I want this to be dont by hand written with showing all calculation step by step.
EconS_305__Homework_5.pdf EconS 305: Intermediate Microeconomics w/o Calculus Homework 5: Repeated Games, Sequential Move Games and Product Di↵erentiation Due: Friday, June 19th, 2020 at 5:00pm via Blackboard - Please submit all homework solutions in the order the questions are presented and as one .PDF. - Please show all calculations as these exercises are meant to refine your quantitative tool set. If I can not follow your calculations or it seems as you just “copy and pasted” answers from the internet, I will be deducting half the points from that solution. 1. A Cournot Game of Competing in Quantities w/ Fixed Costs - A Game Theory Extension with Sustainable Collusion Note: Please re-do all calculations for this exercise, even though Parts A through I are analogous to that of Homework 3 and 4, as it is important that you know how to derive the equilibrium results for this model. This is the base case model for many economic analyses, and it is important for Intermediate Microeconomic extensions. Consider two firms competing a la Cournot in a market with an inverse demand function of p(Q) = a� b(Q), where Q = qi + qj and a > c, and a total cost function of TCi(qi) = F + ciqi. Notice that each firm has the same fixed cost (F ) but their marginal costs (ci) are not equal to each other (i.e. ci 6= cj). This means these homogeneous product producing firms have asymmetric costs, and we can represent the Profit Maximization Problem (PMP) for firm i as: CALCULUS PART: max qi�0 ⇡i = [a� b(qi + qj)] qi � (F + ciqi) @⇡i(qi, qj) @qi = a� 2bqi � bqj � ci = 0 (1) And through symmetry we know that firm j’s PMP is max qj�0 ⇡j = [a� b(qi + qj)] qj � (F + cjqj) @⇡j(qi, qj) @qj = a� 2bqj � bqi � cj = 0 (2) where we now have two equations ((1) and (2)), and two choice variables (qi and qj) to solve for. CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE. (a) Find the optimal equilibrium allocation for each firm when they are competing a la Cournot. That is, find q⇤i and q ⇤ j . Note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (b) Now, consider that the firm’s have symmetric costs (i.e. ci = cj = c) in the competitive equi- librium and for all analyses from here on out. Find the competitive equilibrium quantities (i.e. find q⇤i and q ⇤ j ). Again, note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (c) Find the equilibrium price (i.e. p(Q⇤) = a� b(Q⇤)). Again, note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (d) Find the equilibrium profits (i.e. ⇡⇤). Again, note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (e) Now, assume that both firms are pooling resources and acting as a cartel. Please re-write the equilibrium profits (⇡Cartel) you found in Question 2 in Homework 4. The answer should be exactly the same answer you got in Question 2 of Homework 4. (f) Now, consider that one of the firms in the cartel unilaterally deviates from the cartel to compete in quantities (i.e. sets their quantity at the competitive level instead of the cartel level). Derive the profits from deviating (⇡Devi ) and simplify the expression. Again, please note that this solution is analogous to the answer you derived in Question 3 of Homework 4. The key here is to remember that when deviation occurs, the deviating firm sets their quantities at a level as if they were competing a la Cournot and leaves the other firm operating as a cartel. This implies that the deviating firm’s profits (⇡D) become ⇡Devi = � a� b � qCournoti + q Cartel i �� � qCournoti � � c � qCournoti � � F (g) Similar to part f, we can find the profits of the firm that remains in the cartel while the other firm deviates. We will call these profits ⇡NDevi (i.e. does not deviate) such that ⇡NDevi = � a� b � qCournoti + q Cartel i �� � qCarteli � � c � qCarteli � � F 2 Use this formula to find the profits of the firm that is being deviated upon (i.e. ⇡NDevi ). Again, please note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (h) Take the four di↵erent equilibrium profit functions you found (⇡Cartel, ⇡Cournot, ⇡Dev and ⇡NDev), set all fixed costs equal to zero (i.e. F = 0), and compare them mathematically (i.e. from most profit gained to least profit gained). Once the comparison is done, plug these equations into a matrix following the matrix template given below. This is called a normal form game, and this allows us to determine the “best response” for each firm when trying to decide to participate in a cartel or compete in quantities. Again, please note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (i) Find the Pure Strategy Nash Equilibrium (psNE). Again, please note that this solution is analogous to the answer you derived in Question 3 of Homework 4. (j) Now, consider that both firms are going to engage in an infinitely repeated game where they will need to simultaneously decide if they are going to compete or participate in a cartel at every time t. Please derive the Grim Trigger Strategy (GTS), using our profit notation (i.e. ⇡Dev and ⇡Cournot), where the firm will deviate at the time period t = 0, and then revert to the Nash Equilibrium payo↵ (i.e. ⇡Cournot) in all the time periods after that. In order to do this, recall that we need to consider the discounted stream of payo↵s the firm will receive from t = 0 to t = 1. Please refer to the Chapter 12 Game Theory lecture notes if you are having trouble with deriving the GTS payo↵ stream. (k) Now, please derive the stream of payo↵s an individual firm will receive from participating in the cartel in the infinitely repeated game. Please use the general profit notation instead of the analytical notation. (l) Now, please find the condition on the firm’s patience parameter (i.e. �) in which will can guarantee that the firm’s will both sustain the cartel agreement. Remember that the cartel agreement in the infinitely repeated game will be sustained if the discounted stream of payo↵s from staying in the cartel are greater than the GTS payo↵s. (m) Please plug in the analytical solution to the general profit notation we have been using, and simplify. Simplifying should prove that � = 12 in this infinitely repeated game. 3 2. A Bertrand Model with Product Di↵erentiation Note: I have given you the final answers to these problems, but I want you to show all the work needed to derive these results. Also, please make sure to interpret the results if the question asks you to interpret the results. Consider two firms competing in prices a la Bertrand selling di↵erentiated goods. The de- mand function for every firm i is qi(pi, pj) = a � bpi + pj where i,j 2 {1, 2} and i 6= j. This means that the demand function for firm 1 and firm 2 are q1(p1, p2) = a � bp1 + p2 and p2(q1, q2) = a � bp2 + p1, respectively. In this model b represents the degree of product dif- ferentiation for every firm i. If b > 1 the products are considered to be di↵erentiated (also known as heterogeneous), and if b = 1 the products are identical (also known as homogeneous). Each firm has the same constant marginal cost of c, and all firms’ fixed costs are assumed to be equal to zero (i.e. F = 0). We formulate each firm’s Profit Maximization Problem (PMP) as: CALCULUS PART: max pi�0 ⇡i = pi [a� bpi + pj]� c [a� bpi + pj] @⇡i(pi, pj) @pi = a� 2bpi + pj + bc = 0 (3) And through symmetry we know that firm j’s PMP is max pj�0 ⇡j = pj [a� bpj + pi]� c [a� bpj + pi] @⇡j(pi, pj) @pj = a� 2bpj + pi + bc = 0 (4) where we now have two equations ((3) and (4)), and two choice variables (pi and pj) to solve for. CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE. (a) Please find the Best Response Functions for each firm (i.e. BRFi ⌘ pi(pj) and BRFj ⌘ pj(pi)). How does firm i respond with it’s own price (i.e. pi) with an increase in a, b, c and pj? BRFi ⌘ pi(pj) = (a+ bc) 2b + 1 2b pj BRFj ⌘ pj(pi) = (a+ bc) 2b + 1 2b pi (b) Find the optimal equilibrium allocation for each firm when they are competing a la Bertrand with di↵erentiated products. That is, find p⇤i and p ⇤ j , and please simplify. Are these prices the same? (p⇤i , p ⇤ j) = ✓ (a+ cb) (2b� 1) , (a+ cb) (2b� 1) ◆ 4 (c) Find the optimal quantity demanded for each firm (i.e. q⇤i = a� bp⇤i +p⇤j and q⇤j = a� bp⇤j +p⇤i ). (q⇤i , q ⇤ j ) = ✓ b(a� c(b� 1)) (2b� 1) , b(a� c(b� 1)) (2b� 1) ◆ (d) Find the equilibrium profits of each firm (i.e. ⇡⇤i and ⇡ ⇤ j ). (⇡⇤i , ⇡ ⇤ j ) = ✓ b(a� c(b� 1))2 (2b� 1)2 , b(a� c(b� 1))2 (2b� 1)2 ◆ 5 3. A Cournot Model with Product Di↵erentiation Consider two firms competing in quantities a la Cournot selling di↵erentiated goods. The inverse demand function for every firm i is pi(qi, qj) = a� bqi � dqj where i,j 2 1, 2 and i 6= j. This means that the inverse demand functions for Firm 1 and Firm 2 are p1(q1, q2) = a�bq1�dq2 and p2(q1, q2) = a�bq2�dq1, respectively. In this model b and d represent the degree of product di↵erentiation for every firm i and j. b is assumed to be greater than zero (i.e. b > 0), and for simplicity lets assume that d can take on any value between zero and b (i.e. b � d � 0). The products are considered to be di↵erentiated (also known as heterogeneous goods) if b 6= d, and if b = d the firms’ products are identical (also known as homogeneous goods). Each firm has the same marginal cost of c, and all firm’s fixed costs are assumed to be equal to zero (i.e. F = 0). We formulate