Please read chapter 4 the answer the question 10:

It may seem puzzling that a negative externality may induce a strategic complementarity. But ments over the Nash equilibrium. It is not surprising that Lower has benefitted by being first distributive use of power in economic relationships. think of the phenomenon of conspicuous consumption, first analyzed by Thorsten Veblen (1934 mover; but it is a bit counterintuitive that the Stackelberg follower is better off than in the symmetric Nash equilibrium. The reason is that in the presence of strategic complementarity, the The second important feature of this case is that there is no guarantee that the Nash equilibrium [1899]) over a century ago. The other's luxury consumption not only makes the individual feel less well-off (uA 0, UaA > 0). The result may be a kind of a consumption arms race. Other common increase in the action that will benefit both players. In this case, the self-interested exer- with B > 0, and analogously for Upper. Given this dynamic, figure 4.7 illustrates a stable Nash examples include literal arms races: one country's increased arms reduce the security of the other, and may raise the marginal utility of that country's armaments, thereby inducing a positive cise of power by one player is mutually beneficial. (You may wish to return to the example of equilibrium. But the fact that the best response functions have slopes of the same sign could have produced additional intersections (that is, multiple Nash equilibria). In this case, we could Pareto response. Biology provides many examples of such arms races, with competition for mates leading the fishers and be sure you understand why the exercise of first-mover advantage by one fisher did not benefit the other: the difference arises because in the fishers' activities were strategic sub- rank the stable Nash equilibria (U and u are increasing along the best response functions, and to such otherwise dysfunctional features as peacocks' elaborate tails. Another example of negative both are upward sloping). externalities and complementary strategies are corrupt practices: one's corrupt activities reduces stitutes.) Of course there is no reason in the model why Upper could not have been the others' well-being but may increase the marginal benefit to them of also engaging in corrupt Stackelberg leader (the game is symmetric). In cases like this, the outcome is indeterminate and the An interesting line of inquiry - one inspired by invisible hand reasoning with respect to practices. In these cases the effect of the others' action on the level of one's utility is of opposite model needs to be supplemented by information about military, geopolitical, or other asymmetries institutions discussed in chapter 2 - would be to ask if we have any reason to expect that a sign than the effect on the marginal returns to ones own action. among the nations that may influence their power to make the binding commitments required of system modeled in this way, if perturbed by stochastic influences, would spend most of its time in a first mover. a state near the Pareto-superior high tax equilibrium. The problem is similar to the cases of Positive externalities with strategic substitutes is the converse case. Consider team production multiple equilibria with discrete rather than continuous strategies already encountered in assurance with an equal sharing contract as above, but assume (more realistically than above) that each games (e.g., planting in Palanpur in chapter 1). Without knowing the recent history of the inter- individual's marginal utility of goods is declining in the amount of goods consumed. In this case, Lower's actions and the details of how the players change their strategies when out of equilibrium, one the externality is positive (I benefit from your action, as we both get 1 of the result). But my tax rate, a cannot say much about the likely state of the system. But it seems likely that risk-dominant diminishing marginal utility of goods induces me to reduce my effort when you increase yours equilibria would be more persistent than payoff dominant equilibria, should both exist. We will (your and my effort are strategic substitutes). return to this question in the closing chapters. A final example illustrating positive externalities and complementary strategies is fiscal competition among nations or jurisdictions within nations. Consider two nations in both of which the government (considered as an individual) seeks to maximize a weighted sum of employment CONCLUSION and the level of government expenditure that is financed by a linear tax on profits at the rate a and A. Because firms relocate among nations in response to after-tax profit rate differentials, the level of employment in one of the countries is determined by its own tax rate and the other Any solution to a coordination problem implements not only an allocationtcome - how much fishing each will do, the tax rates of the various countries, and so on - but a distributive outcome country's tax rates. Employment declines in the own-country tax rate and increases in the other as well, the level of well-being for each of the players implied by the allocationtcome and country' tax rates: thus the external effect is positive. If it is also true that the negative responsive whatever redistributive measures are part of the solution (such as the purchase of the fishing of employment to the own-country tax rate is greater, the lower is other countries' tax rates, then permit in the privatization case). The distribution of the benefits of cooperation, should the two countries' tax rates are strategic complements. (Working problem 12 will clarify this cooperation occur, depends on the particular transformation of the game which makes coop- case.) eration possible. An implication is that conflicts may arise about how best to address the For a two-country world (Upper and Lower) the two best response functions are as shown in coordination problems that people face: some participants may prefer a less efficient solution to figure 4.7, with their intersection, labeled N, the Nash equilibrium, and the level of utility of each the allocational problem because it supports a distribution of the benefits of cooperation that nation given by the indifference loci, labeled UN and uN. Preferred indifference loci for Upper are favors them. those above UN (because Upper benefits when Lower's tax rate is higher), and Lower's preferred As a result (and for other reasons as well), differences among the players - in wealth, skills, indifference loci are to the right of un. It can be seen at once that there exists a Pareto-improving Upper's tax rate, A political rights, group identity, information - will influence both the nature of the coordination lens of mutually beneficial higher tax rates defined by the tax rates above UN and to the right of problem and the types of solution that may be implemented. In his classic treatment of collective UN. The proof that this lens exists is identical to the proof that the Nash equilibrium in the Figure 4.7 Fiscal competition: Nash and Stackelberg equilibria. Note: Lower is action problems, Mancur Olson (1965) reasoned that small, highly unequal groups would most fishers case is Pareto inefficient. But here, Pareto improvements require increases in the actions the Stackelberg leader. readily solve these problems. It is easy to see, for example, that if there were decreasing marginal taken by the two agents rather than reductions as was the case with the fishers. The reason is returns to the aggregate level of fishing and one of the fishers had a much larger net than the that the externality is positive, so the countries' actions (taxes) are sub-optimal at the Nash others and so could be assured of catching most of the fish, then his best response would equilibrium. Notice two things about this case. The fact that the first-mover advantage may benefit the second mover (by comparison to the Nash equilibrium of a simultaneous moves game) is a reminder that the exercise of power has approximate the allocation of a single owner of the lake. In this case inequality in wealth among First, were Lower to be in a position to act as first mover, it would of course benefit. But the fishers would attenuate the coordination failure. Similarly, if one among the nations were Upper would also be better off as a result. To see this, recall that in selecting its tax rate, Lower both allocation and distributive effects. In this case, making the first move and the ability to com- would not, as in the Nash case, take Upper's tax rate as exogenous but would take account of mit to it is not just redistributive, it is also productive: power is used to get a larger slice of the much larger than the others, and powerful enough to commit to a given tax rate, it could, as first mover, implement a Pareto improvement over the Nash equilibrium in the simultaneous moves the impact of its choice of a tax rate on Upper's best response. Thus country Lower would vary pie, but its exercise also enlarges the pie. Thus, even when power is exercised in a self-interested way, it may be mutually beneficial. The idea is not new. Thomas Hobbes (1968 [1651]) used it game. to maximize u(a, A) subject to A = A(a). This optimum problem gives us the Stackelberg But inequality may also be an impediment to cooperation. Had the production team members equilibrium (with Lower the leader) labeled S. Notice that S is within the lens of Pareto improve- three-and-a-half centuries ago to justify allocating executive powers to a sovereign ruler, for reasons explained in the epigraph. In chapter 10, I will return to the productive as well as modeled above been of different ethnic groups, or of vastly differing wealth, the altruism andmaximize the Nash equilibrium must be a mutual best response. The Nash equilibrium value of e can thus backward only one period (using only this period's information to determine what to do next u = a(1 - BE)e - e2. be calculated by substituting Upper's best response function into Lower's best response function period), and it does not look forward at all (assuming that the other's action will not change and solving for e, as is illustrated in figure 4.3. Because of the (assumed) symmetry of the between this period and next). It amounts to the following rule: next period, move in the problem, we have, for both Lower and Upper: direction of the action that would have been optimal this period. Letting e' and E' be the fishers Differentiating u with respect to e and setting the result equal to zero to find the optimal level of eN = 7 = EN (4.8) action next period, this rule of thumb gives us effort gives us the first order condition 2 + QB He = a(1 - BE) - 2e = 0, Fishing time by Upper, E Fishing time which clearly requires Lower to equate the marginal (utility) productivity of her labor (the first by Upper, E e "(E) e" ( E) term) with the marginal disutility of her effort (the second term), as is illustrated in figure 4.1. This first order condition gives us a simple closed form best response function: e = a(1 - BE) 2 (4.7) e(e, E) = 0 or/2 E The best-response function for Upper is derived in the same way. EN There is another way to represent the best response function that will be illuminating for what E"( e ) follows. Using the utility functions above we can write Lower's utility function as a function of (e, E) her and Upper's effort levels: v = v(e, E) V(e, E) V = V(e, E) e" ( E) eN a/2 Fishing time by Lower, e Fishing time by Lower, e Presented in (e,E) space, as in figure 4.2, these functions describe familiar indifference loci (only Figure 4.2 Lower's best response function, e* (E). Figure 4.3 Out-of-equilibrium dynamics and a stable Nash equilibrium. Note: Lower's are presented), and by setting the arrows indicate the response to disequilibrium of the two fishers (horizontal movement for Lower, vertical for Upper.) The point z is the Nash equilibrium du = vede + VEdE = 0 we see that What do these values tell us? Without knowing the institutional structure of the interaction between the fishers we have no way of saying what their levels of fishing will be: these Nash Ae = e' - e = y(e* - e) AE = E' - E = [(E* - E) dE equilibrium values might be irrelevant if one of the fishers is the first mover, for example. But it de UE might be an unlikely outcome for an even simpler reason: this Nash equilibrium might be unstable. Disequilibrium Dynamics and Stability. Stability requires that small perturbations of the where y and I are both positive fractions E (0,1] reflecting the speed of adjustment (how much of the gap between desired and actual level of fishing this period is closed by the choice of next Thus, we know that the slopes of the indifference loci (for Lower) are -vdug, and analogously equilibrium values be self-correcting. To see if this is true we need to know something about the period's level of fishing). Of course the speed of adjustment might differ between the two fishers for Upper. The thought experiment that gives the best response function is to hold constant some out-of-equilibrium behavior of the fishers: what do they do when they are not at a Nash (Lower might be a creature of habit with y close to zero, and Upper a lightening responder like level of Upper's fishing time and ask how much fishing Lower would do under these equilibrium? It is sometimes illuminating to think of the figure as a topographical map with e" = Homo economicus with I = 1). The dynamics of the system expressed by these equations say that circumstances. In figure 4.2 this is represented by treating the horizontal dotted line at E (an e*(E) describing a ridge. Lower's optimizing process is a hill-climbing algorithm: for e * e* each moves towards her or his best response function, as indicated by the arrows in figure 4.3. arbitrarily selected level of Upper's effort) as a constraint, and letting Lower maximize her utility, Lower's first order conditions are not satisfied, and for e 2e, or the marginal benefit of fishing exceeds the marginal (disutility) cost of fishing, so But perhaps surprisingly, the fact that each fisher moves towards his or her best response The slope of the constraint is zero, so the optimum requires that the slope of Lower's indifference Lower will choose to fish more. function is not sufficient to insure stability of the Nash equilibrium outcome defined by their locus be zero as well, and this requires that ve = 0, as we saw above. intersection. To see why this is so, suppose that the best response functions were such that if I write Lower's best response function as e" = e*(E), the asterisk indicating a solution to an The out-of-equilibrium dynamics of the system are modeled as follows: consistent with the idea Upper fished one more hour, Lower would fish two fewer hours (de"/dE = -2), and conversely; that people have limited cognitive capacities, we assume that the fishers use a rule of thumb: at and imagine that the two are currently fishing at the Nash equilibrium values. Figure 4.4 gives optimum problem. The representation of e*(E) in figure 4.2 is the locus of points for which ve = the end of this period, change one's behavior in the direction of what would have been optimal 0 and at which Lower would therefore have no incentive to change what she did. We know that given what the other individual did this period. This is shortsighted in both directions: it looks the out-of-equilibrium dynamics: the Nash equilibrium is a saddle, and a perturbation of the Nash values is not self-correcting.twenty-seven. The rules governing the access to the resource for those not excluded were equally common property resources, were a single ownership unit to exit, it could easily be sufficiently diverse (as the competing allocation rules proposed by the northwest coast fishers mentioned in large to preclude effective competition on the relevant markets, thereby inducing familiar market the introduction suggests). The observed rules governing membership, allocation, and other aspects Note how these differ from the first order conditions defining the individual best responses in the noncooperative interaction above: they are identical except for the last term, which captures the failures associated with the exercise of market power. In this case a government or some other of commons governance in combination generate literally thousands of hypothetical commons external party may be able to improve on the Nash equilibrium of the noncooperative game Q governance institutions. Many hundreds are observed in practice. effect of Lower's fishing on Upper's well being (in the first equation) and conversely (in the second). Solving for the level of fishing of each, we have: described above. Privatization. Suppose one of the fishers, Lower, say, owned the lake and as owner could As with privatization, two alternatives suggest themselves. First, the planner (the government), exclude Upper or could regulate the amount that Upper fished. In this case, Lower will maximize e= 2+ 2QB = E (4.11) knowing all the relevant information, could select e and E to maximize total surplus. The planner her utility by varying both e and E. Assume that Upper's options are such that his utility is zero in the next best alternative. An obvious constraint on Lower's optimization problem is the might then implement this outcome by direct regulation, simply issuing a fishing permit allowing requirement that if Upper is to do any fishing at all, Upper must receive at least as much as his each fisher to fish a given number of hours. Thus point w in figure 4.5 is the planner's optimal allocation. Assuming the planner had no reason to favor one fisher over the other from the next best alternative. This restriction is termed Upper's participation constraint (if it is violated, which is obviously less than the Nash equilibrium level (a/(2 + aB), from eq. (4.8)) for the noncooperative interaction modeled in the previous section. Notice that as B goes to zero, standpoint of distribution, w would be both the allocation and the distributional plan. Notice the Upper will not participate; if it is satisfied even weakly (as an equality) we assume that Upper eliminating the overfishing interdependence, the Nash equilibrium becomes the joint surplus maxi- same point represents the allocational outcome (but not the distributional outcome) for the participates). I will consider below why it is not optimal for Lower to exclude Upper from privatization case. fishing entirely. mizing solution, as one would expect. The joint surplus maximizing allocation is indicated by Two types of interaction among the fishers might take place under privatization. Lower might point w in figure 4.5. Rather than implementing the optimal allocational plan by fiat, however, the planner might The optimal allocation plan is based on the assumption that the participation constraint had to desire to let the fishers each decide how much to fish but to alter the incentives facing them in issue a permit allowing Upper to continue to fish independently but to catch not more than a given number of fish, requiring Upper to pay for the permit a sum that does not violate the be met. But why would it not be optimal for Lower to simply select E = 0 and have exclusive such a way as to avert the coordination failure that occurs without government intervention. This access to the lake? The reason (in this case) is that the marginal cost of compensating Upper's is the approach to welfare economics pioneered by early twentieth century economists Alfred participation constraint. Alternately, Lower might offer Upper an employment contract under which Upper would fish under Lower's direction and the fish caught by Upper would be Lower's fishing effort goes to zero as E goes to zero, so some positive level of E will be optimal. Marshall and A. C. Pigou (1877-1959); the modern form of this approach is implementation property, Upper's compensation being a wage (paid in the fish caught by the two of them) (Alternative reasonable specifications of the model would have Lower exclude Upper from theory, mentioned in chapter 1. According to this approach, the planner proposes a tax on fishing sufficient to offset the disutility of Upper's labor (and thus to satisfy the participation constraint). fishing - for example, if Upper had a very advantageous next-best alternative, making it expensive designed to eliminate the discrepancy between the social and private marginal costs and benefits In the permit case, Lower determines both optimal levels of fishing effort (e" and E-) and then for Lower to satisfy his participation constraint. ) of fishing. Assume that the proceeds will be given back to the fishers as a lump sum, and that issues Upper a permit to fish at level E" in return for Upper paying a permit price of F. To take Instead of issuing a permit, Lower might have employed Upper. This case differs because they ignore this lump sum in their calculations (as they would were there two thousand rather Lower now owns the fish that Upper catches but must devote some of this fish to paying a wage than just two fishers, as in a more realistic case.) The problem is thus for the planner to select a account of the participation constraint, we express Lower's offer to Upper as the solution of a W to Upper sufficient to satisfy Upper's participation constraint. Knowing that the participation tax that will maximize the sum of the fishers utilities when the fishers choose how much to fish, standard constrained maximization problem, namely, to vary e and E to maximize constraint is satisfied as an equality allows us to use the fact that the wage paid must just offset given the tax. w = a(1 - BE)e - e2 + F subject to a(1 - Be)E - E2 2 F Uppers disutility of effort or W = E. Lower now must choose e and E to maximize the What is the optimal tax? The problem can be posed this way: find the tax that would expression transform the objective functions of the two fishers so that their individual best response functions would be identical to those implied by the first order conditions of the joint surplus We know that satisfying Upper's participation constraint will be costly to Lower (the two are not a(1 - BE)e - ez + a(1 - Be)E - W, maximum problem, namely, satiated, nor do they love work so much that providing for the other is costless), so the constraint e = a(1 - 2BE) will be satisfied as an equality. We can use this expression to eliminate F from the above 2 expression. Thus Lower should select e and E to maximize which (substituting in the value of W given by the participation constraint) is identical to the problem solved in the permit case. The basic structures of the permit and the employment cases w = a(1 - BE)e - ez + a(1 - Be)E - E2 are thus indistinguishable: because in both cases Upper will gain only an amount equal to the E = a(1 - 2Be) disutility of labor, Lower chooses e and E to maximize the joint surplus, recompenses Upper for 2 the disutility of Upper's labor, and keeps the rest. Note that this is just the joint surplus (total catch minus the total disutility of labor). The solution Privatization produces Pareto-efficient outcomes because the decision maker optimizes subject to to this problem (e" and E ) is Lower's allocation plan, which is implemented along with a the other's binding participation constraint. The utility gained by the other is simply given by his Working backward from the desired first order conditions to the implied individual payoffs and distribution plan that requires Upper to pay a fee of F - = a(1 - Be)E- Ez for permission to next best alternative, so the question of distribution between the two is settled in advance. As a hence the tax rate, we see that the transformed utility function a would have to have the form fish E- hours. Because the participation constraint is satisfied as an equality, the solution will be result, the owner -as residual claimant on the joint surplus - maximizes her utility by choosing (for Lower) Pareto efficient (it is one of the points on the efficient contract locus). an allocation which maximizes the total utility of the two. The key here is that the owner is u" = a(1 - BE)e - e2 - Te Lower's allocation plan is determined by setting e and E according to the first order conditions: powerful enough to determine the distribution of gains independently of the allocation of fishing we = a(1 - BE) - 2e - aBE = 0 times and so has no incentive to adopt any but the most efficient allocation. In chapter 5 I will WE = a(1 - Be) - 2E - aBe = 0 show that this is not generally the case and that when the independence of distribution and allocation fails, private allocations tend to be inefficient. and that if Lower's first order condition is to mimic that implied by joint surplus maximization, External Regulation. It is often impossible for a single party to own an entire common property namely, resource (imagine establishing property rights in fish in the open ocean). And for many suchteam nature of production; groups of producers - often employees of a given firm, sometimes are thus a constraint - called the incentive compatibility constraint-on the team's optimizing numbering in the hundreds-contribute to production and share in the resulting output. The team problem. Of course, the contract must give the team members a level of utility not less than their team members, independently maximizing their utility, will choose the action according to Crusoe's might also be a group of professionals sharing a practice (common among doctors and lawyers) or fallback position, thereby satisfying their participation constraint. The team as a whole has the first order condition, namely, wyg + u. = 0, thereby mimicking Robinson Crusoe and surmounting the 1 problem. This contract implements the efficient outcome because it induces each member a cooperative firm owned by its workers. role of first mover (and is also the principal in a single-principal multi-agent problem of the type to take account of her entire (marginal) contribution to production (rather than just one nith of Q Suppose members of a team of n members jointly produce a good, the level of output analyzed at length in chapter 8). it). Arrangements like this, which implement Pareto-optimal allocations, are termed optimal depending on an action (call it "work effort") taken by each of the n members, a; e [0,1], according to the production function Suppose the members consider a proposal that shares net income equally, offering each member contracts. a per period income of Pleased with his clever idea, the inventor of the optimal contract is sure that his teammates q = ga - k (4.12) y = 9- x will endorse it. But they do not. To see why, introduce some real world risk to the problem. Let n output now be q = {ga - k}(1 + =) where a = Zai, summed over the n team members and g and k are positive constants (known to the team members). As team members are identical, I will drop the subscripts, except where they where x 2 0 is whatever amount of income the team decides to allocate to common projects, and are necessary to avoid ambiguity. There are evidently no inputs other than the actions of the team is selected to satisfy the team members participation constraint or members (maybe this is a dance company that performs in public places). The identical utility where & is a stochastic influence on production (with zero mean and variance o known to the functions of each of the producers are u = u(y, a), where y is the income of the worker and u is team members). Were & observable (and verifiable) then the previous contract written in terms of decreasing and convex in a and increasing and concave in y. Team members' reservation utility is expected rather than realized output could be implemented as long as the firm could borrow Z . when necessary to allow the required payments of ga - k - v to each member. But if & is nonverifiable, then the contract would necessarily be written in terms of actual output. Suppose The members of the team seek to devise a method of allocating the income produced by the The asterisks indicate the equilibrium levels of team member effort and resulting output under the the optimal contract ensured that team members received an expected income sufficient to satisfy team among its members, recognizing that members may seek to free ride on the efforts of their contract. How would this work? A given member's optimizing problem is to vary a; to maximize their participation constraint. Given the stochastic nature of output, however, for teams of any teammates. To provide an efficiency benchmark, the team members engage in a thought significant size each member's realized income in any period could be a large multiple of that experiment, dusting off the ever-useful Robinson Crusoe, who as a social isolate does not have to worry about coordination failures. They know that if production could be carried out by a single u , = 3 g( @1 + ...+ a,,) -x) figure of either sign. This is because each member is residual claimant on the entire team's realized output, and shocks to total output would realistically dwarf any individual's reservation producer who also owned the resulting output, the producer-owner would select a level of effort position. A contract under which a team member would be required in some periods to pay the to maximize utility, giving the first order conditions team a substantial amount is not likely to be attractive for any but risk-neutral members or those Myg + Wa = 0 (4.13) with virtually unlimited access to credit. As a result, for all but very wealthy team members or (Here I have retained the subscript i for the member in question, as it is essential to remember very profitable teams, no contract of this type could satisfy the participation constraint. that while the members are - for analytical convenience-assumed to be identical, each acts The members try another approach: peer monitoring. While the actions taken by each are not or g = -ualuy, equating the marginal productivity of the action to the marginal rate of independently and takes the actions of the others as exogenous when making her own decision.) verifiable, each member has some information about what his or her teammates are doing and substitution between effort and goods in the producer's utility function. The team members then Setting du;/ da; = 0, we have the first order condition: could use this information to implement an agreed-upon level of effort, through the use of seek to implement the allocation (the level of the a's) implied by this first order condition for each y8 +u = 0 informal sanctions such as social disapproval or perhaps even fines imposed by the members on member. They first consider disbanding the team so that each may work alone as Robinson did. n those who contribute less than the stipulated amount. It might seem at first glance that if it is But there is a reason why the team exists: I assume that due to the fixed costs k, the level of costly (either materially or psychologically) to the members to sanction one another, they would effort implementing the above first order condition, a*, is such that u(ga* - k, a*) = 0.3, y = 0.6, 6 = 0.6, and c = 0.75. Then a = 0.5; that is, members implement contributing. The term -1 + 4(1 + B;;) gives the marginal cost of contributing and the marginal the common contribution norms, and as a result, they experience no shame or guilt and do not from i's contribution norm: if j has contributed to their joint project more than i's norm, and increment both to one's own material payoffs, and to the other as well, the latter valued by i's punish one another. As a result, both gain 0.1 in material benefits net of their contribution from > 0, then i experiences good will toward j and positively values his payoffs. But if j and the project (that is, 0.6(0.5 + 0.5) - 0.5). contributed less than a," then i may experience malevolente toward / (B;; ; in the denominator of the expression in chapter 3.) I do not include in i's in shame occasioned both by more closely approximationg one's norm and by invoking less equilibrium they do not experience shame, guilt, or benevolence toward the other does not imply valuation of j's payoffs, the costs to , of punishing i, because it seems implausible that i will punishment. Recalling that Bij = a; + A,(a; - a,"), for A; > 0 total differentiation of the first order increase his contribution because he cares about j and realizes that j will have to bear the costs condition reveals that da;/da; > 0, so i's contribution increases with /'s contribution. It is also true that these motives are unimportant. As confirmation, consider the same two individuals in a that for a;" > a;, da;/dy; > 0 and da;/do> 0, so an increase in guilt motives and the susceptibility disequilibrium state, at which / is contributing 0.4 and i is contributing only 0.1. By shirking, i of punishing him if he (i) contributes too little. captures 0.2 in net material benefits from the project (that is, 0.6(0.1 + 0.4) - 0.1). But j would condition. to shame raise i's contribution. Member j's utility maximization yields the analogous first order experience strong malevolente toward i (B; 0. The project is producing a public good if c > 0. (It is a value of the resource times the individual's share. If y' 0.) The good is nonexcludable because be; + cy > 0 may occur when e; = 0 (i.e., maximized by setting e = 0 for each member. But unless the individual's share of the degradation taken by others. of the resource, y's;, is large, noncooperative determination of effort levels will result in The fact that we derive the best response function this way does not imply that individuals when member j is free riding on the contributions of others). The good is nonrival because the benefit enjoyed by j conditional on the level of the public good produced, namely, c, is overexploitation. This is because s,y + y's; will be positive (even with y' 0 and b = 0, we have a pure public good; if c > a positive level of effort being expended. 0 and b > 0, the project is producing an impure public good. (Of course, if c = 0 and b > 0, it is When the actions open to individuals are limited to a set of distinct strategies, both public and models as analytical tools does not require that the models be accurate descriptions of way that common property goods problems take the form of n-person Prisoners' Dilemma Games with a individuals arrive at decisions, as long as the individuals act as if they were solving such problems. a pure private good.) Pareto-inferior dominant strategy equilibrium, introduced in chapter 1. In this chapter I will In many, perhaps most, cases a reasonable assumption about humans is that we act like the Public goods are underprovided (and public bads overprovided) because c # 0, so individuals acting noncooperatively do not take account of the benefits their effort confers on others, namely, analyze a more general case in which actors may vary their strategies continuously in two generic adaptive agents modeled in chapters 2 and 3; that is, we occasionally observe what others like us models of a coordination problem. I call it generic because it encompasses the underlying reason are doing and tend to copy those who seem to be doing better. We may consciously decide on a cy'. To see this, assume b = 0 (a pure public good) and, ignoring subscripts (because the members' for coordination failures - incomplete contracts-and yet includes the "invisible hand" interaction behavioral rule of thumb designed to work well on the average and then abide by it unless it utility functions are identical), the sum of their sum of utilities, w, is as a limiting case. Virtually all interesting common property or public goods problems involve produces unsatisfactory results. Adapting one's behaviors in this way will lead the fishers to act as w = n(cy - 8(e)) (4.2) large numbers of people, but the underlying structure of incentives and possible resolutions of the if they were maximizing, at least on the average and in the long run. problem are more transparently introduced in the two-person example (returning to the fishers), with which I will begin in the next section. I then present an n-person version of the same Setting e to maximize w requires cny' = 8', thereby equating the marginal benefit of effort model, illustrating it with the problem of team production. I show how social preferences such as devoted to the public good to the marginal disutility of effort. Each individual, selecting e to shame, guilt, and reciprocity may allow coordination of the actions of large numbers of people in maximize utility (eq. 4.1) non-cooperatively, will, however, set cy' = 8', and will thereby their mutual interest. I close with a taxonomy of coordination problems based on the nature of contribute suboptimally (this is a maximum only if cy"