# BPP Field Exam 2010 Answers

### Question B1: Production in Teams

Relevant papers:

#### Question B1.1

• The prompt does not explicitly say that everyone must receive the same share. Nonetheless, many of the relevant papers assume that this is the case. As such, will solve the problem both ways.
• The prompt also does not explicitly require a balanced budget. However, no source of funding for the agents is mentioned besides the combined output V of the individual agents. As such, I will assume that the budget balance restriction must hold.
• The prompt also does not address whether work decisions are made cooperatively or non-cooperatively, or whether transfers or contracts are possible between employees. I will assume that no transfers or contracts between employees are possible, and that work decisions are made non-cooperatively.

Each agent's maximization problem

$\max_{e_{i}}[s_{i}\sum_{j\neq i}z(e_{j})+s_{i}z(e_{i})-e_{i}]$

Note that if the agent chooses to work, his utility is $s_{i}\sum_{j\neq i}z(e_{j})+s_{i}z-1$

If he chooses to shirk his utility is $s_{i}\sum_{j\neq i}z(e_{j})$.

Therefore, he'll work if $s_{i}z-1\gt 0 \iff s_{i}z\gt 1 \iff s_{i}\gt \frac{1}{z}$.

Note that not everyone can have $s_{i}\gt \frac{1}{z}$ because $\sum s_i$ in this world $=\frac{N}{z}\gt 1$.

(a) My answer, assuming that all shares must be equal.

If all shares must be equal, no contract scheme can get any of the workers to work. This is because a worker will only work if $s_{i}\gt \frac{1}{z}$, but we know that not everyone can have this contract because $\sum s_i$ in this world $=\frac{N}{z}\gt 1$.

(b) My answer, assuming that some shares can be different.

If some shares can be different, then the optimal contract is where some number $M\lt N$ workers get $s_{i}=\frac{1}{z}$, and the remainder get $s_{i}=0$. $M$ is the largest number such that $\frac{M}{Z}\leq 1$. $M$ workers will provide effort, and $N-M$ workers will shirk.

#### Question B1.2

In the previous scenario, the CEO got $M$ workers to work. The question now is whether burning some of the output $V$ will motivate the remaining $N-M$

#### 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 $1/N$ 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 $\frac{1}{N}$ chance of getting 100% of the combined output and a $1-\frac{1}{N}$ 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 $u(c)=1-e^{-\rho c}$. An employee i's utility from working (if all others work) is $A=1-\exp[-\rho(\frac{1}{N}\sum_{i\neq j} z(e_{j})+\frac{1}{N}z(e_{i})-1)] =1-\exp[-\rho(z-1)]$.

If employee works but others aren't, the lottery is triggered and employee i's utility is $B=\frac{1}{N}(1-\exp[-\rho(\sum_{i\neq j} z(e_{j})+z(e_{i})-1)])$.

If employee i does NOT work, the lottery is triggered and his utility is: $C=\frac{1}{N}(1-\exp[-\rho(\sum_{i\neq j} z(e_{j}))])$

I will now show that $A\gt C$ and $B\gt C$ -- in other words, working is better than shirking no matter what the other players do.

First, note that $B\gt C$. Within the algebra, note that the utilities are identical except for the exponents. Note that $\sum_{i\neq j} z(e_{j})+z-1\gt \sum_{i\neq j} z(e_{j})$ because $z-1\gt 0$. As such, we know that the inequality always holds. As for the intuition: Note that the lotteries are identical except for the payoff in $\frac{1}{N}$ of the time. If he works, this value is higher, so he prefers to work.

With regards to $A\gt C$, note that if all other workers are working, then $C=\frac{1}{N}(1-\exp[-\rho z (N-1)])$.

Now, consider that $A\gt C \iff N\gt \frac{1-\exp[-\rho z (N-1)]}{1-\exp[-\rho(z-1)]}$.

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 $N\gt 1$, 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 $x^2$.
• 2. Buyer observes $v$
• 3. Seller makes take-it-or-leave-it (TOILI) offer without seeing $v$.
• 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 $x+v-x^2-P$ where $P$ 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 $P$ (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 $2x$. Marginal benefits are $1$. These are equal where $x=1/2$.

(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)

(i)

(ii)

(iii)

(iv)