advanced microeconomics, imperfect competition Question:=Consider two duopolists whose cost functions are Ci 10qi, iwhere C₁ and qi are their costs and outputs, respectively.duopolists have identical products and face a market demand function:p=40 - 2q, where q=Compare the outputs and profits in the following three cases: (i) simulta-neous move Nash equilibrium, (ii) sequential move (leader/follower), (iii)cooperative solution.Use below methods to answer above question:3 Collusion in Quantity Competition• Assume that the two duopolists collude in their choice of quantities.● That is, they solve the joint maximization problem:Thus, we have:• Let the solution be given by: q*,92*.. In other words, we have:: 91 +92.Thus, dividing the two equations, we get:Thus, from equation (2) it follows that:Similarly, for firm 2, we have:Max 91,92 {π¹ (91, 92) + π²(91, 92)}3.1 Example - Linear Case:• Do this as an exercise.Əπ¹ (q*, q2*)Əq₁C*On¹ (qi*, q2*)əq2● Cost:● Thus, profits are given by:π}(9₁*, 92*)π (9₁*, 92*)But, note that the slopes of the iso-profit curves are given by:• Firm 1's problem is:● This yields:. These are shown in the diagram.. Given the symmetry, the solution is:● or simply:+Ən¹ (91, 92)əqi• Case 1: Simultaneous game.• The corresponding best reply functions are:+dq2dq₁SELdq2dq₁● That is, the collusive solution lies on the contract curve.It is, therefore, Pareto efficient.. But, is it sustainable?• Assume that we have the collusive solution.● Since we usually have,dq2dq₁q²*9₁ ==• Thus, for firm 1, the first-order condition above implies that at the collusive solution, we have:π}(qî*, q2*) = −π}(qî*, q2*) 0||π2• Quantity competition and linear demand - strategic substitutes.• Demand:=• In other words, as far as both firms are concerned, they each increase their profits by "cheating" anddeviating from the collusive solution.• Namely, since ¹(qî*, 92*) 0 and ²(q*, 92*) 0, both firms can do better by increasing their outputs.² (¶¶*, 92*)Əq1² (q¶*, 92*)Əq2qi* = q³*9²π²(qi*, q2*) = −π²(q¶*, 92*) 0-||=p = a - b(q² + q¹)a -

Question

advanced microeconomics, imperfect competition Question:=Consider two duopolists whose cost functions are Ci 10qi, iwhere C₁ and qi are their costs and outputs, respectively.duopolists have identical products and face a market demand function:p=40 - 2q, where q=Compare the outputs and profits in the following three cases: (i) simulta-neous move Nash equilibrium, (ii) sequential move (leader/follower), (iii)cooperative solution.Use below methods to answer above question:3 Collusion in Quantity Competition• Assume that the two duopolists collude in their choice of quantities.● That is, they solve the joint maximization problem:Thus, we have:• Let the solution be given by: q*,92*.. In other words, we have:: 91 +92.Thus, dividing the two equations, we get:Thus, from equation (2) it follows that:Similarly, for firm 2, we have:Max 91,92 {π¹ (91, 92) + π²(91, 92)}3.1 Example - Linear Case:• Do this as an exercise.Əπ¹ (q*, q2*)Əq₁C*On¹ (qi*, q2*)əq2● Cost:● Thus, profits are given by:π}(9₁*, 92*)π (9₁*, 92*)But, note that the slopes of the iso-profit curves are given by:• Firm 1's problem is:● This yields:. These are shown in the diagram.. Given the symmetry, the solution is:● or simply:+Ən¹ (91, 92)əqi• Case 1: Simultaneous game.• The corresponding best reply functions are:+dq2dq₁SELdq2dq₁● That is, the collusive solution lies on the contract curve.It is, therefore, Pareto efficient.. But, is it sustainable?• Assume that we have the collusive solution.● Since we usually have,dq2dq₁q²*9₁ ==• Thus, for firm 1, the first-order condition above implies that at the collusive solution, we have:π}(qî*, q2*) = −π}(qî*, q2*) > 0||π2• Quantity competition and linear demand - strategic substitutes.• Demand:=• In other words, as far as both firms are concerned, they each increase their profits by &#34;cheating&#34; anddeviating from the collusive solution.• Namely, since ¹(qî*, 92*) > 0 and ²(q*, 92*) > 0, both firms can do better by increasing their outputs.² (¶¶*, 92*)Əq1² (q¶*, 92*)Əq2qi* = q³*9²π²(qi*, q2*) = −π²(q¶*, 92*) > 0-||=p = a - b(q² + q¹)a -< 0, i, j = 1,2, i ‡ j(a - c)q²:26=NEN26π² = q¹ [a − b(q² + q¹)] − cqi=Ci = cqi-π² (qi*, 92*)-π²² (9₁*, 92*)C*dq2dq₁c)T=Case 2: Sequential game.• The solution to Firm 2's second stage problem is given by its best reply function:(a - c) 1262&#34;q² +q¹==(a1-max{q¹ (a − b(q¹ + [a= c) — 2¹-0¹cq²¹261== 0NI(a − c)36*92= 0¹=4b• The solutions for the simultaneous and sequential games are shown in the diagrams above.• Using this example, we can also derive the contract curve as:-(a — c)26(a - c)261,2,The two• i.e., along the line such that total output equals monopoly output ((a-c)).26. This is also shown in the diagrams above.(3)(4)(5)(6)(7)(8)

Accepted Answer

Demand functionP=40-2qC=cqThere are 2 firmsFirm1 and firm 2