Lecture Notes 3.1 Static Oligopoly (PDF)

4: STATIC OLIGOPOLY MARTIN CRIPPS In this topic we take the tools of strategic form games (in particular Nash equilibrium and dominance) to analyse models of industries where there are many firms who are in competition with each other. There is no single “co...

4: STATIC OLIGOPOLY MARTIN CRIPPS In this topic we take the tools of strategic form games (in particular Nash equilibrium and dominance) to analyse models of industries where there are many firms who are in competition with each other. There is no single “correct” model of these kinds of industry. Here we will look at four models of Oligopoly, but there are others you should be aware of. 1. Cournot To understand how Cournot’s model of oligopoly works it is best to work through an example. Before doing this we imagine a world where firms are producing exactly the same product (homogenous product) and the decision they must take is how much output to produce. By producing more output they drive down the market price for themselves and all other firms in the industry. (We say this negative effect of one firm on others in the industry is a “negative externality”.) One interpretation of this model is it describes competition in the long run. That is when firms are deciding what factories to build. It is then, presumably, that they make their quantity decisions. 1.1. Example of Two-Firm Cournot Oligopoly. In the example we imagine there are only two firms in the industry each producing exactly the same good (water or oil for example). The demand curve for their product is P = 30 − Q, where Q is the total output produced by all the firms and P is the market price. Thus we could write P = 30−(Q1 +Q2 ), where Q1 and Q2 are the individual output of firm 1 and firm 2 respectively. Each of the two firms has the same costs of production with a constant Marginal Cost of 6: C(Q1 ) = 6Q1 and C(Q2 ) = 6Q2. (A quicker way of writing this is C(Qi ) = 6Qi for i = 1, 2.) With this information it is possible to write down the firms’ profits (or payoffs) in this game: π1 (Q1 , Q2 ) = P Q1 − C(Q1 ) = (30 − (Q1 + Q2 ))Q1 − 6Q1 , π2 (Q1 , Q2 ) = P Q2 − C(Q2 ) = (30 − (Q1 + Q2 ))Q2 − 6Q2. What is different from Monopoly is that each firm’s profit depends on the output of another firm. To summarise the game: Each firm chooses and output level Qi ≥ 0. These choices are made simultaneously. The payoffs these choices are given by the firms’ profits π1 and π2 in the above equation. Our tool to analyse this situation will be to find a Nash equilibrium. And because of the large number of actions the firms have the method of finding the Nash equilibrium will 1 2 MARTIN CRIPPS be to find the firm’s best responses. We called this “Method 4 The Intersection of Best Responses”. 1.1.1. Finding the Best Responses. Here are two ways of calculating Firm 1’s best response to a given output of Firm 2. First write down firm 1’s profit. π1 (Q1 , Q2 ) = (30 − (Q1 + Q2 ))Q1 − 6Q1 Observe that this is a quadratic function of Q1 os its maximum can be found by differen- tiating and setting equal to zero. ∂π1 = 30 − (2Q1 + Q2 ) − 6 = 24 − Q2 − 2Q1 = 0. ∂Q1 Solving this for Q1 gives 1 24 − Q2 = 2Q1 , orQ1 = 12 − Q2. 2 Thus firm 1’s optimal choice of output is a linear and decreasing function of firm 2’s output choice. A second and more Economist-y approach to this would be to say that Firm 1’s Revenue = P Q1 = [30 − Q1 − Q2 ]Q1. So we differentiate this to get its Marginal Revenue dR MR = = 30 − 2Q1 − Q2. dQ1 We also know that the firm’s Marginal Cost = 6 from above. So to find where the firm maximises profit we would set MC equal to MR, that is, 1 MC = 6 = 30 − 2Q1 − Q2 = MR or Q1 = 12 − Q2. 2 This tells us firm 1’s best output as a function of firm 2’s output. This relationship is important and is often plotted by economists. Here’s a picture In#pictures:- Q2" Q1"="12"–"0.5"Q2" Q1" The monopoly output is where Q =0 2 There are three points worth emphasizing about this picture #- (1) It usually slopes down, because as my competitor produces more, there is less demand less for me and as a result of the9" reduction in demand it is optimal to produce less.  4: STATIC OLIGOPOLY 3 (2) The intercepts are informative. When Q2 = 0 and firm 1 produces its optimal response it is alone in the market. Thus Firm 1’s optimal response to Q2 = 0 is the monopoly level of output. In this example 12 is the output of a monopolist. (3) The intercepts are informative (again). When Q2 is very high, Firm 1 wants to produce zero output, because it makes zero profit. (In fact when Q = 24 this is true.) Zero profit is actually a characteristic of Perfect Competition. Thus, in this example Q = 24 is the perfectly competitive output. Notice that everything is about Firm 1 is true for Firm 2. So when we try to calculate Firm 2’s Best Response we will get a very similar kind of equation: 1 Q2 = 12 − Q1. 2 Just the names of the outputs have been swapped. 1.1.2. Finding the Cournot/Nash Equilibrium. We now have two reaction functions of the firms: Q1 = 12 − 12 Q2 and Q2 = 12 − 12 Q1. If we are going to find a Nash equilibrium of this game we are going to!Best findresponse where they intersect. functions This intersect is usually at the Nash called a Cournot, or Cournot-Nash equilibrium, because equilibrium"Cournot found this solution to the model many many years before Nash was even born. Here’s the picture of the two reaction functions: Q2# Cournot#(Nash)#Equilibrium# Q2#=#12'0.5Q1# Q1# 15# To find the Cournot equilibrium we must substitute one reaction function into the other 1 Q1 = 12 − Q2 2 1 1 Q1 = 12 − [12 − Q1 ] 2 2 3 Q1 = 6 4 Q∗1 = 8 Performing a similar calculation for Firm 2 we get Q∗1 = Q∗2 = 8. There is a short cut that we can use in this particular case because the firms have the same costs. In this case we know the equilibrium will be symmetric Q∗1 = Q∗2. So instead of substituting firm 2’s reaction function into firm 1’s reaction function we can just substitute 4 MARTIN CRIPPS Q2 = Q1 and get the right answer: 1 Q∗1 = 12 − Q∗2 2 1 Q∗1 = 12 − Q∗1 2 3 ∗ Q = 12 2 1 Q∗1 = 8 This method of find the Cournot equilibrium can make the algebra a lot easier, but you must remember this only works when firms have the same costs and de- mand! 1.2. Generalisation of this Approach to Many Firms. There is nothing particularly difficult about extending the Cournot model to many firms. (Although drawing the pictures gets quite hard.) Suppose now that we have an industry with N identical firms—we will give them the names i = 1, 2,... , N. Firm i has to choose an output qi ≥ 0 for i = 1, 2,... , N. These choices are made simultaneously. Now we must write down the firms’ profits. Each firm has the marginal cost c, or total costs cqi. The final ingredient we need is the demand function and we will assume that Q units can be sold if the price satisfies Q = D − p (so the total amount demanded is always less than D). Of course there are many firms so the price is a function of the total output N X X p = D − (q1 + q2 + · · · + qN ) = D − qi = D − qi − qj. i=1 j6=i This writes the relationship between prices and output in three identical way. For our purposes the last is the most useful. It says price equals D minus P firm i’s output and the output of all the other firms. It will be useful to write Qi = j6=i qj as the output of all firms apart from firm i. Now we are in a position to write down firm i’s profit. πi (qi , Qi ) = pqi − cqi = (D − Qi − qi )qi − cqi. To find Firm i’s best response we want to maximise this by choosing the best qi. So, as above, we differentiate and set equal to zero. ∂πi = D − Qi − 2qi − c = 0. ∂qi Setting this equal to zero gives qi∗ = (N − Qi − c)/2. If we do this for each firm we will get N equations in N unknowns: q1∗ = (D − Q1 − c)/2, q2∗ = (D − Q2 − c)/2,... ∗ qN = (D − QN − c)/2 This can be solved directly, but it is much easier to use the trick we talked about in the earlier section. Here all the firms are the same and have the same costs. We would expect that the Nash equilibrium had all the firms producing the same output q1∗ = q2∗ = · · · = qN ∗.  4: STATIC OLIGOPOLY 5 So let us substitute this into Firm 1’s reaction function. With this assumption, the output of all firms apart from firm 1 can Q1 = N ∗ ∗ P i=2 i = (N − 1)q1 can be calculated. Hence we q get q1∗ = (D − Q1 − c)/2 = (D − (N − 1)q1∗ − c)/2. This solves for q1∗ to give D−c qi∗ =. N +1 Hence as N increases each firm’s output tends to 0, there are more and more firms and each becomes vanishingly small relative to the size of the market. We would hope this would look something like perfect competition, and we can check this by finding out how the price behaves as N increases. The price will satisfy X D−c Nc + D p∗ = D − qi∗ = D − N = →c as N → ∞. N +1 N +1 Here the price tends to Marginal Cost as the number of firms grows. Thus Cournot competition starts to look a lot like perfect competition when there are a lot of firms. 2. Bertrand Competition This model of competition is similar to Cournot’s in that it assumes that firms are producing exactly the same product. However, it assumes that firms choose the price of their product (not how much to produce). This model of price competition is a good description of what goes on in financial markets or of the price wars that many industries experience from time to time. One interpretation of this is that it describes what happens in the short run, rather than the long run. To be precise there are three assumptions in Bertrand competition: (1) The lower-priced firm always claims the entire market. (Firms produce identical products.) (2) All competition is in prices. (3) If the firms set equal prices they will share the market. Again it is easiest to understand Bertrand competition by thinking about an example. 2.1. An Example of Bertrand Competition. There are two ferry companies (it is more usual to consider airlines but I am quite find of boats). They serve the same route offering identical service, the only difference between the two companies is the price they charge. The cost per customer is 30 for both companies. There are 1000 customers who are willing to pay up to 50 to make the crossing. Company 1 charges a price P1 and company 2 charges P2. First let us write down the profits of Firm 1. If it is the low price firm it gets all 1000 customers and has profit π1 = 1000(P1 − 30), P1 < P2. If it has the same price as Firm 2 it gets half of 1000 customers and has profit π1 = 500(P1 − 30), P1 = P2. 6 MARTIN CRIPPS If it has a higher price than Firm 2 it gets no customers and has profit π1 = 0(P1 − 30) = 0, P1 > P2. B.#Nash#Equilibrium:#Bertrand#Competition8 Here is a picture of this function (remember M C = 30). Firm#1;s#Profit8 Firm 1's Profit Firm 1's Price MC P2 The most important thing we notice from this is a problem. Firm 1’s profit is highest if it just undercuts firm 2’s price. But it also wants to charge as high a price as possible. Thus what Firm 1 would like to do is to charge the highest30# price just below firm 1’s price, but no such price exists. Thus, in this game (with continuous prices) the best response function of firm 1 does not exist! The question then is: can we find a Nash equilibrium if we cannot find where the best responses intersect? Well the answer is yes, by telling a story. Suppose, the firms set equal prices of 50 and share the market (which is clearly a good situation). The profit of each firm is 500(50 − 30) = 10, 000. If firm 1 cut its price to 49, then it can attract all the customers and make profit 19, 000 = 1000(49 − 30). In response firm 2 can undercut firm 1 by asking the price 48 and attracting the whole market. This process continues until both firms are charging the price 30 and making zero profit. At this point no firm benefits by undercutting their rival and the firms do actually have a best response. Summary: The Nash equilibrium is at (P1 , P2 ) = (30, 30), although there is no best response function. 2.2. General Properties of Bertrand Competition. Here is a list of points that should be noted: Pure price competition drives oligopoly to look very much like perfect competition. The Nash equilibrium of the game has the firms setting prices equal to marginal costs. This is why we believe markets with price competition such as financial markets may be quite efficient. Price competition does not expand the market in the above example—the demand was always 1000 for low prices. As firms cut prices, one would usually expect more customers to want to buy the good. This expansion of the size of the market will increase the temptation of the firms to undercut their rival’s price. When prices are not allowed to vary continuously the problem of the non-existence of a best response goes away (see the next section). 2.3. Extensions to the Basic Model of Bertrand Competition.  4: STATIC OLIGOPOLY 7 2.3.1. Integer Prices. A lot of the problems with Bertrand competition came from the fact that firms were allowed to continuously vary their prices. If we assume firms are only able to charge whole number prices then there is no problem with finding the biggest price less than some number. It also increases the number of Nash equilibria of the game. Here are the firms profits for the prices 30, 31, 32, 33, in the model of ferry competition above. P2 = 30 P2 = 31 P2 = 32 P2 = 33 P1 = 30 (0,0) (0,0) (0,0) (0,0)... P1 = 31 (0,0) (500,500) (1000,0) (1000,0)... P1 = 32 (0,0) (0,1000) (1000,1000) (2000,0)... P1 = 33 (0,0) (0,1000) (0,2000) (1500,1500).................. If you apply the underlining method to this game you will find that it has 3 Nash equilibria: (P1 , P2 ) = (30, 30), (P1 , P2 ) = (31, 31), (P1 , P2 ) = (32, 32). This increase in the number of Nash equilibria arises because now you really have to make a big change in your price if you are going to undercut your rival. This big change in the price might hurt your profits more than the increase in customers you experience from being the low-price firm. 2.3.2. Price Guarantees. If firms advertise deals like “if you find this good cheaper any- where else we will refund twice the difference” then there is a strange effect. Suppose the firms set prices (40, 41) where would the customers go? If they go to the firm with the sticker price of 40, that is what they will pay. If they go to the firm with the sticker price of 41 they can claim a refund of twice the difference in prices and in fact pay only 39. Thus all the customers would prefer to go to the higher price firm and claim a refund. The sensible response of the firm setting the low price (40) is to raise its sticker price, so the customers come to it and claim the refund. If such guarantees are in place we would expect prices to increase rather than decrease. These guarantees are collusive, although they look like they are good for the consumer. 2.3.3. Capacity Constraints. If firms are unable to fit all 1000 customers on their ferry it never makes sense to cut prices down to Marginal Cost. Instead prices seem to go around in circles. These circles are called Edgeworth cycles. Consider the following slight change in our example. Suppose that only 800 customers will fit on a ferry. So that when you are the low-price firm your profits are 800(P − 30) and when you are the high-price firm your profits are 200(P − 30). The high price firm always gets to serve those who cannot get on the low-price ferry. Suppose now that firm 1 sets the price of 35. Firm 2 can undercut this and set the price of (say) 34 making the profit 800(34 − 30) = 3200. Or firm 2 can embrace being the high-price firm and charge the price of 50 and make the profit 200(50 − 30) = 4000. Clearly the best response is to set the price P2 = 50. Now (of course) firm 1 will respond by raising prices to P1 = 49 and competition will drive prices back down again to 35 where the cycle begins again. 8 MARTIN CRIPPS 3. Differentiated Product Duopoly If firms are not producing identical products, then we say their products are differen- tiated. When products are differentiated it will not be the case that the low-price firm gets all the market, because some consumers just prefer the product of the high-price firm. However, if products are substitutes we would expect their prices to have an effect of the demand for each other. Thus the firms will be playing a game in the choice of price they make. This is what we study here. Again we will do this by working through an example. 3.1. Example of Price Competition with Differentiated Products. There are two firms producing different goods. P1 is the price chosen by Firm 1 and P2 is the price chosen by Firm 2. We will first describe the demand functions of each firm. Firm 1 : Q1 = 12 − 2P1 + P2 Firm 2 : Q2 = 12 + P1 − 2P2 Notice that each firm’s demand is decreasing in its own price but increasing in its rival’s price—they are producing substitute goods and are in competition with each other. We will ignore costs here and suppose that all the costs are fixed costs. Fixed Costs = 20. The first step in defining this game is to write down the Firms’ actions. These are the prices they choose (P1 , P2 ). The next step is to write down the Firms’ payoffs or profits. Firm 1’s Profit : = P1 Q1 − Cost = P1 (12 − 2P1 + P2 ) − 20 Firm 2’s Profit : = P2 Q2 − Cost = P2 (12 − 2P2 + P1 ) − 20 This completes the formal description of the game that is being played here. Now we must find a Nash equilibrium of this game and again we will use the Reaction Function method. We begin by finding the best P1 for firm 1 given it knows the price of firm 2. To do this we maximize firm 1’s profit by differentiating and setting equal to zero. dπ1 = 12?4P1 + P 2 = 0 dP1 This solves to give 1 P1 = 3 + P2. 4 A similar process for firm 2 will also find its optimal choice of P2 as a function of P1 : 1 P2 = 3 + P1. 4 Note here these reaction functions are upward sloping. As my rivals price goes up it is optimal for me to increase my price too. (I can still undercut by raising my price and an increase in my rivals price has a positive effect on my demand.) To find the Nash equilibrium (P1∗ , P2∗ ) we must substitute one reaction function into the other. 1 1 P1 = 3 + (3 + P1 ). 4 4  C.#Nash#Equilibrium:#Differentiated#Product# 4: STATIC OLIGOPOLY 9 Duopoly< This solves to give (P1∗ , P2∗ ) = (4, 4). Here is a picture of what we have just done. The#Firms’#Reaction#Curves#or#Optimal#Responses< #< #< P2# P1 = 3 + 0.25P2 #< #< #< #< P2 = 3 + 0.25P1 #< #< #< # < < P1# < The#Nash#equilibrium#is#at#(P1,P2)=#(4,4).##< 47# 4. A Model of a Single-Unit Auction All the previous models were games where a few sellers compete to sell to a set of buyers. Now we will look at the reverse position where a few buyers compete to acquire a good from a single seller. First we will describe the buyers. The buyers? values for the good are written as (v1 , v2 ,... , vN ) where buyer I’s value, vi is in the interval 0 ≤ vi ≤ 1. We will suppose these values are random and that the buyers only know their value but not the value of the others. The seller does not know the values (v1 , v2 ,... , vN ) and so has a random or unknown demand curve. How the (v1 , v2 ,... , vN ) are determined has a big effect on the nature of competition that among the buyers. Here are some different assumptions that might be made: Independent Symmetric Private Values: vi is drawn independently from the den- sity f (v) on [0, 1]. All buyers have the same distribution of values for the good. Independent Private Values: vi is drawn independently from the density fi (v) on [0, 1]. Buyers have distinct views about the good. Here is the order of events in the auction (1) The players observe their own values and no-one else?s. (2) Then they submit a bid. (3) The rules of the auction determines payoffs. Here a strategy for a buyer is to describe how they should bid for each different value they observe. Thus players’ strategies are bidding functions that takes the player’s value and maps it to a bid. bi : Observed Value → Bid bi : [0, 1] → [0, ∞) vi 7→ bi (vi ) Here is a picture of the bidding function 10 D:#Model#of#a#Single/Unit#Auction5 MARTIN CRIPPS #5 #b# bi#(.)# bi#(vi)# v## 0# vi## 1# We have to find a whole function to describe a player’s equilibrium strategy. To make this easier we will only look for certain kinds of equilibria. That is, equilibria where bids are strictly increasing functions of values. 4.1. Equilibrium in Second-Price Auctions. The rules of the auction determine who wins and who pays what. The easiest and simplest set of auction rules to analyse are second-price auctions. These are auctions where the person submitting the highest bid gets the object, but they pay a price equal to the second highest bid (not their own highest bid). In an auction where the price paid is the second highest bid, the strategy bi (vi ) = vi (that is submit a bid equal to your value), weakly dominates all other strategies. This is a famous result due to Vickrey and is much used in Economics. We now explain why this is true. We first consider the possibility of bidding above your value. That is, overstating how you feel about the good. Suppose you have a value vi and consider your payoffs for a bid b0 > vi. Your payoff only depends on the highest bid from the other players call this B. We will deal separately with the cases where: (1) the highest bid from everyone else is above your new contemplated bid B > b0. (In which case you always lose the auction whether you bid truthfully or exaggerate to b0 ). (2) The highest bid from everyone else falls between your value and your increased bid b0 > B > vi. (In which case bidding truthfully causes you to lose the auction and get zero while bidding b0 causes you to win the auction and pay B > vi so you get a negative payoff.) (3) The highest bid from everyone else is below your value vi > B. (In which case you win the object whether you bid vi or b0 and the price you pay B is independent of your bid. This gives the matrix of payoffs below. You can see that the top row is weakly dominated by the bottom row. B > b0 B ∈ (vi , b0 ) B < vi 0 Bid b Lose = 0 Win = vi − B < 0 Win = vi − B > 0 Bid vi Lose = 0 Lose = 0 Win = vi − B > 0 Now we consider the possibility of bidding below your value. That is, understating how you feel about the good. Suppose you have a value vi and consider your payoffs for a bid  4: STATIC OLIGOPOLY 11 b00 < vi. We will deal separately with the cases where: (1) the highest bid from everyone else is above your value B > vi. (In which case you always lose the auction whether you bid truthfully or understate your bid). (2) The highest bid from everyone else falls between your value and your understated bid b00 < B < vi. (In which case bidding truthfully causes you to win the auction and pay a price B < vi giving positive profit while bidding b00 causes you to lose the auction and get nothing.) (3) The highest bid from everyone else is below your understated bid b00 > B. (In which case you win the object whether you bid vi or b00 and the price you pay B is independent of your bid. This gives the matrix of payoffs below. Again you can see that the top row is weakly dominated by the bottom row. B > vi B ∈ (b00 , vi , ) B < b00 Bid b00 Lose = 0 Lose = 0 Win = vi − B > 0 Bid vi Lose = 0 Win = vi − B > 0 Win = vi − B > 0 Hence we can conclude: it is a weakly dominating strategy to bid truthfully in a second price auction.