Esercizi per prima prova in itinere

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Exercises for the Midterm Exam - Class 21.4.2015 April 22, 2015 0.1 Bertrand duolopy Consider two firms competing on prices, 1 and 2. The produce with con- stant marginal costs MC 1(y) = MC 2(y) = 10 and face the market demand Y (p) = 6000 − 60p. 0.1.1 Find the Bertrand equilibrium. [ p1 =p2 =p∗ = 10,y 1 =y2 =y∗ = 2700,Y ∗ = 5400,π 1 =π2 =π∗ = 0] 0.1.2 Assume the two firms have asymmetric marginal costs, MC 1(y) = 10 and MC 2(y) = 12. Find the new equilibrium. [p∗ 1 = 12 −ε,p ∗ 2 = 12,y ∗ 1 ≈ 5280,y ∗ 2 = 0,π ∗ 1 ≈ 10560,π ∗ 2 = 0] 0.1.3 Assume now that MC 1(y) = 10 and MC 2(y) = 60. How does the equilibrium change? [p∗ 1 =pm 1 = 55,p ∗ 2 = 60,y ∗ 1 =ym 1 = 2700,y ∗ 2 = 0,π ∗ 1 = 121500,π ∗ 2 = 0] 0.1.4 Now the two firms have the same marginal cost functionsMC 1(y) =MC 2(y) = 10, but they operate under capacity constraints, ¯y1 = 1000 and ¯y2 = 1400. [p1 = p2 = p∗ = 60,y 1 = 1000,y 2 = 1400,Y ∗ = 2400,π 1 = 50000,π 2 = 70000] [Check that, given that one firm is playing the equilibrium strategy, being a monopolist on the residual demand is not optimal for the other firm.] 1 0.1.5 Finally, assume that the two firms produce two differentiated goods at zero costs. More specifically, demands faced by each firm are y1 = 300 −p1 + 1 2p2 and y2 = 300 −p2 + 1 2p1, respectively. [p1 =p2 =p∗ = 200,y 1 =y2 =y∗ = 200,Y ∗ = 400,π 1 =π2 =π∗ = 40000] 0.2 Theory of production and cost minimization Consider the production function y =f(x1,x 2) =x1x2 2. Which kind of returns to scale does this technology exhibit? [increasing] Assume that y = 128 units of good need to be produced. Input prices are ω1 = 4 and ω2 = 2. Solve the cost minimization problem and find the optimal combination of inputs. [x∗ 1 = 2,x ∗ 2 = 8] 0.3 Game theory The two-player game considered has three Nash equilibria in pure…