# “Direct” argument for Nash equilibrium

If each firm charges the price of c then the other firm can do no better than charge the price of c also (if it raises its price is sells no output, while if it lowers its price is makes a loss), so (c, c) is a Nash equilibrium.

# No other pair (p_{1}, p_{2}) is a Nash equilibrium since

•

if p

_{i }< c then the firm whose price is lowest (or either firm, if the prices are the same) can increase its profit (to zero) by raising its price to c•

if p

_{i }= c and p_{j }> c then firm i is better off increasing its price slightly•

if p

_{i }≥ p_{j }> c then firm i can increase its profit by lowering p_{i }to some price between c and p_{j }(e.g. to slightly below p_{j }if D(p_{j}) > 0 and to p^{m }if D(p_{j}) = 0).

Note: to show a pair of actions is not a Nash equilibrium we need only find a better response for one of the players—not necessarily the best response.

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