Public-good productivity differentials and non-cooperative public-good provision by Eduardo Ley*

DOCUMENTO DE TRABAJO 97-02

Enero, 1997

FEDEA.

Eduardo Ley FEDEA, Jorge Juan 46, ES-28001 Madrid

Version: January 13, 1997

Abstract. In this paper we present two related results: one positive and one negative. The positive result is the extension of Konrad and (1995)'s Rotten Spouse Theorem to n agents. The negative result deals with general technologies. As it turns out, the theorem does not hold for more general technologies. We discuss some of the implications for emissions models.

Keywords. Rotten Spouse, -abatement, climate change

JEL Classification System. Q20, H4, D62

Address. FEDEA, Jorge Juan 46, ES-28001 Madrid. Fax: (+34-1) 577 9575. Phone: (+34-1) 435 0401. Internet address: edley@fedea.es

Konrad and Lommerud (1995), proposition 3, establish a remarkable Rotten Spouse Theorem. In a non-cooperative model of the family, they show that income transfers to the spouse with comparative advantage in producing the household public good are Pareto improving. Even a rotten egoistic spouse (who has comparative advantage in market activities) will want to transfer income to the other spouse raising the welfare of both in the process. While the theorem generalizes to n agents — i.e., strategic interactions do not get in its way when more agents come into play—, it fails to hold when the technologies for producing the public good are strictly concave instead of linear.

1. Non-Cooperative Public-Good Provision

Consider n agents, indexed by i, who are each endowed with , which can be allocated to the private production of a pure public good, G, or to their private consumption, . By private production we mean that each agent has available an individual and, possibly unique, technology that transforms private resources, , into a public good

\[g _ {i} = f _ {i} (y _ {i}),\]

with

\[f _ {i} (0) = 0, \quad 0 < f _ {i} ^ {\prime} (\cdot) < \infty , \text { and } \quad f _ {i} ^ {\prime \prime} (\cdot) \leq 0.\]

The total amount of the public good is given by . This formulation is most relevant in a -emissions context, where agents (countries) are endowed with wealth which can be either consumed, , or used to abate emissions, ; G being the “atmosphere’s quality.” The interpretation then is that is what country i can obtain (consume) by maximizing output when there are no climate-change concerns ( , and ). If the theorem applied there, cost-effectiveness problems in carbon abatement would not be an issue since cost-efficiency would always be achieved through voluntary wealth transfers among countries.

We assume that agents have preferences over that can be represented by a twice-differentiable strictly quasi-concave utility function, . It is assumed that both goods are strictly normal. To avoid dealing with corner solutions on the private-good axis, we shall assume that

\[S _ {i} (x _ {i}, G) \equiv \frac {\frac {\partial U _ {i} (x _ {i} , G)}{\partial x _ {i}}}{\frac {\partial U _ {i} (x _ {i} , G)}{\partial G}} \to \infty\]

I thank Dallas Burtraw, Ian Parry, and two anonymous referees for valuable comments. This terminology is due to Ted Bergstrom.

as

We focus here on the non-cooperative decentralized provision of G. One classic reference for models of private contributions to public goods is Bergstrom, Blume and Varian (1986) (BBV henceforth). The agents in BBV's model all share a linear technology for producing the public good —i.e., in BBV the is an identity function for all i. In the -emissions context, Hoel (1991) has studied cooperative and non-cooperative outcomes in a global emissions game using a model essentially similar to BBV —the public good is provided by a single concave production process. Sandler (1996) has also used this BBV's framework to study various issues related to global carbon emissions. All these models use a common technology for all agents. Buchholz and Konrad (1994) study the strategic choice of (a linear) technology with different emission-reduction costs in a two-stage game. Ihori (1996) investigates the welfare effects of changes in the public-good productivity differentials in an international public-good context that can easily be applied to emissions. Chichilnisky, Heal and Starrett (1993), Chichilnisky and Heal (1994) and Lin and Heal (1994) depart from linear technologies and instead endow each agent (country) with a specific concave technology to study international emission permit markets.

Agents are modelled as participants in a one-shot non-cooperative game. Agent 's problem is given by

\[\max _ {x _ {i}} \quad U _ {i} (x _ {i}, f _ {i} (w _ {i} - x _ {i}) + G _ {\sim i}) \quad \text { s.t. } \quad 0 \leq x _ {i} \leq w _ {i}.\tag{1}\]

where . That is, each agent takes the other agent's contributions to the public good as given and, furthermore, assumes they will be unaffected by their own choices.

A Nash equilibrium (NE) is a consumption vector which solves (1) for all i when is replaced by . The first-order conditions of problem (1) imply that, in a NE, we must have

\[\left[ S _ {i} \left(x _ {i} ^ {e}, G ^ {e}\right) - f _ {i} ^ {\prime} \left(w _ {i} - x _ {i} ^ {e}\right) \right] \left(w _ {i} - x _ {i} ^ {e}\right) = 0,\]

for all ; where ; together with

\[S _ {i} (x _ {i} ^ {e}, G ^ {e}) \geq f _ {i} ^ {\prime} (w _ {i} - x _ {i} ^ {e}),\]

and

\[w _ {i} \geq x _ {i} ^ {e}.\]

In an interior solution, if , we then have .

The existence, uniqueness, and inefficiency of NE follow from proofs similar to those used to establish the corresponding results in the literature of private provision of public goods —see, e.g., BBV. A few words regarding inefficiency. Provided that an agent is contributing towards the public good, the NE will be Pareto inefficient. Nevertheless it is possible to have a Pareto efficient NE where no agent is devoting any resources to the production of the public good. (However, not all NE with for all i are necessarily Pareto efficient.) This inefficiency of the NE is partly just the well-known result on public good underprovision. Here, however, there is an additional source of inefficiency since, at the NE, the marginal costs of producing the public good might differ among agents.

1.1. Konrad-Lommerud's Rotten Spouse Theorem

In the literature of private provision of public goods, Warr-type redistributions are 'small enough' redistributions performed between agents which are in interior solutions to their individual optimization problems (1). Warr-type redistributions lead to Warr-type neutrality results when they do not affect the total provision of the public good and the private consumption vector at equilibrium —see Warr (1982) and BBV. In this model, unless constant in the redistribution range, then Warr-type redistributions involving agents k and l are non-neutral. In general, then, wealth redistributions among agents will lead to different equilibria and to a different provision of the public good. Redistributions towards the most efficient agents in the production of the public good not only will enhance productive efficiency but they also may sometimes lead to Pareto-superior allocations. However, in the particular case where the technologies are linear in a neighbourhood around the NE then there always exist Pareto-improving transfers.

Proposition 1. (Existence of Welfare-Enhancing Redistributions with Linear Technologies) Suppose that in a neighbourhood of when and to the right of zero when . Provided that , if , a small redistribution of wealth from k to l will increase the provision of G, and the welfare of all agents.

Proof. We only need to show that the level of G in the equilibrium after the redistribution must increase (since that implies larger consumption of the private goods for agents in interior solutions and no smaller for agents in corner solutions, and, hence, increased welfare). Suppose . Then, since and G are normal goods, we must have

\[\Delta x _ {i} \leq 0\]

for all agents. By the individual budget constraints, it follows that

\[\Delta y _ {i} \geq 0\]

for all agents except, possibly, country which gives the transfer. Differentiating the sum of and 's resource constraints we get

\[\Delta x _ {k} + \Delta x _ {l} + \Delta y _ {k} + \Delta y _ {l} = 0\]

which (using the inequalities discussed above) implies

\[\Delta y _ {l} = - \left(\Delta x _ {k} + \Delta x _ {l} + \Delta y _ {k}\right) \geq - \Delta y _ {k}.\]

On the other hand, by assumption,

\[\Delta G = \sum_ {i} \theta_ {i} \Delta y _ {i} \leq 0,\]

which implies

\[\Delta y _ {l} \leq - \frac {\theta_ {k}}{\theta_ {l}} \Delta y _ {k} - \sum_ {i \neq k, l} \frac {\theta_ {i}}{\theta_ {l}} \Delta y _ {i} \leq - \frac {\theta_ {k}}{\theta_ {l}} \Delta y _ {k} < - \Delta y _ {k},\]

which establishes the contradiction. Therefore, we must have .

This proposition — which is a straightforward n-agent generalization of Proposition 3 in Konrad and Lommerud (1995)— has great significance. If all the agents increase their welfare by virtue of these transfers, not only will they be tolerated but they will continue until they are not possible anymore. As a result, the marginal costs of producing the public good will be equalized across all agents in interior solutions when they all have linear technologies.

The game agents play changes now as we allow for wealth transfers. In the first stage, agents choose how much to transfer to other agents. In the second stage agents solve the same problem (1) they did before with their wealth, , properly adjusted for the transfers. In a subgame-perfect equilibrium of this game, the marginal costs producing the public good must be the same for all agents which engage in public good provision. If all agents have linear technologies (so that proposition 1 applies everywhere) then the marginal rates of transformation of all agents engaged in the production of the public good must be equalized when agents are allowed to do voluntary transfers of wealth. (In a -emissions context, the marginal costs of carbon abatement would be equalized across countries.)

Unfortunately proposition 1 does not hold for general production functions — i.e., concave technologies. In the next section, we shall provide a counter-example with a strictly concave production function. The reason why proposition 1 does not hold when we replace linear technologies with strictly concave technologies is that the marginal productivity differential can easily be reversed when the agents readjust their production levels. The change in marginal products induces substitution effects which did not exist before — in the Rotten Spouse case we only have paralell displacements of the budget lines. We discuss the role of this substitution effect in the final section.

1.2. A Counter-Example with a Concave Technology

Consider two agents with identical Cobb-Douglas utility functions, . Total wealth is 1 and represents agent one's share. In the contribution game, agent one takes as given and solves, ; while agent two solves a similar problem, , taking, in turn, as given.

Let and . Agent one is more productive than agent two at the margin when . The NE, for all possible distributions of the total wealth, are given by:

\[x _ {1} ^ {*} = \left\{ \begin{array}{l l} \{w _ {1} - (1 - w _ {1}) [ (1 - w _ {1}) - \sqrt {1 + w _ {1} ^ {2}} ] \} / 2 & \text {if} w _ {1} < 3 / 4 \\ \frac {2}{3} w _ {1} & \text {otherwise} \end{array} \right.\]

and

\[x _ {2} ^ {*} = \left\{ \begin{array}{l l} (1 - w _ {1}) / 2 + \frac {\sqrt {2}}{8} \big \{w _ {1} + (1 - w _ {1}) [ (1 - w _ {1}) - \sqrt {1 + w _ {1} ^ {2}} ] \big \} ^ {\frac {1}{2}} & \text {if} w _ {1} < 3 / 4 \\ (1 - w _ {1}) & \text {otherwise} \end{array} \right.\]

The question is whether, at wealth allocations that lead to interior NE where the agents produce the public good with different marginal costs, both agents will improve their welfare by transferring resources from the marginally less productive agent to the more productive one before the one-shot game is played. The answer is “not always”.

Fig. 1. General Technologies: Rotten Spouse Theorem does not hold. Left: Utilities at the Nash equilibria associated with different wealth distributions. Right: Utility Possibilities Frontier (dashed) and Nash-Equilibria utility pairs (solid).

The left panel in figure 1 shows the utility levels at all NE associated with every possible distribution of wealth. There is a region to the left of where both graphs decrease simultaneously which means that wealth redistributions towards agent two will increase both agents' welfare. However, as an example, to the right of the origin, both agents are in interior solutions and agent one is more efficient at the margin so we should see both graphs increasing if a generalized version of proposition 1 held true. However, the utility of agent two goes down as wealth gets transferred to agent one. The reason is that now there is also a substitution effect playing a role when redistributions of wealth take place. The Pareto-Efficient allocations are given by and . The right panel in figure 1 shows the utility-possibility frontier (dashed line) and the utility pairs at all possible NE (solid graph). Pareto-improving wealth redistributions are possible along the positive-sloped portion of the graph.

Fig. 2. (solid) and cost-efficient (dashed) given .

Fig. 2 shows the level of public good provided at different NE (solid) and the amount that could be provided if production took place in an efficient way. Thus, the dashed line is obtained by efficiently transforming into . Note that the level of production inefficiency can be substantial when provision and production are linked together. (The point of tangency corresponds to the NE where and therefore .)

The reader might want to check out the case when and the Rotten Spouse Theorem holds. In that case, both agents would prefer to redistribute wealth from agent one to agent two whenever , which is the range of wealth distributions that lead to interior NE.

In the context, some countries have realized that provision and production need not go hand in hand. Often, an industrial country interested in providing G contracts the production in a less developed country where the production is cheaper. The concept of joint implementation implies joint ventures involving both industrial countries and developing countries: Norway and Mexico (replacement of small electric appliances in Mexico); The Netherlands and Poland and India (substitution of coal by natural gas). Private firms and foundations are also active participants in joint implementation projects: a Netherlands foundation, Forest Absorbing Carbon Emissions (FACE) is paying for sequestration in selected Latin American countries; the New England Electric System has supported reduced impact logging in Malaysia, and Applied Energy Systems has funded agroforestry in Guatemala.

1.3. Piece-Wise Linear Technologies

What about piece-wise linear technologies? The theorem holds when everybody stays away from the kinks. In that case it is always possible to find redistributions which are small enough that lead to NE where the agents stay positioned inside the same line segments. Once any agent moves to another segment with a different slope, there will be a substitution effect playing a role and anything is possible.

To understand the intuition behind the theorem and why it does not work in the previous example, let us consider first the linear case of Konrad and Lommerud (1995)'s paper; refer to fig. 3. Agent one is able to transform 1 unit of the private good in units of the public good, while agent two obtains units; with . At the initial NE, when agent two contributes , agent one chooses , contributing towards the public good which amounts to .

Suppose that prior to playing the contribution game, a transfer of is made from agent two to agent one so that one's initial wealth moves from to . In order to characterize the new NE, it is useful to do the following thought experiment. Assume that, after the transfer, agent two simply reduces his contribution by . Then agent one's new budget line moves from to . If agent one increased his contribution by exactly , two's budget line would be and his choice . However, facing , one's choice would be to the north-east of by the strict normality of both goods. This means that one would increase his contribution by more than two's decrease ---i.e., one would be choosing an amount of public good greater than . The new NE will look like . Note that one's budget line will move further out since agent two will react to one's larger contribution by increasing his own and .

Let us now assume that agent one's technology is piece-wise linear and that is located at a kink: if ; with . Assume again, for a moment, that after the transfer, agent two reduces his contribution by . If , it is possible to prove that the new equilibrium will necessarily have a lower level of , and while agent one could end up better or worse off, agent two will always be worse off.

However, if , agent one's new budget set expands outwards along all its borders and Pareto-improving transfers are possible. The crucial difference relative to the Rotten Spouse case is that now, if agent one wanted to provide , his marginal product has decreased from to . This happens because he needs to use his technology more intensively in order to offset the decrease in agent two's contribution and therefore arrives to his less productive region earlier. This change in 'relative prices' that agent one would now be facing introduces a substitution effect that was not present in the Rotten Spouse case. This substitution effect goes against the income effect due to the transfer. Tastes and technology parameters will determine whether the transfer is Pareto improving —or, similarly, whether the final effect on the provision of G is positive or negative.

If then d will be aligned with ab, and will also be the NE after the redistribution. Both agents would be consuming the same amounts of the two goods before and after the transfer. This is Warr's neutrality result.

2. Concluding Remarks

We have examined the extent and limitations of Konrad and Lommerud (1995)'s Rotten Spouse Theorem in an n-player non-cooperative public-good provision game. While the theorem generalizes to n agents, it fails to hold for general technologies for producing the public good. When the theorem holds, the production of the public good is achieved in the most efficient way. In a emissions model, this means that the costs of abatement are equalized across countries. In this context, it is possible that the linear assumption is a reasonable approximation for the relevant range of projects being considered.

References

1. Bergstrom, T., L. Blume and H. Varian (1986): “On the Private Provision of Public Goods,” Journal of Public Economics, 29, 25–49.

References

1. Buchholz, W. and K.K. Konrad (1994): “Global Environmental Problems and the Strategic Choice of Technology,” Journal of Economics, 60:3, 299–321.

References

1. Chichilnisky, G. and G. Heal (1994): “Who should abate carbon emissions? An international viewpoint,” Economics Letters, 44:4, 443–450.

References

1. Chichilnisky, G., G.M. Heal and D.A. Starrett (1993): “International emission permits: equity and efficiency,” mimeo, Stanford University.

References

1. Hoel, M. (1991): “Global Environmental Problems: The Effects of Unilateral Actions Taken by One Country,” Journal of Environmental Economics and Management, 20:1, 55–70.

References

1. Ihori, T. (1996): “International public goods and contribution productivity differentials,” Journal of Public Economics, 61:1, 139–154.

References

1. Konrad, K. and K. Lommerud (1995): “Family Policy with Non-cooperative Families,” Scandinavian Journal of Economics, 97:4, 581–601.

References

1. Lin, Y. and G.M. Heal (1994): “Equilibrium and Efficiency: International Emission Permit Markets,” mimeo, Columbia University.

References

1. Sandler, T. (1996): “A Game-Theoretic Analysis of Carbon Emissions,” in The Political Economy of Environmental Protection: Analysis and Evidence, ed. R. Congelton. Ann Arbor: University of Michigan Press.

References

1. Warr, P.G. (1982): “Pareto Optimal Redistribution and Private Charity,” Economics Letters, 19, 131–138.

3. Appendix

Problem (1) is more commonly stated as:

\[\begin{array}{r l} \max _ {x _ {i}, y _ {i}} & U _ {i} (x _ {i}, a _ {i} + G _ {\sim i}) \\ \mathrm{s.t.} & x _ {i} + y _ {i} = w _ {i} \\ & y _ {i} = f _ {i} (y _ {i}) \\ & x _ {i}, y _ {i} \geq 0 \end{array}\]

As a benchmark, Pareto efficient allocations, , with , must satisfy, for all ,

\[\left(\sum_ {i} \frac {1}{S _ {i} \left(x _ {i} ^ {*} , G ^ {*}\right)} - \frac {1}{f _ {i} ^ {\prime} \left(y _ {i} ^ {*}\right)}\right) y _ {i} ^ {*} = 0\]

\[\sum_ {i} \frac {1}{S _ {i} (x _ {i} ^ {*} , G ^ {*})} - \frac {1}{f _ {i} ^ {\prime} (y _ {i} ^ {*})} \geq 0, \quad \mathrm{and} \quad y _ {i} ^ {*} \geq 0\]

along with the aggregate feasibility constraint . Note that if we obtain the familiar Samuelson condition, , which, as expected, mandates the equalization of the marginal rates of transformation across individuals — i.e., for all , when and .

3.1. Some Results

Proposition 2. (Existence) A NE exists.

Proof. Given an initial distribution of wealth, , the set is compact and convex. By Berger's Maximum theorem, the solution to (1), , is a continuous function from X to (remember that ; i.e., for any x in X we can obtain ). We can then construct a continuous function from X onto itself by just assigning to each coordinate of x. By Brouwer's theorem there must exist a fixed point, which corresponds to a Nash equilibrium.

Proposition 3. (Uniqueness) There will be a unique NE associated with each initial distribution of wealth, ; with a unique quantity of G and a unique set of contributing countries.

Proof. Let us examine first the possibility of two different equilibria with the same amount of . The only way that this could happen is with at least some country dedicating less resources to abatement activities and some other country contributing more. But since the country dedicating less resources would be able to afford a higher consumption of the private good and since both goods are strictly normal for all countries, its demand for A would be bigger. Therefore we cannot have two equilibria with the same level of abatement activities.

Let us assume that we have two different equilibria with two different levels of : . At least one country must be allocating more resources to the abatement activities in the second equilibrium. However, by the strict normality of both demands, every country's demand for will be higher in the second equilibrium. The individual budget constraint prevents this two things from occurring simultaneously.

Proposition 4. (Inefficiency) If for some —i.e., if at least some agent is dedicating some resources to the production of the public good— then the NE is not Pareto Efficient.

Proof. By the FOC to (1) we have . While, since Pareto Efficiency implies . Which establishes the contradiction.

Proposition 5. (Redistributions are non-neutral) Unless constant in the redistribution range, then Warr-type redistributions involving agents k and l are non-neutral.

Proof. Since both agents must be in interior solutions, we must have for before and after the redistribution. A Warr-type results requires that (i) ; and (ii) and . Conditions (i) and (ii) can only be satisfied when along the pertinent range.

3.2. An Example with Linear Technologies

We shall have two agents with identical Cobb-Douglas utility functions, . Total wealth is 1 and represents agent one's share. Since in this example the utility functions are of the form , the optimal level of G is going to be independent of the distribution of the private good—see Bergstrom and Cornes (1981). To figure out we can simply give all the wealth to any one of the agents and solve her optimization theorem (1).

As in the counter-example in the paper; in the contribution game, agent one takes as given and solves,

\[\max _ {x _ {1}} \quad x _ {1} (f _ {1} (w _ {1} - x _ {1}) + g _ {2});\]

while agent two solves a similar problem,

\[\max _ {x _ {2}} \quad x _ {2} \left(g _ {1} + f _ {2} \left(\left(1 - w _ {1}\right) - x _ {2}\right)\right),\]

taking, in turn, as given.

We shall examine an example with linear technologies first to see the Rotten Spouse Theorem at work.

Let and . The Nash equilibria, for all possible distributions of the total wealth, are given by:

\[x _ {1} ^ {*} = \left\{ \begin{array}{l l} w _ {1} & \text {if} w _ {1} < 1 / 2 \\ (2 - w _ {1}) / 3 & \text {if} w _ {1} \geq 1 / 2 \text {and} w _ {1} \leq 4 / 5 \\ w _ {1} / 2 & \text {if} w _ {1} > 4 / 5 \end{array} \right.\]

and

\[x _ {2} ^ {*} = \left\{ \begin{array}{l l} (1 - w _ {1}) & \text {if} w _ {1} > 4 / 5 \\ (2 - w _ {1}) / 6 & \text {if} w _ {1} \geq 1 / 2 \text {and} w _ {1} \leq 4 / 5 \\ (1 - w _ {1}) / 2 & \text {if} w _ {1} < 1 / 2 \end{array} \right.\]

Fig. 4. General Technologies: Provision of G at different NE.

The left panel in figure 5 shows the agents' utility levels at different Nash equilibria associated with all possible distributions of wealth. Note that both agents would prefer to redistribute wealth from agent one to agent two whenever . This is an illustration of the remarkable general result due to Konrad and Lommerud (1995), proposition 3 (our proposition 1).

The Pareto-Efficient allocations are given by and , . The right panel in figure 5 shows the utility-possibility frontier (dashed line) and the utility pairs at all possible Nash equilibria (solid graph).

Fig. 5. Linear Technologies: Rotten Spouse Theorem holds. Left: Utilities at the Nash equilibria associated with different wealth distributions.

Right: Utility Possibilities Frontier (dashed) and Nash-Equilibrium utility pairs (solid).

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