Two firms are producing thingies. It costs Firm 1 g1 dollars to produce q thingies. It costs Firm 20.572 dollars to produce 42 thingies. If Q = 91 +42 thingies are produced in total by the two firms, customers will pay 200 - Q dollars per unit. (a) Suppose the two firms want to collude in an attempt to maximize the sum of their profits. Formulate and solve (without the use of software) a nonlinear programming model to determine how many thingies should each firm produce. (b) Suppose collusion is illegal. Formulate and solve (without the use of software) a nonlinear programming model to determine how many thingies Firm 1 should produce as a function of 42. Call this amount B (92). Next, formulate and solve (without the use of software) a nonlinear programming model to determine how many thingies Firm 1 should produce as a function of q. Call this amount B2(91). Finally, simultaneously solve the system of equations 91 = B(92) 92 = B2(91) to determine optimal values of q and q2 when collusion is NOT allowed. (c) Suppose Firm 1 is the market leader. This means that once Firm 2 has determined B2(91), Firm 1 replaces 92 with B2(91) in its profit function then optimizes the re- sulting expression. Once q, is determined, Firm 2 inputs this value into B2(91) to determine 92. Determine the optimal values of q and q2 in this scenario. (d) Compare and contrast the Firms' outputs and profits among the three scenarios. Two firms are producing thingies. It costs Firm 1 g1 dollars to produce q thingies. It costs Firm 20.572 dollars to produce 42 thingies. If Q = 91 +42 thingies are produced in total by the two firms, customers will pay 200 - Q dollars per unit. (a) Suppose the two firms want to collude in an attempt to maximize the sum of their profits. Formulate and solve (without the use of software) a nonlinear programming model to determine how many thingies should each firm produce. (b) Suppose collusion is illegal. Formulate and solve (without the use of software) a nonlinear programming model to determine how many thingies Firm 1 should produce as a function of 42. Call this amount B (92). Next, formulate and solve (without the use of software) a nonlinear programming model to determine how many thingies Firm 1 should produce as a function of q. Call this amount B2(91). Finally, simultaneously solve the system of equations 91 = B(92) 92 = B2(91) to determine optimal values of q and q2 when collusion is NOT allowed. (c) Suppose Firm 1 is the market leader. This means that once Firm 2 has determined B2(91), Firm 1 replaces 92 with B2(91) in its profit function then optimizes the re- sulting expression. Once q, is determined, Firm 2 inputs this value into B2(91) to determine 92. Determine the optimal values of q and q2 in this scenario. (d) Compare and contrast the Firms' outputs and profits among the three scenarios
