# Midterm Exam: Sketch of Solutions - Kellogg School of Management

## Midterm Exam: Sketch of Solutions - Kellogg School of Management

Midterm Exam: Sketch of Solutions Information Economics (Ec 515) · George Georgiadis Prepared by Lingfeng Que Problem 1. Application of a Principal-...

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Midterm Exam: Sketch of Solutions Information Economics (Ec 515) ·

Prepared by Lingfeng Que Problem 1. Application of a Principal-Agent Problem to Credit Markets (9-13-13 points) A risk-neutral entrepreneur has a project that requires capital K to initiate. The project will either succeed, in which case it generates profit R, or it will fail, in which case it generates profit 0. The probability of success is equal to 2

the entrepreneur’s effort level e 2 [0, 1], and his effort cost is e2 . He seeks outside funding from an investor. The investor receives a repayment r if the project is successful and 0 otherwise. Therefore, the entrepreneur’s and the investor’s expected profit is P E = e ( R

r)

e2 2

and P I = e r

K, respectively. Assume that the entrepreneur

makes a take-it-or-leave-it offer to an investor, whose outside option is 0. (This is equivalent to assuming that the investor offers the contract and he operates in a perfectly competitive market.) Furthermore, assume that R2 > 4K. 1. Assume that effort level is contractible. Derive the first-best outcome e FB and r FB . 2. Now suppose that effort is not contractible. Find the optimal solution e⇤ and r ⇤ . 3. Assume that the entrepreneur has initial wealth w 2 [0, K ]. Therefore, he will invest his own wealth w in the project, and borrow K w from an investor. Compute the entrepreneur’s expected utility V (w). Show that it is increasing and concave in w. Provide some intuition and interpretation. Solution of Problem 1: Part 1

Since effort level is contractible, the entrepreneur’s problem is

max

e (R

e, r

s.t.

er

r)

e2 2

K

The entrepreneur’s IR constraint binds, and so the entrepreneur’s problem can be re-written as ⇢

max eR e

where r = Part 2

K e.

The FOC gives e f b = R and so r f b =

K

e2 2

,

K R.

Since the effort level is not contractible, the entrepreneur’s optimal effort level must maximize her payoff

given payment r: n e = arg max e( R e

1

r)

e2 2

o

The FOC gives e = R

r. Therefore, the entrepreneur’s problem is max

e (R

e, r

s.t.

er

e2 2

r) K

e=R

r

or equivalently r )2

(R

max r

s.t.

(R

2

r) r

K

The entrepreneur’s IR constraint can be re-written as

(R

r) r

K () r

2

Rr + K  0 () r 2

"

p

R

R2 2

4K R + ,

p

R2 2

4K

#

The principal’s objective function decreases in r (for r  R), and hence the optimal solution has ⇤

r = Part 3

R

p

R2 2

4K

and e =

R+

p

R2 2

4K

The only difference here relative to part 2 is the IR constraint: er

Using the same approach as in part 2, we get r = V (w) = It’s straightforward to show that V 0 (w)

R

r )2

(R 2

p

K

w

R2 4( K w ) 2

=

[R +

p

and so the entrepreneur’s profit function becomes

R2

4( K

w)]2

8

0 and V 00 (w)  0 for all w 2 [0, K ]; i.e., it is an increasing, concave

function in the entrepreneur’s wealth. This implies that the poorest population would benefit the most from access to credit markets. Problem 2. Moral Hazard in Teams and Different Types of Implementation (15-15 points) A risk-neutral firm employs two identical, risk-neutral workers. Each worker’s utility is w

e where w is his wage

and e is his effort level. Each worker i can either work or shirk; i.e., ei 2 {0, 1}, and efforts are not contractible. Assume that the firm wants to incentivize both workers to work at the lowest possible cost. 1. The firm decides to monitor its workers by group performance. In particular, it observes eˆ = min {e1 , e2 }. If eˆ = 1, then both workers receive wage w g . Otherwise, they both receive 0. Find the optimal wage w g . What is a potential problem with this incentive scheme? 2. The firm now turns to a monitoring technology which detects a worker shirking with probability q 2 [0, 1]. q2 In this case, the worker receives 0. Otherwise, he receives wage w M . The cost of this technology is c (q) = 2 . Assume that the firm first commits to a monitoring level q, and each worker observes the q before choosing his effort level. Find the optimal wage w M and monitoring level q M . Hint: For part 1, you may assume either partial or full implementation. 2

Solution of Problem 2: Part 1 The optimal wage w g is the smallest wage such that there exists an equilibrium in which both workers work; i.e., e1 = e2 = 1. Given that agent i works, the IC constraint of workers j 6= i is wg

1

0

Therefore, the optimal wage is w g = 1. The problem with this incentive scheme is that there exists another Nash equilibrium in which both workers shirk. To see why, suppose that worker i shirks. Then worker j’s best response is to also shirk Part 2

The firm’s problem is min

w M ,q

s.t.

wM +

q2 2

wM

1

q)w M

(1

Note that the worker’s IC constraint asserts that the worker prefers to work and incur the cost of effort rather than shirk and receive wage w M if he isn’t caught (which occurs with probability 1 re-written as w M q

constraint binds in the optimal solution. By substituting w M = min q

The FOC gives

1 q2

q). The IC constraint can be

1, and by noting that the firm’s objective increases in both w M and q, it must be that the IC ⇢

1 q

into the objective function, we have

1 q2 + q 2

+ q = 0 =) q M = 1, and hence w M = 1.

Problem 3. Free-riding in Teams (13-13-9 points)

(1 e1 e2 )2 , where ei denotes agent i’s effort level. Agent i’s cost of effort is given by ei2 . The total profit is equal to yp, where p > 12 is the per-unit price of the product. Two risk-neutral agents collaborate in production. Output y = 1

1. Assume that the agents can coordinate perfectly so that their efforts to maximize their joint n they choose o fb fb 2 2 surplus S = yp e1 e2 . Find the optimal effort levels e1 , e2 , and the corresponding joint surplus S.

2. Now suppose that the workers cannot coordinate, so that each agent will choose his effort level to maximize his profit given his expectations about the other agent’s effort level. Assume that each agent receives 12 yp (i.e.,

1 2

of the profit). Find the equilibrium effort levels {e1 , e2 }, and the corresponding joint surplus. How does it compare to your answer from part 1? Provide some intuition. 3. In the pursuit of coordination, the workers can hire a manager. The manager will assign an effort level to each e22 ), and the agents must follow her instruction. Assume that each agent receives 12 yp, and the manager receives her outside option u¯ = 0. Under what agent to maximize profits (i.e., yp, not surplus S = yp

e12

conditions do the agents prefer to hire a manager (relative to the case from part 2). Provide some intuition. Hint: You may restrict attention to symmetric strategies; i.e., strategies that have e1 = e2 .

3

Solution of Problem 3: Part 1

The efforts that maximize the joint surplus of the two agents solve n h max p 1 e1 ,e2

(1

e2 ) 2

e1

i

e12

e22

o

The FOCs are

fb

fb

which implies that e1 = e2 =

p 1+2p ,

2p (1

e1

e2 )

= 2e1

2p (1

e1

e2 )

= 2e2

and by substituting this into the joint surplus function, we get S f b =

2p2 2p+1 .

Part 2 Now each agent chooses his effort to maximize his own profit while anticipating the effort of the other agent. Therefore, agent i solves max ei

np h 2

1

(1

ei

e j )2

i

ei2

o

where e j is the effort that agent i anticipates from agent j 6= i. The FOC of agent i gives p 1 2

e j = 2ei

ei

Solving for a symmetric equilibrium (i.e., for an equilibrium where e1 = e2 ), we obtain e1 = e2 = corresponding joint surplus is S =

p2 (2p+3) . 2( p +1)2

p 2+2p ,

and the

It is straightforward to show that S < S f b . Intuitively, when the agents choose their efforts non-cooperatively, they exert less effort due to the free-rider problem. Part 3 Denote e1M , e2M , S M as the effort levels and joint surplus when a manager is hired. In this case, the manager ⇥ ⇤ assigns effort levels to maximize the profit yp = p 1 (1 e1 e2 )2 . Note that the manager does not take the agents’ effort costs into account. Observe that the profit is maximized when e1 + e2 = 1, in which case yp = p. Assuming symmetric strategy, we have e1M = e2M = 21 , and the corresponding joint surplus is S M = p the manager receives her outside option which is 0, each agent’s payoff is a manager if and only if SM 1 () p 2 This inequality can be re-written as

2p 1 2p+3

⌘⇣

⌘ p +1 2 p

1 M 2S .

1 2.

Since

Therefore, the agents prefer to hire

S p2 (2p + 3) 2( p + 1)2 1. First observe that the LHS is equal to 0 when p =

1 2,

and so the desired inequality is not satisfied. Moreover, one can show that the LHS increases in p, and the LHS

approaches 1 as p ! •. Therefore, this inequality is never satisfied and so the agents never find it optimal to hire a manager in this scenario.

4