EconS Oligopoly - Part 3

Transcription

1 EconS Oligopoly - Part 3 Eric Dunaway Washington State University eric.dunaway@wsu.edu December 1, 2015 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

2 Introduction Yesterday, we looked at how rms compete when they simultaneously choose quantities. Today, we ll nish up our unit on Oligopoly by looking at a few extensions of our models. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

3 Asymmetric Firms So far, in all of our analysis of oligopoly, we have assumed that the rms are identical. This makes the math quite easy, and allows us to have the same solution for each rm. Unfortunately, in the real world, this rarely happens. Firms are typically di erent, either in the structure of their demand curves or in the costs they face. Let s look at what happens when rms face di erent costs. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

4 Asymmetric Firms What would happen if the rms faced di erent marginal costs? Now, they are no longer symmetric, so we should expect that they produce di erent quantities in Cournot competition. Intuitively, the rm with lower costs should have a higher output level and higher pro ts. The good news is that nothing changes mathematically. We follow exactly the same steps we used before to solve the problem. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

5 Asymmetric Firms Let s use the same inverse demand function that we have been using thus far p = 70 q 1 q 2 where now rm 1 faces a constant marginal cost of MC 1 = 10 and rm 2 faces a higher constant marginal cost of MC 2 = 20. Which rm should have the higher output level (quantity)? Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

6 Asymmetric Firms p = 70 q 1 q 2 MC 1 = 10 MC 2 = 20 Let s start with rm 1 s best response function. First, we need their marginal revenue, which we get from the total revenue equation. TR 1 = pq 1 = (70 q 1 q 2 )q 1 = 70q 1 q1 2 q 1 q 2 MR 1 = 70 2q 1 q 2 (Note that it s exactly the same as in the Cournot problem) Next, we set rm 1 s marginal revenue equal to its marginal cost. MR 1 = MC q 1 q 2 = 10 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

7 Asymmetric Firms 70 2q 1 q 2 = 10 Now, we solve this expression for q 1 to obtain rm 1 s best response function, 2q 1 = 60 q 2 q 1 = q 2 (Again, it s exactly the same as in the Cournot problem) Intuitively, nothing about rm 1 has changed. They face the same marginal revenue and same marginal cost as before, so they will not want to change anything about their behavior. What about rm 2, though? Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

8 Asymmetric Firms q 1 q 1 = q 2 30 BR 1 (q 2 ) 60 q 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

9 Asymmetric Firms p = 70 q 1 q 2 MC 1 = 10 MC 2 = 20 Now, we do the same thing for rm 2. Starting with their marginal revenue, TR 2 = pq 2 = (70 q 1 q 2 )q 2 = 70q 2 q 1 q 2 q 2 2 MR 2 = 70 q 1 2q 2 (Again, it s the same as in the Cournot problem. The marginal revenue hasn t changed) And then we set it equal to rm 2 s marginal cost, MR 2 = MC 2 70 q 1 2q 2 = 20 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

10 Asymmetric Firms 70 q 1 2q 2 = 20 Next, we solve this expression for q 2 to obtain rm 2 s best response function, 2q 2 = 50 q 1 q 2 = q 1 which is di erent from what we saw in the Cournot problem. Intuitively, rm 2 is facing higher costs now. No matter what rm 1 produces, rm 2 will want to produce less because of those costs. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

11 Asymmetric Firms q 1 q 1 = q 2 50 BR 2 (q 1 ) 25 q 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

12 Asymmetric Firms Now, we do the same thing as before, and nd our solution where the two best response functions intersect. This is because if the best response function for rm 1 and the best response function for rm 2 align, we are in equilibrium. Graphically, we just nd where our two best response functions intersect in our gure. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

13 Asymmetric Firms q 1 q 1 = q 2 50 BR 2 (q 1 ) BR 1 (q 2 ) 60 q 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

14 Asymmetric Firms Notice this time that our intersection does not lay on the q 1 = q 2 line, as it has before. This is a direct result of the asymmetry between the rms. Mathematically, we can solve our system of two equations and two unknowns to nd that intersection point Rearranging terms a bit, q 1 = 30 q 2 = q q 1 q q 2 = q 1 + q 2 = 25 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

15 Asymmetric Firms q q 2 = q 1 + q 2 = 25 Let s multiply the botton equation by 2 q q 2 = 30 q 1 2q 2 = 50 and now add the two equations together, q q 2 q 1 2q 2 = q 2 = 20 q 2 = Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

16 Asymmetric Firms q 2 = Lastly, we can plug q2 into rm 1 s best response function to get our answer q = 30 2 q 2 = 30 = Wow, rm 1 produces quite a bit more than rm 2. Their cost advantage lets them take a much larger share of the market! Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

17 Asymmetric Firms q 1 q 1 = q 2 50 BR 2 (q 1 ) BR 1 (q 2 ) q 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

18 Asymmetric Firms We can also calculate pro ts for each rm. This rst requires that we calculate the price, though p = 70 q 1 q 2 = 70 Starting with rm 1, = TR 1 = p q1 = 33.33(23.33) = TC 1 = MC 1 q1 = 10(23.33) = π1 = TR 1 TC 1 = = and for rm 2, TR 2 = p q2 = 33.33(13.33) = TC 2 = MC 2 q2 = 20(13.33) = π2 = TR 2 TC 2 = = Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

19 Asymmetric Firms As we can see, rm 1 had both a higher output and pro t level than rm 2 had. In fact, rm 1 did even better than it did under the standard Cournot model, since rm 2 reduced their output by a lot. Firm 2 ended up worse o, produce much less and earning much less in pro ts. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

20 Stackelberg Competition The Stackelberg model was developed in the mid twentieth centry by German economist Heinrich Freiherr von Stackelberg. He started with the Cournot model, and let one rm move rst (The leader) and the rest of the rms act as followers. In our model, we will assume that rm 1 is the Stackelberg leader rm and gets to move rst, setting its level of output. Then, rm 2 is able to observe rm 1 s output level and respond with its own output level. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

21 Stackelberg Competition Why does one rm get to move rst? Typically, the leader rm ( rm 1 in our case) is much larger than the other rms, or it has some kind of control over the market that the other rms wait to see what the leader does. We are going to see that the leader rm gets a huge advantage from moving rst. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

22 Stackelberg Competition Remember from our Game Theory unit that we want to use backward induction to nd an equilibrium. We start at the bottom of the game tree and gure out the best response for rm 2. Then, we substitute that best response in for rm 1 and solve it again. Since we are dealing with a whole spectrum of quantities, we really can t draw the game tree in this case, but mathematically, we are going to follow the same process. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

23 Stackelberg Competition Let s return to our same example, p = 70 q 1 q 2 MC = 10 In order to use backward induction, we want to solve for rm 2 s best response function. To get this, we rst need his marginal revenue, which we get by applying the power rule to rm 2 s total revenue. TR 2 = pq 2 = (70 q 1 q 2 )q 2 = 70q 2 q 1 q 2 q 2 2 MR 2 = 70 q 1 2q 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

24 Stackelberg Competition Next, we set rm 2 s marginal revenue equal to its marginal cost, and solve for q 2 to get rm 2 s best response function. MR 2 = MC 70 q 1 2q 2 = 10 2q 2 = 30 q 1 q 2 = q 1 Notice that this best response function is identical to what we found under Cournot competition. This should be the case for the follower. From their perspective, nothing has changed. They are going to observe a quantity from rm 1 and then optimally respond to it. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

25 Stackelberg Competition If we were working with a game tree, now we would substitute these results up the tree and get a reduced form game. Firm 1 would just pick which of its strategies gave the highest payo, knowing how rm 2 is going to respond. We can still do this mathematically. Firm 1 needs to calculate its marginal revenue. Starting with total revenue, TR 1 = pq 1 = (70 q 1 q 2 )q 1 = 70q 1 q 2 1 q 1 q 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

26 Stackelberg Competition Before rm 1 calculates its marginal revenue, it will want to incorporate what it knows about rm 2. It knows that rm 2 s quantity is going to be based o its best response function. q 2 = q 1 This gives rm 1 an advantage. It can nd its optimal quantity based on this information. We can substitute this best response function in for q 2 in the total revenue equation. TR 1 = 70q 1 q1 2 q 1 q 2 = 70q 1 q1 2 1 q q 1 = 70q 1 q q q2 1 = 40q q2 1 and, applying the power rule, MR 1 = 40 q 1 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

27 Stackelberg Competition A little bit of intuition on this step: There are some subtle di erences between how the variable q 1 behaves before and after we apply the power rule. Before we apply the power rule (The Total Revenue Level), q 1 is just a regular variable, and rm 1 wants as much information about the variable as possible. This is why the rm wants to substitute the best response function in at this level. After we apply the power rule, q 1 is now an equilibrium variable. Think about it as before and after the rm makes its decision. Firm 1 needs the information about the best response function before it chooses its optimal level of output. Thus, we need to insert the best response function before we apply the power rule. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

28 Stackelberg Competition From here, we just need to set rm 1 s marginal cost equal to its marginal revenue. MR 1 = MC 40 q 1 = 10 q 1 = 30 Then, we plug this value in to rm 2 s best response function q 2 = q 1 = 30 1 (30) = 15 2 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

29 Stackelberg Competition Firm 1 produces double the amount of rm 2! In actuality, rm 1 is producing the monopoly market quantity. This is no coincidence, as rm 1 will use its market power to take full advantage of its pro ts. After taking the lion s share, rm 2 is left with the scraps, and produces half as much as rm 1. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

30 Stackelberg Competition To nd the market price, we just plug our equilibrium values back in to the inverse demand curve, p = = 25 And now, we can calculate the total revenue, total costs, and total pro ts for each rm: TR 1 = p q 1 = 25(30) = 750 TC 1 = 10q 1 = 10(30) = 300 π 1 = TR 1 TC 1 = = 450 TR 2 = p q 2 = 25(15) = 375 TC 2 = 10q 2 = 10(15) = 150 π 2 = TR 2 TC 2 = = 225 Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

31 Stackelberg Competition Let s compare this with our results from yesterday. For rm 1, the pro t level is the same as in the cartel. This also means that by being the leader, rm 1 receives more pro ts than it would by moving simultaneously in Cournot Competition. This is known as rst mover advantage. For rm 2, it s bad. Its pro t level is lower than both the cartel and Cournot levels. The price is lower than both the cartel and Cournot levels, which is nice for the consumers. This also implies that total pro ts are less under Stackelberg than they are for both the cartel and Cournot. More consumer surplus! Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

32 Welfare Comparisons Let s look at some welfare comparisons before we move on to our last topic for the day. For simplicity, I will just be showing the results graphically, but all of these values for consumer surplus, producer surplus, and dead weight loss could easily be calculated using triangle and rectangle formulas. I have included the actual values on the gures. Practice calculating them on your own. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

33 Welfare Comparisons p 70 D Cartel / Monopoly Cournot Stackelberg Bertrand / Competitive S Q Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

34 Welfare Comparisons p 70 D CS 450 PS 900 DWL 450 Cartel / Monopoly S Q Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

35 Welfare Comparisons p 70 D CS 800 PS Cournot DWL S Q Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

36 Welfare Comparisons p 70 D CS PS Stackelberg DWL S Q Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

37 Welfare Comparisons p 70 D CS 1800 Bertrand / Competitive S Q Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

38 Welfare Comparisons As we can see, with increased competition, the consumer surplus rises and both the producer surplus and dead weight loss fall. The consumers are able to take advantage of the in ghting among the rms. At the perfectly competitive level, both the producer surplus and dead weight loss are eliminated, and the consumers reap all of the surplus from the market. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

39 Price Match Guarantee To conclude this unit, I wanted to talk about an interesting application: the price match guarantee. Several companies in the US o er to match their competitors prices, guaranteeing that their customers will always get the best deals! But does this actually make consumers better o? Let s go back to Bertrand competition and see what happens when both rms o er a price match guarantee. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

40 Price Match Guarantee A few things to remember about Bertrand competition: The best response for each rm is to undercut the other rm by one penny (as long as the price is between marginal cost and the monopoly price). The equilibrium price is equal to marginal cost. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

41 Price Match Guarantee Now, we implement the price match guarantee. The consumer now is able to receive the lowest price between the two rms regardless of which rm he buys from. Let s see how the best response functions change. We ll look at rm 1 s perspective. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

42 Price Match Guarantee Using the same three segments as before. If rm 2 s price is above the monopoly price, it is best for rm 1 to respond with the monopoly price. This is due to pro ts being able to be increased by lowering the price. Both rm 1 and rm 2 get half of the market since rm 2 s customers just price match down to the monopoly price. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

43 Price Match Guarantee If rm 2 s price is between marginal cost and the monopoly price, rm 1 has 3 options. It can charge a price higher than rm 2 s price, and then rm 1 s customers will just price match to rm 2 s price, and both rms get half of the market. It can charge a price equal to rm 2 s price, and both rms get half of the market. It can charge a price lower than rm 2 s price, and then rm 2 s customers will price match to rm 1 s price, and both rms get half of the market. As opposed to the typical Bertrand model, undercutting is now the worst option. Firm 1 doesn t gain any customers, but it loses pro ts by charging a lower price! Firm 1 s best response is to charge any price between rm 2 s price and the monopoly price. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

44 Price Match Guarantee If rm 2 s price is below marginal cost, rm 1 is best to just not sell the product. If it sold the product at any price above rm 2 s price, customers could match it to rm 2 s price and rm 1 would take a loss. If it undercut rm 2, it would also take a loss. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

45 Price Match Guarantee It s hard to plot how the best functions are going to look, but we can describe them. If rm 2 starts with a price between marginal cost and the monopoly price, rm 1 will charge at least as high of a price as rm 2, if not higher. If rm 1 charges a price higher than rm 2 s, rm 2 will respond by raising their price to at least as high as rm 1 s price, if not higher. This "price creep" will continue until the market nds equilibrium with both rms charging the monopoly price! Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

46 Price Match Guarantee Price Match Guarantees lead to higher prices? Yes they do. By preserving their customer base regardless of their price, rms lose all incentive to undercut one another. They are able to bring in much higher pro ts at expense of the consumers. They also create signi cant dead weight loss. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

47 Summary When a rm is able to set their quantity rst, they can claim a large portion of the market at the expense of the other rm. Competition increases consumer surplus and lowers dead weight loss. Price Match Guarantees are meant to look like they help consumers, but really just discourage competition. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

48 Preview for Friday Risk and Uncertainty How do insurance premiums work? Perlo, Chapter 17. Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49

49 Assignment 7-5 (1 of 1) 1. Consider a two rm duopoly that faces an inverse demand curve of p = 150 q 1 q 2 and constant marginal costs of MC = 60. a. If rm 1 behaves as a Stackelberg leader ( rst mover) and rm 2 behaves as a Stackelberg follower (second mover), what are the equilibrium quantities, prices and pro ts for each rm? (Hint: They will not be equal!) Eric Dunaway (WSU) EconS Lecture 33 December 1, / 49