Please help me solve it. Suppose you are looking at a particular used car. Given its brand, age and overall condition, its true value to you could be 40, 50 or 60 (these are the three possible types), whatever your true value is, the sellers true value is 20% lower (so there is some room for trade). The seller knows his value but you do not know yours. All you think is that each of the three values is equally likely.
(a) Consider the best possible equilibrium in that as many car types as possible are being transacted in this equilibrium. What car types are being transacted and what is the range of possible market prices?
(b) consider the worst possible equilibrium in that only one car type is being transacted in this equilibrium. What car type is being transacted and what is the range of possible market prices?
c) consider an intermediate equilibrium in which more car types than in b) are being transacted. What are these car types are being transacted and what is the range of possible market prices?
d) if the seller can commit to publicly burning some amount of sale revenue (charity for example) before car is sold.Can this type of signaling make sure that all car types are being transacted? Why?
(a) In the best possible equilibrium where as many car types as possible are being transacted, the car types being transacted are the ones that yield a positive trade surplus for both the buyer and the seller.
Let's analyze each car type individually:
Car type 40:
- Buyer's valuation: 40
- Seller's valuation: 0.8 * 40 = 32
- Trade surplus: 40 - 32 = 8 (positive)
Car type 50:
- Buyer's valuation: 50
- Seller's valuation: 0.8 * 50 = 40
- Trade surplus: 50 - 40 = 10 (positive)
Car type 60:
- Buyer's valuation: 60
- Seller's valuation: 0.8 * 60 = 48
- Trade surplus: 60 - 48 = 12 (positive)
From the analysis, all three car types have a positive trade surplus, indicating that they can all be transacted in the best possible equilibrium. The range of possible market prices would be from the seller's lowest valuation (32 for car type 40) to the buyer's highest valuation (60 for car type 60).
(b) In the worst possible equilibrium where only one car type is being transacted, the car type being transacted would be the one with the highest potential trade surplus.
From the analysis in part (a), the car type with the highest trade surplus is car type 60, with a surplus of 12. Therefore, in the worst possible equilibrium, only car type 60 would be transacted. The range of possible market prices would be from the seller's lowest valuation for car type 60 (48) to the buyer's highest valuation for car type 60 (60).
(c) In an intermediate equilibrium where more car types than in part (b) are being transacted, we need to consider the trade surpluses for each car type again.
From the analysis in part (a), we know that all three car types have a positive trade surplus. To include more car types, we can add the car types with the next highest trade surpluses.
Adding car type 50 to the mix:
- Buyer's valuation: 50
- Seller's valuation: 0.8 * 50 = 40
- Trade surplus: 50 - 40 = 10 (positive)
From the analysis, both car types 50 and 60 have positive trade surpluses, indicating that they can be transacted in the intermediate equilibrium. The range of possible market prices would be from the seller's lowest valuation for car type 50 (40) to the buyer's highest valuation for car type 60 (60).
(d) If the seller can commit to publicly burning some amount of sale revenue before the car is sold, it can potentially signal the quality of the car to the buyer. By burning a certain amount of money, the seller is sacrificing some of their revenue, which indicates that the car is of higher quality.
This type of signaling can make sure that all car types are being transacted because it allows the buyer to infer the quality of the car based on the amount of money the seller is willing to burn. This additional information can lead to a more efficient trade where both parties are willing to transact a wider range of car types, including those with lower trade surpluses.
To analyze this problem, we can use game theory and the concept of Bayesian Nash equilibrium. Let's break down each part of the question:
(a) In the best possible equilibrium where as many car types as possible are being transacted, we need to find the range of possible market prices. Since you, as the buyer, do not know your true value, we need to consider all possible cases.
Let's denote the true value for you as X, where X can be 40, 50, or 60 with equal probability. The seller's true value is 20% lower, so their true value, denoted as Y, would be 0.8X.
To determine which car types are being transacted, we need to compare your potential willingness-to-pay (WTP) with the seller's willingness-to-accept (WTA). If WTP ≥ WTA, the transaction will occur.
Considering all possible values for X and Y, we have:
For X = 40, WTP = 40, WTA = 0.8*40 = 32. Transaction occurs if 40 ≥ 32.
For X = 50, WTP = 50, WTA = 0.8*50 = 40. Transaction occurs if 50 ≥ 40.
For X = 60, WTP = 60, WTA = 0.8*60 = 48. Transaction occurs if 60 ≥ 48.
In this best equilibrium, all three car types can be transacted since there is some overlap in the willingness-to-pay and willingness-to-accept ranges. The range of possible market prices would be between 32 and 60.
(b) In the worst possible equilibrium where only one car type is being transacted, we need to find which car type that is and the range of possible market prices.
Let's analyze each case separately:
For X = 40, WTP = 40, WTA = 0.8*40 = 32. Transaction occurs if 40 ≥ 32. Car type 40 can be transacted.
For X = 50, WTP = 50, WTA = 0.8*50 = 40. Transaction occurs if 50 ≥ 40. Car type 50 can be transacted.
For X = 60, WTP = 60, WTA = 0.8*60 = 48. Transaction occurs if 60 ≥ 48. Car type 60 can be transacted.
In this worst equilibrium, only one car type is being transacted at a time (either 40, 50, or 60). The range of possible market prices depends on the specific car type being traded but will be within the range of 32 to 60.
(c) In an intermediate equilibrium where more car types than in (b) are being transacted, we need to identify which car types are being traded and the range of possible market prices.
To have more car types traded, there should be more overlap in the willingness-to-pay and willingness-to-accept ranges. Let's analyze each case:
For X = 40, WTP = 40, WTA = 0.8*40 = 32. Transaction occurs if 40 ≥ 32. Car type 40 can be transacted.
For X = 50, WTP = 50, WTA = 0.8*50 = 40. Transaction occurs if 50 ≥ 40. Car type 50 can be transacted.
For X = 60, WTP = 60, WTA = 0.8*60 = 48. Transaction occurs if 60 ≥ 48. Car type 60 can be transacted.
In this intermediate equilibrium, all three car types (40, 50, and 60) can be transacted, similar to the best equilibrium. The range of possible market prices would again be between 32 and 60.
(d) If the seller can commit to publicly burning some amount of sale revenue before the car is sold, it is called signaling. This type of signaling can potentially ensure that all car types are being transacted.
By publicly burning some amount of sale revenue, the seller can signal a lower willingness-to-accept (WTA) price, indicating a higher true value. This signal can influence the buyer to offer a higher price, increasing the likelihood of a transaction for higher car types.
For example, if the seller publicly burns 10% of the sale revenue before the transaction, it indicates a lower WTA price for the car. This can incentivize the buyer to offer a higher price, leading to increased transaction possibilities for higher car types (i.e., 50 and 60). However, the transaction for car type 40 may still occur based on the buyer's lower WTA.
In summary, by using signaling through burning some amount of sale revenue, it is possible to increase the likelihood of transactions across different car types and potentially ensure that all car types are being transacted.