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→Question B1: Production in Teams
*This page is included under the section [[BPP Field Exam]]
*This page is provides answers to the [[BPP Field Exam 2010]]
=== Question A===
[http://www.edegan.com/repository/EEganMorningSessionPartA.pdf Proposed solution here]
===Question B1: Production in Teams===
[http://www.edegan.com/repository/EEganMorningSessionPartB.pdf Proposed solution]
<b>Relevant papers</b>:
====Question B1.3====
The CEO can design a scheme that exploits the risk aversion of the agents using chance. The contract would work like this: If all employees exert work, each worker will get an equal share <math>1/N</math> of the effort. However, if any single worker does NOT work, then the payoffs will be determined by a lottery in which each employee gets a <math>\frac{1}{N}</math> chance of getting 100% of the combined outputand a <math>1-\frac{1}{N}</math> chance of getting zero. I will now show that irrespective of what other players are doing, the dominant strategy is to work.
Note that CARA utility is <math>u(c)=1-e^{-\rho c}</math>. An employee i's utility from working (if all others work) is <math>A=1-\exp[-\rho(\frac{1}{N}\sum_{i\neq j} z(e_{j})+\frac{1}{N}z(e_{i})-1)]=1-\exp[-\rho(z-1)]</math>.
If employee works but others aren't, the lottery is triggered and employee i's utility is <math>B=\frac{1}{N}(1-\exp[-\rho(\sum_{i\neq j} z(e_{j})+z(e_{i})-1)])</math>.
If employee i does NOT work, the lottery is triggered and his utility is: <math>C=\frac{1}{N}(1-\exp[-\rho(\sum_{i\neq j} z(e_{j}))])</math>
I will now show that <math>A>C</math> and <math>B>C</math> -- in other words, working is better than shirking no matter what the other players do.
First, note that <math>B>C</math>. Within the algebra, note that the utilities are identical except for the exponents. Note that <math>\sum_{i\neq j} z(e_{j})+z-1>\sum_{i\neq j} z(e_{j})</math> because <math>z-1>0</math>. As such, we know that the inequality always holds. As for the intuition: Note that the lotteries are identical except for the payoff in <math>\frac{1}{N}</math> of the time. If he works, this value is higher, so he prefers to work.
With regards to <math>A>C</math>, note that if all other workers are working, then <math>C=\frac{1}{N}(1-\exp[-\rho z (N-1)])</math>.
Now, consider that <math>A>C \iff N> \frac{1-\exp[-\rho z (N-1)]}{1-\exp[-\rho(z-1)]}</math>.
In the above RHS expression, we know that the numerator is smaller than the denominator, so the fraction is less than 1. We know that <math>N>1</math>, so the inequality always holds.
===Question B2: Relationship Specific Investments===
First some clarification of my interpretation of the problem.
Timing:
* 1. Buyer makes investment, costing <math>x^2</math>.
* 2. Buyer observes <math>v</math>
* 3. Seller makes take-it-or-leave-it (TOILI) offer <i>without</i> seeing <math>v</math>.
* 4. Buyer accepts or rejects.
Utility functions: The problem does not make reference to utility functions. I will assume that the buyer's utility is <math>x+v-x^2-P</math> where <math>P</math> refers to the price of the widget -- if the buyer chooses to buy. Otherwise his utility is zero. As for the seller: I will assume his utility is simply <math>P</math> (the price of the widget) if it is sold, and otherwise is zero. Note that both agents are risk neutral in this setup.
(a) Socially optimal level will be where marginal cost equals marginal benefits, or where social welfare is maximized. Marginal costs of investment are <math>2x</math>. Marginal benefits are <math>1</math>. These are equal where <math>x=1/2</math>.
(b) This solution requires backwards induction starting with step 4 above.
* First, note that there is a cutoff price at which the buyer will accept or not.
* Next, note that seller will correctly infer this (in expectation) and make a corresponding offer that will leave the buyer indifferent between accepting and rejecting the offer.
* Lastly, note that buyer will correctly anticipate seller's step 3 behavior and make corresponding investment decision.
(c)
===Question C1: Agenda Control and Status Quo===
(iv)
=== Question C2: Retrospective Voting ===
[http://www.edegan.com/repository/2ndyearexamquestion2010.pdf Proposed solution here]
[http://www.edegan.com/repository/EEganAfternoonSessionPartC.pdf Another attempt here]
=== Question D ==
[http://www.edegan.com/repository/EEganAfternoonSessionPartD.pdf Suggestion here]
