Changes
Jump to navigation
Jump to search
<b>====Question B1.1</b>====
<b>====Question B1.2</b>====
Reminder: CARA utility (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>u(c) x= 1 − e^{−\alpha c}/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 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>:
* [http://www.edegan.com/wiki/index.php/Holmstrom_%281999%29_-_The_Firm_As_A_Subeconomy Holmstrom (1999), Firm as Subeconomy], Not on 2010 exam's reading list.
<b>Notes about assumptions</b>:
If some shares can be different, then the optimal contract is where some number <math>M<N</math> workers get <math>s_{i}=\frac{1}{z}</math>, and the remainder get <math>s_{i}=0</math>. <math>M</math> is the largest number such that <math>\frac{M}{Z}\leq 1</math>. <math>M</math> workers will provide effort, and <math>N-M</math> workers will shirk.
In the previous scenario, the CEO got <math>M</math> workers to work. The question now is whether burning some of the output <math>V</math> will motivate the remaining <math>N-M</math>
====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 output and 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 <bmath>N>1</math>, so the inequality always holds. ===Question B1B2: 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</bmath>(the price of the widget) if it is sold, and otherwise is zero. Note that both agents are risk neutral in this setup.
===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]
