EC220 Problem Set 4
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
EC220 Problem Set 4
November 19, 2020
1 Adverse Selection [will be covered in seminar]
Adverse selection refers to the situation in which one side of a market has better infor- mation that allows it to trade selectively, at the expense of the other side of the market.
This can result in mutually beneicial transactions not going through. Classic examples of this are life insurance, where the customers most willing to buy insurance are exactly those who know they are likely to die soon; and the used car market, where those most willing to sell their car are exactly those owners who know their car is a lemon. This exercise concerns the health insurance market, which, when left unregulated (as in the US prior to 2010), faces severe problems of adverse selection.
Consider a potential insuree with a type x 2 [0; 1]. The cost to provide insurance to an insuree is x/2. The beneit of insurance to the insuree is x.
1.1 Consider frst a benevolent social planner (let’s say, the NHS) who knows x and wants to provide insurance for free whenever the benefts outweigh the costs. For what values of x should the social planner provide insurance? Is knowledge of x essential here?
Instead of a social planner, we now have a health insurer. Assume that irst the insurer decides on a price for insurance p 2 [0; 1], and then the insuree decides to acceptor reject that ofer. An insuree accepting an ofer gets x - p whereas the insurer gets p - x/2. If the ofer is rejected, both parties get 0. From now on, for simplicity we are only interested in solutions where the potential insuree accepts when indiferent.
1.2 For a given price p, what types of the potential insuree will purchase insurance?
1.3 Suppose x is commonly known. Find all SPNEs. For what values of x is insurance issued?
1.4 Suppose the common prior on x is that it is uniformly distributed on [0; 1] . The exact value of x is known only to the potential insuree. Find all PBEs.
1.5 Which consumers are underserved in the private market of 1.4?
One might think adverse selection limits the market only because we’ve constructed it as a monopoly - with a single irm setting prices. So let’s consider a duopoly. Let two insurers, 1 and 2, set their prices p1 and p2 simultaneously. A potential insuree buying from insurer i gets x - pi. You can assume a consumer indiferent between purchasing from either irm randomizes with equal probabilities.
1.6 What are the insurers’ profts as a function of p1 , p2 and (a commonly known) x?
1.7 Find all PBEs when the prior on x is a uniform distribution on [0, 1] and the exact value is known only to the potential insuree.
1.8 Which consumers are underserved in the duopoly market of 1.7? Do you think fur- ther competition would ameliorate this problem?
The Afordable Care Act requires that all US residents purchase insurance, or face a ine. That is, the potential insuree now get x -pi by purchasing insurance from insurer i, or alternatively -k if they go without insurance, where k is the amount of the ine. 1.9What is the smallest fne such that all types of potential insuree buy insurance in a PBE?
2 Perfect Bayesian Equilibrium in Stackelberg with informed irst-mover [will be covered in seminar, time permitting]
Questions
Two irms costlessly produce a homogeneous good, choosing quantities q1 , q2 2 f1, 2g, which we will denote low (L) and high (H) production. Price in the market is given by
P = maxf0, x - q1 - q2 g
where x is a random variable, distributed according to the common prior giving P (x = 3) = p and P (x = 6) = 1 - p. Firm 1 is established: it knows x privately, and chooses q1 irst (and observably). Once irm 2 observes q1 , it chooses q2 .
2.1 Draw this game’s Extensive Form.
2.2 For p = 3/1, find the pure BNEs and PBEs of the game.
3 Costly Signaling: No Rest in the Nest [will not be covered in seminar]
A nestling (that is, a baby bird who has yet to leave the nest) is attended to by its mother. It may be starving or healthy. It can choose to either squawk or stay silent. The mother then observes the nestling’s choice and may choose to either feed it a worm or keep the worm for herself. The nestling gets 6 utility from the worm if starving, or x utility if healthy. Squawking, however, may attract predators; in expectation, squawking costs 2 utility due to the possibility of being eaten. The mother bird likes her nestling, and gets the same utility as the nestling, with the exception that if she gets to keep the worm she gets an additional 4 utility. Suppose the common prior puts a probability :4 on the nestling being of the starving type.
3.1 Draw this problem as the extensive form of a dynamic game of incomplete infor- mation.
3.2 For x = 1, fnd the set of pure PBEs. Which are pooling and which are separating?
3.3 For x = 3, fnd the set of pure PBEs. Which are pooling and which are separating?
3.4 For x = 5, fnd the set of pure PBEs. Which are pooling and which are separating?
3.5 The cost of squawking is identical across nestling types and, in itself, squawking is destructive. Yet it may serve a purpose as a signaling device. Why?
3.6 The example in this exercise is known as ‘costly signaling’ to biologists. Can you think of any other examples of costly signaling in real life?
2024-01-26