? Microeconomics

FINAL EVALUATION - PART 1: PROBLEMS TO

SOLVE

Problem 1:

We consider the market of used cars. On this market, there are sellers that offer cars of high quality q1 and sellers that offers cars of low quality q2. Buyers cannot discriminate between the two types of cars, as they do not directly observe the car quality. Still, they can observe the number of months of warranty offered by sellers. Buyers therefore use the length of the warranty as a signal for the quality of cars. Buyers have rational beliefs (based on the length of the warranty) and a priori beliefs about the type of a given car:

? µ(g) is the probability for the buyer that the car is of high quality (q1) when the buyer offers g months of warranty. (1- µ(g)) is therefore the probability for the buyer that the car is of quality q2 when the seller offers g months of warranty.

? µ0 is the ex ante probability (not based on the number of months of warranty) for the buyer that the car is of quality q1. The same, (1-µ0) is the ex ante probability for the buyer that the car is of type q2.

1. Let’s first assume that there is no asymmetric information: buyers can directly observed the quality of cars (this is the benchmark case). What is the price a buyer is willing to pay for a quality car and for a low quality car in such a case?

2. Let’s now consider the case of asymmetric information as explained above. Let’s also consider an initial situation in which all sellers offer a warranty of 6 months.

a) What is now the price that a buyer is willing to pay for a car?

b) How is this price compared to the prices found in question 1.? What is the type of cars that are offered on the market?

3. Use the graph below to explain why the initial situation, where all sellers offer the same number of months of warranty, is not a sustainable pooling equilibrium. To do so, answer the following questions :

a) What is the isoprofit curve of the sellers of high quality cars and what is the isoprofit curve of the sellers of low quality cars?

b) In which direction isoprofit curves need to move for the seller to reach higher profits?

c) For whom is it worth to deviate from the initial situation? Sellers of low quality cars or sellers of high quality cars? Why?

d) In which range will be the number of months of warranty offered by sellers who deviate from the initial situation? Why?

e) What is the reaction of the other type of sellers to the deviation from the initial situation ? Put differently, what is the number of months of warranty offered by the other type of sellers in case we are in a separating equilibrium?

6 8 10 Mois de garan

Months of warranty

Problem 2:

Let’s consider the problem of a risk-neutral employer who cannot observe the efforts of his employee and wants to determine the optimal contract to offer his employee.

The employee decides to exert an effort or not. If he decides to exert efforts, e=1 and the worker pays a disutility equal to ?=20. If he decides to exert no effort, e=0 and the worker suffers no disutility ?=0.

The production level q is perfectly observable and verifiable and is equal to either 0 or 1. It equals 1 with probability p, with 0 p 1, if the worker exerts effort. Probability of success (ie of getting q1=1) equals 0.5 if the worker exerts no effort. Put differently, production level depends on the effort made by the worker, but also on a shock.

S(q) is the value of production of level q for the employer. S(1)=200 and S(0)=0. The reservation utility of a worker is u=50.

1.What is the type of asymmetric information here? Which party is the agent and why party is the principal?

2.We first assume that both the worker and the employer are risk-neutral. The employer cannot monitor the salesman, but can pay him wages contingent on success (meaning that the employer offers a pay-for-performance scheme).

a) why does such a compensation scheme make sense here? What could be the downsides of a pay-for-performance scheme and in which context could a pay-for-performance scheme by counterproductive?

b) What is the program that the employer needs to solve to find the optimal contract to offer to a worker that exerts an effort? The same, what is the program the employer needs to solve to find the optimal contract to offer to a worker that exerts no effort ? Explain your answer by explaining what are the choice variables, the objective function and the constraints.

c) What is the optimal contract for each p in (0,1)? Hint: you need to find the threshold probability p* staring from which the optimal contract is the one in which the workers exerts no effort.

d) Is this contract socially optimal? Why?

FINAL EVALUATION - PART 1: PROBLEMS TO

SOLVE

Problem 1:

We consider the market of used cars. On this market, there are sellers that offer cars of high quality q1 and sellers that offers cars of low quality q2. Buyers cannot discriminate between the two types of cars, as they do not directly observe the car quality. Still, they can observe the number of months of warranty offered by sellers. Buyers therefore use the length of the warranty as a signal for the quality of cars. Buyers have rational beliefs (based on the length of the warranty) and a priori beliefs about the type of a given car:

? µ(g) is the probability for the buyer that the car is of high quality (q1) when the buyer offers g months of warranty. (1- µ(g)) is therefore the probability for the buyer that the car is of quality q2 when the seller offers g months of warranty.

? µ0 is the ex ante probability (not based on the number of months of warranty) for the buyer that the car is of quality q1. The same, (1-µ0) is the ex ante probability for the buyer that the car is of type q2.

1. Let’s first assume that there is no asymmetric information: buyers can directly observed the quality of cars (this is the benchmark case). What is the price a buyer is willing to pay for a quality car and for a low quality car in such a case?

2. Let’s now consider the case of asymmetric information as explained above. Let’s also consider an initial situation in which all sellers offer a warranty of 6 months.

a) What is now the price that a buyer is willing to pay for a car?

b) How is this price compared to the prices found in question 1.? What is the type of cars that are offered on the market?

3. Use the graph below to explain why the initial situation, where all sellers offer the same number of months of warranty, is not a sustainable pooling equilibrium. To do so, answer the following questions :

a) What is the isoprofit curve of the sellers of high quality cars and what is the isoprofit curve of the sellers of low quality cars?

b) In which direction isoprofit curves need to move for the seller to reach higher profits?

c) For whom is it worth to deviate from the initial situation? Sellers of low quality cars or sellers of high quality cars? Why?

d) In which range will be the number of months of warranty offered by sellers who deviate from the initial situation? Why?

e) What is the reaction of the other type of sellers to the deviation from the initial situation ? Put differently, what is the number of months of warranty offered by the other type of sellers in case we are in a separating equilibrium?

6 8 10 Mois de garan

Months of warranty

Problem 2:

Let’s consider the problem of a risk-neutral employer who cannot observe the efforts of his employee and wants to determine the optimal contract to offer his employee.

The employee decides to exert an effort or not. If he decides to exert efforts, e=1 and the worker pays a disutility equal to ?=20. If he decides to exert no effort, e=0 and the worker suffers no disutility ?=0.

The production level q is perfectly observable and verifiable and is equal to either 0 or 1. It equals 1 with probability p, with 0 p 1, if the worker exerts effort. Probability of success (ie of getting q1=1) equals 0.5 if the worker exerts no effort. Put differently, production level depends on the effort made by the worker, but also on a shock.

S(q) is the value of production of level q for the employer. S(1)=200 and S(0)=0. The reservation utility of a worker is u=50.

1.What is the type of asymmetric information here? Which party is the agent and why party is the principal?

2.We first assume that both the worker and the employer are risk-neutral. The employer cannot monitor the salesman, but can pay him wages contingent on success (meaning that the employer offers a pay-for-performance scheme).

a) why does such a compensation scheme make sense here? What could be the downsides of a pay-for-performance scheme and in which context could a pay-for-performance scheme by counterproductive?

b) What is the program that the employer needs to solve to find the optimal contract to offer to a worker that exerts an effort? The same, what is the program the employer needs to solve to find the optimal contract to offer to a worker that exerts no effort ? Explain your answer by explaining what are the choice variables, the objective function and the constraints.

c) What is the optimal contract for each p in (0,1)? Hint: you need to find the threshold probability p* staring from which the optimal contract is the one in which the workers exerts no effort.

d) Is this contract socially optimal? Why?

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