Exam Rules

The rules for the exam are as follows: You can bring whatever notes you like to class. You are not allowed to bring books nor technology that substitutes for books including web surfing technology. Unfortunately, I cannot allow calculators except for "old-school" versions that only do basic operations. What this means is that you are free to use a calculator if you like, but only calculators that have no internet/kindle etc capability are allowed. So if you use your smartphone as a calculator, I'm afraid you'll have to spring the $3 or so for a basic one if you want to use it on this exam.

Extra Study Materials

1) Reminder: if you are planning to come to office hours, please try to email me the past exam questions you want to go over beforehand so that I can read through them.

2) Here is a list of previous exam questions that I would not spend too much time on because they are moral hazard questions (or include a moral hazard red herring).
2011 #1,2
2010 #2
2007 #1
2006 #1
2004 #2


3) Below is a problem set and solution key from last year.  The last two problems are cheap talk problems.

problem set
solutions

4) Professor Morgan has provided a few extra problems to practice with:

Question 1: The Modigliani-Miller theorem provides conditions where a firm’s capital structure is irrelevant to its investment decisions. Yet, in practice, a firm’s capital structure does seem to profoundly affect investment decision making as well as the valuation placed on the company in capital markets. Using tools and concepts from the field of contract theory, offer some explanations for the following questions.

1. What determines the optimal stake in a company for outside equity to take in an owner-operated firm? How does the structure of the stake (debt versus equity) affect this decision?

2. What determines when a firm should make a seasoned equity offering (i.e. sell additional equity in the capital market after previously issuing equity at some point in the past)? How should markets respond to seasoned equity offerings in determining the valuation of a firm?

3. A firm is competing with several other firms in trying to acquire some other company that is on the market. What determines the optimal bidding strategy? How should the market respond to a successful acquisition in terms of the share price for the acquirer and the target?

4. Suppose that a firm is selling some assets and faces bidders offering equity stakes. When would it prefer to sell for equity versus selling in exchange for cash or debt? 

In formulating your answers, please describe one or more of the key economic tradeoffs associated with each of the questions and describe how these tradeoffs are affected by some of the details of the situation. In particular, your answers should focus on how the organization will react to changes in its balance sheet in terms of effort undertaken in the organization, projects pursued or not pursued, and acquisitions pursued or not pursued.

In doing so, you may, at your discretion, provide a “toy model” to illustrate your answer. You may also, at your discretion, highlight key existing work illuminating the tradeoffs you have identified.

And

Question 2: (All work and no play makes Jack an investment banker) There are two types of investment banking analysts, high or low with probabilities .8 and .2 respectively. Analysts can choose two levels of work intensity, normal and psychotic. Partners in investment banks observe levels of intensity (but not types) and then decide between the actions fire or promote. The analysts payoffs are the sum of two elements: she obtains 2 units if she is promoted and 2 units if she works at her preferred level of intensity. Low types prefer to work at a normal intensity level; whereas high types prefer to work at a psychotic intensity level. The partner receives a payoff one for firing low types and a payoff of one for promoting high types. Otherwise, the partner’s payoff is zero.

Use the notion of weak perfect Bayesian equilibrium to answer the following questions.

            a. Characterize the set of separating equilibria (if any) in this game. If there are no such equilibria, then prove that this is the case.

            b. Characterize the set of pooling equilibria (if any) in this game. If there are no such equilibria, then prove that this is the case.

            c. Use belief-based refinements to argue that at least one of the above equilibria is unreasonable. (You can be somewhat informal in these arguments should you wish.)

This week

By popular request, the review session that was scheduled to take place during regular section time on Tuesday has been replaced by extended office hours.  Office hours for Megan this week will be from 3:30-5:00 on Tuesday and 3:10-4:00 on Wednesday in 204 Giannini.    


Final Exam Solution Sets:

2004

2006

2007

2008

2010

2011

Point-by-point maximization

For those who were in section today, here is why point-by-point maximization works.  As discussed in class, maximizing the integral of f(s(x))dx by choosing s(x) is the same as maximizing f(s(x)) by choosing s(x).  The concern that was brought up in class was that you might choose an s*( ) that would maximize f for some values of x but not for others.  However this would contradict s*( ) being the argmax of f(s(x))!  We set no limitations on what s( ) can be - it can be any sort of function at all, piecewise, discontinuous, whatever - and s*( ) is defined as the function s( ) which maximizes f at every point.

The function s*( ) is a rule chosen for every single value of x such that for every single value of x, we have maximized f (and therefore maximized the integral of f with respect to x.)

If you are still confused come grab me.

Problem Set 2 Solutions

PS 2 solutions can be found here.

Next week in section I will be covering moral hazard.

Class #9 Highlights

Whew. At long last we finally get to signaling. Here, we studied an education game with two types where education is a pure costly signal--it has no productivity value. In this class, we looked for a pooling equilibrium, this is an equilibrium in which all types choose the same signal and therefore the posteriors of the receiver are the same as the priors.

To find such an equilibrium, we needed to specify strategies as well as beliefs on the part of the receiver. To determine the full set of such equilibria, we relied on pessimistic beliefs. In our setting, these beliefs are a bit like a grim trigger strategy in repeated games--anything not supportable by these beliefs cannot be supported by any other beliefs.

With this specification, we determined that the upper bound occurred where the low type was indifferent between the equilibrium education level, e*, and choosing e = 0 and being branded as a low type hence making r_L.

These beliefs are a WPBE and, with an appropriate specification of perturbations, a sequential equilibrium as well. To support these as an SE, suppose that L types choose a random e with probability epsilon. H types follow the same strategy with probability epsilon^2. Taking limits as epsilon -> 0, it then follows that the posterior beliefs of the receiver put infinitely more weight on low types for all off equilibrium choices of e. But these are precisely the pessimistic beliefs we specified.

In a sense, these beliefs are goofy. Why would low types be infinitely more likely to choose higher, and hence more costly, education levels than high types?

To deal with this, one refinement that is commonly used is the intuitive criterion. It asks the following question: Suppose a given type chose an education level (i.e. a signal) e and got to pick what it would like the receiver to believe. Given these beliefs, the receiver would then choose an optimal action. If the result is that this type still earns less than her equilibrium payoffs, then the receiver should never believe that such an e came from that type.

In terms of our game, choose an education level e' such that, if the receiver R believed that it came from a high type for sure, an L type would be indifferent between and the equilibrium e*. Formally, e' solves:

r_H - e' theta_L = r_bar - e* theta_L.

Rewriting:

r_H - r_bar = theta_L (e' - e*)

Clearly, all education levels above e' could not have come from low types. But notice that, for a high type,

r_H - r_bar > theta_H (e' - e*)

since theta_H < theta_. But this means H types could profitably deviate from e* if they are recognized as H types.

The intuitive criteria requires that beliefs following choices e > e' must place all weight on H types. As a consequence, none of the pooling equilibria satisfy the intuitive criterion.


Key things should be able to do after this class:

1. Know how to construct a pooling equilibrium.
2. Make an argument using the intuitive criterion to rule out any pooling equilibrium.

Class #8 Highlights

In this class, we studied equilibrium concepts in sequential games with private information. The logical extension of Nash equilibrium to private information settings is Bayes-Nash equilibrium. This is simply a best response to the expectation of a rival's strategy, where the expectation is formed using Bayes' Rule where possible. The two key difficulties with this solution concept are:

1. It places no restriction on out of equilibrium actions.
2. It places no restriction on out of equilibrium beliefs.

Problem 1 is a familiar one from sequential games of complete information. The usual issue is that non-credible out of equilibrium actions can be used to affect on equilibrium actions. We saw a private information version of this idea in the entry game used to motivate the study of solution concepts. The point here was that the incumbent could threaten to fight out of equilibrium even though this was a dominated strategy.

Weak Perfect Bayesian Equilibrium takes care of the first objection. It requires that all strategies be sequentially rational; that is, on and off equilibrium actions must be best responses to beliefs, which are again formed using Bayes' rule where possible.

But this does not deal with the second objection. One could still use "goofy" beliefs to support sequentially rational out of equilibrium actions.

Sequential Equilibrium attempts to address this concern. This is a bit like the private information analog of trembling hand equilibrium. The idea here is to consider perturbations of the equilibrium strategies that are completely mixed. This forces the use of Bayes' rule on and off the equilibrium path. If there exists a sequence of such perturbations that converges to a WPBE, then we have found a sequential equilibrium.

In terms of our entry game, if the entrant deviated and entered the game, we saw that it would rationally choose a fight strategy and hence one could not justify the beliefs supporting the fight strategy by the incumbent following a sequence of such perturbed strategies.

As we will see in class #9, even sequential equilibrium does not rule out all goofy beliefs. We might still be clever in constructing sequences of deviations satisfying this equilibrium concept yet not satisfying common sense (or empirical testing in the lab or the field).

Technical footnote: Sequential equilibrium is only formally defined for games with finite actions, types, etc. For games with continuous types, the right extension s a matter of some debate. The notion of perfect Bayesian equilibrium, which is a strengthening of WPBE is one way to deal with this.


Key things should be able to do after this class:


1. Know what a BNE, WPBE, and SE are and use these concepts to analyze sequential games of private information.
2. Recognize that the "correct" solution concept depends on the game being studied. For many games, WPBE works just fine. Use your judgment rather than being dogmatic.