Monitoring, Management Compensation

Question 1

Relationship between managers and owners empirically

Answer 1

Hard to measure - data availability etc

Question 2

What happens if managers goals do not align to those of owners

Answer 2

Owners need to provide incentives to exert high effort (principle agent problem)

Question 3

Set up for principle agent model
Principal is restaurant owner
Agent is waiter
If waiter does not exert hard effort, customers may not return and revenues fall.

First, owner writes contract with waiter compensation based on revenue observed. Waiter decides to accept or not, and if accepts, choose what effort to exert ( 2 levels of effort)

What is reservation utility

Answer 3

Utility if he does not accept (10 in this instance, from going to work at another place)

Question 4

So what is waiters 2 possible utility functions

Answer 4

U= w - e (If he devotes a effort level e)
Or if not accept contract, reservation utility:
U = 10 ( working at another place)

Question 5

What is principals 2 possible revenue expressions R(e)

B) so with that being revenue what is profit?

Answer 5

R(e) (revenue is a function of effort)

R(e) = H, if e=2 (i.e high revenue if waiter works hard)
Or
R(e) = L, if e=0 (low revenue if waiter exerts low effort)

So R(2) > R(0)

B) π = R(e) - w (simple, revenue - wage cost of worker)

Question 6

So what do owners offer waiter if want revenue high vs low

Answer 6

Wh is revenue high
Wl if revenue low

Question 7

Agent’s participation constraint (I.E when will they accept the contract

Answer 7

U = W - e
Wh - 2 >= 10

(utility from high effort > reservation utility)
I.e participate when utility from contract >= reservation utility

Question 8

Agent incentive constraint

Answer 8

This refers to their decision on which effort to choose
U = w - e
Wh - 2 >= Wl - 0

Incentive to work hard when this holds

Question 9

Equality case - given our participation and incentive constraint, what values of Wh, Wl, πh, πl do we get (this is for the optimal contract)

Answer 9

Wh = 12
Wl = 10

πH = H - 12, when e=2
πl = L - 10, when e=0

And πH>πL which is
H - 12 > L - 10
Rearrange to get
H - L > 2

Question 10

So why is to know good for manager

Answer 10

There is perfect monitoring, owner can observe if revenue high or low, then know which effort level the waiter has exerted.

So they can design contract to get desireable outcome. In reality we add some level of uncertainty.

Question 11

So in reality add uncertainty.

Let when,
R(2) = H with Pr=0.8 , or L with Pr=0.2
And
R(0) = H with Pr=0.4 , or L with Pr=0.6

I.e when e=2, high effort, so probability of high revenue is 0.8, and 0.2 chance of low revenue with high effort.

Moving from low effort (0) to high (2), increases probability of earning high revenue from 0.4 to 0.8

Answer 11

so basically can still have high revenue with low effort, and can still get low revenue even if put in high effort! i.e introducing uncertainty

Question 12

Utility function with uncertainty (2 possible)

Answer 12

U = E(w) - e, if takes offer and exerts effort

Or reservation utility
10 if does not take offer

E(w) is expected wage (since now have probabilities)
So utility is expected wage - effort
Previously with perfect monitoring utility was wage - effort U = w - e

Question 13

So what are our 2 expected wages E(w)

Answer 13

when e=0 (low effort)
E(w) = 0.4Wh + 0.6Wl

when e=2 (high effort)
E(w) = 0.8Wh + 0.2Wl

Question 14

Given this, what is their partipation constraint (decision to take the offer)

Answer 14

0.8Wh + 0.2Wl - 2 ⩾10
(utility from high effort > reservation utility, just like previous)

Question 15

What about the incentive constraint with uncertainty (on which effort level to pursue)

Answer 15

0.8Wh + 0.2Wl - 2 ⩾ 0.4Wh + 0.6Wl - 0

in order to exert high effort

Question 16

How to find optimal contract i.e values of Wl, Wh

Answer 16

Rearranging incentive constraint to get Wl = Wh - 5 (or Wh=Wl+5)

sub that into participation constraint to get value of the other one. then sub that answer into other one to get the last Wh/Wl.

We get Wh=13 Wl=8

Question 17

So with perfect monitoring Wh = 12, Wl = 10.
under uncertainty, Wh = 13, Wl = 8.

What would expected wage E(w) be then with high effort?

b) what can we notice? how is the owner and worker affected

Answer 17

0.8(13) + 0.2(8) = 12

b) expected wage is the same as wage under perfect monitoring! so the expected bill is not costly for onwer

however bad for workers, who are not guaranteed the high wage since there is a probability 0.2 of earning low wage 8.

Question 18

Now add risk to the model. Assume waiter is risk adverse. What does this do to the model pg6

Answer 18

Waiters thinks probability of high revenue is lower than what the owner thinks, as wear of risk.

So while owners estimate
Ro(2) = H with probability 0.8 (when e=2)

Ro(2) = H with probability 0.7 (when e=2)

(still have same belief for probabilities of revenue with low effort i. 0.4 chance of high rev, 0.6 chance of low revenue)

Question 19

What does waiters’ risk adversity mean for their expected wage under high effort compared to the owners expected wage (which is the cost to them)

b) expected wage comparison inequality expression (owner vs workers)

c) what is needed for the waiter?

Answer 19

Expected wage under high effort is lower for the waiter, compared to the owner. (they expect to get less)

b)
0.8Wh + 0.2Wl > 0.7Wh + 0.3Wl

c) waiter needs greater compensation to match the expected wage of owner.

Question 20

So as waiters are more risk adverse, their expected wage is less than the owners expectation of the wage that they will pay.

thus need a greater compensation to match expected wage of owner. Solve this by using the PC and IC of waiter.

Start with PC first

Answer 20

PC: 0.7Wh + 0.3Wl - 2 >= 10

rearrange to get Wh = 12-0.3Wl/ 0.7

Put in diagram from showing participation constraint

Question 21

Diagram form of participation constraint pg 6

Answer 21

Y axis is Wh, X axis is Wl.

Wh = 12-0.3Wl / 0.7
thus
Y intercept is 12/0.7

Above PC line, contract will definitely be accepted, along the line is the minimum amount to be accept contract (rather than not taking contract, working elsewhere)

Question 22

So that was PC.

What about the incentive constraint for the worker?

b) diagram form

Answer 22

0.7Wh + 0.3Wl - 2 >= 0.4Wh+ 0.6Wl - 0

Rearrange to find Wh = 2/0.3 + Wl

b) IC line is upward sloping, intersect 2/0.3

Above (left) of line, exert high effort, right of line exert low effort

Question 23

How can we then find the optimal contract Wh and Wl.
(working pg 7)

b) Plot the IC and PC curves together in one diagram, we find equilibrium give us optimal contract (values of Wh and Wl) pg 7

Answer 23

Since we have rearrange both the PC and IC to make Wh subject, equate each one and solve.

get Wh=14, Wl = 22/3 (7.33)

B) intersection is where these values are.

Question 24

So owners Wh=13, Wl=8

Workers who are risk adverse Wh=14 Wl=22/3 (7.33)

2 opposing dynamics we see on their decision to take contract

Answer 24

The gap between Wh and Wl for workers is larger,

They are disincentised to take the contract as Wl is lower,

however incentivised as higher Wh.

Question 25

Owner then chooses Wh and Wl to minimise the expected wage. pg 7 express this b) key finding

Answer 25

put workers’ new optimal wages into the OWNERS expected wage. Eo(w) = 0.8(14) + 0.2(22/3) = 12.66 b) expected wage (which is a cost for owners) is now higher (12.66>12) , due to the waiter being risk adverse, they need a premium to compensate the risk.