u = u(c"', b) = 14\"th vc\"), b) where u is increasing and concave in its rst argument and decreasing and convex in the second. Leisure and consumption are complements so u\T? 9:53 seen from the effect of punishment on utility, namely, 1 0',- (a,"' 51,-). Third, when one or more members have reciprocal preferences, the interaction will exhibit positive feedbacks, with the actions of one of the team inducing changes in the actions Of the others. Figure 4.6 describes a unique stable Nash equilibrium in the presence of these feedbacks. But it is not difcult to conceive of interactions with multiple stable equilibria, some with high contributions and some with low, separated by unstable equilibriatipping points dening the boundary of the basins Of attraction Of the stable equilibria. A TAXONOMY OF COORDINATION PROBLEMS The underlying structure of both the shers and the team production problems THE 2164 / 7837 28% 10 Bad Chemistry (4) Consider the generic coordination problem given by eqs. 4.19. 10.1 10.2 10.3 10.4 For the Nash equilibrium (i.e., 61"" and A\"), give conditions under which the external effect is positive or negative and the two strategies at and A are substitutes or complements. What is the rst order condition for a symmetric Pareto-efcient alloca- tion? Using this first order condition (assuming the second order condi- tion holds) and your expression for the Nash equilibrium above to show that 51* and A\" exceed the Pareto-efcient levels if and only if the external effect is negative. Explain Why this is so. Assuming that the Nash equilibrium is in pure strategies, show that there will always be a rst mover advantage, and that the second mover will do Worse (than in the Nash equilibrium) if strategies are substitutes and better if strategies are complements. Explain Why this is so. Two adjacent farmers (Lower and Upper) choose Whether to use a chemical intensive anti-pest strategy or a less chemical-intensive ap- proach that uses natural predators to control the pests which threaten their crops (integrated pest management or 1PM). The use of chemicals generates negative external effects (the chemicals kill the natural preda- tors as well), while IPM generates positive external effects (the natural T? 9:53 interaction among the members of the population. The activity is joint because 14., 0: what the others do, directly affects the individual's well-being. The outcome of a noncooperative interaction among these individuals is likely to be Pareto inefcient, because the direct effects on one's actions on the others utility (that is, ua) are not accounted for in the individuals\" optimization. One solution to the problem would be to transform it from a noncooperative to a cooperative game, perhaps by letting a state determine the values of a for each individual. The reasons Why this solution may be infeasible or undesirable have already been mentioned. Within the noncooperative game framework, there are three generic ways to avert the coordination failures that may arise in joint activities. None are practical ways of averting the problem entirely, but understanding their logic will help clarify some of the relevant institutional options. THE 2174 / 7837 28% 10.5 predators do not respect the farmer's property boundaries and prey on the pests throughout the area). Specically, increased use of chemicals by one raises output of the user and lowers the output and raises the marginal productivity of chemical use in the other farm for any given level of other inputs. Letting a and A be the level of chemical use by the two, give the values of the parameters of the above utility functions that describe this interaction. Conspicuous Consumption. Suppose individuals differ in some trait that inuences hourly wages and that they choose their hours of work (5) to maximize a utility function, the arguments of which are leisure (which we normalize as 1K?) and what we term effective consumption, 6", dened as their own consumption level (6) minus a constant 12 {for Veb- len) times the consumption level of some higher income reference group (6"). The individual's reference group might be the very rich or it might be an intermediate group. The reference group's rank in the income distribution is taken as exogenous, as is the Veblen constant :2. It may be convenient to think of each individual as belonging to a homoge- neous income class, each member of which takes the next highest in- come class as its reference group (the richest class have no reference group). Together, the reference group and 0 measure the nature and intensity of the relevant social comparisons. Individuals do not save, so 6 = wb, where w is the wage rate. Thus for some individual not in the richest group we have T? 9:53 The rst idealized solution is to alter the institutional setup so that individual utility is maximized subject to a binding participation constraint for each of the others. The allocation resulting from this maximum problem must be a Pareto optimum (by denition). To see this, suppose an allocation is such that the chooser's own indifference locus is not tangent to the indifference locus representing the participation constraint of one of the others. This allocation cannot be a solution to the stated constrained optimizing problem, for in that case the chooser could do better by adopting a different allocation. The privatization solution to the shers' problem, by establishing residual claimancy on the lake's entire output and control of its use by a single individual, while constraining the owner to satisfy the other's participation constraint as an equality, made a single person the owner THE 2179 / 7837 28% T? 9:53 of all of the consequences of his actions, a kind of ctive Robinson Crusoe. I'll call this the binding participation constraint solution. A second way of averting a coordination failure is to alter the under lying interaction so that the actions of others affect each individual only through the price vector, so no, = 0 . The Pigouvian taxes in the shers example approximated this result by imposing a price (in the form of a tax) on one's own shing equal to the costs that it imposed on others. In this case the utility function a(a; p(a)), and the individual takes the price vector as an becomes a exogenous constraint on the optimizing process. The resulting allocation will be such that for every individual the common price vector is tangent to their indifference locus (the arguments of which are the various proximate determinants of their utility, such as work effort, goods, and the like mentioned HE 2184 / 7837 28% T? 9:53 above). But this of course means that the indifference loci of all members of the population have a common slope (all marginal rates of substitution are equal among all pairs of goods), thus implementing a Pareto optimum. This is the complete contracting solution. A third way of averting the coordination failure is the simplest: it may be possible to structured the interaction so that social preferences can substitute for complete contracts. In the shers case we saw that complete altruism by all individuals (each caring about the others as much as about themselves) would implement a social optimum. While this utopian approach has little practical relevance, it is sometimes the case that the peer monitoring and sanctioning by a minority of group members who are motivated by other-regarding preferences can induce other individuals to act as if they cared about the others. The public goods game fig 2189 / 7837 28% T? 9:53 with punishment introduced in chapter 3 is an example. This is the social preferences solution. While sharing a common structure and a common set of possible institutional responses, coordination problems also differ in two important ways: the sign of the direct effect of the others' actions on one's utility (positive or negative externalities) and the sign of the effect of others' actions on one's own actions (determining whether strategies are complements or substitutes). These two distinctions will be claried by a two-person example in which we abstract from the price effects represented by the p vector, above. Consider two symmetric individuals (Lower and Upper, again) with identical utility functions: M [[(daA) U=&w (EE 2193 / 7837 28% T? 9:53 where a and A are the actions taken by the two individuals, and the f function is concave in its rst argument. (Symmetry allows us to use the same function f() for the two individuals, but with the arguments reversed.) The coordination problem arises because of the direct effect of the action of each on the utility of the other: that is, f2, the derivative of f with respect to the second argument, is not zero. Suppose these two functions take the following form: ot+[3a+yA+8aA+Mu2 0t+(3141+ya+8ai1+)tA2 u U (4.19) where )\\