Problem Set 2

Problem set 2 has been revised since we will not have time to cover the material necessary for two of the questions before the due date.  The new revised problem set has been posted at 12:45pm on April 3rd, so if you downloaded it before that date, please download it again.  The changes made are as follows:

Questions 5 & 6 on cheap talk have been removed and replaced with a new question on a public works project.   The problem set is now 5 questions long.


Problem set 2 is due April 10.  You can view it here.

Class # 5 Highlights

In this class, we studied general solutions to the information problem. We can think about the "game" designed by the principal (seller, employer, etc.) as a message game with an arbitrary message space. The vector of messages is then translated into allocations. For example, in a first-price auction, the bids are messages, and the allocation gives the good to the person sending the highest "message" and obliges them to pay the amount of their message. Viewed in this light, all such situations are indirect mechanisms.

A direct mechanism is a message game with the restriction that the message space is equal to the type space.

The Revelation Principle states that any BNE of any indirect mechanism is a BNE of the direct mechanism under the additional constraint that all agents report their types truthfully.

The intuition for the revelation principle is that reporting untruthfully corresponds to deviating from the equilibrium message in the indirect mechanism.But since we are implementing a BNE of the indirect mechanism then such a deviation must not be profitable, otherwise we didn't have a BNE in the first place.

We then specialized preferences to the simple specification where

u_i = type_i x_i - T_i

In that setting, we derived an equilibrium condition of a direct mechanism. This is simply the FOC evaluated at truth-telling. Then, using the envelope theorem, we know that the change in equilibrium utility with respect to type is simply x_i.. 

A lemma shows that any incentive feasible allocation must have the property that x_i is non-decreasing for all i. Hence, using the fundamental theorem of calculus, we can show that the equilibrium utility under a direct mechanism is simply

u[type_i] = u[type_0] + Integrate[x(s), {s, type_0, type_i}]

(using Mathematica notation) and where type_0 is the lowest possible type. 

Equilibrium utility can also be expressed as

u[type_i] = type_i x_bar[type_i] - T_bar[type_i]

or, equivalently

 T_bar[type_i] = type_i x_bar[type_i] - u[type_0] - Integrate[x(s), {s, type_0, type_i}]

Thus, the payment to the principal depends only on the allocation schedule x and the utility of the lowest type. 

This yields the Revenue Equivalence Theorem: Any mechanism resulting in the same payoff to the lowest type and the same allocation schedule also yields the same expected revenue to the principal. 

In terms of our two auctions, notice that the x schedule was the same for both--in either case, the object was given to the person with the highest value. Moreover, the payment to the lowest type was the same: In each case, that type earned zero. Thus, the two mechanisms are revenue equivalent. But now we know that any auction that produces an efficient outcome and has free-opt out, so the lowest type earns the same amount, will produce the same expected revenue as either of these auctions.

Key things should be able to do after this class:

1. Explain how the revelation principle works--especially why it is without loss of generality to restrict attention to truth-telling strategies.
2. Understand the revenue equivalence theorem and know when it does or does not apply. 

Class #4 Highlights

In this class, we studied decentralized solutions to the information problem. Specifically, we considered a situation where sellers were on the short side of the market, buyers had independent private valuations and unit demand. We studied the case where there was a single unit available though this is easily extended to the case where there are k units and n > k potential bidders.

Our solution concept for this game was a Bayes-Nash equilibrium. This is a strategy that is optimal in expectation and where expectations are formed using Bayes rule whenever possible. Specifically, given private information, v, a bidder's strategy is a schedule b(v).

In a first-price auction, we saw that bid shading (bidding below value) is optimal, where the degree of shading depends on the number of rival bidders. Specifically, we showed that a symmetric BNE consists of a bidding function:

b(v) = E[v_1:n-1 | v_1:n-1 <= v]

The expression v_1:n-1 is an order statistic, it denotes the highest value out of n-1 draws from the value distribution F. 

We performed a similar analysis for the second-price (Vickrey) auction. Here, we could use the stronger equilibrium concept of dominance to establish that b(v) = v is an equilibrium in weakly dominant strategies. The reason that truthful bidding is optimal is that an individual's bid only determines circumstances for winning or losing, it does not determine the price paid. 

Given this bidding strategy, the expected payment of the winning bidder is simply the highest rival value conditional on this being below the value of the winning bidder. That is, the expected payment of the winner whose valuation is v is:

p(v) = E[v_1:n-1 | v_1:n-1 <= v]

Notice that, under each auction, the payment of the winning bidder is the same. It therefore follows that the seller earns the same expected revenue under each auction form. Moreover, since the individual with the highest value wins in both cases, the allocation is efficient--no reallocation can produce a Pareto improvement. 

In a sense, the auction "solves" the information problem completely. Note, however, that we have assumed that the auctioneer simply chooses an auction with no reserve price. That is, there is no profit maximization motive imposed on the choice of auction. One can easily show that, given the option to impose a positive reserve price, a profit maximizing auctioneer can produce higher revenue by choosing a reserve. To see this, notice that, evaluated at the lowest possible bid, the efficiency losses from an infinitesimal increase in the reserve price are zero while the gains are strictly positive. Therefore, under profit maximization, we still obtain the familiar inefficiency of the solution to the information problem. As in the pricing case, this stems from market power on the part of the seller rather than the private information per se. Determining the optimal reserve price depends on the assumption that the seller knows the exact distribution of values, F.

Key things should be able to do after this class:

1. Compute a Bayes-Nash equilibrium of an auction.
2. Understand why truthful bidding is a dominant strategy in a Vickrey auction.

Section 3

Section this week is going to be based on the paper "Securites Auctions under Moral Hazard: An Experimental Study" by Shimon Kogan and John Morgan.  We will be going over both the theory and experimental results.  I strongly recommend taking a glance at the paper before section, particularly the theory covered in section 2, so that you are familiar with the approach before we go over it in class.  You can find it here:

http://faculty.haas.berkeley.edu/rjmorgan/Securities%20auctions.pdf

Slides from section can be found here.

Section 2

You can find the slides for last section here.

You can also download the slides for the experiment: section with the dragon fruit and section with the chocolate.

Youssef

Class # 3 Highlights

In this class, we studied how using a combination of price and quality, a firm can alleviate the information problem. Here, we considered a situation of a monopoly firm choosing price and quantity when there are two types of consumers who differ in their valuation of a unit of quality.

In an equilibrium where both types of consumers are served in the market, the monopolist must pay attention to individual rationality (IR) conditions, so that consumers find it in their interest to purchase the item as well as incentive compatibility (IC) conditions, so that each type finds it optimal to buy the bundle designed for them.

We saw that the IR condition was binding for the lowest type. The reason is that these guys cannot profitably pretend to have a higher willingness to pay (WTP). The IC condition binds for higher types since they can pretend to have lower WTP.

Combining these conditions yields the optimal quality strategy:
1. High types are offered the socially efficient quality, but at a discounted price.
2. Low types are offered a downward distorted quality at a price that leaves them indifferent between buying or not.

Intuitively, the firm distorts the quality for the lower types to save on the "information rent" paid to high types. This rent is the amount paid to get them to tell the truth.

While this equilibrium partially solves the information problem, it involves distortions, so it is not efficient. In principle, the firm could perfectly solve the information problem by offering the socially optimal quality to both types and discounting sufficiently for the high type. This, however, is not profit maximizing. Here, the difficulty lies in the market power exerted by the monopoly firm leading to an efficiency loss. Clearly, a policy maker could do better by simply implementing this outcome.


Key things should be able to do after this class:
1. Compute an optimal pricing scheme when buyers have private information about their willingness to pay.
2. Understand why optimal schemes distort the allocations of lower types but not the highest type and how this relates to the prices charged to each.

Class #2 Highlights

In this class, we studied the information problem in a competitive equilibrium setting. Here, we took a standard labor market setting and tweaked it by adding private information about worker productivity and outside options. Under complete information, the market equilibrium was, of course, efficient as per the first welfare theorem.

Adding private information complicates the computation of equilibrium by adding beliefs on the part of firms as to the average productivity of the labor pool attracted by a given wage offer. In our setting, demand is infinite if beliefs > wage. Demand is zero if beliefs < wage. Demand is anything if beliefs = wage. Therefore, in any competitive equilibrium, beliefs = wage.

Beliefs are correct if, ex post, the average worker productivity is equal to beliefs. A wage offer of w will attract workers of types [theta_0, r-1(w)]. Thus, beliefs must be E[theta | theta <= r-1(w)] when wage w is offered. Combining this with the zero profit condition yields a competitive equilibrium. That is, any wage w such that

w = E[theta | theta <= r-1(w)]

is a competitive equilibrium. 

The key properties of competitive equilibrium are:

1. Allocations can be inefficient. 

2. Under some conditions, the market can completely unravel. 

3. There can be multiple equilibria, which are Pareto rankable. The "best" equilibrium is the one with the highest wage. 

Finally, we examined whether a policy maker could do better absent knowledge of thetas and absent the ability to compel workers to accept employment. Under these conditions, a policy maker can do no better than to choose the highest equilibrium wage provided that it must satisfy budget balance.

Key things should be able to do after this class:

1. Compute a competitive equilibrium in a market with private information.
2. Determine whether there is an efficiency loss in such an equilibrium and compare the outcome to a first-best benchmark. 

Problem Set 1

You can download problem set 1 here.

It will be due on March 22nd.
Youssef

The Winner's Curse

You can download the slides (updated) from section 1 here.

Good luck with your metrics exam next week!

Youssef

Welcome to Econ 201b - Information Economics

Welcome to the second half of Econ 201b. Here, we will cover information economics. Please check out the course syllabus here.

Syllabus