The problem is how to maximize the efficiency of collective decision making.

The term "efficiency" means a ratio of output to inputs.

The adjective "efficient" therefore refers to the efficiency of a given process

relative to some norm;

it can refer to a system that has the highest possible efficiency.

Vilfredo Pareto 1848-1923 Italian economist, pioneered the optimum conditions for


The term "optimal" means the best or most favorable.

An "optimum" is a situation which is optimal.

"Optimal" in economics thus means the most efficient process of a system;

the terms "technically optimal" and "(most) technically efficient" are synonymous.

In economics, efficiency has two meanings:

1) technical efficiency, the ratio of the volume of the output of some process

relative to the inputs (or costs) into that process, per unit of time;

2) Pareto efficiency, the situation in which the technical efficiency (of goods or utility) of a system or process at some particular moment cannot be increased without reducing one the products of the system.

The Pareto principle, was explained in Chapter 2, p. 33.

Pareto optimal: We can't make someone better off without making another worse off.

A Pareto improvement: a reallocation of resources which improves the welfare of some

without reducing that of others

Contract curve, the locus of all Pareto efficient points in the Edgeworth box, Fig. 2.2 p. 34.

Also called the "core."

Along the contract curve, indifference curves are tangent. Slopes are equal.

The |slope| of an indifference curve is the ratio at which people are willing to exchange goods, the marginal rate of substitution.

Pareto efficiency requires that MRS are equal for all.

An economy with perfect competition is Pareto optimal.

Operating at maximum efficiency.

In 1959, Kenneth Arrow proved that a perfectly competitive economy is Pareto optimal.

It is a theoretical benchmark.

Real-world markets exhibit many of the effects of perfect competition,

as an approximation, which is then called atomistic competition.

If the utility of all the persons in a system is one of the products, then Pareto efficiency

implies that technical efficiency cannot be increased without making one person worse off.

"Pareto optimal" (or "optimum") and "Pareto efficient" are related, "efficiency" often

used for products and "optimal" for the utility of persons.

Pareto optimality is therefore related to unanimity; a Pareto-optimal situation is one whose

efficiency cannot increase without the objection of one person.

The economist Paul Samuelson not only described mathematically the distinction between individually used and collective goods, but also their Pareto-efficient criteria.

For collective goods, the marginal rate of transformation among the goods equals

the sum of the users' marginal rates of substitution.

Production possibility curve - maximum production of output from inputs.

|Slope|: marginal rate of transformation = Mcx/MCy

Marginal cost is the extra cost of producing one more unit. The increase in total cost.

Marginal cost of one good is the opportunity cost of not producing the other good.

Opportunity cost: foregone option.

With variable production, the outcome is Pareto efficient when MRT = MRS for all

Therefore also, the ratio of MC = the MRS of all.

The first fundamental theorem of welfare economics.

If there is atomistic competition in production and consumption,

and there are markets for all goods, the outcome is Pareto efficient.

Firms produce at minimum average cost.

Price = marginal cost - criterion for economic efficiency.

Thus, market economies are productive relative to command economies.

Pareto efficiency requires that prices be in the same ratio as marginal costs,

and competition guarantees that this condition is obtained.

The Second Fundamental Theorem of Welfare Econmics states that society can obtain

any efficient allocation of resources with particular initial endowments.

This is not a redistribution of resources, but an equitable initial distribution, which can

be an ongoing distribution.

The question posed by the Samuelson criteria is then whether these Pareto-efficiency conditions can be achieved via market processes.

Samuelson's answer is that "no decentralized pricing system can serve to determine optimally these levels of collective consumption."

By "optimal," Samuelson means Pareto-optimal.

As a reason, Samuelson points to the problem of knowing the collective demand.

It is in the selfish interest of each user to indicate less interest in the public good than he really has, to avoid payment for it.

The implication is that the share of expenditure for each person can only be determined by

that person's communicated expression of it.

But this premise does not logically follow purely from the definition of a collective good or

from the Pareto-efficiency criterion.

Several premises of Samuelson's logic are implied:

1) the public good is non-excludable;

2) the individuals providing or affected by the good are necessarily selfish in the narrow sense;

3) the public good is not provided as a by-product or tie-in to private goods or excludable public goods;

4) the transaction costs and any impossibility of obtaining the relevant information regarding users’ utilities (in order to determine the demand and hence quantity to supply) are excluded from the production and hence of the rates of transformation among the goods.

None of the first three premises are necessarily true in general, although they might well apply in

some situations.

The fourth premise is an arbitrary exclusion of a category of actual costs.

Regarding the first premise, a public good can be excludable by its nature or by its limited

domain of users.

As an example of the former, the public characteristics of a water supply are excludable by the nature of the severable supply of the quantity of the water.

An example of the latter is a local public good, whose domain is limited to some geographical area. Relative to a larger region, the good, supplied separately by many local communities, is excludable.

The optimality proposition presumes that the good in question is a global collective good, universally non-excludable. Few civic goods fit that category.

As indicated above, a system is "efficient" relative to some norm or standard, and the

realistic alternative to voluntary provision is imposed provision.

If people will not reveal their true demands in a voluntary system, they will not do so under an imposed governance as well, for even though they may suffer penalties for not telling the truth, the governors may have no means of determining what the truth is; prisoners may lie under duress or say what they feel their captors want them to say.

Hence, a decentralized provision of collective goods is no less Pareto optimal

a priori than centralized provision.

The market fails only relative to an unrealizable ideal system, not relative to governmental provision.

Samuelson himself seems to acknowledges this, stating that even taxing according to a benefit theory does not solve the computational problem, given his premises.

Market failure relative to an imaginary ideal renders the Pareto criterion of no value.

With the common goods associated with a condominium unit, identical units will tend to sell in a

fairly narrow price range, and the owners will have differing consumer surpluses.

Purchasers of condominium units would very likely consider it a gross injustice if they were

somehow forced to pay different prices for the goods according to their personal utilities.

Goods that are private in individual use are typically collective in their various qualities.

If two or more units of a good are produced with certain common quality characteristics,

such as color, aroma, size, shape, beauty, and texture, those qualities collective.

There is no quality rivalry among the purchasers of the goods.

This argument can go even further, to the price of severable goods.

Suppose there are 100 pencils of an identical type being sold in one location at the same price. That price is a public characteristic associated with the good.

The price of oil, for example, is an important macroeconomic collective good.

On the other hand, the Lindahl-type payments or contributions by individuals (based on differing marginal valuations) for a good whose quantity-consumption is collective are private characteristics associated with the good, the total cost being the sum of the individual payments.

Private goods have public payment characteristics, while public goods have private payment characteristics.

Few goods, if any, can be purely private or public in all characteristics.

Since the quality, let alone the price, of a good produced for more than one person is a

collective characteristic, the inability of persons to obtain their subjective valuations of public goods is universal to all marketed goods.

Since goods come as a set of many different types of qualities, and because there are limited sets of available quality mixtures, an impossibility theorem can be applied to obtaining the optimal amount of each quality.

In practice, marginal valuations are second-best, equating marginal utilities subject to the qualities available in a market.

To sum up, the optimality proposition, the argument that collective goods are inherently less

optimally supplied than private goods, is countered by its inapplicability to all collective goods,

the failure of government provision to be any more optimal, the inclusion of transaction costs in

production and thus in the marginal conditions, and the universality of public characteristics..

The failure of the optimality proposition refers to Pareto optimality;

questions regarding technical efficiency can be meaningfully posed and answered.

An entrepreneur creating a development can estimate the most profitable mixture and amounts of civic goods to produce.

Pareto also wrote about land, which he called “natural capital.”

He said that the possessors of natural resources are in a better position than other owners to secure extraordinary benefits in the case of increasing demand;

while the other capitals, in the same case of increasing demand for their services,

can only secure economic profits for more or less short periods,

because attracted by the high profits,

new capitals will come and the competition will tend to reduce the price and profits.

In contrast, the holders of territorial capitals "enjoy a more concrete monopoly, which in certain cases can be an absolute one. They will be able to obtain substantial gains.

Pareto also wrote that from ancient times until nowadays,

political power has belonged, with rare exception, to the owners of the land…

and so there must be some reason to explain the privileges

which the possessors of territorial capitals enjoy.

Pareto wrote that one remedy is to tax that advantage.

The Pareto-optimality principle implies unanimity.

P. 133: Decisions made by majority vote contain an inherent externality.

The decision may be inefficient, in contrast to the method of demand revelation.

We do require unanimity in some cases, such as for a jury in a criminal case,

when there is a very high cost of an incorrect decision.

A club is voluntary at the constitutional level of choice, of joining.

Afterwards, operational decisions can be made by majority vote or demand revelation.

It is voluntary even if you don’t like the decision, because you agreed to the rules.

P. 135: Decision-making costs rise as the required proportion of members rises.

Externality decision cost: the cost on those opposed to the decision.

P. 136: Optimal decision rule: the minimum of the sum of decision-making and externality costs.

Supermajority: more than 50 percent.

The 50 percent rule is more moral than economic.

Optimal club size. How can we determine the technical optimal?

Two variables to optimize: the amount of the public good, and the number of members.

Club theory concerns the size of the organization and the selection of the amount of the collective goods to provide, as well as the nature of the club revenues.

It depends on the effect of the good on crowding and on the cost.

Illust. p. 138, 139.

If the marginal cost is constant, then the optimal size is proportional to the number of members.

If a good is never congested, the optimal size is the population of the earth.

James Buchanan's theory deals with a club providing collective goods and seeks to determine the size of club which its members would consider optimal in providing utility for the members. Such club theory usually concerns a cooperative, a club in which each member (rather than share of stock) has one vote and whose property is commonly owned by the membership.

In Buchanan's model, the utility an individual obtains from using a club good is a function of the good as well as of the number of members in the club.

Buchanan excludes camaraderie as a good, and so the collective good is excludable and congestible.

Since the cost of the good is divided among the members as an equal per-capita fee, the cost of the good to a member decreases with club size.

Buchanan extends the Samuelson model to include the utility from the number of members:

(1) Ui = Ui[(Xi1,Ni1),(Xi2,Ni2),....,(Xin+m,Nin+m)],

where U is utility, superscript i refers to an individual, X is an X good,

N the number of persons in the club, n the number of private goods,

and m the number of collective goods.

The production function also includes N,

since additional members may affect the cost of the good per member:

(2) F = Fi[(Xi1,Ni1),(Xi2,Ni2),....,(Xin+m,Nin+m)].

For each individual, the optimal marginal rate of substitution between goods

equals the marginal rate of transformation between the goods in production,

and the (negative) marginal utility of club size equals the (negative number) cost of the good

with an additional member.

The equilibrium marginal conditions with respect to the consumption of goods X are met when

(3) uij/fij = uir/fir = uiNj/fiNj,

where the lower case u and f represent partial derivatives, j is a representative good,

r signifies a numeraire good, and the subscripts with Nj refer to

the utility and cost of the number of members N of the club.

An individual will have an optimal quantity of collective goods X and share this quantity optimally over a group of the determined size.

If the optimum size is infinite, the second equal sign is replaced by "greater than," e.g. if uiNj equals zero, there is no cost per additional member.

Then, of course, any finite size is smaller than optimal.

Buchanan adds the restriction that all club members are identical, noting that the results regarding optimal club size do not apply otherwise.

Costs being subjective, without identical utilities, the problem of demand revelation and the charge per member once again has no solution in this model.

A deeper problem concerns the nature of crowding, which itself is presumed but not analyzed. Under what situations would a public characteristic be in the congestible and excludable category?

Once a characteristic is in fact congested, then it is no longer public, as discussed in Chapter 1. But if it is not yet congested, what would make a characteristic capable of becoming congested?

The domain of the good, the field in which the good has a common availability, has a lower utility for some persons in the domain when additional persons enter the domain.

If the domain is space, which seems to be the most common application regarding civic goods, then the impact of goods in space itself needs to be accounted for, i.e. the induced rent, which the Buchanan model does not take into account.

If the domain is not space, then, being an excludable good, some entry fee proportional to congestion may be charged offsetting the marginal cost.

But the Buchanan model posits a fixed fee divided by the membership.

A comprehensive theory of clubs, which as Buchanan states, is also a theory of optimal exclusion, needs to take into account the alternatives of user charges and rents.

Crowding may yield a vector of utilities rather then one value.

Members may not like the quality rivalry itself, but may like some effects of crowding.

For many clubs, the domain is capacious, inherently incapable of becoming crowded, but there may still be a marginal cost of adding a member.

A political association, for example, typically desires to have as many members as possible.

A new member does not crowd the domain, but he does add some maintenance cost to the club. The members want entrants, but only if they at least pay their way.

Such clubs are funded by an equal assessment of dues plus voluntary donations that the members may care to send.

The point here is that such a club is neither infinite nor does it have an finite optimal size; the collective good is in the category of excludable and capacious, which does not fit the mold of the Buchanan model.

Typically, clubs such as churches and political groups seek to convert others into their domain; their goods are not necessarily welcomed by those not in the club.

Ellickson states that, contrary to the Buchanan model, the determination of the optimal club size requires a global comparison among alternative clubs.

A choice within a particular allocation is not sufficient to establish optimality.

Equilibrium is relevant to the assignment of individuals to jurisdictions.

This relates to Samuelson's statement that a determination of the optimum optimorum requires computing over all possible goods to determine the total optimum mix - an unattainable Pareto ideal.

The model of Buchanan does not deal with the site rents generated by a club that owns land and provides territorial collective goods.

If a good such as a bridge is provided by a club, it will generate a territorial rent which would be reduced but not necessarily eliminated by a congestion fee.

For example,a bridge that is never crowded could generate rent.

A model of a territorial club

Buchanan's theory of clubs will be extended here by incorporating the rent R generated by a territorial collective good G, affecting some area A, with N number of units in the club.

Such a club could be a residential association, with G consisting of goods such as swimming pools, parks, boats, trails, and community meeting facilities.

The members do not need to have identical utility functions, since payment by rent reveals demand.

Besides the amount of the public good and the size of the club, the amount of rent an individual is willing to pay for the use of some collective good is a function of his income or wealth.

Like the Buchanan model, this model abstracts from differences in wealth, just as it abstracts from individual tastes, cultural standards, and the many other factors influencing the desire for goods.

The area A is divided up into residential lots.

The utility of G for the users of the lots is independent of lot size (e.g. the use of a common swimming pool is independent of the size of one's dwelling unit).

Let the size of the lots be uniform and equal to some constant S; the number of units (lots) N is therefore A / S.

Assume furthermore that there is some known average number of residents per residential unit, so that the total population in A is a multiple of the N lots.

Suppose there is an entrepreneur who proposes to provide G within A.

To maximize his profits, he needs to please the potential membership.

His two choice variables are A and G, similar to the choice variables in Buchanan's model, i.e. the public good and club membership number.

For the user of lot i, the utility derived from G is

(4) Ui = ui(G, N) = ui(G, A/S)

The first derivative of U with respect to G is positive; additional amounts of G yield greater utility.

The first derivative of U with respect to N or A is negative, as in the Buchanan model, since camaraderie is excluded from the utility function, and crowding is assumed to have negative utility for all N > 1.

In this territorial case, there is a rationale for this negativity, since a larger N implies a larger A, which increases transportation costs within A and also increases the number of persons and thus the crowding of goods such as swimming pools.

A site is rented to the highest bidder during each time period (current residents either pay the rent or else are replaced).

The rental revenue due to G is a function of the utility U derived from a lot by holder i:

(5) Ri = r(Ui) = r(ui[G, N])

            There is a cost C in producing G,

(6) C = c(G)

which is a function of G and not directly dependent on A.

For example, the cost of a swimming pool does not depend on the area of the resort.

Included in C is the increasing governance costs of managing a greater amount of G.

The entrepreneur's objective is to maximize the present value of a future profit stream.

It is assumed that once the club and G is created, the size of A and G will not change.

With constant A and G in all time periods, profits are maximized by maximizing the profit P within each time period, where the profit consists of the total rental revenue from (5) minus the cost in (6).

Assuming that the utility and thus the rent for each lot is identical,

(7) P = N.Ri - c(G) = N.r(ui[G, N]) - c(G).

            Profit is maximized where the first derivatives are zero:

(8) ∂P/∂G = N.∂Ri/∂G - ∂C/∂G = 0, given some A,

            where ∂Ri/∂G = (∂Ri/∂ui).(∂ui/∂G) per the chain rule, and

(9) ∂P/∂N = N.∂Ri/∂N + r(ui[G, N]) = 0, given some G.

            Hence, from (8),

(10) N.∂Ri/∂G = ∂C/∂G,

and from (9),

(11) r(ui[G, N]) = -N.∂Ri/∂N.

            Since ∂R/∂N is negative, the right hand term in (11) is positive.

            As area and the number of units increase, the marginal renter is willing to pay ever less rent.

From (11), the optimal club membership N, and equivalently the optimal area A, is that at which the per-unit rent equals the negative of the number of units times the decrease in rent from the marginal unit.

From (10), the optimal supply of G, for a given A or N, is that amount where the marginal cost of G equals N times the marginal unit rent induced by the marginal amount of G.

Since the units are identical, this is simply the amount of G in which the marginal cost equals the total marginal rent.

Equations (10) and (11) provide two equations and two unknowns, G and N.

Solving (11) for N, the optimal number of units equals the per-unit rent divided by the negative of ∂Ri/∂N:

(12) N = r(ui[G, N])/(-∂Ri/∂N).

The optimal number varies directly with rent and inversely with the negative of ∂Ri/∂N.

This suggests that high-rent fancy hotels or residential associations, with luxurious public areas such as lobbies, would be correlated with a greater number of units than less fancy ones, a proposition one could test if one could control for ∂Ri/∂N.

Equation (12) also suggests that a greater decrease in marginal rent per extra unit would decrease the number of units, not surprisingly.

At the extreme, families who greatly value privacy or exclusivity have a very high negative ∂Ri/∂N and if that value is about equal to R, then N = 1, a single family residence.

If the decrease in rent per extra unit is zero for all values of N, the good is capacious, the optimal club size A is infinite, and G is not a territorial good.

            Substituting for N in (10),

(13) ∂C/∂G = ∂Ri/∂G.(r(ui[G, N])/[-∂Ri/∂N]).

            At the optimal supply of G, the marginal cost of G equals the marginal per-unit rent times the unit rent divided by the negative of the decrease in rent from the marginal unit.

A solution to G and N will exist if along some interval of G the marginal cost curve crosses the marginal rent curve from below and then remains above it with increasing G.

            Solving (13) for r(uiG, N]),

(14) r(ui[G, N]) = {∂C/∂G.(-∂Ri/∂N)}/(∂Ri/∂G).

            The greater the marginal cost of the good and the marginal decrease in rent for additional units, the greater the optimal rent must be.

The greater the marginal rent induced by the good, the less the rent has to be.

If these marginal schedules are known, then r(ui[G, N]) or Ř is determined, and that value can be entered in (12) to determine N as Ň, and therefore A, if A has not been set exogenously.

Since Ř is a function of G and Ň, G can now be determined:

(15) Ř = r(ui[G, Ň]),

with Ř and Ň now known.

If functions r and u (or their combination) are known, the inverse function g of the combined functions r and u can be derived.

That is, if we know the amount of rent induced by an amount of G, then we can know the amount of G required to induce a given amount of rent. Given the optimal rent and unit numbers, the entrepreneur will produce that level of G which induces that much rent for those units:

(16) G = g(Ř, Ň).

            With G now determined as Ğ, the total cost C is determined by (6).

From (7), profit is now determined:

(17) P = Ň.Ř - c(Ğ).

If the profit P is greater than zero and if the profit divided by the cost is greater than other opportunities offer, the club entrepreneur proceeds to develop the community.

He acquires space of size A = Ň/S and produces the quantity and quality of G equal to Ğ.

This completes the model.

The assumption in the model of a land-owning club that area A is a choice variable has some empirical validity.

The Walt Disney Company deliberately obtained a large area for Walt Disney World in order to capture the rents generated by the large G they were planning.

Both Columbia and Reston were chosen for the large available areas for communities of 50,000 to 100,000 persons, capturing the site value generated by the large economies of scale, but they also included smaller villages and clusters appropriate to local governance for the provision of congestible and lower-scale goods.

Condominiums are an example of smaller-scale developments (compared to large towns), due to the costs that a large membership and area would incur, with no offsetting economies of scale.

Back to Holcombe

P. 140: Economic efficiency is maximized when there are several levels of government rather than one unified level.

Decision-making costs rise with more members.

So smaller group size is better.

Second, a large number of small governments provides more choice,

and competition induces greater efficiency.

P. 140: Representative democracy

Lower decision-making costs.

Benefits from specialization.

An economic model of democracy.

To keep government effective, its tasks should be specified in advance,

as was done in the original US Constitution.

p. 142: Optimal constitutional rules

A voluntary organization can operate efficiently by agreeing to operational rules

that involve majority rule.

Operational rules are also called post-constitutional.

The constitutional rules are agreed to unanimously.

The veil of ignorance:

What would people choose, not knowing personal information,

but knowing economics, ethics, governance, psychology.

P. 145: We can judge justice by outcomes or by process.

With procedural justice, if the initial endowments are just and the process is just,

then the outcomes are just.

Like in sports, where whoever wins by the rules, wins fairly.

p. 147: Robert Axelrod’s tournament

Computer tournamen, repeated prisoner’s dilemma.

The players remember their history.

Tit-for-tat won; cooperate with those who cooperate.

After defecting, one could cooperate and be forgiven.

In a repeated game, the dominant strategy is cooperation.

The players get a reputation for cooperation.

No government direction needed in this case.

The market economy: the emergence of cooperation

as a result of human action, but not of a planned design.

Attempts to design a system contrary to the market, have failed.

The problem of knowledge and incentive.