Suppose two duopolists produce given differentiated goods. The prices that each obtains in equilibrium depends on the quantities output by each (or, alternatively, the equilibrium quantities demanded for each depends on the prices each charges), according to the inverse demand functions
Pi = A - X1 - bxz and Pz = A -X z - bx1,
for constants A and b. The constant b is restricted to the interval (-1, 1), where b > 0 means that the goods in question are "substitutes" and b 0 means they are "complements." (We use quote marks here because we are not working with precise definitions of these terms; cf. chapter 2.)
(a) Give Cournot, von Stackelberg and Bertrand equilibria for this model.
If you find the algebra getting a bit too thick, solve for equilibrium values (quantities, prices, profits) for the parameters A = 10, k = 1 and b = .9, .5, .1, and -.5. Note that the two firms do better in Bertrand than in Cournot for the case b 0. Can you give an intuitive expla;nation of this?
(b) In the von Stackelberg equilibrium of part (a), the follower has Cournot conjectures and the leader optimizes given the follower's reaction function. We could equally well imagine a case in which the follower had Bertrand conjectures and the leader optimizes given the reaction function so en tailed. This is called the Bertrand-van Stackelberg equilibrium. (The von Stackelberg equilibrium in part (a) is then called the Cournot-von Stackel berg equilibrium.) Solve for it, at least for the four parameterizations given previously.
(c) In both sorts of von Stackelberg equilibria, the leader has higher profits than if there were no leader at all. Prove that this is so. (It will suffice to demonstrate a weak inequality.)
(d) In Cournot-von Stackelberg for b > 0 and Bertrand-von Stackelberg for b 0, the follower has smaller profits than if there is no leader, while for Bertrand for b > 0 and Cournot for b 0, the follower has larger profits than if there is no leader. What intuitive explanations can you give for these observations?