Two phenomena are widespread in politics: policies with externalities are often required to be harmonized, and side payments are often not allowed. That is, two EU countries are forced to implement the same environmental regulation, without regard to local preferences. Also, one country or state is not allowed to request a payment from the other agent to go along with the scheme. More generally, you can think of “side payments” as a form of horse-trading: I will support environmental policy X which is optimal for you if you support trade policy Y which is optimal for me. Negotiation rules often limit the amount of such horse-trading. At first glance, these phenomena seem suboptimally restrictive. They also seem to have little in common with each other.
This 2007 AER by Harstad explains both phenomena. Consider a bargaining game, where each of two states chooses to buy a certain amount of the public good. The public good is not necessarily pure, but rather I get a percentage x weakly above 50 of the public good that I buy, and I get 100-x of the public good that you guy. Utility from the public good is linear in the amount of the good multiplied by (privately known) bounded coefficients v(i). One unit of the public good costs 1. The total amount of public good is assumed to be capped at 1, and the coefficients v(i) are high enough that in the social optimum, each agent prefers a total amount of the public good equal to 1. Bargaining occurs over who pays for it. We consider the cases both with and without side payments.
At time 0, agent 1 makes an offer specifying what percentage of the public good should be bought by him, and what percentage by agent 2. Both agents discount at common discount factor delta. At any time after time 0, agent 2 can accept or reject the offer. If she rejects, then she proposes a new split. Following that second offer, agent 1 can that, after any delay of his choosing, reject or accept, and make a new offer if he rejects. This continues until an offer is accepted. The equilibrium concept is sequential equilibrium satisfying the Cho-Kreps intuitive criterion. (A technical note: though time is continuous, an assumption is made that offers can only be made at discrete times with arbitrarily small intervals between them is necessary for the sequential equilibrium concept to have meaning. Subgames in continuous time without such an assumption are often a giant mathematical mess.)
There is a unique equilibrium. If agent 1 has the highest value possible for the public good, he proposes at time 0 an equal split. This is accepted immediately by agent 2 if agent 2 is also of the highest type. Otherwise, agent 2 will delay a sufficiently long time, and then propose a split where agent 1 pays a greater share of the good. This is credible because since agent 2 discounts time, she will only delay if she in fact has a lower value for the public good than agent 1. Likewise, if agent 1 has the lowest possible type, he will propose that agent 2 pay for everything after waiting a sufficiently long time after time 0 to make the offer. After the offer is made, an agent 2 of the highest possible type will accept immediately, and otherwise agent 2 will delay long enough that her proposal for a “more just split” is seen as credible. In any case, the final split is precisely what would be achieved in the unique equilibrium if there was no imperfect information; in some sense, this means that no one will want to renegotiate. But note that with perfect information, we would agree on the split immediately and would not have any delay, so welfare would be higher.
What if the law required harmonized policy where each agent contributes equally? In that case, agent 1 proposes and agent 2 accepts right away. There is no point in delaying (if we are using the intuitive criterion) since when the final agreement is reached, each agent will pay for half of the public good anyway. The tradeoff then is less delay in exchange for less payment by the agent who values the public good more, and therefore would prefer more parks, for instance, in his town than in the neighboring town. This tradeoff argues in favor of harmonization when preferences are fairly similar (the range of v(i) is small) and spillovers are large.
What about when side payments are allowed? If we require harmonized policy, the only reason to delay with side payments is for the low value agents to extract money from the high value agents, since the loss from delay is higher for the high-value agents. If policy does not need to be harmonized, the high value type in equilibrium will provide more of the public good, but depending on parameter values, he may either pay the low value type a side payment (to agree to a split more quickly) or get paid by the low value type (in exchange for paying for more of the public good in a classic gains-from-trade scenario). In either case, agreement is reached quicker and the distribution of who buys public goods is more efficient than in the case with side payments alone. That is, legality of side payments and unharmonized policy are in some sense complementary goods! Again, if preferences are fairly similar and spillovers are large, we are better off requiring both no side payments and harmonization.
One final comparison: what about allowing differentiated policies but not allowing side payments? We noted above when differentiated policies alone are better or worse than harmonization with no side payments. How, though, does differentiation alone compare to allowing differentiation and side payments? If the possible “gains from trade” are large, then side payments allow those gains to be more efficiently and more quickly reached. If the possible gains from trade are small, then side payments are allow low value types to extract rents through delay, and therefore side payments should not be allowed.
You may wonder how robust these results are to the particular bargaining game chosen. It turns out they are in a sense very robust. In particular, consider any mechanism mapping revealed types into required public good purchases (with or without side payments). The equilibria solved for above implement the outcomes of the most efficient dominant strategy mechanisms that are “fair” in the sense that no one wants to renege ex-post.
http://www.kellogg.northwestern.edu/faculty/harstad/htm/harmonization.pdf (Final WP – final version published in AER 2007)
A Fine Theorem 