Bjârn Andrén 🐻
    AI2156 - Contract Theory with Application to Property Management
    πŸ“

    AI2156 - Contract Theory with Application to Property Management

    ELI5 - Theories & Concepts

    β€£
    🚧 Prisoner’s Dilemma
    πŸ€”
    Prisoner's dilemma is like a game where two people have to make a choice without talking to each other. They can either cooperate or betray each other. If both cooperate, they both get a good outcome. But if one betrays while the other cooperates, the betrayer gets a better outcome while the other person gets a worse outcome. If both betray, they both get a not-so-good outcome. It's tricky because each person wants the best outcome for themselves, but sometimes working together would be better for both.
    β€£
    πŸ‘Β Strategic complements

    Strategic complements mutually reinforce one another - Upward sloping reaction functions.

    🧠
    Smartphones and mobile apps have a symbiotic relationship. As smartphones become more advanced and widely adopted, they create a larger market for mobile apps. At the same time, the availability of a diverse range of mobile apps enhances the value and utility of smartphones for users. This mutually reinforcing cycle drives the growth of both the smartphone and mobile app industries, as each component's development and popularity fuel the demand and innovation of the other.
    β€£
    πŸ‘ŽΒ Strategic substitutes

    Strategic substitutes mutually offset one another - Downward sloping reaction functions

    🧠
    An example of strategic substitutes that mutually offset each other is the competition between ride-sharing platforms like Uber and Lyft. When one platform lowers prices or offers promotions to attract customers, the other platform responds by implementing similar measures to prevent losing market share. As a result, the actions of each platform offset the potential gain of the other, leading to a competitive equilibrium in the ride-sharing market.
    β€£
    πŸ”‚Β Folk Theorem
    πŸ€”
    The Folk theorem tells us that when people play a game many times, they can find ways to cooperate even in situations where they would usually act selfishly. It means that in games like the prisoner's dilemma, where both players can benefit by cooperating but might be tempted to act selfishly, if they play the game repeatedly, they can learn to cooperate and get better outcomes together.
    β€£
    πŸ—£οΈΒ Coase Theorem
    πŸ€”
    The Coase theorem says that if people can freely negotiate and there are no transaction costs, they can solve their problems and reach efficient agreements, no matter who has the initial rights. It means that if people can talk to each other and there are no barriers to making deals, they can figure out the best solution, even if things start off in a messy way.
    β€£
    πŸͺ–πŸ’΅Β Bertrand Competition
    πŸ€”
    Bertrand competition is like a price war between companies. When two or more companies sell the same thing, they try to lower their prices to attract customers. It's like a race to offer the lowest price. The competition is intense because customers will choose the cheapest option, so the companies keep lowering their prices until they reach the lowest possible price, which is called the competitive equilibrium.
    β€£
    πŸ¦„Β Separating Equilibrium

    In aΒ separating equilibrium, senders of different types always choose different signals. This means that the signal always reveals the sender's type, so the receiver's beliefs become deterministic after seeing the signal.

    πŸ€”
    A separating equilibrium is like a special rule in a game where different players make different choices to reveal something about themselves. It's like a way to show their true colors. By making different choices, they help others learn more about them and make better decisions. It's a strategy to separate themselves from others and stand out in the game.
    β€£
    πŸ₯‹Pooling Equilibrium

    In aΒ pooling equilibrium, senders of different types all choose the same signal. This means that the signal does not give any information to the receiver, so the receiver's beliefs are not updated after seeing the signal.

    πŸ€”
    A pooling equilibrium is like a rule in a game where different players make the same choices to appear similar, even if they are different on the inside. It's like wearing the same uniform to blend in and hide individual differences. By doing this, they create an equal perception among others and make it harder for others to differentiate them. It's a strategy to pool together and appear as a group rather than stand out as individuals.
    β€£
    πŸ€·β€β™‚οΈΒ Semi-separating equilibrium

    In aΒ semi-separating equilibriumΒ (also calledΒ partial-pooling), some types of senders choose the same message and other types choose different messages.

    β€£
    πŸ“Β A Complete Contract

    In a complete contract, while the terms and conditions are explicitly stated and agreed upon by the involved parties, it does not necessarily mean that everything is observable to anyone. Complete contracts aim to specify the rights, obligations, and contingencies of the parties involved, but there can still be unobservable information or private knowledge that affects the implementation or outcomes of the contract.

    β€£
    🦾 Dominant Strategy

    A dominant strategy provides the player the highest payoff at every subgame played.

    β€£
    🎯 Cournot Oligopoly

    In a Cournot oligopoly, a group of companies are selling similar products and they make their decisions at the same time. It's like a race where everyone is trying to sell as much as they can without knowing what the others are doing.

    Imagine you and your friends are selling lemonade. You all set up your lemonade stands and decide how much lemonade to make without talking to each other. You have to guess how much lemonade the others will make and try to sell more than them. You want to make enough lemonade to earn a profit, but not so much that you have leftover unsold lemonade.

    So, in simple terms, Cournot oligopoly is like a lemonade-selling competition where each seller tries to guess how much the others will sell and make their own decisions to maximize their profits. It's a game of strategy where each player tries to find the right balance between selling enough and avoiding excess supply.

    β€£
    πŸ₯‡Stackelberg Oligopoly

    In a Stackelberg oligopoly, the leading company makes its production or pricing decisions first, and the other companies have to adjust their strategies accordingly. The leader has an advantage because it gets to set the terms of competition, and the other companies have to react and adapt to what the leader does.

    Think of it as a group of friends deciding what game to play. One friend takes charge and decides on the game they will play, and the rest of the friends have no choice but to go along with that decision. They can't change the game or choose something different because the leader has already made the decision.

    In simple terms, Stackelberg oligopoly is like a game where one player gets to make the rules, and the others have to play by those rules.

    β€£
    πŸ—ΊοΈΒ Property Rights
    πŸ€”
    Property rights refer to the legal rights and privileges that individuals or entities have over a particular asset or resource. It grants the owner the exclusive authority to use, control, transfer, and benefit from the asset. These rights provide individuals with the legal framework to claim ownership, possess, and make decisions regarding the use and disposition of the property.
    • Complete Exam AI2156 - June 2022
    • 1. Determine an individuals risk based on a utility function of wealth
    • 2. Deterring incumbents to enter a Market
    • 3. Identify a sub-game of perfect equilibrium
    • 4. If two variables are strategic complements, then…
    • 5. The Classical Microeconomic Model
    • 6. How do you minimise risk of equilibrium with adverse selection?
    • 7. A important parameter in the Folk Theorem
    • 8. Define a separating equilibrium?
    • 9. Hidden actions causes what?
    • 10. Pricing in the Bertrand oligopoly model
    • 11. Identify the dominant strategy and Nash Equilibria in a payoff matrix
    • 12. Two firms, Compute Price and Quantity in the Cournot & Stackelberg Model
    • 13. Interpret indifference curves for two types of insurance customers
    • 14. Fixed-Fee, Hire and Sharing Contracts
    • 15. Deadweight loss on environment
    • 16. Determine the property developer’s certainty
    • Complete Exam AI2156 - August 2022
    • 1. Identify the coefficient of absolute risk aversion given by a equation
    • 2. When is entry accommodated by the incumbent firm?
    • 3. Identify the subgame of perfect equilibrium
    • 4. Strategic Substitutes and their reaction functions
    • 5. The main drawback with the principal-agent approach to explain the boundary of a firm? ⚠️
    • 6. The theorem behind cooperative equilibrium prisoner’s dilemma?
    • 7. What is a property right?
    • 8. Important parameter in model of monopolistic competition
    • 9. What is a pooling equilibrium?
    • 10. A hidden characteristic is typically the reason
    • 11. Identify the dominant strategy and Nash Equilibria in a payoff matrix
    • 12. Determine the price and quantity each firm produces in a Cournot oligopoly model
    • 13. Avoid Adverse Selection by providing a menu of Contracts
    • 14. Difference between a complete and incomplete contract
    • 15. Examples of Principal agent problems
    • 16. Certainty Equivalent of this project
    • Exercise 2 - Oligopoly Theory
    • Incomplete Exam AI2156 - June 2021

    Complete Exam AI2156 - June 2022

    1. Determine an individuals risk based on a utility function of wealth

    1️⃣
    An individual with utility function 𝑒(π‘₯) = 2 + 5π‘₯ is... a) Risk-neutral

    b) Risk-averse

    c) Risk-loving

    d) None of the above

    β€£
    Answer and Explanation βœ…

    a) Risk-neutral

    Risk Type
    Utility Functions
    Optimization for
    Risk-Neutral
    𝑒(π‘₯) = π‘Ž + 𝑏π‘₯
    Maximizing overall wealth
    Risk-Averse
    𝑒(π‘₯) = π‘Ž ln(π‘₯)
    Balancing risk and return, avoiding losses
    Risk-Averse
    𝑒(π‘₯) = π‘Žπ‘₯^𝑏
    Balancing risk and return, diminishing marginal utility
    Risk-Loving
    𝑒(π‘₯) = 1 βˆ’ 𝑒^βˆ’π‘π‘₯
    Pursuing high returns, embracing risk
    Risk-Loving
    𝑒(π‘₯) = π‘Ž + 𝑏π‘₯ + 𝑐π‘₯^2
    Seeking both high returns and additional wealth

    2. Deterring incumbents to enter a Market

    2️⃣
    In an entry game with one incumbent firm and potential entrants, when is entry deterred?

    a) When there is no need for the incumbent firm to take any action.

    b) When an action by the incumbent firm prevents potential entrents from entering.

    c) When the incumbent firm is risk-neutral.

    d) When every action taken by incumbent firm is observable.

    β€£
    Answer & Explanation βœ…

    b) When an action by the incumbent firm prevents potential entrents from entering.

    Entry deterrence occurs when the incumbent firm adopts strategies or takes actions to discourage potential entrants from entering the market.
    Entry Deterrence Strategies
    Explanations
    Barriers to Entry
    Incumbent firms establish significant barriers, such as high capital requirements, exclusive resources, or legal restrictions, to make it difficult for potential entrants to enter the market.
    Preemptive Actions
    Incumbent firms adopt preemptive measures, like aggressive pricing strategies or offering discounts, to signal potential entrants about the intense competition and reduced profitability they would face upon entry.
    Capacity Expansion
    Incumbent firms strategically expand their capacity beyond current market demand, indicating to potential entrants the presence of excess capacity and intensified competition, deterring them from entering due to concerns about low market share and reduced profitability.
    Reputation and Branding
    Incumbent firms with strong reputations or established brands create customer loyalty and trust, making it challenging for potential entrants to compete against their well-known and trusted image.
    Strategic Investments
    Incumbent firms make strategic investments or commitments that increase costs or risks for potential entrants, such as research and development for new technologies, long-term contracts with suppliers or distributors, or securing exclusive partnerships, creating barriers that are difficult for new entrants to replicate.

    3. Identify a sub-game of perfect equilibrium

    3️⃣
    Consider the following game tree.
    image

    Firm A starts by choosing strategy A1 or A2, and then firm B chooses strategy B1 or B2. Here (xy) means that firm A receives payoff x and firm B payoff y. Which of the payoffs is the result of a subgame perfect equilibrium?

    β€£
    Answer & ExplanationΒ βœ…

    c) 5,3

    Maximum payoff for A = 5 β†’ A2, if A chooses A2 then B will choose B1 for a maximum of 3.

    4. If two variables are strategic complements, then…

    4️⃣
    If two variables are strategic complements, then…

    a) an investment makes the incumbent firm tough.

    b) an investment makes the incumbent firm soft.

    c) the reaction functions are upward sloping.

    d) the reaction functions are downward sloping.

    β€£
    Answer & Explanation βœ…

    c) the reaction functions are upward sloping.

    Strategic complements mutually reinforce one another - Upward sloping reaction functions.

    5. The Classical Microeconomic Model

    5️⃣
    In a classical microeconomic model of the firm, the firm is seen as

    a) a principal

    b) a black box

    c) a nexus of contracts

    d) an agent

    β€£
    Answer & Explanation βœ…

    b) a black box

    Players
    Seen as
    Explanation of actor
    Households
    Consumers and Workers
    Modeled by consumption patterns, labor supply decisions, savings behavior, and expectations about income and prices.
    Firms
    Black Box
    Assumed to optimize production decisions to maximize profits, without explicit modeling of their internal workings. Outputs and pricing decisions are important variables in the model.
    Government
    Active Participant
    Includes fiscal and monetary policy, government spending, taxation, and regulation. Viewed as an entity that can influence economic variables through policy choices.
    Financial Sector
    Variable Treatment
    Modeled to capture the role of financial institutions, banks, and markets in intermediating funds, providing credit, and influencing investment decisions.
    Foreign Sector
    Variable Treatment
    Captures international trade and capital flows, including exports, imports, exchange rates, and foreign investment. May be treated as an external entity impacting the domestic economy.
    Central Banks
    Variable Treatment
    Modeled as the authority responsible for monetary policy, controlling money supply, setting interest rates, and maintaining stability in the financial system.

    6. How do you minimise risk of equilibrium with adverse selection?

    6️⃣
    One way of minimising the risk of ending up in an equilibrium with adverse selection is to use...

    a) Contractor

    b) Well defined property rights

    c) Reaction functions

    d) Screening

    β€£
    Answer & Explanation βœ…

    d) Screening

    πŸ€”
    For example, let's say you're hiring a new employee for a job. You have many resumes from different candidates, and you want to narrow down your choices. You start by looking at their qualifications and experience to see if they meet the basic requirements. This is like screening their resumes to find out who might be a good fit for the job.

    7. A important parameter in the Folk Theorem

    7️⃣
    An important parameter in determining the equilibrium in a setting described by the Folk theorem is.. a) the discount factor. b) the number of players. c) the coefficient of absolute risk aversion. d) none of the above.
    β€£
    Answer & Explanation βœ…

    a) the discount factor

    πŸ€”
    Folk theorem is a repeated game and in a repeated game, the discount factor influences the players' decision-making process by determining the relative importance of future outcomes. A higher discount factor means that players place more weight on future payoffs, encouraging cooperative behavior and the emergence of sustained cooperative outcomes as equilibria. On the other hand, a lower discount factor reduces the relative importance of future payoffs, making non-cooperative outcomes more likely.

    8. Define a separating equilibrium?

    8️⃣
    A firm is offering its customers a menu of different products. What is a separating equilibrium? a) A situation where no equilibrium exists. b) An equilibrium where different groups choose the same item from the menu. c) An equilibrium where different groups choose different items from the menu. d) The same as a subgame.
    β€£
    Answer & Explanation βœ…

    c) An equilibrium where different groups choose different items from the menu.

    πŸ€”
    The basket of product offerings caters two or more different groups, eg. there exists a separating equilibrium

    9. Hidden actions causes what?

    9️⃣
    A hidden action is typically the reason for...

    a) adverse selection. b) transaction costs. c) moral hazard. d) complete contracts.

    β€£
    Answer & Explanation βœ…

    c) moral hazard.

    πŸ€”
    Hidden actions can contribute to moral hazard because when someone engages in activities that are not easily observed or known by others, they may feel more inclined to take risks or behave in ways that they wouldn't if their actions were transparent.

    10. Pricing in the Bertrand oligopoly model

    πŸ”Ÿ
    Which of the following statements is true in a Bertrand oligopoly model with two identical firms producing the same good?

    a) One of two firms sets the price first, and then the second firm sets it price.

    b) The price is set in order to maximize the total profit of the firms.

    c) The strategic variable is quantity.

    d) The equilibrium price is the same as the price-taking competitive equilibrium price.

    β€£
    Answer & Explanation βœ…Β 

    d) The equilibrium price is the same as the price-taking competitive equilibrium price.

    πŸ€”
    The Bertrand oligopoly modelΒ considers firms that make an identical product but compete on price and make their pricing decisions simultaneously.

    11. Identify the dominant strategy and Nash Equilibria in a payoff matrix

    ❓
    Given is the following payoff matrix.
    image

    Here (x,y) means that Player A receives payoff x and Player B payoff y. a) Is there a dominant strategy for any of the players? Motivate your answer. (3 p) b) Determine the Nash equilibria in pure strategies for this game. (3 p)

    β€£
    Solution a) and b) βœ…

    a) For player A there exists a dominant strategy. Player A can play S2 at every subgame and get the highest pay-off.

    Strategy A
    Strategy B
    If B plays:
    If A plays:
    S1 β†’ S2
    S1 β†’ S1
    S2 β†’ S2
    S2 β†’ S1
    S3 β†’ S2
    S3 β†’ S2

    b) The Nash Equilibrium in pure strategies for this game is (S2,S1) β†’ (3,5) eg. A plays their dominant strategy (S2) and player B responds (S1) or player B plays S1 and player A will then respond with their dominant strategy S2.

    12. Two firms, Compute Price and Quantity in the Cournot & Stackelberg Model

    ❓
    Two firms, Firm 1 and Firm 2, are producing an identical good. They both have the same cost function

    𝐢(π‘ž) = 2π‘ž

    The demand function for the product the firms produce is

    𝑄 = 20 βˆ’ 𝑝

    Determine the price of the good and the quantity each firm produces in the following two cases: a) In a Cournot oligopoly model. (3 p)

    b) In a Stackelberg oligopoly model when Firm 2 is the leader. (3 p)

    β€£
    a) Solution βœ…Β - Cournot Oligopol Model

    Step 1. Derive Cost function and Price function (dc/dQ and dp/dQ)

    𝐢(π‘ž) = 2π‘ž β†’ 𝐢’(π‘ž) = 2

    𝑄 = 20 βˆ’ 𝑝 β†’ 𝑝 = 20 - 𝑄

    𝑝 = 20 - 𝑄 β†’ 𝑝’ = -1

    Step 2. Solve for quantity for each Firm

    Firm 1 (Ο€1Ο€_1Ο€1​)

    Ο€1=pq1βˆ’cΟ€_1 = pq_1-cΟ€1​=pq1β€‹βˆ’c

    Ο€1β€²=pβ€²q1+pq1β€²βˆ’cβ€²Ο€_1' = p'q_1 + pq_1'-c'Ο€1′​=pβ€²q1​+pq1β€²β€‹βˆ’cβ€² β†’ βˆ’q1+20βˆ’Qβˆ’2=βˆ’q1+20βˆ’(q1+q2)βˆ’2-q_1+20-Q-2 = -q_1+20-(q_1+q_2)-2βˆ’q1​+20βˆ’Qβˆ’2=βˆ’q1​+20βˆ’(q1​+q2​)βˆ’2 β†’ In equilibrium this is =0

    β†’ βˆ’q1+20βˆ’(q1+q2)βˆ’2=0-q_1+20-(q_1+q_2)-2 =0βˆ’q1​+20βˆ’(q1​+q2​)βˆ’2=0 β†’ βˆ’2q1+18βˆ’q2=0-2q_1+18-q_2 = 0βˆ’2q1​+18βˆ’q2​=0 β†’ q1=9βˆ’(q2/2)q_1= 9-(q_2/2)q1​=9βˆ’(q2​/2)

    Firm 2 (Ο€2Ο€_2Ο€2​)

    Ο€2=pq2βˆ’cΟ€_2 = pq_2-cΟ€2​=pq2β€‹βˆ’c

    Ο€2β€²=pβ€²q2+pq2β€²βˆ’cβ€²Ο€_2' = p'q_2 + pq_2'-c'Ο€2′​=pβ€²q2​+pq2β€²β€‹βˆ’cβ€² = Ο€2β€²=(βˆ’1)q2+20βˆ’Qβˆ’2Ο€_2' = (-1)q_2 + 20 -Q -2Ο€2′​=(βˆ’1)q2​+20βˆ’Qβˆ’2 β†’ q2+20βˆ’(q2+q1)βˆ’2q_2 + 20 -(q_2+q_1) -2q2​+20βˆ’(q2​+q1​)βˆ’2

    q2=9βˆ’(q1/2)q_2= 9-(q_1/2)q2​=9βˆ’(q1​/2) πŸ€”Β We can conclude q2=q1q_2=q_1q2​=q1​

    Step 3 Solve for and q2q_2q2​ and q1q_1q1​

    (1)βˆ’2q1βˆ’q2+18=0(1) -2q_1 -q_2 + 18 = 0(1)βˆ’2q1β€‹βˆ’q2​+18=0

    (2)βˆ’2q2βˆ’q1+18=0(2) -2q_2 -q_1 + 18 = 0(2)βˆ’2q2β€‹βˆ’q1​+18=0 ← Take (2)*(-2) and add to first equation = 2q1+4q2βˆ’36βˆ’2q1βˆ’q2+18=02q_1 +4q_2 -36 -2q_1-q_2+18 = 02q1​+4q2β€‹βˆ’36βˆ’2q1β€‹βˆ’q2​+18=0

    3q2βˆ’18=0β†’q2=18/3β†’q2=63q_2 -18 = 0 \rightarrow q_2=18/3 \rightarrow q_2=63q2β€‹βˆ’18=0β†’q2​=18/3β†’q2​=6 and q1=9βˆ’q2/2β†’q1=9βˆ’6/2β†’q1=q2β†’q1=6q_1 = 9-q_2/2 \rightarrow q_1=9-6/2 \rightarrow q_1=q_2 \rightarrow q_1 = 6q1​=9βˆ’q2​/2β†’q1​=9βˆ’6/2β†’q1​=q2​→q1​=6

    Step 4. Solve for price

    𝑝 = 20 - 𝑄 β†’ p=20βˆ’(q1+q2)β†’p=20βˆ’(6+6)β‡’p=8p=20-(q_1+q_2) \rightarrow p=20-(6+6) \Rightarrow p=8p=20βˆ’(q1​+q2​)β†’p=20βˆ’(6+6)β‡’p=8

    Answer:Β In the cournot model the price is 8 and the quantities of each firm is 6.

    β€£
    b) Solution βœ…Β - Stackelberg Oligopoly Model

    Step 1. Firm 2 is the leader - They set the rules of the game

    Ο€2=pq2βˆ’c=(20βˆ’Q)q2βˆ’2q2β‡’20βˆ’(q1+q2)q2βˆ’2q2Ο€_2 = pq_2-c = (20-Q)q_2-2q_2 \Rightarrow 20-(q_1+q_2)q_2-2q_2Ο€2​=pq2β€‹βˆ’c=(20βˆ’Q)q2β€‹βˆ’2q2​⇒20βˆ’(q1​+q2​)q2β€‹βˆ’2q2​

    ⚠️ where q1=9βˆ’(q2/2)q_1= 9-(q_2/2)q1​=9βˆ’(q2​/2) ⚠️

    Step 2. Exchange q1q_1q1​ for q1=9βˆ’(q2/2)q_1= 9-(q_2/2)q1​=9βˆ’(q2​/2)

    Ο€2=20βˆ’((9βˆ’(q2/2))+q2)q2βˆ’2q2Ο€_2 = 20-((9-(q_2/2))+q_2)q_2-2q_2Ο€2​=20βˆ’((9βˆ’(q2​/2))+q2​)q2β€‹βˆ’2q2​

    Ο€2=9q2βˆ’q22/2Ο€_2 = 9q_2-q_2^2/2Ο€2​=9q2β€‹βˆ’q22​/2

    Step 3. Derive Ο€2Ο€_2Ο€2​ and compute Ο€2β€²=0Ο€_2' = 0Ο€2′​=0

    Ο€2β€²=9βˆ’q2Ο€_2' = 9-q_2Ο€2′​=9βˆ’q2​

    Ο€2β€²=0:q2=9Ο€_2' =0 : q_2 = 9Ο€2′​=0:q2​=9

    Step 4. Compute q1q_1q1​ and price

    q1=9βˆ’(9/2)q_1= 9-(9/2)q1​=9βˆ’(9/2) β†’ q1=4,5q_1=4,5q1​=4,5

    p=20βˆ’(q1+q2)=20βˆ’(9+4,5)=6,5p=20-(q_1+q_2)=20-(9+4,5)=6,5p=20βˆ’(q1​+q2​)=20βˆ’(9+4,5)=6,5

    Answer: Β In the stackelberg model 6,5 and the quantities of firm-one 4,5 and firm-two is 9.

    13. Interpret indifference curves for two types of insurance customers

    ❓
    The indifference curves of two groups of insurance company customers are given by π‘…βˆ’5𝐷 = 𝑒0 and (1) 𝑅 βˆ’ 𝐷 = 𝑒0 (2) respectively, where 𝑅 is the reduction in insurance premium, 𝐷 is the deductable and 𝑒0 is the utility level. Which of the two groups is the high risk group? Motivate your answer. (3 p)
    β€£
    Answer βœ…
    image

    Deductable = Upfront cost

    Insurance Premium = Monthly payment

    Step 1. Re-write functions

    R1=u0+5DR_1=u_0+5DR1​=u0​+5D

    R2=u0+DR_2=u_0+DR2​=u0​+D ← Step 2. Draw the functions & Motivate

    High-risk utility functions are usually steeper β†’ R1 is the high risk group

    Hence the High-risk group is represented by R1. These customers are more willing to accept higher deductibles for greater reductions in insurance premiums, indicating a higher tolerance for risk.

    The low risk group R2. These customers are less willing to accept higher deductibles for greater reductions in insurance premiums, indicating a lower tolerance for risk.

    🧠
    Basically, High-risk individuals get the most utility from low up-front cost and a low monthly payment. Low risk individuals want to pay a high up-front cost for a low reduction to their monthly payment. ( I think)

    14. Fixed-Fee, Hire and Sharing Contracts

    ❓
    What is meant by a fixed-fee contract, a hire contract and a sharing contract respectively?
    β€£
    Answer βœ…
    Type of Contract
    Summary
    Explanation
    +Pro’s -Con’s
    Fixed Fee Contract
    Payment is a predetermined fixed amount, regardless of performance.
    A fixed-fee contract is an agreement where a fixed amount of money is paid upfront or periodically, regardless of the actual performance or outcome of the contracted work. It establishes a predetermined fee that remains unchanged regardless of the level of effort or resources expended.
    + Fixed Cost + Less Risk - No adjustments
    Hire Contract
    Employment agreement where the employee receives a wage or salary.
    In this type of contract, the employee is hired to perform certain tasks or provide services in exchange for a specified wage or salary. The compensation is usually based on factors such as time worked, skills, qualifications, and job responsibilities.
    - Principal Agent - Unions + Screening πŸ‘
    Sharing Contract
    Agreement to share revenue or profits from a venture among involved parties.
    Two or more parties agree to share the revenue or profits generated from a particular venture or project. The distribution of the shared revenue or profits is typically based on predefined terms and can be proportional to the contributions, investments, or other agreed-upon criteria of the involved parties.
    - Rev.Share - Complex + Aligned interests

    15. Deadweight loss on environment

    ❓
    A monopolist is polluting the environment when producing its good. The inverse demand function for the good the firm produces is 𝑝(𝑄) = 20 βˆ’ 𝑄, and the monopolist’s cost function is 𝐢(𝑄) = 𝑄^2. The total cost for the society is given by 𝐢𝑆(𝑄) = 𝑄^2 + 2𝑄. a) How large is the deadweight loss? (5 p) b) Describe how well defined property rights can be used to make the quantity produced equal to the socially optimally level. (3 p)
    β€£
    Answer a) and b) βœ…

    Inverse Demand function: p(Q)=20βˆ’Qp(Q)=20-Qp(Q)=20βˆ’Q

    Monopolist Cost Function: Cm(Q)=Q2C_m(Q)=Q^2Cm​(Q)=Q2

    Total Cost for society: Cs(Q)=Q2+2QC_s(Q)=Q^2+2QCs​(Q)=Q2+2Q

    Step 1. Derive the cost functions

    Csβ€²(Q)=2Q+2C_s'(Q)=2Q+2Cs′​(Q)=2Q+2

    Cmβ€²(Q)=2QC_m'(Q)=2QCm′​(Q)=2Q

    Step 2. Set them equal to the inverse demand function and calculate P for each

    Society β†’ 2Q+2=20βˆ’Qβ‡’Q=62Q+2=20-Q \Rightarrow Q=62Q+2=20βˆ’Qβ‡’Q=6 β†’ p = 14

    Monopolist β†’ 2Q=20βˆ’Qβ‡’Q=20/32Q=20-Q \Rightarrow Q=20/32Q=20βˆ’Qβ‡’Q=20/3 β†’ p=13,3

    Step 4. Cry because you couldn’t figure out this question 😒

    b) How well defined property rights can be used to make production into the socially optimal level

    1. The Coase Theorem states that under ideal economic conditions, where there is a conflict of property rights, the involved parties can bargain or negotiate terms that will accurately reflect the full costs and underlying values of the property rights at issue, resulting in the most efficient outcome. With well defined property rights actors in society or the government can own the rights to a thing. The owner of said rights can then accept payment from society to combat negative externalites.

    16. Determine the property developer’s certainty

    ❓
    A property developer has utility function 𝑒(π‘₯) = √π‘₯, and is considering investing in a project with the following possible payoffs (in millions Euro):
    Payoff (mEUR)
    Probability (%)
    10
    0,4
    50
    0,4
    100
    0,2

    Determine the property developer’s certainty equivalent of this project. (4 p)

    β€£
    Solution βœ…

    Utility function: u(x)=xu(x)=\sqrt{x}u(x)=x​

    Expected Utility E(x) = (10βˆ—0,4)+(50βˆ—0,4)+(100βˆ—0,2)=6,0933(\sqrt{10}*0,4)+(\sqrt{50}*0,4)+(\sqrt{100}*0,2) = 6,0933(10β€‹βˆ—0,4)+(50β€‹βˆ—0,4)+(100β€‹βˆ—0,2)=6,0933

    Certainty: 6,0933=cβ‡’c=6,09332=37,1286,0933=\sqrt{c}\Rightarrow c=6,0933^2 = 37,1286,0933=c​⇒c=6,09332=37,128

    Complete Exam AI2156 - August 2022

    1. Identify the coefficient of absolute risk aversion given by a equation

    1️⃣
    An individual has utility function is 𝑒(π‘₯)=1βˆ’π‘’βˆ’2x𝑒(π‘₯) = 1 βˆ’ 𝑒^{-2x}u(x)=1βˆ’eβˆ’2x The value of this individual’s coefficient of absolute risk aversion at π‘₯ = 10 is given by a) βˆ’2. b) 0. c) 1. d) 2.
    β€£
    Answer & Explanation βœ…

    d) 2 Calculate by taking (-U’’(x))/U’(x)

    πŸ€”
    𝑒(π‘₯) = 1 βˆ’ 𝑒^βˆ’π‘π‘₯ β†’ The individual’s coefficient of absolute risk aversion is b in this equation, which is 2.
    Risk Type
    Utility Functions
    Optimization for
    Risk-Loving
    𝑒(π‘₯) = 1 βˆ’ 𝑒^βˆ’π‘π‘₯
    Pursuing high returns, embracing risk
    Risk-Loving
    𝑒(π‘₯) = π‘Ž + 𝑏π‘₯ + 𝑐π‘₯^2
    Seeking both high returns and additional wealth
    Risk-Neutral
    𝑒(π‘₯) = π‘Ž + 𝑏π‘₯
    Maximizing overall wealth
    Risk-Averse
    𝑒(π‘₯) = π‘Ž ln(π‘₯)
    Balancing risk and return, avoiding losses
    Risk-Averse
    𝑒(π‘₯) = π‘Žπ‘₯^𝑏
    Balancing risk and return, diminishing marginal utility

    2. When is entry accommodated by the incumbent firm?

    2️⃣
    In an entry game with one incumbent firm and potential entrants, when is entry accommodated?

    a) When there is no need for the incumbent firm to take any action. b) When an action by the incumbent firm prevents potential entrants from entering. c) When every action taken by incumbent firm is observable. d) None of the above.

    β€£
    Answer & Explanation βœ…

    d) None of the above.

    🧠
    Entry is accommodated: The incumbent firm accepts that the entrant will enter the market. Typically it is too expensive to deter the entrant, and the market becomes a duopoly. 𝐾 > 0 (Lecture 4)

    3. Identify the subgame of perfect equilibrium

    3️⃣
    Consider the following game tree.
    image

    Firm A starts by choosing strategy A1 or A2, and then firm B chooses strategy B1 or B2. Here (x,y) means that firm A receives payoff x and firm B payoff y. Which of the payoffs is the result of a subgame perfect equilibrium? a) (6,2) b) (5,7) c) (1,8) d) (2,3)

    β€£
    Answer & Explanation βœ…

    b) (5,7)

    Player A will go for (6,2) or (5,7) = A1 Dominant strategy β†’ Leading Player B to choose B2 β†’ (5,7) is the subgame of perfect equilibrium.

    4. Strategic Substitutes and their reaction functions

    4️⃣
    If two variables are strategic substitutes, then...

    a) an investment makes the incumbent firm tough. b) an investment makes the incumbent firm soft. c) the reaction functions are upward sloping. d) the reaction functions are downward sloping

    β€£
    Answer & Explanation βœ…

    d) the reaction functions are downward sloping

    🧠
    An example of strategic substitutes that mutually offset each other is the competition between ride-sharing platforms like Uber and Lyft. When one platform lowers prices or offers promotions to attract customers, the other platform responds by implementing similar measures to prevent losing market share. As a result, the actions of each platform offset the potential gain of the other, leading to a competitive equilibrium in the ride-sharing market.

    5. The main drawback with the principal-agent approach to explain the boundary of a firm? ⚠️ 

    5️⃣
    What is the main drawback with the principal-agent approach to explain the boundary of a firm?

    a) It is a black box. b) It disregards the fact that it is costly to write contracts. c) It does not take into account the fact that there is a nexus of contracts. d) It violates the conditions of the Coase theorem.

    β€£
    Answer & Explanation βœ…

    b) It disregards the fact that it is costly to write contracts.

    🧠
    The principal-agent approach focuses on the relationship between the principal and the agent. It assumes that contracts can be written to align the interests of the principal and the agent, ensuring efficient and optimal decision-making.

    However, in reality, writing and enforcing contracts can be costly and challenging. Transaction costs such as negotiating, monitoring, and enforcing contracts can significantly affect the efficiency and feasibility of contracting arrangements. The principal-agent approach tends to overlook these transaction costs associated with writing and enforcing contracts, thereby overlooking the practical limitations of using contracts alone to govern relationships and determine the boundaries of a firm.

    6. The theorem behind cooperative equilibrium prisoner’s dilemma?

    6️⃣
    The fact that the cooperative equilibrium can exist in a prisoner’s dilemma game if it is played several times is known as...

    a) the Coase theorem. b) the property rights solution. c) Bertrand competition. d) a Folk theorem.

    β€£
    Answer & Explanation βœ…

    d) a Folk theorem

    🧠
    Folk theorem suggests that if the game is repeated over multiple rounds, players have the opportunity to establish cooperative agreements and achieve better outcomes. By playing the prisoner's dilemma game repeatedly, players can use strategies that incorporate punishment or reward mechanisms based on the other player's previous actions. This allows for the emergence of cooperative equilibria where both players cooperate rather than solely pursuing their dominant strategy of defection.

    7. What is a property right?

    7️⃣
    A property right is... a) an asset preferred by a risk averse individual. b) an example of a Nash equilibrium. c) an exclusive privilege to use an asset. d) the same as a hidden action.
    β€£
    Answer & Explanation βœ…

    c) an exclusive privilege to use an asset.

    πŸ€”
    Property rights refer to the legal rights and privileges that individuals or entities have over a particular asset or resource. It grants the owner the exclusive authority to use, control, transfer, and benefit from the asset. These rights provide individuals with the legal framework to claim ownership, possess, and make decisions regarding the use and disposition of the property.

    8. Important parameter in model of monopolistic competition

    8️⃣
    An important parameter in determining the equilibrium in a model of monopolistic competition is... a) the discount factor. b) the number of firms. c) the coefficient of absolute risk aversion. d) none of the above.
    β€£
    Answer & Explanation βœ…

    b) the number of firms.

    πŸ€”
    Monopolistic competition is a market structure characterized by a large number of firms that produce differentiated products, meaning each firm offers a slightly different product from its competitors. The number of firms in this market structure plays a crucial role in determining the equilibrium outcome

    9. What is a pooling equilibrium?

    9️⃣
    A firm is offering its customers a menu of different products. What is a pooling equilibrium? a) A situation where no equilibrium exists. b) An equilibrium where different groups choose the same item from the menu. c) An equilibrium where different groups choose different items from the menu. d) The same as a subgame.
    β€£
    Answer & Explanation βœ…

    b) An equilibrium where different groups choose the same item from the menu.

    🧠
    In aΒ pooling equilibrium, senders of different types all choose the same signal. This means that the signal does not give any information to the receiver, so the receiver's beliefs are not updated after seeing the signal.

    10. A hidden characteristic is typically the reason

    πŸ”Ÿ
    A hidden characteristic is typically the reason for... a) adverse selection. b) transaction costs. c) moral hazard. d) complete contracts.
    β€£
    Answer & Explanation βœ…

    A hidden characteristic is typically the reason for a) adverse selection.

    πŸ€”
    Adverse selection refers to a situation where one party in a transaction possesses more information or knowledge about a particular characteristic than the other party. When there is a hidden characteristic, such as private information or an asymmetry of information, it creates a situation where one party has an advantage in knowing the true value or quality of a product, service, or action. This hidden characteristic can lead to adverse selection, as the party with less information may make decisions based on incomplete or inaccurate information, resulting in undesirable outcomes or imbalances in the transaction.

    11. Identify the dominant strategy and Nash Equilibria in a payoff matrix

    ❓
    Given is the following payoff matrix.
    image

    Here (x,y) means that Player A receives payoff x and Player B payoff y. a) Is there a dominant strategy for any of the players? Motivate your answer. (3 p) b) Determine the Nash equilibria in pure strategies for this game. (3 p)

    β€£
    Solution βœ…
    Best response Strategy A
    Best response Strategy B
    B: S1 β†’ A: S1 (3,4)
    A: S1 β†’ B: S1 (3,4)
    B: S2 β†’ A: S2 (5,8)
    A: S2 β†’ B: S2 (5,8)
    B: S3 β†’ A: S3 (7,4)
    A: S3 β†’ B: S2 (2,5)

    a) We can conclude from table above that there is no dominant response strategy for A or B.

    b) Notice the high-lighted pairs above S1 and S2 for A and B respectively.

    12. Determine the price and quantity each firm produces in a Cournot oligopoly model

    ❓
    Two firms, Firm 1 and Firm 2, are producing an identical good. They have cost functions 𝐢1(π‘ž)=π‘žπΆ_1(π‘ž) = π‘žC1​(q)=q and 𝐢2(π‘ž)=3π‘žπΆ_2(π‘ž) = 3π‘žC2​(q)=3q respectively. The demand function for the product the firms produce is Q=20βˆ’pQ=20-pQ=20βˆ’p Determine the price of the good and the quantity each firm produces in a Cournot oligopoly model. (4 p)
    β€£
    Solution βœ…

    Step 1. Derive Cost and price functions based on Q

    Cost function firm 1: C1(q)=qβ†’C1β€²(q)=1C_1(q)=q \rightarrow C'_1(q)=1C1​(q)=qβ†’C1′​(q)=1

    Cost function firm 2: C2(q)=3qβ†’C2β€²(q)=3C_2(q)=3q \rightarrow C_2'(q)=3C2​(q)=3qβ†’C2′​(q)=3

    Demand Function Q=20βˆ’pQ=20-pQ=20βˆ’p

    Price Function Q=20βˆ’pQ=20-pQ=20βˆ’p β†’ p(Q)=20βˆ’Qp(Q) = 20-Qp(Q)=20βˆ’Q β†’ pβ€²(Q)=βˆ’1p'(Q)=-1pβ€²(Q)=βˆ’1

    Step 2. Solve for quantity for each Firm

    Firm 1

    Ο€1=pq1βˆ’C1(q)Ο€_1 = pq_1-C_1(q)Ο€1​=pq1β€‹βˆ’C1​(q)

    Ο€1β€²=pβ€²q1+pq1β€²βˆ’C1β€²(q)βŸΉΟ€_1' = p'q_1 + pq_1'-C_1'(q) \LongrightarrowΟ€1′​=pβ€²q1​+pq1β€²β€‹βˆ’C1′​(q)⟹ βˆ’1βˆ—q1+1(20βˆ’Q)βˆ’1β‡’-1*q_1 +1(20-Q)-1 \Rightarrowβˆ’1βˆ—q1​+1(20βˆ’Q)βˆ’1β‡’ βˆ’q1+(20βˆ’(q1+q2)βˆ’1=0-q_1 +(20-(q_1+q_2)-1 = 0βˆ’q1​+(20βˆ’(q1​+q2​)βˆ’1=0

    19βˆ’2q1βˆ’q2=0β‡’q1=19/2βˆ’q2/219-2q_1-q_2 = 0 \Rightarrow q_1=19/2 - q_2/219βˆ’2q1β€‹βˆ’q2​=0β‡’q1​=19/2βˆ’q2​/2

    Firm 2

    Ο€2=pq2βˆ’C2(q)Ο€_2 = pq_2-C_2(q)Ο€2​=pq2β€‹βˆ’C2​(q)

    Ο€2β€²=pβ€²q2+pq2β€²βˆ’C2β€²(q)βŸΉβˆ’1q2+(20βˆ’Q)βˆ’3β‡’βˆ’q2+20βˆ’q1βˆ’q2βˆ’3=0Ο€_2' = p'q_2+pq_2'-C_2'(q) \Longrightarrow -1q_2+(20-Q)-3 \Rightarrow -q_2+20-q_1-q_2-3 =0Ο€2′​=pβ€²q2​+pq2β€²β€‹βˆ’C2′​(q)βŸΉβˆ’1q2​+(20βˆ’Q)βˆ’3β‡’βˆ’q2​+20βˆ’q1β€‹βˆ’q2β€‹βˆ’3=0

    17βˆ’2q2βˆ’q1=0β‡’q2=17/2βˆ’q1/217-2q_2-q_1 = 0 \Rightarrow q_2=17/2 - q_1/217βˆ’2q2β€‹βˆ’q1​=0β‡’q2​=17/2βˆ’q1​/2

    Step 3. Solve for and q2q_2q2​ and q1q_1q1​

    q1=19/2βˆ’q2/2q_1=19/2 - q_2/2q1​=19/2βˆ’q2​/2

    q2=172βˆ’q12⟹q2=172βˆ’19+q222β‡’q2=5q_2=\frac{17}{2} - \frac{q_1}{2} \Longrightarrow q_2=\frac{17}{2} - \frac{\frac{19+q_2}{2}}{2} \Rightarrow q_2=5q2​=217β€‹βˆ’2q1β€‹β€‹βŸΉq2​=217β€‹βˆ’2219+q2​​​⇒q2​=5

    q1=19/2βˆ’q2/2β‡’q1=19/2βˆ’5/2β‡’q1=7q_1=19/2 - q_2/2 \Rightarrow q_1=19/2 - 5/2 \Rightarrow q_1=7q1​=19/2βˆ’q2​/2β‡’q1​=19/2βˆ’5/2β‡’q1​=7

    Step 4. Solve for price

    Quantity Q=(q1+q2)=7+5Q=(q_1+q_2) =7+5Q=(q1​+q2​)=7+5

    Price P=20βˆ’Q=20βˆ’12P = 20-Q =20-12P=20βˆ’Q=20βˆ’12

    Answer: q1 = 7 and q2=5, both with a price of 8 according to Cournot Oligopoly Model

    13. Avoid Adverse Selection by providing a menu of Contracts

    ❓
    Explain in detail how introducing a menu of contracts can result in an equilibrium where adverse selection is avoided. (4p)
    β€£
    Solution βœ…
    πŸ€”
    A menu of contracts reveals the preferences of the bidders. You could look at it like an inverted screening, where rather than trying to screen the candidates for contextually approproiate signals, they bid based on their preferences and risk-profiles. If there is multiple bidders and pooling occurs there is a market segmentation within these contracts.
    Adverse Selection Mitigation
    Description
    Examples
    Keywords
    Self-Selection
    Individuals choose contracts that align with their preferences and risk profiles, revealing their private information.
    Insurance policies with different coverage limits and premiums based on risk profiles.
    Preferences, risk profiles, private information
    Price Differentiation
    Pricing contracts based on risk or quality, charging higher premiums to higher-risk individuals and lower premiums to lower-risk individuals.
    Health insurance premiums based on age, driving insurance premiums based on driving history.
    Pricing, risk, premiums
    Pooling
    Segmenting the market into different contract options to create risk pools with individuals who have similar characteristics.
    Group life insurance policies for specific professions, such as firefighters or pilots.
    Risk pools, segmentation, characteristics
    Signal of Quality
    Offering a variety of contract options as a signal of knowledge and trustworthiness, attracting customers who are more likely to reveal accurate private information.
    Financial investment firms offering different portfolios with various risk levels.
    Trustworthiness, knowledge, signal, portfolios

    14. Difference between a complete and incomplete contract

    ❓
    What is the difference between a complete contract and an incomplete contract? (3p)
    β€£
    Answer βœ…
    Complete Contract
    Incomplete Contract
    Define
    A contract that specifies all possible contingencies and terms
    A contract that lacks specific details or provisions for certain scenarios
    Level of Detail
    Provides Explicit instructions for various situations
    May leave certain terms and conditions open to negotiation or interpretation
    Complexity
    Increased complexity due to comprehensive terms.
    May be simpler due to fewer details and provisions to consider.
    Transaction Costs
    Increased transaction costs due to a larger contract scope and through details
    Less complexity leads to decreased transaction cost due to flexibility and simplicity.
    Enforcement
    Easier to enforce since all terms are clearly stated.
    Requires additional negotiations or legal processes for enforcement.
    Flexibility
    Less flexible as all scenarios are covered
    Allows for flexibility in adapting to unforeseen circumstances.

    15. Examples of Principal agent problems

    ❓
    There are many examples of principal-agent relationships. In this question you are asked to give examples of some specific situations. a) Give an example of a situation where it is reasonable to assume that the principal is risk neutral and the agent is risk averse, and motivate your answer. (3 p) b) Give an example of a situation where it is reasonable to assume that the principal is risk averse and the agent risk neutral, and motivate your answer. (3 p) c) Give an example of a situation where a firm can be both a principal and an agent, and motivate your answer. (3 p)
    β€£
    Solution βœ…
    Question
    Principal
    Agent
    Example
    a)
    Risk Neutral Principal: Large Company
    Risk Averse Agent: Employee
    For the risk neutral employer, the agent can easily be switched out by the company. The agent does not play a crucial role for the companys success. However, the agent might rely on this paycheck. Eg. That paycheck is crucial for the agent.
    b)
    Risk Averse ”Privatperson” / Individual
    Risk Neutral ”Contractor”
    A physical person looking for a contractor to re-model their kitchen. For the contractor, they do not rely on this single contract to stay in business. They also do not need to live with potential wrong-doings made to the kitchen eg. low skin in the game. While the individual buying their service, both expend capital and have to live with consequences of how the kitchen turns out.
    c)
    Principal = Agent ”Big Pipes LLC”
    Agent = Principal ”Long Pipes LLC”
    Water has flooded the basement of a property. The landlord contacts their usual plumbing contractors but at the site β€œBig Pipes LLC” understand that this a job for β€œLong Pipes LLC”. Therefore Big Pipes hires Long Pipes as a subcontractor for the problem. In this scenario β€œBig Pipes LLC” is an agent towards the landlord, but the principal towards the sub-contractor.

    16. Certainty Equivalent of this project

    ❓
    A real estate company has utility function 𝑒(π‘₯)=ln(π‘₯+5) , and is considering investing in a project with the following possible payoffs (in millions Euro):
    Payoff (mEUR)
    Probability (%)
    -3
    0,55
    12
    0,3
    24
    0,15

    Determine the property developer’s certainty equivalent of this project. (4 p)

    β€£
    Solution βœ…

    Utility function: u(x)=ln(x+5)u(x)=ln(x+5)u(x)=ln(x+5)

    For the certainty equivalent, U(c)=EU(x)U(c)= E_U(x)U(c)=EU​(x) holds.

    Expected Utility E(x) = 0.55(ln(βˆ’3+5))+0.3(ln(12+5))+0.15(ln(24+5))0.55(ln(-3+5))+0.3(ln(12+5))+0.15(ln(24+5))0.55(ln(βˆ’3+5))+0.3(ln(12+5))+0.15(ln(24+5))

    EU(x)=0.55(ln(2))+0.3(ln(17))+0.15(ln(29))=1,736E_U(x)=0.55(ln(2))+ 0.3(ln(17))+0.15(ln(29)) = 1,736EU​(x)=0.55(ln(2))+0.3(ln(17))+0.15(ln(29))=1,736

    Certainty: U(c)=ln(5+c)U(c)= ln(5+c)U(c)=ln(5+c)

    : 1,736=ln(5+c)⟹e1,736=(5+c)⇔e1,736βˆ’5=cβ‡’c=0,6761,736 = ln(5+c)\Longrightarrow e^{1,736}=(5+c) \Leftrightarrow e^{1,736}-5=c \Rightarrow c=0,6761,736=ln(5+c)⟹e1,736=(5+c)⇔e1,736βˆ’5=cβ‡’c=0,676

    β€£

    Exercise 2 - Oligopoly Theory

    Bertrand Model

    In a Bertrend Model, the demand functions for two goods are given by D1 and D2

    D1(p1,p2)=10βˆ’p1+2p2D_1(p_1,p_2) = 10-p_1+2p_2D1​(p1​,p2​)=10βˆ’p1​+2p2​

    D2(p1,p2)=5+p1βˆ’p2D_2(p_1,p_2) = 5+p_1-p_2D2​(p1​,p2​)=5+p1β€‹βˆ’p2​

    How much is produced of each of the goods, if both firms have the cost function C(q)=qC(q)=qC(q)=q

    β€£
    Method to Solve the Bertrand Model πŸ‘

    Step 1. Calculate Profit, Derive for Maximum Profit to get Reaction function

    Firm 1:

    Profit:

    Ο€1=p1Q(p1)βˆ’C(q1)Ο€_1 = p_1Q(p_1)-C(q_1)Ο€1​=p1​Q(p1​)βˆ’C(q1​)

    Ο€1=p1(10βˆ’p1+2p2)βˆ’(10βˆ’p1+2p2)β‡’βˆ’p12+10p1+2p1p2+p1βˆ’10βˆ’2p2Ο€_1 = p_1(10-p_1+2p_2)-(10-p_1+2p_2)\Rightarrow -p_1^2+10p_1+2p_1p_2+p_1-10-2p_2Ο€1​=p1​(10βˆ’p1​+2p2​)βˆ’(10βˆ’p1​+2p2​)β‡’βˆ’p12​+10p1​+2p1​p2​+p1β€‹βˆ’10βˆ’2p2​

    Maximum profit FOC: Ο€1β€²(p1)=2(p1+p2)+11β‡’2(p1+p2)+11=0Ο€_1'(p_1) =2(p_1+p_2)+11 \Rightarrow2(p_1+p_2)+11=0Ο€1′​(p1​)=2(p1​+p2​)+11β‡’2(p1​+p2​)+11=0

    p1=11/2+p2p_1=11/2+p_2p1​=11/2+p2​

    Reaction function Firm 1: R1(p2)=11/2+p2R_1(p_2)=11/2+p_2R1​(p2​)=11/2+p2​

    Firm 2:

    Profit: Ο€2=p2Q(p2)βˆ’C(q2)Ο€_2 = p_2Q(p_2)-C(q_2)Ο€2​=p2​Q(p2​)βˆ’C(q2​) Ο€2=p2(5+p1βˆ’p2)βˆ’(5+p1+p2)β‡’βˆ’p22+5p2+p1p2+p2βˆ’5βˆ’p1Ο€_2 = p_2(5+p_1-p_2)-(5+p_1+p_2)\Rightarrow -p_2^2+5p_2+p_1p_2+p_2-5-p_1Ο€2​=p2​(5+p1β€‹βˆ’p2​)βˆ’(5+p1​+p2​)β‡’βˆ’p22​+5p2​+p1​p2​+p2β€‹βˆ’5βˆ’p1​

    Maximum profit FOC:

    Ο€2β€²(p2)=p1βˆ’2p2+6β‡’p1βˆ’2p2+6=0Ο€_2'(p_2) =p_1-2p_2+6 \Rightarrow p_1-2p_2+6 =0Ο€2′​(p2​)=p1β€‹βˆ’2p2​+6β‡’p1β€‹βˆ’2p2​+6=0

    p2=3+p1/2p_2=3+p_1/2p2​=3+p1​/2

    Reaction function Firm 2: R2(p1)=3+p1/2R_2(p_1)=3+p_1/2R2​(p1​)=3+p1​/2

    Step 2. Find Equilibrium

    The prices in the Bertrand Equilbrium satisfies

    R1(p2)=p1⇔R2(p1)=p2R_1(p_2)=p_1 \Leftrightarrow R_2(p_1)=p_2R1​(p2​)=p1​⇔R2​(p1​)=p2​

    p2=3+11/2+p22⇔p2=11,5p_2 = 3+\frac{11/2+p_2}{2} \Leftrightarrow p_2=11,5p2​=3+211/2+p2​​⇔p2​=11,5

    p1=112+p2⇔p1=5,5+11,5=17p_1 = \frac{11}{2}+p_2 \Leftrightarrow p_1=5,5+11,5 =17p1​=211​+p2​⇔p1​=5,5+11,5=17

    Step 3. Calculate the Optimal Quantities

    D1(p1,p2)=10βˆ’17+2βˆ—11,5=14D_1(p_1,p_2) = 10-17+2*11,5 = 14D1​(p1​,p2​)=10βˆ’17+2βˆ—11,5=14

    D2(p1,p2)=5+17βˆ’11,5=11,5D_2(p_1,p_2) = 5+17-11,5 = 11,5D2​(p1​,p2​)=5+17βˆ’11,5=11,5

    Optimal Quantities: Q1 = 14 and Q2=11,5

    Cournot Model

    Two firms are producing identical products. These are Firm 1 and Firm 2’s cost functions

    Firm 1: C1(q1)=100+20q1C_1(q_1)=100+20q_1C1​(q1​)=100+20q1​

    Firm 2: C2(q2)=50+5q2C_2(q_2)=50+5q_2C2​(q2​)=50+5q2​

    The demand function for the product both firms produce is Q=200βˆ’2pQ=200-2pQ=200βˆ’2p

    What are their quantity strategies and, the price in Cournot Equilibrium?

    β€£
    Method to Solve Cournot Model πŸ‘

    Step 1. Derive Reaction Function

    Q=q1+q2Q=q_1+q_2Q=q1​+q2​

    Q=200βˆ’2p=>p=100βˆ’Q/2Q=200-2p => p=100 -Q/2Q=200βˆ’2p=>p=100βˆ’Q/2

    Profit Firm 1:

    Ο€1=pq1βˆ’C1(q1)Ο€_1 = pq_1-C_1(q_1)Ο€1​=pq1β€‹βˆ’C1​(q1​)

    Ο€1=(100βˆ’(q1+q2)/2)q1βˆ’(100+20q1)β‡’=βˆ’q12/2+80q1βˆ’q1q2/2βˆ’100Ο€_1 = (100-(q_1+q_2)/2)q_1-(100+20q_1) \Rightarrow =-q_1^2/2+80q_1-q_1q_2/2-100Ο€1​=(100βˆ’(q1​+q2​)/2)q1β€‹βˆ’(100+20q1​)β‡’=βˆ’q12​/2+80q1β€‹βˆ’q1​q2​/2βˆ’100

    Maximum Profit FOC

    Ο€1β€²(q1)=βˆ’q1βˆ’q2/2+80β‡’q1=80βˆ’q2/2Ο€_1'(q_1) =-q_1-q_2/2 +80 \Rightarrow q_1=80-q_2/2Ο€1′​(q1​)=βˆ’q1β€‹βˆ’q2​/2+80β‡’q1​=80βˆ’q2​/2

    Reaction function Firm 1:

    R1(q2)=80βˆ’q2/2R_1(q_2)=80-q_2/2R1​(q2​)=80βˆ’q2​/2

    Profit Firm 2:

    Ο€2=pq2βˆ’C2(q2)Ο€_2 = pq_2-C_2(q_2)Ο€2​=pq2β€‹βˆ’C2​(q2​)

    Ο€2=(100βˆ’(q1+q2)/2)q1βˆ’(50+5q2)β‡’=βˆ’q22/2+95q2βˆ’q1q2/2βˆ’50Ο€_2 = (100-(q_1+q_2)/2)q_1-(50+5q_2) \Rightarrow =-q_2^2/2+95q_2-q_1q_2/2-50Ο€2​=(100βˆ’(q1​+q2​)/2)q1β€‹βˆ’(50+5q2​)β‡’=βˆ’q22​/2+95q2β€‹βˆ’q1​q2​/2βˆ’50

    Maximum Profit FOC

    Ο€2β€²(q2):βˆ’q2βˆ’q1/2+95=0β‡’q2=95βˆ’q1/2Ο€_2'(q_2):-q_2-q_1/2 +95 =0 \Rightarrow q_2=95-q_1/2Ο€2′​(q2​):βˆ’q2β€‹βˆ’q1​/2+95=0β‡’q2​=95βˆ’q1​/2

    Reaction function Firm 2:

    R2(q1)=95βˆ’q1/2R_2(q_1)=95-q_1/2R2​(q1​)=95βˆ’q1​/2

    Step 2. Compute Equilibrium

    Cournot Equilibrium is when reaction functions meet

    q2=95βˆ’q1/2q_2=95-q_1/2q2​=95βˆ’q1​/2

    q1=80βˆ’q2/2q_1=80-q_2/2q1​=80βˆ’q2​/2

    q2=95βˆ’(80/2βˆ’q2/4)=>55+q2/4=q2=>3q2/4=55=>q2=55βˆ—4/3q_2= 95-(80/2-q_2/4) => 55+q_2/4 =q_2 => 3q_2/4=55 => q_2=55*4/3q2​=95βˆ’(80/2βˆ’q2​/4)=>55+q2​/4=q2​=>3q2​/4=55=>q2​=55βˆ—4/3

    Answer: q2 =220/3 and q1 = 130/3

    p=100βˆ’q1+q2/2=41,67p=100-q_1+q_2/2=41,67p=100βˆ’q1​+q2​/2=41,67

    Step 3. Calculate the Optimal Quantities

    D1(p1,p2)=10βˆ’17+2βˆ—11,5=14D_1(p_1,p_2) = 10-17+2*11,5 = 14D1​(p1​,p2​)=10βˆ’17+2βˆ—11,5=14

    D2(p1,p2)=5+17βˆ’11,5=11,5D_2(p_1,p_2) = 5+17-11,5 = 11,5D2​(p1​,p2​)=5+17βˆ’11,5=11,5

    Optimal Quantities: Q1 = 14 and Q2=11,5

    Stackelberg Model - Quantity Leadership

    Two firms are producing a similar good, the demand is:

    P=100βˆ’0,01QP=100-0,01QP=100βˆ’0,01Q

    Q=q1+q2Q=q_1+q_2Q=q1​+q2​

    The cost function is C=40qC=40qC=40q

    Where firm is the leader, and firm 2 is the follower. What is the profit for both firms?

    β€£
    Method to Solve Stackleberg πŸ‘

    As firm 1 has the priority to make the decision on quantity according to firm 2’s strategy, this we can use backward induction.

    Step 1. - Start with firm 2 profit:

    Ο€2=pq2βˆ’C2(q2)=>(100βˆ’0,01(q1+q2))q2βˆ’40q2Ο€_2 = pq_2-C_2(q_2) => (100-0,01(q_1+q_2))q_2-40q_2Ο€2​=pq2β€‹βˆ’C2​(q2​)=>(100βˆ’0,01(q1​+q2​))q2β€‹βˆ’40q2​

    Ο€2=100q2βˆ’0,01q1q2βˆ’0,01q22βˆ’40q2Ο€_2 =100q_2-0,01q_1q_2- 0,01q_2^2-40q_2Ο€2​=100q2β€‹βˆ’0,01q1​q2β€‹βˆ’0,01q22β€‹βˆ’40q2​

    Ο€2β€²=100βˆ’0,01q1βˆ’0,02q2βˆ’40=>q2=3000βˆ’q1/2Ο€_2' =100-0,01q_1- 0,02q_2-40 =>q_2=3000-q_1/2Ο€2′​=100βˆ’0,01q1β€‹βˆ’0,02q2β€‹βˆ’40=>q2​=3000βˆ’q1​/2

    Step 2. - Insert in demand function

    p=100βˆ’0,01(q1+3000βˆ’q1/2)β‡’70βˆ’0,005q1p=100-0,01(q_1+3000-q_1/2) β‡’ 70-0,005q_1p=100βˆ’0,01(q1​+3000βˆ’q1​/2)β‡’70βˆ’0,005q1​

    Step 3. Firm 1 profit

    Ο€1=pq1βˆ’C1(q1)=>(70βˆ’0,005q1))q1βˆ’40q1=>70q1+0,005q12βˆ’40q1Ο€_1 = pq_1-C_1(q_1) => (70-0,005q_1))q_1-40q_1 => 70q_1+0,005q_1^2-40q_1Ο€1​=pq1β€‹βˆ’C1​(q1​)=>(70βˆ’0,005q1​))q1β€‹βˆ’40q1​=>70q1​+0,005q12β€‹βˆ’40q1​

    Ο€1β€²=70+0,01q1βˆ’40=>30+0,01q1=>q1=3000Ο€_1' =70+0,01q_1-40 =>30+0,01q_1 => q_1=3000Ο€1′​=70+0,01q1β€‹βˆ’40=>30+0,01q1​=>q1​=3000

    Step 4. Conclusion

    By setting 3000 as a quantity strategy, firm 1 will have maximum profit. Based on this q1=3000 and q2=1500 and p=55

    Profit firm 1: 55βˆ—3000βˆ’40βˆ—3000=4500055*3000-40*3000=4500055βˆ—3000βˆ’40βˆ—3000=45000

    Profit firm 2: 55βˆ—1500βˆ’40βˆ—1500=2250055*1500-40*1500=2250055βˆ—1500βˆ’40βˆ—1500=22500

    Price Leadership model

    β€£
    Method to Solve Leadership Model πŸ‘
    β€£

    Incomplete Exam AI2156 - June 2021

    1. Effectiveness of Risk-sharing as well as efficiency in production

    1️⃣
    Having efficiency in risk-sharing as well as efficiency in production is... a) impossible to have. b) usually desirable. c) typically existing in a complete contract. d) bad. e) often the case. f) None of the above.
    β€£
    Answer & Explanation βœ…

    b) usually desirable.

    🧠
    Having both efficiency in risk-sharing and efficiency in production is generally desirable because it allows for the optimal allocation of resources, minimizes waste, and promotes overall economic efficiency. When risks are shared effectively, it reduces the potential negative impact of risks on individuals or organizations, while efficiency in production ensures that resources are used effectively to generate the maximum possible output.

    2. Three Player Game Tree - Find a subgame of perfect equilibrium

    2️⃣
    Consider the following game tree
    image

    Firm A starts by choosing strategy A1 or A2. Firm B can either choose strategy B1, which mean that it uses Firm C to determine a strategy for Firm B, and Firm C will choose the strategy that is best for Firm B, alternatively, Firm B can use strategy B2. In the tree, (xy) means that Firm A receives payoff x and Firm B payoff y. Which of the payoffs is the result of a subgame perfect equilibrium? a. (1,4). b. (2,5). c. (3,4). d. (4,5). e. (3,7). f. (4,2).

    β€£
    Answer & Explanation βœ…

    e. (3,7)

    Maximum Payoff Player A = 4,2 and 4,5 β†’ Firm C plays for Firm B β†’ The choose C2 to maximize payoff for player B, leading the subgame of perfect equilbrium to be (3,7)

    3. How to determine what students are good PhD candidates

    3️⃣
    In order to determine which students to accept for a PhD course interviews are undertaken. What is this an example of? a) Moral hazard. b) Signaling. c) Screening. d) Adverse selection. e) Risk neutrality. f) None of the above.
    β€£
    Answer & Explanation βœ…

    c) Screening

    🧠
    Screening refers to a process where an informed party (in this case, the institution offering the PhD course) gathers information to differentiate between different types of individuals (in this case, students) with varying characteristics or abilities. By conducting interviews, the institution aims to gather information about the students' qualifications, skills, research interests, and overall fit for the program. This helps them make more informed decisions about accepting or rejecting applicants based on the collected information.

    4. Calculate the value of a coefficient of absolute risk aversion

    4️⃣
    An individual has utility function π‘ˆ(π‘₯) = ln(100 + π‘₯) The value of this individual’s coefficient of absolute risk aversion at π‘₯ = 100 is given by a) 0.2 b) 0.5 c) 0.75 d) 1 e) 1.5 f) None of the above.
    β€£
    Answer & Explanation βœ…

    𝑒(π‘₯) = π‘Ž ln(π‘₯) β†’ None of the above, it can be calculated see drop down below

    β€£
    Super-complex solution to the question

    The coefficient of absolute risk aversion (CARA) measures an individual's willingness to bear risk and is derived from their utility function. To find the CARA at π‘₯ = 100 for the given utility function π‘ˆ(π‘₯) = ln(100 + π‘₯), we need to take the second derivative of the utility function with respect to π‘₯ and evaluate it at π‘₯ = 100.

    First, let's calculate the second derivative:

    π‘ˆ''(π‘₯) = dΒ²/dπ‘₯Β² [ln(100 + π‘₯)] = (1 / (100 + π‘₯)) * d/dπ‘₯ [1 / (100 + π‘₯)] = (1 / (100 + π‘₯)) * (-1 / (100 + π‘₯)Β²) = -1 / ((100 + π‘₯) * (100 + π‘₯)Β²) = -1 / ((100 + π‘₯)Β³)

    Now, let's evaluate the second derivative at π‘₯ = 100:

    π‘ˆ''(100) = -1 / ((100 + 100)Β³) = -1 / (200Β³) = -1 / 8,000,000

    The coefficient of absolute risk aversion (CARA) is the negative value of the second derivative:

    CARA = -π‘ˆ''(100) = 1 / 8,000,000

    None of the given options matches the calculated value. Therefore, the correct answer is f) None of the above.

    5. Out-sourcing legalwork creates what type of principal agent problem?

    5️⃣
    You go to a lawyer for legal help. The lawyer in turn lets an assistant lawyer work on your case. Which of the following statements is true? a) You are an agent and the assistant lawyer is an agent. b) The lawyer is an agent and the assistant lawyer is a principal. c) There is no principal-agent relation in this example. d) The assistant lawyer is a principal and you are an agent. e) The risk is borne by the assistant lawyer. d) None of the above.
    β€£
    Answer & Explanation βœ…

    6. Indifference curves of two employees

    6️⃣
    The indifference curves of two employees are given by π‘€βˆ’π‘’2=𝑒0π‘€βˆ’π‘’^2=𝑒_0wβˆ’e2=u0​ (1) and π‘€βˆ’40𝑒2=𝑒0π‘€βˆ’40𝑒^2 =𝑒_0wβˆ’40e2=u0​ (2) respectively, where 𝑀 is the salary, 𝑒 is the effort put down by the employee and 𝑒0𝑒_0u0​ is the utility level. The company is offering two different wage schemes: (A) Put down the effort 𝑒 = 1 and get the salary 𝑀 = 100. (B) Put down the effort 𝑒 = 2 and get the salary 𝑀 = 200. Which of the following is true? a) Employees with preferences described by Equation (1) are high-ability workers and will choose wage scheme (A), while employees with preferences described by Equation (2) are low-ability workers and will choose wage scheme (B). b) Employees with preferences described by Equation (1) are high-ability workers and will choose wage scheme (B), while employees with preferences described by Equation (2) are low-ability workers and will choose wage scheme (A). c) Employees with preferences described by Equation (1) are low-ability workers and will choose wage scheme (A), while employees with preferences described by Equation (2) are high-ability workers and will choose wage scheme (B). d) Employees with preferences described by Equation (1) are low-ability workers and will choose wage scheme (B), while employees with preferences described by Equation (2) are high-ability workers and will choose wage scheme (A). e) The employees are indifferent between the two wage schemes. f) None of the above.
    β€£
    Answer & Explanation βœ…

    e) The employees are indifferent between the two wage schemes.

    7. Complete Contracts are what?

    7️⃣
    Which of the following statements is not correct? a) Hidden action can lead to moral hazard in equilibrium. b) One way of trying to avoid adverse selection is to use screening. c) Asymmetric information can lead to opportunistic behavior. d) In a complete contract, everything is observable to anyone, and the observable information is also verifiable. e) Hidden characteristics is a form of ex post asymmetric information. f) None of the above.
    β€£
    Answer & Explanation βœ…

    d) In a complete contract, everything is observable to anyone, and the observable information is also verifiable.

    8. Game Theory Behind Construction of Houses between two Developers

    8️⃣
    The cost for a construction company to build a multi-family house is 1. There is vacant land to build at most four of these multi-family houses. Two identical construction companies compete and the total revenue is always 12, given that at least one house is built (if no house is built, then the total revenue is 0). The revenue is shared among the builders, so if in total 𝑛 β‰₯ 1 houses are built, then each house gives a revenue of 12/𝑛 to the builder of the respective house(s). The number of houses each company builds is either zero, one or two, and the number of houses each company builds can be seen as a strategy in a static game. Which of the following statements is true? a) The is no pure Nash equilibrium in this game. b) The choice of building one house each is a Nash equilibrium. c) The choice of building two houses each is a Nash equilibrium. d) The choice of building houses such that the total quantity built is two is a Nash equilibrium. e) The choice of building houses such that the total quantity built is three is a Nash equilibrium. 7) None of the above.
    β€£
    Answer & Explanation βœ…

    b) The choice of building one house each is a Nash equilibrium.

    🧠
    Since the total revenue is allways 12, each company wants to build the least amount of houses to get a share of the total revenue, which is one house each.

    9. I DONT KNOW THIS QUESTION

    9️⃣
    A number of identical firms produces an identical good. Each firm has a constant marginal cost of 5 and a fixed cost of 8. If there is free entry and exit to the market and the market’s inverse demand function is given by 𝑝=25βˆ’2𝑄𝑝 = 25 βˆ’ 2𝑄p=25βˆ’2Q How many firms will be in the market in equilibrium? a) 1 b) 2 c) 3 d) 4 e) 5 f) None of the above

    10. Calculate The amount of effort an employee puts down.

    πŸ”Ÿ
    A risk-neutral employee that puts down the effort 𝑒 is producing the output π‘ž=2𝑒+π‘‹π‘ž = 2𝑒 + 𝑋q=2e+X, where 𝑋 is an uncertain outcome with mean 0 and variance o2=4o^2=4o2=4. The salary of the employee is given by 𝑀(π‘ž)=1+3π‘žπ‘€(π‘ž) = 1 + 3π‘žw(q)=1+3q, the cost for the employee to put down the effort 𝑒 is given by 𝑐(𝑒)=𝑒2𝑐(𝑒) = 𝑒^2c(e)=e2 and the certainty equivalent for the employee of not working is 5. Which of the following is true? a) The employee will work and put down the effort 3. b) The employee will work and put down the effort 3.5. c) The employee will work and put down the effort 3.75. d) The employee will work and put down the effort 4. e) The employee will choose not to work. f) None of the above.
    β€£
    Answer & Explanation βœ…

    15. Argue for higher salary for based on a arbitrary course

    A construction company is revising its employees’ salary. One group of employees has taken and passed a course in theoretical philosophy. Argue to convince the manager that this group should be given a higher salary than the group that did not take and pass the course in theoretical philosophy. (3 p)

    β€£
    Answer & Explanation βœ…
    β€£
    Perfect Bayesian equilibrium

    A perfect Bayesian equilibrium is a combination of beliefs and strategies for each player

    • Sequential rationality: each strategy should maximize a player's expected utility, given their beliefs.
    • Consistency: each belief should be updated according to the equilibrium strategies, the observed actions, and Bayes' rule on every path reached in equilibrium with positive probability. On paths of zero probability, known as "off-equilibrium paths", the beliefs must be specified but can be arbitrary.

    The kinds of perfect Bayesian equilibria that may arise can be divided in three different categories:Β pooling equilibria, separating equilibria and semi-separating. A given game may or may not have more than one equilibrium.

    • In aΒ pooling equilibrium, senders of different types all choose the same signal. This means that the signal does not give any information to the receiver, so the receiver's beliefs are not updated after seeing the signal.
    • In aΒ separating equilibrium, senders of different types always choose different signals. This means that the signal always reveals the sender's type, so the receiver's beliefs become deterministic after seeing the signal.
    • In aΒ semi-separating equilibriumΒ (also calledΒ partial-pooling), some types of senders choose the same message and other types choose different messages.