Unrooted binary tree

Conjectural variation is a term used in oligopoly theory referring to the beliefs that one firm has about the way its competitor(s) may react if it varies its output (or price). The firm forms a conjecture about the variation in the other firm's output that will accompany any change in its own output. For example, in the classic Cournot model of oligopoly, it is assumed that each firm treats the output of the other firms as given when it chooses its output. This is sometimes called the "Nash conjecture" as it underlies the standard Nash equilibrium concept. However, alternative assumptions can be made. Suppose you have two firms producing the same good, so that the industry price is determined by the combined output of the two firms (think of the water duopoly in Cournot's original 1838 account). Now suppose that each firm has what is called the "Bertrand Conjecture" of -1. This means that if firm A increases its output, it conjectures that firm B will reduce its output to exactly offset firm A's increase, so that total output and hence price remains unchanged. With the Bertrand Conjecture, the firms act as if they believe that the market price is unaffected by their own output, because each firm believes that the other firm will adjust its output so that total output will be constant. At the other extreme is the Joint-Profit maximizing conjecture of +1. In this case each firm believes that the other will imitate exactly any change in output it makes, which leads (with constant marginal cost) to the firms behaving like a single monopoly supplier.

History

The notion of conjectures has maintained a long history in the Industrial Organization theory ever since the introduction of Conjectural Variations Equilibria by Arthur Bowley in 1924 and Ragnar Frisch (1933) (a useful summary of the history is provided by Giacoli). Not only are conjectural variations (henceforth CV) models able to capture a range of behavioral outcomes - from competitive to cooperative, but also they have one parameter which has a simple economic interpretation. CV models have also been found quite useful in the empirical analysis of …firm behavior in the sense that they provide a more general description of …firms ’behavior than the standard Nash equilibrium.

As Stephen Martin has argued: There is every reason to believe that oligopolists in different markets interact in different ways, and it is useful to have models that can capture a wide range of such interactions. Conjectural oligopoly models, in any event, have been more useful than game-theoretic oligopoly models in guiding the specification of empirical research in industrial economics.

Consistent conjectures

The CVs of firms determine the slopes of their reaction functions. For example, in the standard Cournot model, the conjecture is of a zero reaction, yet the actual slope of the Cournot reaction function is negative. What happens if we require the actual slope of the reaction function to be equal to the conjecture? Some economists argued that we could pin down the conjectures by a consistency condition, most notably Timothy Bresnehan in 1981. Bresnehan's consitency was a local condition that required the actual slope of the reaction function to be equal to the conjecture at the equilibrium outputs. With linear industry demand and quadratic costs, this gave rise to the result that the consistent conjecture depended on the slope of the marginal cost function: for example, with quadratic costs of the form (see below) cost = a.x², the consistent conjecture is unique and determined by a. If a=0 then the unique consistent conjecture is the Bertrand conjecture $\phi ^{*}=-1$, and as a get bigger, the consistent conjecture increases (becomes less negative) but is always less than zero for finite a.

The concept of consistent conjectures was criticized by several leading economists. Essentially, the concept of consistent conjectures was seen as not compatible with the standard models of rationality employed in Game theory.

However, in the 1990s Evolutionary game theory became fashionable in economics. It was realized that this approach could provide a foundation for the evolution of consistent conjectures. Huw Dixon and Ernesto Somma showed that we could treat the conjecture of a firm as a meme (the cultural equivalent of a gene). They showed that in the standard Cournot model, the consistent conjecture was the Evolutionarily stable strategy or ESS. As the authors argued "Beliefs determine Behavior. Behavior determines payoff. From an evolutionary perspective, those types of behavior that lead to higher payoffs become more common". In the long-run, firms with consistent conjectures would tend to earn bigger profits and come to predominate.

Mathematical Example 1: Cournot model with CVs

Let there be two firms, X and Y, with outputs x and y. The market price P is given by the linear demand curve

$P=1-x-y$

so that the total revenue of firm X is then

$xP=x(1-x-y)=x-x^{2}-xy$

For simplicity, let us follow Cournot's 1838 model and assume that there are no production costs, so that profits equal revenue $\Pi =x-x^{2}-xy$.

With conjectural variations, the first order condition for the firm becomes:

${\frac {d\Pi }{dx}}=(1-2x-y)-x{\frac {dy}{dx}}=0$

where ${\frac {dy}{dx}}=\phi$ is the firms conjecture about how the other firm will respond, the conjectural variation or CV term. This first order optimization condition defines the reaction function for the firm, which states, for a given CV, the optimal choice of output given the other firm's output.

$x=R(y,\phi )={\frac {1-y}{2+\phi }}$

Note that the Cournot-Nash Conjecture is $\phi =0$, in which case we have the standard Cournot Reaction function. The CV term serves to shift the reaction function and most importantly later its slope. To solve for a symmetric equilibrium, where both firms have the same CV, we simply note that the reaction function will pass through the x=y line so that:

$x={\frac {1-x}{2+\phi }}$ so that in symmetric equilibrium $x^{*}=y^{*}={\frac {1}{3+\phi }}$ and the equilibrium price is $P^{*}={\frac {1+\phi }{3+\phi }}$.

If we have the Cournot-Nash conjecture, $\phi =0$, then we have the standard Cournot equilibrium with $P^{*}={\frac {1}{3}}$. However, if we have the Bertrand conjecture $\phi =-1$, then we obtain the perfectly competitive outcome with price equal to marginal cost (which is zero here). If we assume the joint-profit maximizing conjecture $\phi =+1$ then both firms produce half of the monopoly output and the price is the monopoly price $P^{*}={\frac {1}{2}}$.

Hence the CV term $\phi$ is a simple behavioral parameter which enables us to represent a whole range of possible market outcomes from the competitive to the monopoly outcome, including the standard Cournot model.

Mathematical example 2: Consistency

Take the previous example. Now let the cost of production take the form: cost = a.x². In this case, the profit function (revenue minus cost) becomes (for firm X and analogously for firm Y):

$\Pi =(x-x^{2}-xy)-{\frac {a.x^{2}}{2}}$.

The first-order condition then becomes:

${\frac {d\Pi }{dx}}=(1-2x-y)-x{\frac {dy}{dx}}-ax=0$

which defines the reaction function for firm X as:

$x=R(y,\phi )={\frac {1-y}{2+a+\phi }}$

This has slope (in output space)

$R_{y}=-{\frac {1}{2+a+\phi }}$

and analogously for firm Y which (we assume) has the same conjecture. To see what consistency means, consider the simple Cournot conjecture $\phi =0$ with constant marginal cost a=0. In this case the slope of the reaction functions is -1/2 which is "inconsistent" with the conjecture. The Bresnehan consistency condition is that the conjectured slope $\phi$ equals the actual slope $R_{y}$ which means that

$\phi =-{\frac {1}{2+a+\phi }}$

This is a quadratic equation which gives us the unique consistent conjecture

$\phi ^{*}=-(1+{\frac {a}{2}})+{\sqrt {\frac {4a+a^{2}}{4}}}$

This is the positive root of the quadratic: the negative solution would be a conjecture more negative than -1 which would violate the second order conditions. As we can see from this example, when a=0 (marginal cost is horizontal), the Bertrand conjecture is consistent $\phi ^{*}=-1$. As the steepness of marginal cost increases (a goes up), the consistent conjecture increases. Note that the consistent conjecture will always be less than 0 for any finite a.

Notes

 43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to  destinations like Camino Real de Tierra Adentro.

External links

• Conjectural variations and competition policy Office of Fair Trading Report, 2011.
• Series on Mathematical Economics & Game Theory, Volume 2: Theory Of Conjectural Variations by Charles Figuières, Alain Jean-Marie, Nicolas Quérou, Mabel Tidball.

Earlier than you determine whether or not stainless steel cookware is price buying, lets first discuss what stainless-steel cookware is. Stainless-steel is product of an alloy, or a mixture of metals. Most commonly, fundamental iron with chromium, nickel or another minor metals. The chromium gives rust safety and gives your cookware sturdiness. The nickel gives rust protection as effectively, and adds a elegant look. Most properly made chrome steel cookware has copper or aluminum added to the underside of the pan or pot. This is carried out to increases the power of the pot or pan to conduct heat.
The best stainless steel cookware is the main category, but nonetheless it is divided into a number of subcategories based on the standard and the price vary. It may be complicated to choose the best stainless-steel cookware out of the categories that can meet your necessities. That is where we took a step forward to clarify you all the data that might be useful for you to know the way to decide on the most effective stainless-steel cookware. The very best stainless-steel cookware set is manufactured from cheap to costly and quality constructed pots and pans.
You'll find magnetic chrome steel in the layer on the skin of some high quality pieces of stainless steel. This is to make it compatible with induction stovetops, which contain using a rapidly charging electromagnetic field to warmth cookware. Excessive-quality chrome steel, like All-Clad , uses three layers of steel—the austenite layer of metal on the within, ferrite metal on the outside, and a layer of aluminum sandwiched between the 2 for optimum heat conductivity (metal alone does not conduct heat evenly). Lesser-quality stainless-steel is often just one layer of austenitic stainless-steel.
Aesthetically speaking, stainless steel is a wise choice for those who choose to show or cling pots or pans. The clean, crisp look of all chrome steel kitchenware can remodel a mishmash of cookware into a sophisticated décor assertion. Stainless steel kettles, such as the Cuisinart Tea Kettle will mix individual kitchenware right into a cohesive and pleasant entity. Consider buying chrome steel utensils as effectively. Already acquired a beautiful stainless steel cookware collection? The Cuisinart Chef’s Collection stainless pot rack might be the completion for a kitchen, liberating up area and making these pots and pans readily accessible. Get the stainless-steel cookware of your culinary desires at Macy’s!
Arduous-anodized aluminum cookware is without doubt one of the hottest varieties of material, even though many individuals don't fairly understand the construction. Onerous-anodized aluminum is plain aluminum that has been processed in a collection of chemical baths charged with an electrical present. The result is a fabric that has the identical superior warmth conductivity as aluminum however is non-reactive with acidic foods, resembling tomatoes, and twice as laborious as chrome steel. Two drawbacks to laborious-anodized cookware are that it's not dishwasher-protected and, as a result of it's not magnetic, it won't work with induction vary tops.
The enamel over metal approach creates a chunk that has the heat distribution of carbon steel and a non-reactive, low-stick floor. Such pots are much lighter than most other pots of similar dimension, are cheaper to make than stainless steel pots, and wouldn't have the rust and reactivity problems with cast iron or carbon steel. quotation wanted Enamel over steel is right for big stockpots and for different large pans used mostly for water-based cooking. Due to its gentle weight and easy cleanup, enamel over steel can also be in style for cookware used whereas tenting. Clad aluminium or copper edit
Unique specialty cookware pieces served a la carte to compliment any cookware set are constructed of a durable Stainless Steel with a brushed exterior end. Designed with an influence bonded, aluminum disk encapsulated base which distributes heat shortly and evenly to permit exact temperature control. Handles are riveted for sturdiness and efficiency. The New Specialty Cookware is appropriate for all vary sorts including induction. Along with the multi use perform, another distinctive feature is bottom to top inside quantity markings in each quarts and metric measurement; and every bit comes with a tempered glass lid, oven protected to 350°F.
Whether you are a cooking fanatics, a professional chef or just cooking for your family you understand the significance of having a totally stocked kitchen. Not only do you want the precise elements, however you also need the proper instruments to get the job performed. In any kind of fundamental cooking training lesson, you will be taught that stainless-steel is your new best pal in terms of kitchen cookware. What you will also learn is that high quality cooking gear does not often come at a discounted price. For that reason, you will need to take good care of your cookware! Here are some basics for stainless steel care.
To fight the uneven heating drawback, most stainless-steel pans are laminations of aluminum or copper on the underside to spread the heat around, and stainless steel inside the pan to provide a cooking surface that's impervious to no matter you might put inside. In my experience, this stainless steel surface remains to be too sticky to fry on, and in case you ever burn it you get a permanent bother spot. However, generally a chrome steel cooking surface is useful when you can't use aluminum (see under) so I maintain some around. Select one thing with a reasonably thick aluminum layer on the underside.
Properly, except you’re a metals professional and go examine the manufacturing unit the place the steel is made to see whether or not or not their manufacturing course of creates a pure austenite without corrosive materials formed, you’re not going to know for certain whether or not the craftsmanship of your stainless is of the best high quality. I feel your greatest guess is to easily buy high-quality stainless steel from the start, from a brand with a popularity for good high quality. However, I believe I have figured out a technique that you can determine if the stainless cookware you already have is doubtlessly reactive. Earlier than you decide whether stainless steel cookware is worth buying, lets first discuss what stainless steel cookware is. Chrome steel is manufactured from an alloy, or a mix of metals. Mostly, basic iron with chromium, nickel or another minor metals. The chromium supplies rust safety and gives your cookware durability. The nickel supplies rust safety as effectively, and adds a elegant look. Most properly made chrome steel cookware has copper or aluminum added to the bottom of the pan or pot. That is performed to increases the ability of the pot or pan to conduct heat.
The perfect stainless-steel cookware is the principle category, but nonetheless it is divided into several subcategories primarily based on the standard and the value range. It can be confusing to choose the best chrome steel cookware out of the classes that may meet your necessities. That is where we took a step ahead to explain you all the data that will be useful for you to understand how to choose one of the best stainless steel cookware. The most effective chrome steel cookware set is manufactured from cheap to expensive and high quality constructed pots and pans.
You can find magnetic chrome steel in the layer on the outside of some quality pieces of stainless steel. This is to make it suitable with induction stovetops, which involve the use of a quickly charging electromagnetic area to warmth cookware. Excessive-quality stainless-steel, like All-Clad , makes use of three layers of steel—the austenite layer of metal on the within, ferrite steel on the skin, and a layer of aluminum sandwiched between the 2 for optimum warmth conductivity (metal alone does not conduct heat evenly). Lesser-quality stainless steel is usually just one layer of austenitic chrome steel.
Aesthetically talking, stainless steel is a smart alternative should you prefer to display or hang pots or pans. The clean, crisp look of all stainless-steel kitchenware can remodel a mishmash of cookware into a classy décor assertion. Chrome steel kettles, such as the Cuisinart Tea Kettle will mix particular person kitchenware into a cohesive and nice entity. Take into account purchasing stainless-steel utensils as well. Already acquired a stunning stainless-steel cookware collection? The Cuisinart Chef’s Assortment stainless pot rack is perhaps the of completion for a kitchen, liberating up house and making those pots and pans readily accessible. Get the stainless-steel cookware of your culinary desires at Macy’s!
Arduous-anodized aluminum cookware is likely one of the hottest varieties of materials, despite the fact that many people don't quite perceive the construction. Onerous-anodized aluminum is plain aluminum that has been processed in a collection of chemical baths charged with an electric present. The result's a fabric that has the same superior heat conductivity as aluminum but is non-reactive with acidic foods, such as tomatoes, and twice as exhausting as stainless-steel. Two drawbacks to laborious-anodized cookware are that it is not dishwasher-protected and, because it is not magnetic, it is not going to work with induction vary tops.
The enamel over steel technique creates a bit that has the warmth distribution of carbon steel and a non-reactive, low-stick floor. Such pots are much lighter than most different pots of similar measurement, are cheaper to make than stainless-steel pots, and don't have the rust and reactivity problems with cast iron or carbon steel. citation wanted Enamel over metal is ideal for big stockpots and for other giant pans used largely for water-based cooking. Because of its mild weight and simple cleanup, enamel over metal is also common for cookware used while camping. For more about stainless steel cookware reviews look at our web site. Clad aluminium or copper edit
Distinctive specialty cookware pieces served a la carte to go with any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior end. Designed with an impact bonded, aluminum disk encapsulated base which distributes warmth quickly and evenly to allow precise temperature control. Handles are riveted for durability and performance. The New Specialty Cookware is compatible for all range sorts including induction. Along with the multi use operate, another distinctive characteristic is bottom to prime interior volume markings in both quarts and metric measurement; and every bit comes with a tempered glass lid, oven safe to 350°F.
Whether or not you're a cooking enthusiasts, knowledgeable chef or just cooking for your family you already know the importance of having a totally stocked kitchen. Not solely do you need the appropriate substances, however you additionally need the proper tools to get the job done. In any sort of fundamental cooking training lesson, you will learn that stainless steel is your new greatest friend with regards to kitchen cookware. What you will also study is that high quality cooking tools does not often come at a discounted price. Because of this, it is very important take good care of your cookware! Listed below are some fundamentals for chrome steel care.
To fight the uneven heating downside, most stainless steel pans are laminations of aluminum or copper on the underside to spread the heat around, and stainless steel inside the pan to provide a cooking floor that is impervious to no matter you would possibly put inside. In my expertise, this stainless-steel floor continues to be too sticky to fry on, and should you ever burn it you get a everlasting bother spot. However, sometimes a stainless steel cooking floor turns out to be useful when you can't use aluminum (see under) so I maintain some around. Choose one thing with a fairly thick aluminum layer on the underside.
Effectively, until you’re a metals professional and go inspect the factory where the steel is made to see whether or not their manufacturing process creates a pure austenite with out corrosive materials fashioned, you’re not going to know for certain whether or not the craftsmanship of your stainless is of the best quality. I believe your finest guess is to easily purchase excessive-quality stainless-steel from the beginning, from a brand with a status for good quality. But, I think I have discovered one way which you could determine if the stainless cookware you have already got is doubtlessly reactive.