Monitoring, Management Compensation Flashcards
(23 cards)
Set up for principle agent model
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). If not accept, works elsewhere with utility of 10
What is reservation utility
Utility if he does not accept (10 in this instance, from going to work at another place)
So what is waiters 2 possible utility functions
U= w - e (If he devotes a effort level e)
Or if not accept contract, reservation utility:
U = 10 ( working at another place)
What is principals 2 possible revenue expressions R(e)
Assume e can either e=2 (high effort) e=0 (low)
B) so with that being revenue what is profit?
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)
So what do owners offer waiter if want revenue high vs low
Wh is revenue high
Wl if revenue low
Agent’s participation constraint (I.E when will they accept the contract
When U (which is w-e) > Reservation utility
Wh - 2 >= 10
(utility from high effort > reservation utility)
I.e participate when utility from contract >= reservation utility
Agent incentive constraint
This refers to their decision on which effort to choose
U = w - e
Wh - 2 >= Wl - 0
Incentive to work hard when this holds
Equality case - given our participation and incentive constraint, what values of Wh, Wl, πh, πl do we get (this is for the optimal contract)
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
So why is to know good for manager
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.
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
so basically can still have high revenue with low effort, and can still get low revenue (0.2 chance) even if put in high effort! i.e introducing uncertainty
Utility function with uncertainty (2 possible)
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
So what are our 2 expected wages E(w)
when e=0 (low effort)
E(w) = 0.4Wh + 0.6Wl
when e=2 (high effort)
E(w) = 0.8Wh + 0.2Wl
Given this, what is their partipation constraint (decision to take the offer)
0.8Wh + 0.2Wl - 2 ⩾10
(utility from high effort > reservation utility, just like previous)
What about the incentive constraint with uncertainty (on which effort level to pursue)
0.8Wh + 0.2Wl - 2 ⩾ 0.4Wh + 0.6Wl - 0
in order to exert high effort
How to find optimal contract i.e values of Wl, Wh
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
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
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.
Now add risk to the model. Assume waiter is risk adverse. What does this do to the model pg6
Waiters thinks probability of high revenue is lower than what the owner thinks, as wear of risk.
So while owners estimate
Rowner(2) = H with probability 0.8 (when e=2)
Rwaiter(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)
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?
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.
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
B) then rearrange to find Wh, as gives values for diagram
PC: 0.7Wh + 0.3Wl - 2 >= 10
rearrange to get Wh = 12-0.3Wl/ 0.7
We arranged to Wh since Y axis is Wh
Put in diagram from showing participation constraint
Diagram form of participation constraint pg 6
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)
So that was PC.
What about the incentive constraint for the worker?
b) diagram form
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
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
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.
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
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.
So effects of being risk adverse is ambiguous, but assume generally worse unless compensated
Owner then chooses Wh and Wl to minimise the expected wage. pg 7
express this
b) key finding
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.
So risk adverse waiters is bad for owner and worker is ambiguous but generally worse without compensation i.e compensation was the new higher expected wage (risk premium 0.66) 12.66!
