AI2156 - Contract Theory with Application to Property Management

ELI5 - Theories & Concepts

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🚧 Prisoner’s Dilemma

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👍 Strategic complements

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👎 Strategic substitutes

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🔂 Folk Theorem

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🗣️ Coase Theorem

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🪖💵 Bertrand Competition

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🦄 Separating Equilibrium

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🥋Pooling Equilibrium

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🤷‍♂️ Semi-separating equilibrium

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📝 A Complete Contract

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🦾 Dominant Strategy

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🎯 Cournot Oligopoly

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🥇Stackelberg Oligopoly

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🗺️ Property Rights

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 𝑢(𝑥) = 2 + 5𝑥 is... a) Risk-neutral

b) Risk-averse

c) Risk-loving

d) None of the above

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Answer and Explanation ✅

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.

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Answer & Explanation ✅

3. Identify a sub-game of perfect equilibrium

3️⃣

Consider the following game tree.

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?

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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

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Answer & Explanation ✅

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

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

10. Pricing in the Bertrand oligopoly model

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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.

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Answer & Explanation ✅

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

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Given is the following payoff matrix.

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)

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Solution a) and b) ✅

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

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

𝐶(𝑞) = 2𝑞

The demand function for the product the firms produce is

𝑄 = 20 − 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)

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a) Solution ✅ - Cournot Oligopol Model

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b) Solution ✅ - Stackelberg Oligopoly Model

13. Interpret indifference curves for two types of insurance customers

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The indifference curves of two groups of insurance company customers are given by 𝑅−5𝐷 = 𝑢0 and (1) 𝑅 − 𝐷 = 𝑢0 (2) respectively, where 𝑅 is the reduction in insurance premium, 𝐷 is the deductable and 𝑢0 is the utility level. Which of the two groups is the high risk group? Motivate your answer. (3 p)

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Answer ✅

14. Fixed-Fee, Hire and Sharing Contracts

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What is meant by a fixed-fee contract, a hire contract and a sharing contract respectively?

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Answer ✅

15. Deadweight loss on environment

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A monopolist is polluting the environment when producing its good. The inverse demand function for the good the firm produces is 𝑝(𝑄) = 20 − 𝑄, and the monopolist’s cost function is 𝐶(𝑄) = 𝑄^2. The total cost for the society is given by 𝐶𝑆(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)

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Answer a) and b) ✅

16. Determine the property developer’s certainty

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A property developer has utility function 𝑢(𝑥) = √𝑥, 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)

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Solution ✅

Complete Exam AI2156 - August 2022

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

1️⃣

An individual has utility function is $𝑢(𝑥) = 1 − 𝑒^{-2x}$ The value of this individual’s coefficient of absolute risk aversion at 𝑥 = 10 is given by a) −2. b) 0. c) 1. d) 2.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

3. Identify the subgame of perfect equilibrium

3️⃣

Consider the following game tree.

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)

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Answer & Explanation ✅

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

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

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.

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Answer & Explanation ✅

10. A hidden characteristic is typically the reason

🔟

A hidden characteristic is typically the reason for... a) adverse selection. b) transaction costs. c) moral hazard. d) complete contracts.

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Answer & Explanation ✅

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

❓

Given is the following payoff matrix.

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Solution ✅

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

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Two firms, Firm 1 and Firm 2, are producing an identical good. They have cost functions

𝐶_1(𝑞) = 𝑞

and

𝐶_2(𝑞) = 3𝑞

respectively. The demand function for the product the firms produce is

Q=20-p

Determine the price of the good and the quantity each firm produces in a Cournot oligopoly model. (4 p)

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Solution ✅

13. Avoid Adverse Selection by providing a menu of Contracts

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Explain in detail how introducing a menu of contracts can result in an equilibrium where adverse selection is avoided. (4p)

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Solution ✅

14. Difference between a complete and incomplete contract

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What is the difference between a complete contract and an incomplete contract? (3p)

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Answer ✅

15. Examples of Principal agent problems

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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)

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Solution ✅

16. Certainty Equivalent of this project

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A real estate company has utility function 𝑢(𝑥)=ln(𝑥+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)

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Solution ✅

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Exercise 2 - Oligopoly Theory

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Incomplete Exam AI2156 - June 2021

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Perfect Bayesian equilibrium