Policy Example: How should the government Have comment to huge Oil agency Mergers?
Exploring the policy QuestionWhat room the strategic incentives for banks to take risks?What plan solutions present themselves native this analysis?
18.1 Cournot design of Oligopoly: quantity Setters
Learning target 18.1: define game theory and also they species of situations it describes.
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18.2 Bertrand model of Oligopoly: Price Setters
Learning target 18.2: describe normal type games and identify optimal strategies and equilibrium outcomes in such games.
18.3 Stackelberg model of Oligopoly: first Mover Advantage
Learning objective 18.3: explain sequential relocate games and explain just how they space solved.
18.4 Policy Example: exactly how Should the federal government Have responded to the Banking crisis of 2008?
Learning objective 18.4: describe how video game theory deserve to be offered to understand the banking situation of 2008.
18.1 Cournot design of Oligopoly: quantity Setters
Learning objective 18.1: define game theory and they types of cases it describes.
Oligopoly sectors are markets in which only a couple of firms compete, wherein firms create homogeneous or differentiated products and also where barriers to entry exist that might be organic or constructed. There room three key models that oligopoly markets, each take into consideration a slightly various competitive environment. The Cournot model considers firms that make an similar product and make calculation decisions simultaneously. The Bertrand design considers firms the make and also identical product however compete ~ above price and also make your pricing decision simultaneously. The Stackelberg design considers quantity setup firms v an similar product that make output decisions simultaneously. This module considers all three in order start with the Cournot model.
Table 18.1: Metrics of the Four basic Market structures
|Number the Firms||Similarity the Goods||Barriers come Entry or Exit||Module|
Identical or Distinct
Oligopolists face downward sloping demand curves which means that price is a duty of the complete quantity created which, in turn, implies that one firm’s output affects not just the price the receives because that its output but the price its rivals receive together well. This creates a strategic atmosphere where one firm’s benefit maximizing calculation level is a function of their competitors’ calculation levels. The version we use to analyze this is one an initial introduced through French economist and mathematician Antoine Augustin Cournot in 1838. Interestingly, the solution to the Cournot version is the exact same as the more general Nash equilibrium principle introduced by man Nash in 1949 and the one provided to resolve for equilibrium in non-cooperative gamings in Module 17.
We will start by considering the most basic situation: only two companies who make an the same product and who have the same cost function. Later we will check out what happens when we be safe those assumptions and allow an ext firms, identified products and also different price functions.
Let’s start by considering a instance where there are two oil refineries situated in the Denver, Colorado area who room the only two carriers of petrol for the Rocky Mountain regional wholesale market. We’ll call them commonwealth Gas and National Gas. The gas they create is identical and also they every decide independently, and without understanding the rather choice, the quantity of gas to create for the week at the beginning of every week. Us will call Federal’s calculation choice qF and National’s output choice qN , wherein q to represent liters the gasoline. The weekly need for everyone gas in the Rocky Mountain an ar is P=A – BQ, where Q is the total quantity that gas supplied by the two firms or, Q=qF+qN. Automatically you have the right to see the strategy component: the price the both receive for your gas is a function of each company’s output. We will certainly assume that each liter the gas produced costs the agency c, or that c is the marginal price of developing a liter the gas because that both companies and also that there room no addressed costs.
With these assumptions in place, we deserve to express Federal’s profit function:
Note the we have actually now described a game finish with players, Federal and National, strategies, qF and qN, and payoffs πF and πN. Now the task is to find for equilibrium of the game. To execute so we have to begin with a best solution function. In this situation the best solution is the firm’s profit maximizing output. This will depend on both the firm’s own output and the contending firm’s output.
If the profit function is
If πF = qF ( A – B ( qF + qN ) -c ) then us can expand to find
Taking the partial derivative that this expression v respect to qF
If we re-arrange this we can see that this is simply an expression of MR=MC.
The marginal revenue look at the exact same as a monopolist’s MR duty but v one additional term, -BqN.
Solving for qF yields:
This is federal Oil’s best solution function, your profit maximizing output level given the output choice of their rivals. The is the same ideal response role as the persons in Module 17. By symmetry, nationwide Oil has an similar best response function:
We know from Module 15 the the monopolists marginal revenue curve when encountering an inverse demand curve P=A-BQ
is MR(q)=A-2Bq. That runs the end in this duopolist example that the firms’ marginal revenue curves include one extra term:
The profit-maximizing preeminence tells united state that to find profit maximizing output we must set the marginal revenue to the marginal cost and solve. Doing for this reason yields
Now that we understand the best solution functions addressing for equilibrium in the model is fairly straightforward. Us can begin by graphing the best response functions. These graphical illustrations of the best response functions are referred to as reaction curves. A Nash equilibrium is a correspondence of best an answer functions i m sorry is the exact same as a cross of the reaction curves.
Figure 18.1.1: Nash Equilibrium in the Cournot Duopoly Model
In number 18.1.1, we have the right to see the Nash equilibrium the the Cournot duopoly model as the intersection the the reaction curves. Mathematically this intersection is uncovered by solving the system of equations,
simultaneously. This is a mechanism of two equations and two unknowns and also therefore has actually a distinctive solution as lengthy as the slopes are not equal. We can solve this by substituting one equation into the other which returns a single equation v a single unknown:
Solving by steps:
And by symmetry we know:
The Nash equilibrium is:
Let’s take into consideration a certain example. Suppose in the above example the weekly need curve because that wholesale gas in the Rocky Mountain region is p = 1,000 – 2Q, in thousands of gallons, and both firm’s have consistent marginal prices of 400. In this case A = 1,000, B = 2 and also c = 400. So
To analysis this from the beginning we can collection up the total revenue duty for federal Oil:
The marginal revenue role that is linked with this is:
We know marginal price is 400, so setting marginal revenue equal to marginal expense results in the adhering to expression:
This is the ideal response function for commonwealth Oil. By symmetry we understand that national Oil has the exact same best solution function:
Solving for the Nash equilibrium:
We can insert the systems for
18.2 Bertrand model of Oligopoly: Price Setters
Learning objective 18.2:.
In the previous section we learned oligopolists that make an identical good and who complete by setting quantities. The example we used in that ar was wholesale gasoline whereby the sector sets a price that translates supply and also demand and also the strategic decision the the refiners was how much oil come refine right into gasoline. In this section we turn our fist to a different case in which the oligopolists contend on price. The instance here are the sleeve gas stations that bought the all gas from the refiners and are now ready to offer it come consumers. We still have identical goods, for consumers the gas the goes right into their car is every the same and also we will certainly assume away any type of other distinctions like cleaner station or the existence of a mini-mart.
Lets imagine a basic situation where there 2 gas stations, rapid Gas and Speedy Gas on either next of a busy main street. Both station have huge signs that display screen the gas prices the each station is offering for the day. Consumers room assumed to be indifferent around the gas or the stations, so they will certainly go to the terminal that is supplying the reduced price. Therefore an separation, personal, instance gas station’s demand is conditional ~ above its loved one price through the other station
Formally we deserve to express this through the adhering to demand duty for rapid Gas:
Speedy Gas has an equivalent demand curve:
In words, these demand curves say that if a station has a reduced price than the other, they will certainly get all of the demand at the price and the various other station will acquire no demand. If they have actually the same price, climate each will obtain one fifty percent of the need at that price.
Let’s assume that rapid Gas and also Speedy Gas both have the same consistent marginal price of c, and also will i think no fixed prices to keep the evaluation simple. The inquiry we now have to answer is what space the best an answer functions because that the 2 stations? Remember that best response functions are one player’s optimal strategy an option given the strategy an option of the other player. Therefore what is one rapid Gas’s best an answer to the Speedy Gas’s price?
If Speedy Gas chargesPS > c , quick Gas can set PF > PS and castle will gain no customers at all and also make a profit of zero. They could instead set PF=PS and get ½ the demand at that price and make a hopeful profit. Or they might set PF = PS – $0.01 , or collection their price one cent below Speedy Gas’s price and get all of the customers at a price the is one cent below the price at which lock would get ½ the demand. Clearly, this third option is the one that returns the most profit. Now we just have to consider the situation where PS = c. In this case, undercutting the price by one cent is no optimal since Fast Gas would certainly get every one of the demand yet would lose money ~ above every gallon of gas sold yielding negative profits. Setting PF = PS = c would give them half the need at a break-even price and would yield precisely zero profits.
The best response function we just defined for fast Gas is the same finest response duty for Speedy Gas. So where is the correspondence of best response functions? As long as the price are over c there is always an impetus for both stations come undercut each other’s price, so there is no equilibrium. But at PF = PS = c both stations are playing your best response to each various other simultaneously. For this reason the distinctive Nash equilibrium to this video game is PF = PS = c.
What is specifically interesting around this is the truth that this is the same outcome that would have developed if they to be in a perfect competitive market due to the fact that competition would have driven prices down to marginal cost. Therefore in a situation where compete is based upon price and the great is fairly homogeneous, as couple of as two firms deserve to drive the sector to an reliable outcome.
18.3 Stackelberg model of Oligopoly: very first Mover Advantage
Learning objective 18.3:
Both the Cournot model and also the Bertrand design assume simultaneous relocate games. This provides sense once one firm needs to make a strategy decision before knowing around the strategy selection of the other firm. But not all cases are prefer this, what happens once one firm makes its strategy decision first and the various other firm chooses second? This is the case described by the Stackelberg design where the firms are quantity setters offering homogenous goods.
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Let’s go back to the instance of 2 oil companies: commonwealth Gas and National Gas. The gas they develop is identical but now they decision their calculation levels sequentially. We will certainly assume that federal Gas set its output an initial and then, after ~ observing Federal’s choice, nationwide Gas decides ~ above the amount of gas they room going to produce for the week. We will again speak to Federal’s calculation choice qF and National’s output choice qN , whereby q represents liters of gasoline. The weekly demand for wholesale gas is still P = A – BQ , wherein Q is the full quantity the gas gave by the two firms or, Q=qF+qN.
We have actually now turn the previous Cournot game into a sequential game and the SPNE solution to a sequential video game is discovered through backward induction. Therefore we need to start at the second move of the game: National’s calculation choice. When national makes this decision, Federal’s output selections is already made and also known to national so that is takes together given. Therefore, we deserve to express Federal’s profit duty as: