Question: A competitive economy may be described as a game with a continuum of players. Concepts such as iterated dominance, rationalizability, and Nash equilibrium can, with minor adjustments, be applied to such situations. Consider the following "wheat market": There is a continuum of farmers indexed by a parameter I distributed with a density f(i) on [i. i], where i > 0. They must choose the size of their crop q(i) before the market for wheat opens. The cost function of farmer i is C(q, i) = q2/2i. The farmer's utility function is thus ui = pq(i) - q(i)2/2i, where p is the price of wheat. Let O(p) denote the aggregate supply function when farmers perfectly predict p:
The demand curve is D(p) = a - bp for 0≤ p ≤a/b and 0 otherwise. The timing is such that the farmers simultaneously choose the size of their crop, then the price clears the market:
Apply iterated strict dominance in this game among farmers. Show that if b > k, the game is solvable by iterated strict dominance, which yields the perfect-foresight equilibrium. When b ≤k, determine the interval of prices that correspond to outputs that survive iterated strict dominance. Draw the link with the stability of the "cobweb" tatnnement in which the market is repeated over time, and farmers have point price expectations equal to the last period's price. (This exercise is drawn from Guesnerie 1989, which also addresses production and demand uncertainty, price floors and ceilings, and sequential timing of crop planting).