MICROECONOMICS II PROBLEM SET III: MONOPOLY


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1 MICROECONOMICS II PROBLEM SET III: MONOPOLY EXERCISE 1 Firstly, we analyze the equilibrium under the monopoly. The monopolist chooses the quantity that maximizes its profits; in particular, chooses the quantity that equates marginal income and marginal costs. Then: Introducing the inverse demand function and the cost function: Now, the monopolist chooses the quantity that maximizes its profits. To do so, it derivates the previous expression with respect to y and equates to The left hand side of the equality is the marginal income, while the right hand side is the marginal cost. Isolating the quantity, we obtain that 10. Introduce the quantity in the inverse demand function to obtain the price of the monopolist, which is: Introducing the quantity in the profit function (the function we have just maximized), we obtain that: Finally, the consumer surplus is given by the following expression: Now, let s analyze the equilibrium under perfect competition. In perfect competition, the inverse supply function results from equating the price and the marginal cost. Previously, we obtained that the marginal cost is 40. Then: 40 Equating both inverse functions of demand and supply: Isolate, and obtain that in perfect competition: 15 Introducing the quantity in each of the inverse functions of demand or supply, we obtain that the price in perfect competition is: Introducing the quantity of perfect competition in the profit function of the firm, we obtain that: Finally, the consumer surplus is given by the following expression: En resumen, los resultados son los siguientes: MONOPOLY PERFECT COMPETITION Y P CS 100 5
2 As we can observe, the monopolist offers a lower quantity at a higher price than in perfect competition. It causes a rise in its profits and a fall in the consumer surplus. Moreover, we see that there is a deadweight loss in the economy, since the rise in the monopolist profits does not compensate the fall in the consumer surplus. Graphically: EXERCISE.a) Firstly, notice that the inverse demand function is (simply isolate the price of the demand function): 10 The monopolist chooses the quantity that maximizes its profits; in particular, chooses the quantity such that its marginal income is equal to the marginal cost. Then: Introducing the inverse demand function and the cost function: Now, the monopolist chooses the quantity that maximizes its profits. To do so, derivate the previous expression with respect to y and equate to 0: 40 4 The left hand side of the expression is the marginal income while the right hand side is the marginal cost. Isolating the quantity, we obtain that 54. Introducing the quantity in the inverse demand function, we obtain the price of the monopolist: Introducing the quantity in the profit function (the one we have previously maximized) we obtain that: Graphically:
3 .b) In perfect competition, 4 (we obtained the MgC in the previous exercise). To obtain the quantity of perfect competition, equate the MgC to the inverse demand function: 10 4 Isolating, we obtain the quantity in perfect competition: 108 Introducing the quantity of perfect competition in the profit function of the firm, we obtain that: We see then that the profit of the firm is negative in perfect competition. Then, in perfect competition the market has no producers, since firms will not remain in it with negative profits..c) Under monopoly, we have that: Under perfect competition, we have that: We see that the efficiency loss of the economy caused by the monopoly is: 48 The loss is given by the following area:
4 .d) Due to the fact that now the technology does not have fix costs, if we break the monopoly rights, the market would have producers, since the profits in perfect competition would be 0 ( 0). Then, it would be recommendable that the government promoted competence in order to maximize the total surplus. EXERCISE Let be the inverse demand function and the cost function, where is the quantity of the good. The profit function of the monopolist is: The monopolist chooses the quantity that maximizes the profit function. Deriving with respect to and equating to 0 we obtain that: And isolating the inverse demand function, we obtain that: Notice that 0 (remember that there is a negative relation among price and quantity in the demand), therefore the price will be equal to the marginal cost plus a positive value. EXERCISE 4 Firstly, we analyze the equilibrium under the monopoly. The monopolist chooses the quantity that maximizes its profits; in particular, chooses the quantity that equates marginal income and marginal costs. Then: Introducing the inverse demand function and the cost function: 100 Now, the monopolist chooses the quantity that maximizes its profits. To do so, it derivates the previous expression with respect to q and equates to
5 The left hand side of the equality is the marginal income, while the right hand side is the marginal cost. Isolating the quantity, we obtain that 1.5. Introduce the quantity in the inverse demand function to obtain the price of the monopolist, which is: Introducing the quantity in the profit function (the function we have just maximized), we obtain that: The consumer surplus is given by the following expression: The social surplus would be: Graphically: Where area A is the consumer surplus, area B is the producer surplus and area C is the loss of efficiency in the economy. 4.c) Now, let s analyze the equilibrium under perfect competition. In perfect competition, the inverse supply function results from equating the price and the marginal cost. Previously, we obtained that the marginal cost is. Then: Equating both inverse functions of demand and supply: 100 Isolate, and obtain that in perfect competition: 0 Introducing the quantity in each of the inverse functions of demand or supply, we obtain that the price in perfect competition is: 40. Introducing the quantity of perfect competition in the profit function of the firm, we obtain that:
6 The consumer surplus is given by the following expression: The social surplus would be: 1000 Then, the loss of efficiency of the economy would be 140. EXERCISE 5 5.a) First, we compute the equilibrium of the monopoly without taxes. We start computing the inverse demand function from the demand function to get: 50 Introducing the inverse demand function and the cost function in the profit function: 50 Now, compute the equilibrium exactly as we did in the previous exercises. The equilibrium of the monopoly is given by: 5 The total surplus is thus, TS= Now, compute the equilibrium with tax/subsidy set by the government. The price paid by the consumer will be the Price received by the monopolist plus the tax, i.e., p+t. Thus: 50 And tehrefore, profits are: 50 So the first order condition yields (Marginal revenue equals marginal cost): 5 Therefore By doing the graph, we can see that the surplus are the area of two triangles
7 Notice that with t=0, the surplus are the same as calculated without taxes. To get the same welfare as under perfect competition, the Price paid by consumers has to equal the marginal cost. Thus, And therefore, the Price paid by the consumer has to be p*=0. Since consumers have to buy 10 units, it has to be the case that the final Price paid by consumers is P=0, so in order for the price received by monopolist to be equal to the marginal cost: p+t=cmg, it has to be that t=5, and thus, t is a subsidy. The price received by the monopolist is thus p=5. CS t = 5, PS t = =150 and the cost of the subsidy is 50. Total surplus (not taking into account the cost of the subsidy ists t =175. Thus, the subsidy has increased TS (this would also be the case even if taking into account the cost of the subsidy.subsidies do not create efficiency losses in monopolies). 5.b) To answer this question, we need to compute the equilibrium in perfect competition. Equating the inverse demand and supply functions: 50 We obtain that the quantity and price in equilibrium: 10, 0 To get the values of, we need to compute the profits in the long run, and check for which values they will be positive: / We obtain that / 0 if and only if 100. It is straightforward to check that if F=104, in perfect competition (regulated market) the zoo would close (since it would have negative profits equal to 4), while if the marked were not regulated, the zoo would open since profits in the long run would be / 104, (recall that 104,17 are the profits obtained previously and 104 are the fix costs). 5.c) If fix costs are lower or equal to 100, the situation of perfect competition maximizes the sum of surpluses and then, the government cannot impose a better tariff. However, perfect competition expulses firms from the market when F>100, since its profits are negative in the long run. Then, we should impose a condition that, independently on the costs, the firm will obtain 0 profits in the long run (we will have a corner solution since the truly maximum cannot be implemented).
8 0 This new expression for the price is the one that the government will use to impose the new tariff and then equates to the demand: / / Therefore, if F=104 we have solutions: 8. and 8 with associated prices of 0. and 1. In both cases the profit of the firm is 0 and therefore 104. If we compute the surpluses of the consumer for (, ) and (, ) we obtain that, =18,77 and, =1. Then the consumer surplus (as the sum of surpluses) is maximized in 0., that would be the price that the government must set. If we compute the sum of surpluses of consumer and producer for this regulation and for the non regulated monopoly, we get: 1) Regulated monopoly: ) Non regulated monopoly: EXERCISE Firstly, we aggregate the demands, since it will be useful during the exercise. Adding up quantities we get that: 40 And isolating p, we obtain that the inverse demand function is: 40 Moreover, since we know that the marginal cost is constant and equal to, integrating the function we obtain the cost function is:.1) In case the monopolist is not discriminating, we take into account both groups. Then, it maximizes: 40 Solving as we did in the previous exercises, we obtain that in equilibrium: Now suppose m=10. In such a case, substituting the previous functions we obtain that:
9 And introducing this price in the individual demand functions we obtain that: Compute also the surplus of the consumers: = Now suppose m=. In this case, substituting in the previous functions we obtain that: As the price is higher than m, the monopolist will not sell to agents of group. Then, it reformulates the problem with only group And solving we obtain that in equilibrium 18, 11. Computing also de consumer surplus for each group: = 81 =0.) Now the monopolist can discrimínate among groups. Then, its problem becomes:, 40 The first order condition with respect to is: 0 Which implies 18 And using the inverse demand function of the first group: 11 Moreover, the consumer surplus of the first group is: For the second group, we see that the first order condition with respect to is: Which implies 1 Now we separate cases: Case 1: m=10 Introducing the value in the corresponding functions we get that:
10 Case : m= Introducing the value in the corresponding functions we get that: 4 4.) We compute the aggregate consumer surplus (we sum groups for all the cases analyzed above separately) so that we have: With no discrimination: With discrimination: In both cases, consumers are better off with discrimination. EXERCISE 7 Firstly we aggregate the demand functions. To do so, we sum quantities (demand functions) and obtain that: 10 And the inverse demand function is: 10 7.a) The problem of the monopolist when there is no discrimination is: 10 4 Solving exactly as we did in the previous exercises, we get that: 9, 19.5 In order to obtain the quantity demanded for each of the countries, plug the obtained price in each demand functions: Graphically:
11 7.b) This is a third degree price discrimination, since the monopolist can identify the different groups. In particular, applying different prices to different countries is called dumping. The problem of the monopolist is:, Solving this problem exactly as we did in the previous exercises, we get that: 9 54 And introducing the values in the respective inverse demand functions we obtain that: 17 7.c) With price discrimination, the efficiency loss is lower, since more consumers get access to the good. Although the surplus is appropriated by the firm, this also computes to measure efficiency.
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