In Bertrand Competition with Identical Örms, We have Assumed that Örms do not Have Capacity Constraints: Microeconomics Assignment, SMU, Singapore

Bertrand Competition with Capacity Choices

In Bertrand competition with identical Örms, we have assumed that Örms do not have capacity constraints. That is, each Örm can serve the whole market, whereas in equilibrium each Örm only serves a half market. That is, half of the production capacity will never be used in equilibrium.

In the real world, building the production capacity is costly and Örms in most cases face the capacity constraint. This problem shows that, when Örms face the capacity constraint in supply, Bertrand competition will lead to the equilibrium outcome of Cournot competition. In this sense, these two models are equivalent.

Consider two identical Örms in a market. The market demand is given by q = D(p) and the inverse demand function is denoted by p = P(q). Each Örm incurs a constant marginal cost c in supplying the good but faces the capacity constraint. However, before competing in the market, an Örm I can invest in its production capacity ki at a per-unit cost (constant marginal cost ). That is, the timing of the game can be illustrated in the following two-stage game:

Stage 1: Each Örm i chooses its capacity ki simultaneously;

Stage 2: Each Örmís capacity is commonly known. Firms compete by setting prices simultaneously.

This is a dynamic game with complete information. The equilibrium can be solved by backward induction.

Suppose Örms have invested the capacity k1, k2 at stage 1. Consider the price competition at stage 2.

(a)  Suppose Örms set prices p1 and p2, what is Örm 1ís demand?
Assume that each Örm shares the market equally when they set the same price.

What is the equilibrium price at stage 2? (You need to show this is indeed an equilibrium!) What are each Örmís proÖts? (Hint: you need to consider several di§erent cases: (1) P(k1 + k2) > c and (2) P(k1 + k2)  c) Now back to stage 1 and consider each Örmís decision in capacity building.

(b). What is each Örmís optimal capacity in equilibrium? Assume D(p) = a p. Solve for the equilibrium capacity and price.

2. Cournot Competition with Mergers

Consider an industry consists of three Örms with identical costs C(q) = 18q + q2 . The Örms face a linear market demand function Q = 150 p. Firms compete in quantity as Cournot oligopoly.

(a). What is the equilibrium outcome of the price, the output and the proÖts for each Örm if the Örms?

Suppose now Firm 1 and Firm 2 propose a merger to be one Örm (call the merged entity as Firm L). Assume there is no extra cost for conducting the merger. After the merger, Örm L will compete with the remaining Firm 3 in quantity as Cournot duopoly.

(b). The merged Örm L can choose to produce using both original Örms plants or just using one Örm and closing another one. Which way leads to lower costs for the merged Örm?

(c). Suppose the merged Örm adopts the lower-cost production after the merger. What are the proÖts of the merged Örm L and the stand-alone Örm 3 after the merger? Comparing Örm Lís proÖt with the aggregate proÖts of Örms 1 and 2 before the merger, would it pay for Firm 1 and Firm 2 to merge?

3. Price Discrimination:

. James opens a pub at Monash University Clayton James faces two types of customers: students and sta§s (adults). Using his economics knowledge, James estimates the demand for drinks (mainly beers) by a typical student is QS = 18 3p, and the demand by a typical adult sta§ is QA = 10 2p. There is a roughly equal number of students and sta§s to the pub, which can be normalized.

1 (that is, assume there is one customer in each group). The marginal cost of each drink is $2:

James wants to practice his knowledge of price discrimination from his honors unit ECC4650. He adopts the following pricing schemes step by step and compares the proÖts under di§erent pricing schemes.

(a). In the beginning, James o§ers a uniform price for all customers. What price should he set? What is his total proÖt?

(b). James is reminded by his lecturer of the economics of two-part tari§s. He launches a new pricing scheme. Customers need to pay a permit fee T; in addition, they need to buy tokens for drinks. Each token is charged a price p and can be used for one drink. What is the optimal price for a token and the permit fee? What is the total proÖt?

(c). James can separate two groups of customers by checking their Monash ID cards. He can set di§erent two-part tari§s for di§erent groups. He issues a blue permit for the student customer with fee T S and a red permit for the sta§ customer with a fee T A. Students need to buy blue tokens for drinks and each token is charged a price p S ; sta§s need to buy red tokens at a per-unit price p A. What fees T S and T A should James charge? What prices for tokens p S and p A should he set? What is his total proÖt?

(d). After running this new pricing scheme in question (c) for two months, James receives complaints from sta§ customers for discrimination. The University then prohibits him to charge di§erent prices for di§erent customers.

After thinking carefully, James decides to serve students only and changes the name of the pub to “Monash Student Bar”. He still issues the permit for students with fee T and charges a price p for each token. What is the proÖt now? How do you think about his decision?

2 Research Question: Predation

Predation is one of the most controversial business conducts that may violate antitrust laws. The following exercise shows how to build simple theories of predation using tools of game theory. Such theories must be very intuitive but robust, and can be easily understood by lawyers and the general audience. Read the following introduction Örst and then complete the related questions. For further references, please read “The Theory of Industrial Organization” by Jean Tirole, Chapter 9.

2.1 Reading Material: Background

An Örm engages in predatory pricing by Örst lowering its price in order to drive rivals out of business and scare o§ potential entrants and then rising its price when its rivals exit the market. In most deÖnitions, the Örm lowers price below some measure of costs. The Örm incurs a short-run loss to obtain long-run gains.

However, if the Örm succeeds in driving out its current rivals and then raises its price, new rivals may enter the market, and the incumbent must again lower its price to drive out those Örms. Thus, the incumbent must convince the potential entrants that it does not pay to enter. Only then can the incumbent raise its price to the monopoly level with no fear of inducing entry.

Predatory pricing is a very controversial business strategy and is subject to antitrust scrutiny. A recent case of predation in European airline markets was recorded as an unsuccessful story of predation. Easyjet is a small airline company which provides low-cost no-frills service between di§erent European cities, using London-Lutton airport as its main hub. Soon after entering the London-Amsterdam segment, KLM, which holds 40% of the market, responded by matching Easyjetís low fares.

For KLM, this amounted to pricing below-cost, implied serious losses for Easyjet on that particular route. It seems plausible to induce Easyjet to exit the market. Low price was the main selling point for EasyJet, thus having KLM match its fares was a big blow to EasyJet. Eventually, EasyJet accused KLM in the court, and also launched a publicity campaign in the media.

This put an end to KLMís aggressive strategy so that easyJet could survive. The key challenge to predatory pricing in antitrust is to identify the intention for pricing below-cost. How can one be sure that KLMí’s price-matching strategy is targeted to exclude the rival? For this purpose, two conditions must be examined.

First of all, one needs to identify that KLMís pricing is below its cost, but which cost? Average cost is commonly adopted in test. Given its lighter structure, EasyJet has lower average cost than KLM. It is thus quite likely that KLMís price matching is well below its average cost and making a loss. However, its loss can be justiÖed only if KLM actually intended to drive EasyJet out of the market, since KLM could then make more proÖts by raising its prices.

Second, the courts must be convinced that the incumbent has the capacity to raise the price and recoup the loss from predation after driving the rivals out of the market.

This ìcapacity of recoupmentîcondition requires the plainti§ to show the evidence that the incumbent is able to maintain its monopoly position for a su¢ ciently long period.

In 1986, the U.S. Supreme Court reached an important decision regarding predatory pricing in the case Matsushita vs. Zenith. A group of manufacturers claimed that certain Japanese Örms had conspired for 20 years to sell electrical products in U.S. at prices below cost to drive the U.S.

producers out of business. The Supreme Court concluded that it was unlikely that any Örm would willingly incur losses upon itself for 20 years in order to drive Örms out of business eventually. It ruled that predation was not the reason for the low prices and that other explanation, such as legitimate competition, better described the reason.

Predation is one of the most di¢ cult areas in antitrust. There is huge disagreement among economists and/or practitioners. First, whether predatory pricing exists in practice is still under dispute. A price reduction by an incumbent in response to entry can be interpreted as a competitive strategy, rather than an attempt to drive the rival out.

However theoretical and empirical developments are creating a consensus on its existence. Why should predatory pricing be illegal? For consumerís welfare, there is a trade-o§. Predatory pricing implies lower prices for consumers at present but could lead to monopolization. Consumers may end up with higher prices in the future.

An extensive economic and legal literature suggests several standards for determining whether an Örm is practicing predatory pricing. Many courts have adopted a rule proposed by Areeda and Turner (1975): A Örmís pricing is predatory if its price is less than its short-run marginal cost. The logic behind this test is that no Örm ever proÖtably chooses to operate if the price is less than short-run marginal cost unless it is motivated by strategic concerns. Unfortunately most of the suggested tests for predation are di¢ cult
to implement.

First, the data needed to determine short-run marginal cost or even average variable costs are often di¢ cult to obtain. Second, other factors having nothing to do with price predation may explain violation of the tests. For instance, price promotions are adopted frequently which involve below-cost pricing for a short period.

In practice, the US Supreme Court has clariÖed two conditions for illegal predatory pricing in the case Brook Group Ltd v. Brown & Williams Tobacco Corp. (1993): First, price must well below its cost. Second, a Örm pricing below cost is su¢ ciently likely to recoup its short-run losses; that is, predation must be a rational strategy. It even requires the plainti§ to show the predatorís capability of recoupment.

At the same time, the Court has repeatedly shown its skepticism about predatory pricing: ìPredatory pricing schemes are rarely tried, and are even more rarely successfulî. However, the economic analysis suggests that the Courtís view is áawed when it implicitly assumes predation is unlikely to happen. Not only we observe predation in practice, but also we Önd convincing rationales and explanations for such practice. But, distinguishing predation from competitive behavior is di¢ cult, and there is few successful cases in US after 1993.

2.2 A Simple Theory of Predation

For successful predation to occur, the predating Örm needs an inherent advantage over its rivals. Many early studies of predatory pricing described the incumbent as a large Örm and its rival as a small Örm, and argued that large Örms can a§ord losses during predatory periods better than smaller Örms. This assumption is questionable. Why wouldnít somebody lend to a small Örm if it is not believable that the large Örm will continue to incur losses forever.

Consider a market with an incumbent and a new entrant. The game runs for two periods: In period 1, the incumbent decides whether to set a below-cost price. If it does, both Örms make a loss L. If the incumbent does not act aggressively, then both Örms make a duopoly proÖt  D > L. At the end of Period 1, the entrant decides whether to exit or stay. In period 2, the incumbent becomes a monopoly if the entrant exits and earns the monopoly proÖt  M.

However, if the entrant stays in the market, then the game in Period 2 runs exactly as in Period 1 and the incumbent then decides whether to predate the entrant again

Suppose both Örms have no constraints of reÖnancing in case of running a loss.

(1).  Draw a game tree to illustrate the game of predation.
(2). Solve for the equilibrium by back-ward induction. If the entrant stays in Period 2, what will be the Nash equilibrium in the subgame?
(3). Expecting the equilibrium result in Period 2, what is the incumbentís optimal decision in Period 1? Will predation arise in equilibrium? What
is the incumbentís total proÖt of two periods (assume no discount of proÖts from period 2)?

Predation under Asymmetric Financial Capacities

The above story relies too much on rationality and perfect information. However, in real world, there are asymmetric Önancial capacities between the incumbent and the entrant. The incumbent may have a deep-pocket and needs not to borrow from the bank, whereas the entrant usually faces a Önancial constraint and is placed a disadvantageous position in competition. When the entrant runs out of cash in Period 1 due to predation, it can survive only when the bank is willing to lend.

However, the bank may be quite cautious due to incomplete information. In this case, the entrant can get the loan only with some probability  < 1. That is, the entrant will have to exit if it fails in Önancing, which happens with probability 1. The incumbent then becomes a monopoly in Period 2 and earns a proÖt 
M.

(4).Draw a game tree to illustrate the game.
(5). Suppose the incumbent chooses to predate the entrant in Period 1, what will be its expected total proÖt of two periods?
(6). Can predation arise in equilibrium? What is the condition for such possible equilibrium?

2.3 Other Theories of Predatory Pricing

More recent models explain that di§erences in Örmsíbeliefs about their rivals can result in successful predation. For example, suppose that a Örm can be either a high-cost Örm or a low-cost Örm and that only the Örm knows its own costs. In response to entry, an incumbent Örm may lower its price for one of two reasons. First, if the incumbent is a low-cost Örm, the price decline might simply represent vigorous price competition.

Second, if the incumbent is a high-cost Örm, it may engage in predatory pricing. The other Örm (the entrant), after observing the incumbentís price behavior, infers whether the incumbent Örm is likely to have low or high costs. The lower its cost, the more likely the incumbent Örm is to meet entry with very low prices.

As a result, an incumbent can acquire a reputation of being a low-cost Örm by responding to entry with very low prices. Its pricing history is used by other potential entrants as an indicator, as to whether the incumbent has low or high costs. Because its pricing history is only a rough indicator, a high-cost Örm might be able to predate and convince potential entrants that it is really a low-cost Örm.

An entrant with no associated pricing history cannot ináuence the incumbentís beliefs about its costs. So there is a natural asymmetry between the Örms, because the entrant has no prior history whereas the incumbent has a history. The incumbentís beliefs about the entrant may di§er from the entrantís beliefs about the incumbent.

Pricing below cost for a high-cost Örm turns to be a rational strategy if it is able to create the illusion that it is a low-cost Örm, and thereby deter entry. By pricing aggressively, the incumbent may acquire a reputation for being tough, so that it can deter potential entry. The following case can support this theory of predation.

Aspartame is a low-calorie, high-intensity sweetener, which was discovered in 1965 by Searle Company. In 1985, Monsanto acquired Searle and its patent and became the monopolist. When the patent expired in 1987, Holland Sweetener Company (HSC) entered into the EU market. When HSC started to sell the generic version of Aspartame, Monsanto dropped its price of Nutrasweet from $70 to $25 per pound.

This means negative proÖts for HSC, but also an enormous drop in Nutrasweetís revenue in EU. Monsantoís reaction might be excessive, since HSCís capacity was only 5% of the world market. However, European is only a small fraction of the world market; the US market alone is ten times the EU one. By Öghting entry in a small market, it may convince the potential entrants not to attempt entering into other large markets.

Moreover, production of Aspartame is subject to a steep learning curve and Monsanto managed to cut costs by 70% over ten years. Monsantoís predation thus slows down HSCís learning process and delays its expansion plan. This restores Monsantoís bargaining power and Önally both Coca and Pepsi singed long-term contracts with Monsanto.

A Örm may practice predatory pricing to force its rivals to sell their Örms to it at a low price. By so doing, a Örm can acquire its rivals cheaply and gain market power. The Tobacco Trust allegedly engaged in predatory pricing against its rivals around the turn of the century. During the period 1881-1906, the Tobacco Trust acquired over 40 rivals and gained control of large shares of sales.

Frequently, the Tobacco Trust would identify a rival that it wished to buy and then introduce a competitive brand at a low price. The low proÖts would induce rivals to sell out to the Tobacco Trust at a low price.

For example, in 1901, the Tobacco Trustís American Beauty brand of cigarettes in North Carolina compete with a similar product of the Wells-Whitehead Tobacco Company of Winston. The American Beauty price was $1.5 per thousand, which was exactly equal to the required tax.

It was therefore deÖnitely below production costs. The Tobacco Trust claimed that the low price was an introductory o§er. In 1903, the Tobacco Trust purchased its rival. A detailed analysis shows that predation lowered the acquisition costs by about 25 percent.

2.3.1 A simple model of limit pricing under asymmetric information

Consider a two-period model with an incumbent and a potential entrant. The incumbent Örm is a protected monopoly in the Örst period (say due to patents protection). It learns its marginal cost c at the beginning of the Örst period and sets its monopoly quantity

(c), which determines the Örst-period price p (c). The second period consists of two stages. In Stage 1, the entrant decides whether to enter and pay a Öxed cost e which is sunk at this point. Denote by ” = 1 if entry takes place and ” = 0 otherwise. The entry strategy is also denoted by “(p1).

In Stage 2, after entering the market, the entrant learns the incumbentís cost. Both Örms compete in quantity (Cournot Competition) in case of entry, whereas the incumbent remains its monopoly position if the entrant stays out. Before the entry decision, the entrant does not know the cost of the incumbent. However, it is estimated as two possible values.

The entrant believes that with probability  the incumbent has high cost cH, and with probability 1  the incumbent incurs a low cost cL < cH, the same as the entrant. Let E (c) denote the entrantís proÖt in the second stage when the incumbentís marginal cost is c.

Assume that the entrantís decision of entry is ináuenced by its beliefs about the incumbentícost, with E (cH) > e > E (cL): Thus, when the incumbentís marginal cost is cL, it is never proÖtable for the entrant to enter in Stage 2. Given this, the low cost type incumbent will always set its monopoly price at pL, which generates the maximum proÖt m L.

Observing this, the entrant will not enter and the incumbent can keep its monopoly position. In contrast, if the incumbentís marginal cost is cH, it should set is monopoly price pH > pL which yields its maximum proÖt m H in the Örst period. If the incumbent wants to mimic the low cost type by setting a low price pL, it will incur a loss
and its proÖt is reduced to m H
. The incumbent hopes to recoup this loss when the entrant is deterred, in which case it can enjoy a monopoly proÖt m H instead of a doupoly proÖt d H. Thus,the high cost incumbent has incentives to mimic the low cost type if

For further illustration, consider a following example. Assume the entrant incurs a marginal cost normalized to zero. Assume also cH < 1=2 and cL = 0. The market demand is given by P (q) = 1 q in each period. We focus on the case where the entrant incurs non-trivial entry cost such that e > 1.

Consider the subgame in the second period.

(7). Suppose the incumbentís marginal cost is commonly known. Then, the entrant can make the entry decision under complete information. Under what conditions of cost e the entrant will enter if it knows the incumbent has high cost?

(8). When the entrant does not know the incumbentís marginal cost, it has to use the prior beliefs to calculate expected proÖts. Under what conditions the entrant will not enter? Explain your result Although the entrant does not know the incumbentís marginal cost, it can observe the incumbentís price in period one, p which can be used for the entrant to update its beliefs about the incumbentís marginal cost.

Expecting this, the incumbent has incentives to ináuence the entry decision by sending the price signal strategically. Consider the following strategies. In period one, the incumbent always sets a low price

(9). Show that this is a Perfect Bayesian Nash equilibrium. What is the equilibrium outcome? (Hint: check if this entry strategy is the best response given its beliefs and check whether the incumbent with high cost has incentives to mimic the low-cost type).