where g*_{ denotes +----b g*_r + g*+1 +----h g*. Substituting g* into (1.2.5), summing over all n farmers' first-order conditions, and then dividing by n yields where G* denotes gj + ■ ■ • + g*. In contrast, the social optimum, denoted by G**, solves
Comparing (1.2.6) to (1.2.7) shows12 that G* > G**: too many goats are grazed in the Nash equilibrium, compared to the social optimum. The first-order condition (1.2.5) reflects the incentives faced by a farmer who is already grazing g, goats but is consider
12Suppose, to the contrary, that G* < G". Then v(G*) > v(G"), since v' < 0. Likewise, 0 > v'(G*) > v'(G,t), since v" < 0. FinaUy, G'/n < G". Thus, the left-hand side of (1.2.6) strictly exceeds the left-hand side of (1.2.7), which is impossible since both equal zero.
0<G<oo the first-order condition for which is v(G**) + G**v'(G**) - c = 0.
ing adding one more (or, strictly speaking, a tiny fraction of one more). The value of the additional goat is v(gi + gl,) and its cost is c. The harm to the farmer's existing goats is v'{gi+g*_i) per goat, or giv'{gi in total. The common resource is overutilized be cause each farmer considers only his or her own incentives, not the effect of his or her actions on the other farmers, hence the presence of GV(G*)/n in (1.2.6) but G**z/(G**) in (1.2.7).
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