# Quiz 15: Imperfect Competition

The demand curve shows the relation between quantity demanded and prices keeping all other things constant. a. The demand curve of the monopolist is given below: The cost of the monopolist is equal to 0. The total revenue is the addition of the revenue earned by the firm at each unit. The formula for the calculation of TR is given below: The total revenue for the firm is shown below: The total cost for firm is zero The profit of the firm is the difference between the total revenue received and total cost incurred. The profit for the monopolist is calculated below: Monopolist will maximize his profit. The output level for maximum profit is calculated below: Equate the differentiation to zero to get the equilibrium level for the output as shown below: The equilibrium price at this quantity is calculated below: The profit calculated at this quantity is shown below: b. In Cournot competition, there are two firms which choose their profit maximizing quantities taking the quantity of the other firm as given. The market demand curve is given below: Q is the total quantity demanded. . Where, Quantity of firm 1 Quantity of firm 2 The inverse demand curve is given below: The profit function for firm 1 is given below: Differentiate the profit function and equate it to zero to maximize the profit as shown below: The profit function for firm 2 is given below: Differentiate the profit function and equate it to zero to maximize the profit as shown below: Substitute the value of q 2 in q 1 as shown below: Put the value of in equation for as shown below: Total quantity is given below: The profit of the firm 1 is calculated below: The profit of the firm 1 is calculated below: c. When the firms are in Bertrand competition instead of choosing quantity simultaneously as in Cournot competition, they choose price simultaneously. While maximizing profit, each firm takes the price of the other good as fixed while choosing its own price. It is more aggressive competition than Cournot competition because each firm wants to undercut the other firm to steal the whole market. This leads to the equilibrium price being equal to the marginal cost. As the firms in this excercise do not have any cost, this means that the firms charge zero prices. The profit maximizing strategy for firm 1 is given below: There are four possible scenarios as given below: 1. Firm 1 earns a negative margin on every unit it sells. Since it sells a positive quantity, it must earn negative pro?ts. It could increase its pro?t by deviating to a higher price. Hence this is not Nash Equilibrium. 2. This cannot be Nash equilibrium either because the firm can increase its price, keeping it lower than that of firm 2 by a marginal amount of .Until price of firm 1 remain lower than price of firm 2, fir, 1 can increase its profit by increasing the price. 3. This cannot be Nash equilibrium either because the firm can decrease its price, undercutting firm 2 by a marginal amount . Then it would capture the entire market and increase its profit. 4. Now this is Nash equilibrium for both firms as they cannot increase their profit given the strategy of the other firm. In Nash equilibrium both firm charge the price that is equal to MC. As MC is zero, price will be zero. Total output supplied by the firms at price equal to zero is calculated below: As the firms are symmetric Therefore, So, and The profit of the firm 1 is calculated below: The profit of the firm 2 is calculated below: d. The market demand curve shows the negative relation between quantity demanded and prices. The market demand curve is shown below. Figure 1 The X axis shows the quantity and Y axis shows the price. The points A, B and C shows the equilibrium points in different market. Here, A: Equilibrium in Monopoly B: Equilibrium in Cournot Competition C: Equilibrium in Bertrand Competition

a) The inverse market demand curve is given by The average and marginal cost of the monopolist is equal to c. So the cost function is given by Hence, profit of the monopolist is given by On differentiating the profit function with respect to quantity we get, For maximizing profit, Hence we get that profit maximizing quantity is given as The price associated with this quantity is equal to The profit of the monopolist is given by b) In Cournot competition, firms choose their profit maximizing quantities taking the quantity of the other firm as given. The inverse market demand curve of is given by Where and are the quantity produced by firm 1 and firm 2 respectively. When deciding its profit maximizing level of output, firm 1 will take output of firm 2 fixed at . Then its profit will be given as The first order condition for the above profit function is given by So we get, ...... (1) As the firms are symmetric, similarly From the above equation we get, ...... (2) As in NE equation (1) and (2) should be equal, hence From the above we get, As the firms are symmetric, Hence, market output is equal to Putting the above value in the inverse-demand function, we get the price The profit of the firms is then given by Industry profits are given by c) When the firms are in Bertrand competition instead of choosing quantity simultaneously as in Cournot competition, they choose price simultaneously. While maximizing profit, each firm takes the price of the other good as fixed while choosing its own price. It is more aggressive competition than Cournot competition because each firm wants to undercut the other firm to capture the whole market. This leads to the equilibrium price being equal to the marginal cost. This can be shown below. We observe the profit maximizing strategy for firm 1 There are four possible scenarios (i) Firm 1 earns a negative margin on every unit it sells. Since it sells a positive quantity, it must earn negative pro?ts. It could increase its pro?t by deviating to a higher price. Hence this is not NE. (ii) . This cannot be NE either because the firm can increase its price , keeping it lower than p 2 by a marginal amount of . It would not lose the market but increase its profit. (iii) This cannot be NE either because the firm can decrease its price, undercutting firm 2 by a marginal amount . Then it would capture the market and increase its profit. iv) Now this is NE for both firms as they cannot increase their profit given the strategy of the other firm. Output the firms is found by using the demand function As the firms are symmetric Hence, From this we get the quantity produced by each firm Profit of each firm is given by The market output is given by ` d) In Cournot competition, firms choose their profit maximizing quantities taking the quantity of the other firm as given. The inverse market demand curve of the monopolist is given by Where and are the quantity produced by firm 1 and firm 2 respectively. As the firms are symmetric, firm 1 expects the same output from firm 2 and the remaining (n-2) firms. Adding the output of firm 2 and the remaining (n-2) firms we get (n-1)Q 2. When deciding its profit maximizing level of output, firm 1 will take output of firm 2 fixed at . Then its profit will be given as The first order condition for the above profit function is given by So we get, ......(3) As the firms are symmetric, similarly From this we get, As the firms are symmetric, Hence, market output is equal to Putting the above value in the inverse-demand function, we get the price So profit of each firm is given by Profit of the industry is then given by e) Setting n=1, we get the following value Quantity produced by the firm is given by Price set by the firm is given by Profit of the firm is given by The above results are identical with the outcomes of the monopolist in part a). Setting n=2, we get the following value Quantity produced by firm 1 is given by Price set by the firm is given by Profit of the firm is given by Profit of the industry is given by The above results are identical with the outcomes of the Cournot duopolist in part b). Setting n=8, we get the following value Total quantity produced is given by Price is given by Profit of the industry is given by The above results are identical with the outcomes of the Bertrand duopolist in part c).

a) In Cournot competition, firms choose their profit maximizing quantities taking the quantity of the other firm as given. The inverse market demand curve is given by Where and are the quantity produced by firm 1 and firm 2 respectively. When deciding its profit maximizing level of output, firm 1 will take output of firm 2 fixed at . Cost function for firm 1 is given by Then its profit will be given as The first order condition for the above profit function is given by So we get, ......(1) Similarly From this we get, ......(2) As in NE equation (1) and (2) should be equal, hence From the above we get, Similarly, Hence, market output is equal to Putting the above value in the inverse-demand function, we get the price The profit of the firms is then given by Similarly Industry profits are given by Consumer surplus is given the are above the price in the demand curve The inverse market demand function is given by Hence, the market demand function is given by The Cournot Competition price is equal to The formula for consumer surplus is given by Thus, the total welfare is given by Profits + CS = b) The best NE can be shown on the graph by mapping the best response functions of both the firms and seeing their intersection points. The best response function are given by the following equations. Similarly The graph is shown below: A decrease in the cost of firm 1 shifts its best response curve out. Hence there is a shift in the NE. Production of firm 1 increases, whereas that of firm 2 decreases. This is shown in the graph below. An iso-profit curve shows the locus of points at which a firm would get equal profit. A representative iso-profit curve for firm 1 is shown below. It should be noted that the highest point of the iso-profit curve lies on the firm's best response function. The graph is shown below: