Production-ONigepody Reaction curves Drawing the reaction curves for both firms on the same graph yields the picture below q1(q2) 41 The curve q(qz)yields the optimal level of gn for any given 42. The curve q2(gn)yields the optimal level of qz for any given qr. These curves will cross. Why? a point at which there is no profitable unilateral deviation is a Nash equilibrium. That is a point which is a best response to a best response(and so on)-where the two curves cross, written(qi. 42) action- Oligopoly Nash Equilibriun What is the value of qi and g?? They could be read off from the graph. Alternatively, solve the two equations. Substituting the value of q2 into the equation for qi yields: Symmetrically solving for gives q*=(a-c)/3. With the same linear demand and constant marginal cost assumptions in place but with n firms in the industry it is (mathematically) simple to show that each of the n firms will produce at:Production — Oligopoly 3 Reaction Curves • Drawing the reaction curves for both firms on the same graph yields the picture below. ................................................................................................................................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................. . . . . ....... ............. ............. ............. 0 q2 q1 q ∗ 2 q ∗ 1 q1(q2) q2(q1) • The curve q1(q2) yields the optimal level of q1 for any given q2. The curve q2(q1) yields the optimal level of q2 for any given q1. These curves will cross. Why? • A point at which there is no profitable unilateral deviation is a Nash equilibrium. That is a point which is a best response to a best response (and so on) — where the two curves cross, written (q ∗ 1 , q ∗ 2 ). Production — Oligopoly 4 Nash Equilibrium • What is the value of q ∗ 1 and q ∗ 2 ? They could be read off from the graph. Alternatively, solve the two equations. • Substituting the value of q2 into the equation for q1 yields: q ∗ 1 = 1 2 ½ a − c − a − c − q ∗ 1 2 ¾ =⇒ q ∗ 1 = a − c 3 • Symmetrically solving for q ∗ 2 gives q ∗ 2 = (a − c)/3. • With the same linear demand and constant marginal cost assumptions in place but with n firms in the industry it is (mathematically) simple to show that each of the n firms will produce at: q ∗ i = a − c n + 1