Premium Essay

Nash Equilibrium Exercises

In: Business and Management

Submitted By josephk
Words 1054
Pages 5
Information k = units of Capital l = units of Labour rK = Cost of Capital in dollars = $20 wL = Labour cost in dollars = $10 Output:

Q=K^(1/4) √L

Solution

Min: C_((K,L))=r_K .k+ w_L .l=20k+10l

Constraint: Q=k^(1/4) √l

Using Lagrangian method

L_((k,l,λ))=20k+10l- λ(k^(1/4) .l^(1/2)-Q)

dL/dk=20- 1/4 λk^(-3/4) .l^(1/2)=0 dL/dl=10- 1/2 λk^(1/4) .l^(-1/2)=0 dL/dλ=- 1/4 λk^(1/4) .l^(1/2)-Q=0

Equating
20- 1/4 λk^(-3/4) .l^(1/2)= 0 20= 1/4 λk^(-3/4) .l^(1/2) λ= (80k^(3/4))/l^(1/2) 10- 1/2 λk^(1/4) .l^(-1/2)=0 10= 1/2 λk^(1/4) .l^(-1/2) λ= (20l^(1/2))/k^(1/4)

(80k^(3/4))/l^(1/2) = (20l^(1/2))/k^(1/4)

80k=20l

4k=l

Substituting Lagrangian relation into Output equation

Q=K^(1/4) √L

Q=k^(1/4) 〖(4k)〗^(1/2)

Q=k^(1/4) 〖2k〗^(1/2)

Q/2=k^(3/4)

k= Q^(4/3)/2^(4/3) = Q^(4/3)/∛16= Q^(4/3)/(2∛2)

Solving for l

4k=l

2^2 (Q^(4/3)/2^(4/3) )=l

l= Q^(4/3)/2^(2/3) = Q^(4/3)/∛4

Minimum total expenditure on capital and labour in terms of Q
C_((K,L))=20k+10l
C_((K,L) )=2^2.5.(Q^(4/3)/2^(4/3) )+10(Q^(4/3)/2^(2/3) )
C_((K,L) )= (5Q^(4/3)+10Q^(4/3))/2^(2/3) = 〖15Q〗^(4/3)/2^(2/3) = 〖2^2.5.Q〗^(4/3)/2^(2/3) = (5Q^(4/3))/∛2

Information x denotes units of good X y denotes units of good Y
Cx denotes the unit cost of good X in dollars = $1
Cy denotes the unit cost of good Y in dollars = $1
0 < β < 1
M denotes the maximum amount of dollars to spend in the two goods Utility Function
U_((x,y))=〖(x^β+ 3y^β)〗^(1/β)

Solution Max: U_((x,y))=〖(x^β+ 3y^β)〗^(1/β)

Constraint: M_((x,y))= C_x x+ C_y y

Using Lagrangian method

L_((x,y,λ) )= (x^β+ 3y^β )^(1/β)- λ(x+y-M)

dL/dx= 1/β (x^β+ 3y^β )^(1/β-1) .(βx^(β-1) )- λ=0

dL/dy= 1/β (x^β+ 3y^β )^(1/β-1) .(3βy^(β-1) )- λ=0

dL/dλ= -x-y+M=0…...

Similar Documents

Premium Essay

Marquet Equilibrium

...MARKET EQUILIBRATION PROCESS ABSTRACT Market Equilibrium is a very important topic people face daily in economy. Understanding how market equilibrium is maintained is essential for business managers. This paper will explain the market equilibrating process. It will include the law of demand and determinants of demand; law of supply and determinants of supply. It will also include examples to better present the topic of surplus and shortage. The market of a certain product is in equilibrium when the amount offered of that product aligns with the amount needed. When demand is higher than what is being offered, businesses increase the offered quantities of their product as well the prices resulting in a decrease on demand. When the offer is higher than demand, businesses promote clearance to reduce inventory. Both scenarios show how companies tend to equilibrate price and quantities in the market. Competition among buyers and among sellers drives the price to the equilibrium price; once there, it remains unless it is subsequently disturbed by changes in demand or supply (McConnell, 2009). The law of demand implies that consumers will buy more of a product at a low price than at a high price. So, other things equal, the relationship between price and quantity demanded is negative or inverse and is graphed as a down-sloping curve. Some determinants of demand include: consumer tastes, the number of buyers in the market, consumer incomes, the prices of related...

Words: 715 - Pages: 3

Free Essay

Nash

...on black board’s, wall and windows. He occasionally left numerology treatises. The students became wary of him and gave him the nick name “Phantom Of Fine Hall” Then in 1970, his delusions cleared up, and he became more aware of the world surrounding him. Although it was unclear to how he became better. In 1994, at the ripe age of 66, John Nash received The Nobel Prize in economics, for his work on game theory. Thirty eight years after his and Alicia’s divorce the decided to remarried. Nash got a job at Princeton, where he continues to study mathematics. In pure mathematics he is know best for what is sometimes called “Nash Embedding Theorm.” It is very long and confusing process. You can get a collage degree in math and not even skim over what he accomplished. It is very hard and long thought out process, which requires a lot of mental and physical ability. He was one of the very few, who over came his disease and made a almost full recovery and moved on to do great things. www.-history.mcs.st-andrews.ac.uk/Biographies/.html www.sritweets.com/who-is-john-nash-and-what-are -his-achivements-john-nash-got-a-noble-prize-and-the-film-based-on-his-biography-wins-oscar/ www.popular-scince.net/nobel/nash.html www.econlib.org/libary/enc/bios/nash.html...

Words: 412 - Pages: 2

Premium Essay

Market Equilibrium

...Market Equilibrium within the Oil Market Professor Uhimchuk Eco 100 November 3, 2013 In the article titled “Oil Market equilibrium fragile, says think tank”, the author speaks about the fragile state that oil prices are reaching and countries within Europe as well as the United States are looking for other means to produce oil rather than paying the high cost of oil barrels. By looking for alternatives to the traditional oil, the oil industries have been raising the price of the barrel to help compensate for the demand leveling out and not rising as it would normally. According to CGES “The world is increasingly burning fuels than oil to generate power, leaving transportation as the only area where there is still no large scale alternative to oil.” With the transportation industries still depending on traditional fuel supplies, the oil industry can depend on the consumers continuing demand for their source of oil. Leo Drollas who is the Chief Economist for CGES, talks about his interpretation of the equilibrium of oil among China, Europe and the United States. While Europe has steadied out and their demand for crude oil hasn’t risen nor has it fallen, China’s demand for it has continued to grow, but he also states that this doesn’t necessarily mean that they are having a growing demand for it, but that they could just be storing it for future use. The consumption and demand between these three powerhouses are “considered the catalysts for a global recovery.”......

Words: 822 - Pages: 4

Premium Essay

Market Equilibrium

...Market Equilibrium Process The importance of understanding the process between two equilibrium points due to changes within a business’ supply and demand of their product is key for any business to be successful. Business owners rely on their supply managers to make decisions and determine the best options based on the company’s fluxuating market. The opportunity costs also are associated with a managers overall decision in determining the supply and demand between two equilibrium points. “Markets bring together buyers (“demanders”) and sellers (suppliers”), and they exist in many forms” (McConnel, Brue, Flynn, 2009 Ch. 3). Therefore, if the demand curve is downsloping and the supply curve is upsloping where the two meet in the middle will be the equilibrium price between the demand and supply (McConnel, Brue, Flynn, 2009 Ch. 3). When either side fluxuates then it effects the other moving the equilibrium price, therefore, decreasing the risk of shortages and surpluses of a particular product. A perfect example of market equilibrium is the medical marijuana dispensaries; in the past few years states have legalized the selling of legal medical marijuana, most recently with Washington and Nevada. However, Clark County will be issuing 4 different types of licenses and business owners can apply for one of the four licenses. Clark County received over 200 applicants along with formal business plans and financial statements proving the applicants had the assest to operate......

Words: 601 - Pages: 3

Free Essay

Nash

...Lecturas Matem´ticas a Volumen 24 (2003), p´ginas 137–149 a John Nash y la teor´ de juegos ıa Sergio Monsalve Universidad Nacional de Colombia, Bogot´ a Al profesor y acad´mico Don Jairo Charris Casta˜ eda e n In memoriam Abstract. In the last twenty years, game theory has become the dominant model in economic theory and has made significant contributions to political science, biology, and international security studies. The central role of game theory in economic theory was recognized by the awarding of the Nobel Price in Economic Science in 1994 to John C. Harsanyi, John F. Nash, & Reinhard Selten. The fundamental works in game theory of John F. Nash together with a brief exposition of them are included in this article. Key words and phrases. John Nash, History of Mathematics, Game Theory 1991 Mathematics Subject Classification. Primary 01A70. Secondary 91A12. Resumen. En los ultimos veinte a˜os, la teor´ de juegos se ha ´ n ıa convertido en el modelo dominante en la teor´ econ´mica y ha ıa o contribuido significativamente a la ciencia pol´ ıtica, a la biolog´ ıa y a estudios de seguridad nacional. El papel central de la teor´ ıa de juegos en teor´ econ´mica fue reconocido con el premio Nobel ıa o en Econom´ otorgado a John C. Harsanyi, John F. Nash & ıa Reinhard Selten en 1994. Se presentan los aportes de John Nash a la teor´ de juegos conjuntamente con una exposici´n ıa o elemental de ellos. 138 SERGIO MONSALVE 1. Introduction La Real Academia Sueca......

Words: 4568 - Pages: 19

Premium Essay

Nash Equalibrium

...and team 2 would invest money until they reach Nash equilibrium and their profits are maximized. Since this model is a symmetric model, both teams have the same incentive to win and therefore at equilibrium it can be assumed that win% for team 1 will equal the win% for team 2. Knowing these givens and the equation given, profits for team 1 can be found as followed: π1 = Vw1 – t1 → V(2-g)/4 π1 = (500)(2-.5)/4 π1 = 187.5 The profit can then be plugged back into the initial equation to determine team 1’s investment: π1 = Vw1 – t1 The win percentages for both teams will be .5 because .5 + .5 = 100% in a symmetric model 187.5 = 500(.5) – t1 t1= 62.5 Since this is a symmetric model, it is assumed that the investments and profits for team 1 will equal team 2’s investments and profits. π1 = π2 187.5 = π2 t1 = t2 62.5 = t2 2. For model 1, profits are larger when g=1/2 because the investments made by both teams are lower and therefore π1 = Vw1 – t1 will result in higher profits. The reason team 1 and team 2’s investments will be lower is because there is less incentive to invest when the advantage for investing goes down, as it does with a lower sensitivity parameter. A lower g (sensitivity parameter) means that teams need to investment less to yield a higher win%. The following graph shows the difference in profits for g=1 and g= ½: 3. To calculate the Nash equilibrium for an asymmetric model win % and team......

Words: 2149 - Pages: 9

Premium Essay

Nash Equilibrium

...In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.[1] If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will's decision, and Will is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others in the game. Contents   [hide]  * 1 Applications * 2 History * 3 Definitions * 3.1 Informal definition * 3.2 Formal definition * 3.3 Nash's Existence Theorem * 4 Examples * 4.1 Coordination game * 4.2 Prisoner's dilemma * 4.3 Network traffic * 4.4 Competition game * 4.5 Nash equilibria in a payoff matrix * 5 Stability * 6 Occurrence * 6.1 Where the conditions are not met * 6.2 Where the conditions are met * 7 NE and non-credible threats * 8 Proof of existence * 8.1 Proof using the Kakutani......

Words: 1390 - Pages: 6

Premium Essay

Market Equilibrium

...During the week’s one and two lectures, we identified the importance of the market equilibrium for the business managers. Market is considered to be in the equilibration when the supply of a product or service is equal to the demand for that product or service. The main goal of every organization is to ensure the output is at the equilibrium, meaning there is no surplus or shortage. Business managers need to recognize how economic principles such as supply, demand, as well as the determents are a part of an ongoing business decisions. This paper examines how economic principles such as concepts of supply, demand, and market equilibrium and their relationships to real-world examples. Efficient market theory holds that the price of a particular product is determined by supply of the product and the demand as well as a consensus in the marketplace (Schroeder, Clark & Cathey, 2005). If the market is in the equilibrium, the price of a product will not change unless some external factor changes the supply or demand. In other words, the equilibrium price and quantity will change with the change in demand or supply. According to McConnell, Brue & Flynn, “other things equal, as the price falls, the quantity demanded rises, and as price rises, the quantity demanded falls. In short, there is a negative or inverse relation-ship between price and quantity demanded. Economists call this inverse relationship the law of demand.” (p. 47). In order to understand how this concept works, we......

Words: 527 - Pages: 3

Premium Essay

Chemistry - Equilibrium

...185 EQUILIBRIUM 185 UNIT 7 EQUILIBRIUM After studying this unit you will be able to • identify dynamic nature of equilibrium involved in physical and chemical processes; • state the law of equilibrium; • explain characteristics of equilibria involved in physical and chemical processes; • write expressions for equilibrium constants; • establish a relationship between Kp and K c ; • explain various factors that affect the equilibrium state of a reaction; • classify substances as acids or bases according to Arrhenius, Bronsted-Lowry and Lewis concepts; • classify acids and bases as weak or strong in terms of their ionization constants; • explain the dependence of degree of ionization on concentration of the electrolyte and that of the common ion; • describe pH scale for representing hydrogen ion concentration; • explain ionisation of water and its duel role as acid and base; • describe ionic product (Kw ) and pKw for water; • appreciate use of buffer solutions; • calculate solubility product constant. Chemical equilibria are important in numerous biological and environmental processes. For example, equilibria involving O2 molecules and the protein hemoglobin play a crucial role in the transport and delivery of O2 from our lungs to our muscles. Similar equilibria involving CO molecules and hemoglobin account for the toxicity of CO. When a liquid evaporates in a closed container, molecules with relatively higher kinetic energy escape the liquid surface into the......

Words: 16204 - Pages: 65

Premium Essay

Nash

...individual for private study. 2 Nash Equilibrium: Theory 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Strategic games 11 Example: the Prisoner’s Dilemma 12 Example: Bach or Stravinsky? 16 Example: Matching Pennies 17 Example: the Stag Hunt 18 Nash equilibrium 19 Examples of Nash equilibrium 24 Best response functions 33 Dominated actions 43 Equilibrium in a single population: symmetric games and symmetric equilibria 49 Prerequisite: Chapter 1. 2.1 Strategic games is a model of interacting decision-makers. In recognition of the interaction, we refer to the decision-makers as players. Each player has a set of possible actions. The model captures interaction between the players by allowing each player to be affected by the actions of all players, not only her own action. Specifically, each player has preferences about the action profile—the list of all the players’ actions. (See Section 17.4, in the mathematical appendix, for a discussion of profiles.) More precisely, a strategic game is defined as follows. (The qualification “with ordinal preferences” distinguishes this notion of a strategic game from a more general notion studied in Chapter 4.) A STRATEGIC GAME D EFINITION 11.1 (Strategic game with ordinal preferences) A strategic game (with ordinal preferences) consists of • a set of players • for each player, a set of actions • for each player, preferences over the set of action profiles. 11 12 Chapter 2. Nash Equilibrium: Theory A very wide range of......

Words: 19938 - Pages: 80

Premium Essay

Nash Equilibrium

...player 1 earns $650 and player 2 earns $0. a. Write the above game in normal form. b. Find each player's dominant strategy, if it exists. c. Find the Nash equilibrium (or equilibria) of this game. d. Rank strategy pairs by aggregate payoff (highest to lowest). e. Can the outcome with the highest aggregate payoff be sustained in equilibrium? Why or why not? a. Write the above game in normal form. | |Player 1 chooses A |Player 1 chooses B | |Player 2 chooses A |A, A |A, B | | |500, 500 |0 , 650 | |Player 2 chooses B |B, A |B, B | | |650, 0 |100, 100 | b. Find each player's dominant strategy, if it exists. The dominant strategy for each player is to choose B. 100>0 & 650>500 c. Find the Nash equilibrium (or equilibria) of this game. Nash equilibrium is for each to choose B. This is because no matter what the player 2 does, if the player 1 picks B, they'll get a better result. d. Rank......

Words: 1598 - Pages: 7

Premium Essay

Nash Equilibrium

...A Nash equilibrium is a pair of strategies, one for each player, in which each strategy is a best response to other. These represent the likely outcome of the game. According to Roger B. Myerson, “If we can predict the behavior of all the players in such a game, then our prediction must be a Nash equilibrium, or else it would violate this assumption of intelligent rational individual behavior. That is, if our predicted behavior does not satisfy the conditions for Nash equilibrium, then there must be at least one individual whose expected welfare could be improved simply by re-educating him to more effectively pursue his own best interests, without any other social change.” The above argument does not prove that Nash equilibrium should be the only methodological basis for analysis of social institutions. But it does explain why studying Nash equilibria should be a fruitful part of the critical analysis of almost any kind of social institution. The prisoners dilemma is one of the best examples of Nash Equilibrium. | | Jack | | | | C | NC | Tom | C | -10,-10 | 0,-20 | | NC | -20,0 | -5,-5 | | | | | *Numbers represent years in prison If every player in a game plays his dominant pure strategy (assuming every player has a dominant pure strategy), then the outcome will be a Nash equilibrium.According to the above game both players know that 10 years is better than 20 and 0 years is better than 5; therefore, C is their dominant strategy and they will......

Words: 405 - Pages: 2

Premium Essay

Market Equilibrium

...determine the rise of prices. According to Fama (2004-2015) “the efficient markets theory is a proposition that the prices of stocks, bonds, and other securities fully reflect all available information at any point in time” (Behind This Chicago Idea). With efficient markets theory business are allowed to increase prices of products and labor based on future predictions. If the business owners suspects any changes that will affect the equilibrium they can increase prices as needed. Business owners must make sure there is a balance or equilibrium between the supply of services as well as materials and the demand for these services and materials. If the business owners have a surplus of supplies they may find it difficult to sell all products or services and may have to lower prices of these products and services until they reach a state of equilibrium. If there is an excess of demand the business owner will find that there is a shortage of products or services and raise prices for consumers. One everlasting product that seems to fluctuate between equilibriums of supply and demand is gasoline. Gasoline in major cities are typically more expensive then in smaller towns. In 2004, people were able to fill the tanks of smaller vehicles (10 gallon tanks) for twenty dollars or less. According to Walton (2008) “the hurricane hit some platforms and refineries in the Gulf of Mexico, so the supply of gas was disrupted.” After losing some access to oil supply in 2008 due to Hurrican Ike,......

Words: 693 - Pages: 3

Premium Essay

Equilibrium

...lifestyle is not only beneficial for our respiratory and circulatory system. It can strengthen our immune system, increase energy level, enhance brain health, improve our appearance, give us a better mood and prolong our existence. It can also help us gain our confidence, have a positive and optimistic attitude, be more focused, and be in our best and finest when it comes to activities. We should start having this practice as early as we can. Though preoccupied with things we should always a lot time for the things that really matters. We should not only advance ourselves in terms of technology and education. We should also advance vessel in which our soul is entrapped. But keep in mind that we should not overdo it. We don’t need to exercise like a body builder and eat all the foods in the food pyramid to have a healthy lifestyle. Remember that balance is the key. So what are you waiting for? Get your life in balance....

Words: 959 - Pages: 4

Free Essay

Ball Like Steve Nash

...Ball Like Steve Nash Standing at 6’3 and weighing in at 178 lbs most people know the infamous Steve Nash from the past fifteen years that he has dedicated his life to professional basketball. Before pursuing the American dream of becoming a professional athlete, Nash was born in South Africa but grew up in Victoria, Canada where he played soccer till his late teens. He later discovered a passion for basketball where he hooped at St. Michaels University School with his younger brother Martin. He comes from a family of athletes including his father John Nash who played minor professional soccer in South Africa. His sister Joann Nash held the title as captain of the soccer team for The University of Victoria for three years. It was only right for him to keep on the family legacy. Nash took his talent to Santa Clara University in California where he led his team to the NCAA tournament as a freshman. He established a name for himself by his sophomore year with his quick feet, fundamentals, and amazing skills. Nba scouts were all over him and at no surprise he was drafted in the first round by the Phoenix Suns in 1996. He played two seasons with the Suns before being traded to the Dallas Mavericks. There Nash made a name for himself in the league with Dirk Nowitzki and Michael Finley by his side. They formed an unstoppable trio that carried the team to the Western Conference Finals. Although they didn’t get much farther Nash participated in the NBA All-Star Game and was......

Words: 640 - Pages: 3