The goats of two villagers graze on a common pasture. The number of goats that villager i = 1, 2 keeps is xi , assumed to be real non-negative number. The payoff of villager i depends directly on xi . It also depends on how much of the pasture is left unused for regeneration. Namely, if X = x1+x2 is the total number of goats, and K is some exogenous capacity of the pasture, then the payoff is increasing in the difference (K X). To be precise, the total payoff of agent i = 1, 2 is ui = (xi )+ ( K X). (If villagers select the number of goats so that K < X, then the pasture is destroyed, all goats die and the payoff drops to zero)

1.Pigouvian tax. Suppose that we impose a tax of per goat. Namely, if villager i keeps xi goats on the pasture, she has to transfer xi goats to the other villager. The consumption of villager i consists of the remaining goats plus the transfer from the other villager, (1 ) xi + xj . The utility, therefore, is ui = ((1 ) xi + xj) + (K X). What must be so that the total number of goats in equilibrium is efficient? It seems that this tax just reshuffles the goats among the villagers, so it does not really change anything. How is it then that it can help the villagers to achieve the efficiency?

2.( no-tax case = 0) Suppose that villager 2 observes how many goats villager 1 buys, and then decides how many goats to buy. Specify the game formally, in particular explain the strategies of each player. What solution concept would you use? Find an equilibrium. Who has an advantage, the first mover or the second mover?