4. (challenging!) Each of n farmers can costlessly produce any quantity of wheat desired. Suppose that the k – th farmer produces Wk, so that the total amount of what produced is W = Wi +W2 + ... + Wn. The price P at which wheat sells is then determined by the demand equation P = e-w. -
Show that that there exists an equilibrium in dominant strategies. (Hint: show that the strategy of producing one unit (Wk = 1 for all k) of wheat (strongly) dominates all of a profit-maximizing farmer’s other strategies by setting up the profit function, deriving the first order condition and solving it for Wk ).
A dominant strategy equilibrium is arrived at when every player picks their own dominant strategy. Since just one of them has a dominant strategy, there is no dominant strategy equilibrium. We should then continue by eliminating dominated strategies. A dominant strategy is one that is best regardless of the move is made by the other party to the game. At the point when the two players have dominant strategies, the result is steady in light of the fact that neither one of the gatherings has an incentive to change.
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