Quiz 15: Imperfect Competition

Business

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: img 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: img The total revenue for the firm is shown below: img 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: img Monopolist will maximize his profit. The output level for maximum profit is calculated below: img Equate the differentiation to zero to get the equilibrium level for the output as shown below: img The equilibrium price at this quantity is calculated below: img The profit calculated at this quantity is shown below: img 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: img Q is the total quantity demanded. img . Where, img Quantity of firm 1 img Quantity of firm 2 The inverse demand curve is given below: img The profit function for firm 1 is given below: img Differentiate the profit function and equate it to zero to maximize the profit as shown below: img The profit function for firm 2 is given below: img Differentiate the profit function and equate it to zero to maximize the profit as shown below: img Substitute the value of q 2 in q 1 as shown below: img img Put the value of img in equation for img as shown below: img Total quantity is given below: img The profit of the firm 1 is calculated below: img The profit of the firm 1 is calculated below: img 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. img 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. img 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 img .Until price of firm 1 remain lower than price of firm 2, fir, 1 can increase its profit by increasing the price. 3. img This cannot be Nash equilibrium either because the firm can decrease its price, undercutting firm 2 by a marginal amount img . Then it would capture the entire market and increase its profit. 4. img 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: img As the firms are symmetric img Therefore, img So, img and img The profit of the firm 1 is calculated below: img The profit of the firm 2 is calculated below: img d. The market demand curve shows the negative relation between quantity demanded and prices. The market demand curve is shown below. img 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 img The average and marginal cost of the monopolist is equal to c. So the cost function is given by img Hence, profit of the monopolist is given by img On differentiating the profit function with respect to quantity we get, img For maximizing profit, img Hence we get that profit maximizing quantity is given as img The price associated with this quantity is equal to img The profit of the monopolist is given by img img 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 img Where img and img 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 img . Then its profit will be given as img The first order condition for the above profit function is given by img So we get, img ...... (1) As the firms are symmetric, similarly img From the above equation we get, img ...... (2) As in NE equation (1) and (2) should be equal, hence img From the above we get, img As the firms are symmetric, img Hence, market output is equal to img Putting the above value in the inverse-demand function, we get the price img The profit of the firms is then given by img img Industry profits are given by img 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) img 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) img . This cannot be NE either because the firm can increase its price , keeping it lower than p 2 by a marginal amount of img . It would not lose the market but increase its profit. (iii) img This cannot be NE either because the firm can decrease its price, undercutting firm 2 by a marginal amount img . Then it would capture the market and increase its profit. iv) img 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 img As the firms are symmetric img Hence, img From this we get the quantity produced by each firm img Profit of each firm is given by img The market output is given by ` img 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 img Where img and img 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 img . Then its profit will be given as img The first order condition for the above profit function is given by img So we get, img ......(3) As the firms are symmetric, similarly img From this we get, img As the firms are symmetric, img Hence, market output is equal to img Putting the above value in the inverse-demand function, we get the price img img So profit of each firm is given by img img Profit of the industry is then given by img e) Setting n=1, we get the following value Quantity produced by the firm is given by img Price set by the firm is given by img Profit of the firm is given by img 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 img Price set by the firm is given by img Profit of the firm is given by img Profit of the industry is given by img 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 img Price is given by img Profit of the industry is given by img 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 img Where img and img 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 img . Cost function for firm 1 is given by img Then its profit will be given as img The first order condition for the above profit function is given by img So we get, img ......(1) Similarly img From this we get, img ......(2) As in NE equation (1) and (2) should be equal, hence img From the above we get, img Similarly, img Hence, market output is equal to img Putting the above value in the inverse-demand function, we get the price img The profit of the firms is then given by img img Similarly img Industry profits are given by img Consumer surplus is given the are above the price in the demand curve The inverse market demand function is given by img Hence, the market demand function is given by img The Cournot Competition price is equal to img The formula for consumer surplus is given by img img img img Thus, the total welfare is given by Profits + CS = img 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. img Similarly img The graph is shown below: img 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. img 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: img