Equilibrium conditions

Every year 1,000 citizens of Peñaranda green their used car and buy a new one. Original used car owners have nowhere to keep a second car and must sell it. The quality of these used cars varies greatly from one to another. Their original owners know exactly what their strengths and weaknesses are, but potential buyers can’t tell the difference just by looking at them. Sadly, while Peñaranda’s used car owners are exemplary citizens in other respects, they have no qualms about lying when it comes to their old gadgets. Each car has a value, V, that a buyer who knows all its qualities would be willing to pay. There are a very high number of potential buyers, each of whom would be willing to pay V euros for a car worth V euros.
It is quite easy to describe the distribution of the values ​​of used cars on the market. In any given year, given a value of Ventre 0 and 2,000 euros, the number of used cars for sale that are worth less than V euros is V euros / 2. Prospective used car buyers are all risk neutral. That is, if they are not sure what a car is worth, they value it at its expected value, given the information they have.
The López de Peñaranda Workshops can test used cars and find out their true V value. Their appraisals are known to be completely accurate and honest. The only problem is that an exact appraisal costs 200 euros. People who have a very bad car are not going to want to pay 200 euros for Talleres López to tell the world how bad it is. But people who have a very good car will be willing to pay you 200 euros to price it and sell it for its true value. Let’s try to figure out how exactly the equilibrium works, which cars are priced, and what is the price at which the ones that are not priced are sold.
(a) If no one had your car appraised, what would the market price of used cars in Peñaranda be? What would be the total income used car owners would get from this?
(b) If all cars worth more than X euros were appraised and all cars worth less than X euros were sold without appraisal, what would be the market price of used cars that were not appraised? (Hint: what is the expected value of a random extraction from the set of cars worth less than X euros?)
(c) If all cars worth more than X euros were appraised and all cars worth less than X euros were sold without appraisal and your car was worth X euros, how much money would you have left if you had appraised it and sold it for its True value? How much money would you receive if you sold it without having appraised it?
(d) In equilibrium conditions, there is a car of marginal quality such that all cars are priced better than this car and none of the worst cars are priced. The owner of this car does not care exactly the same to sell it without appraisal than valuing it. What will be the value of this marginal car?
(e) At equilibrium, how many cars will be sold unpriced and for how much?
(f) In equilibrium conditions, what will be the total net income of all used car owners after their appraisal has been paid to Talleres López?

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