Math Review

1
Math Review

• Probability

2
Random Variable Review

• A Random Variable is a function from a sample
  space S into the real numbers
• Maps outcomes to real numbers
• One way to characterize distribution of an RV is
  through Probability Density Function (PDF) or
  Probability Mass Function (PMF)
• We will focus here on continuous RVs and PDFs

• PDF gives density of RV at each real number
• Sets of outcomes mapped to larger areas are more
  likely

-4 -3 -2 -1 0 1 2 3 4 5
3
PDF, CDF

• Def If X is a continuous random variable, then
  function given by
• where g(t) is probability density function of x
  at t, is called the Cumulative Distribution
  Function or CDF.

PDF
CDF
1
0
4
Example Uniform Distribution

• Probability spread evenly across an interval
• If interval is wide, then outcomes are more
  dispersed

PDF
-3
3
CDF
1
0
5
Theorem

• For constants altb (and given some conditions on
  G and g)

• a-2, b1
• G(a) red
• G(b) blue

-4 -3 -2 -1 0 1 2 3 4 5
6
Chapter 11

• B. Tournaments and Promotions

7
Reading for next class

• http//www.boston.com/news/local/articles/2005/01/
  17/summers_remarks_on_women_draw_fire/
• Also instead of the textbook, we will read
• http//www.econ.ucsb.edu/babcock/BertMul.pdf

8
Question from reading

• Do top golfers play better or worse or the same
  as usual when Tiger Woods is in a tournament?
• A. Better
• B. Worse
• C. Same

9
Clip

• http//www.youtube.com/watch?vO7juJjupBAc

10
Opinion Poll

• If you were on TVs Millionaire quiz show, and
  you were asked the first question (worth 100)
  and you didnt know the answer, how much would
  you be willing to pay for the right answer?
• A. 100
• B. More than 100
• C. Less than 100

11
Promotions as Tournaments

• Think Tennis or Golf
• If 2 workers in workplace, 1 gets promotion or
  title
• Prospect of wage hike increases effort
• Key insight Wage hikes generate effort two
  different ways
• Incentivize workers to whom they are paid
• Paying high wages to senior workers incentivizes
  junior workers

12
What should pay schedule for promotions look like?

• Intuition
• There is noise or luck in the promotion decision
• Typical worker works less hard if there is more
  noise
• We can use random variables to model
  tournaments, where each workers output depends
  on effort and a random shock, epsilon
• (Lazear Appendix)

13
Opinion Poll

• Suppose this course were graded in the following
  way
• Your percentage score on the exams counted for
  one tenth of your grade
• A roll of the dice counted for nine tenths of
  your grade
• (A1, B2, C3, D4, F5, Incomplete6))
• Would you this grading scheme cause you to put in
  more or less time studying?
• A. More
• B. Less
• C. Same

14
Set up

• Two workers
• Worker with larger output in per 1 becomes boss
  and gets w1gtw2, where w1wage of boss
• Output of worker 1 q1.
• Output of worker 2 q2.
• qimiei
• mi effort
• ei noise
• Noise can be or -
• C(mi) effort cost
• Price of output 1

mi
15
Set up

• Utility for worker j
• Expected (wage)-C(mj)
• Increasing marginal disutility of effort
• Max w1Pw2(1-P)-c(mJ)
• Pprobability win
• (function of effort).
• F.O.C.
• marg. Margben cost

mi
16
What is P?

• P Prob(mJeJgtmkek) Prob(ek-eJltmJ-mk )
• Let GCDF of ek-eJ
• Then PG(mJ-mk)
• dP/dmg(mJ-mk)
• F.O.C.
• But mJ mk, because workers identical.

mJ-mk
0
17
Question

• Which distribution of ek-eJ would imply there was
  more luck involved in winning a promotion?
• A.
• B.
• C. (Cant tell)

A
0
B
0
18
Analysis, contd

A

• 1.
• Note RHS increases with mJ
• (Slope of effort cost curve increases the further
  out you go, i.e., cgt0)
• If w1-w2 up, then mJ up
• Bigger pay gap implies higher effort
• if g(0) down then mJ down
• Lots of noise means workers have less incentive
  to work hard

0
B
0
19
Firms Decision

• 2. Max mJmk-(w1w2) w1,w2s.t. E(Ui)0 or3.
  (w1w2)/2c(m)
• Because mmJmk this is
• Max 2m-(w1w2)w1, w2
• Sub in 3
• Max 2m-2c(m)w1, w2

• F.O.C.s
• Therefore C(m)1
• M.C. of effort Marginal benefit to firm

20
Analysis, contd

• c(m)1 (This gives us m)
• Sub into 1
• (w1-w2)g(0)c(m)
• (w1-w2)g(0)1
• 4. w1-w21/g(0)
• The more noise in the process, the lower g(0) and
  the higher should be w1-w2
• Average wage doesnt change if g(0) changes, but
  gap does.
• To solve for w1, w2, combine 4. with
• 3. (w1w2)/2c(m)
• Two equations, two unknowns

A
0
B
0
21
Example

• a) If the firm chooses the profit-maximizing w1
  and w2, how much effort m will each worker
  choose?

• Two workers, j and k, weave wicker baskets.
  Quantity each weaves is
• qi mi ei
• q is number of baskets, m is effort, e is luck
  factor.
• Basket price 24 each..
• x ek ej takes on values between -1/4 and 1/4
  with a uniform prob distribution
• (So g(x), the PDF of ek ej , is 2 at all points
  between - ¼ and ¼
• Worker utility E(wage)3m2 .
• At end of period 1, worker who wove the most
  baskets gets w1 and other gets w2.

22
Example

• b) What w1 and w2 will the firm choose to get the
  workers to put forth this amount of effort?

• Two workers, j and k, weave wicker baskets.
  Quantity each weaves is
• qi mi ei
• q is number of baskets, m is effort, e is luck
  factor.
• Basket price 24 each..
• x ek ej takes on values between -1/4 and 1/4
  with a uniform prob distribution
• (So g(x), the PDF of ek ej , is 2 at all points
  between - ¼ and ¼
• Worker utility E(wage)3m2 .
• At end of period 1, worker who wove the most
  baskets gets w1 and other gets w2.

23
Example

• c) What is the firms profit at this level of
  output and this wage schedule?

• Two workers, j and k, weave wicker baskets.
  Quantity each weaves is
• qi mi ei
• q is number of baskets, m is effort, e is luck
  factor.
• Basket price 24 each..
• x ek ej takes on values between -1/4 and 1/4
  with a uniform prob distribution
• (So g(x), the PDF of ek ej , is 2 at all points
  between - ¼ and ¼
• Worker utility E(wage)3m2 .
• At end of period 1, worker who wove the most
  baskets gets w1 and other gets w2.

24
Example

• d) Suppose instead the firm chose w140 and
  w234. How much effort would each worker choose?
  Why would the firm not choose these wages?

• Two workers, j and k, weave wicker baskets.
  Quantity each weaves is
• qi mi ei
• q is number of baskets, m is effort, e is luck
  factor.
• Basket price 24 each..
• x ek ej takes on values between -1/4 and 1/4
  with a uniform prob distribution
• (So g(x), the PDF of ek ej , is 2 at all points
  between - ¼ and ¼
• Worker utility E(wage)3m2 .
• At end of period 1, worker who wove the most
  baskets gets w1 and other gets w2.

25
Tiger Woods, revisited

• Woods presence in a tournament means
• Lower probability that your effort will matter.
• (Similar to reducing g(0))
• This elicits lower effort
• Evidence appears to back this up

26
Question

• Suppose there are 5 levels in a firm 1)Intern,
  2) Worker 3)Middle-Manager, 4) VP, 5) President
• Each promotion from one level to the next carries
  with it an identical pay raise.
• Which promotion is likely to increase the
  workers utility the most?
• A. 1 to 2
• B. 2 to 3
• C. 3 to 4
• D. 4 to 5
• E. All equal

27
Pay hikes at low and high job levels

• Typically, higher percentage of workers get
  promoted in lower tier promotions
• ½ get promoted from 1 to level 2
• ¼ of those get promoted to level 3
• 1/8 of those get promoted to level 4
• and so on
• Getting a promotion is riskier for a high
  position.
• Based on what we know, this means pay gap must be
  higher to get high effort

28
High pay raise at high level

• Reason 1 Riskier to get high promotion, so pay
  gap must be higher
• Reason 2 Option value
• Part of incentive to work hard in low tiers is
  not just next pay raise but
• later raises that first raise makes possible
• Promotions give workers option to keep getting
  promoted
• e.g. Millionaire game
• But no option value at top

1 2 3 4
29
Other concerns

• Heterogeneous work force
• Create many job tiers, with pay hikes at each
• Reason Want workers prody to be similar within
  each tier
• If not, some give up.

1 2 3 4
30
Tournaments vs. Piece Rates

• If some of the variability in workers output or
  in supervisors perceptions of worker output, is
  correlated across workers, then tournaments may
  be better than piece rates
• Workers tend to be risk averse.
• Dont want pay varying based on what they cant
  control
• Relative pay helps fix this.
• Tournaments remove link between average worker
  pay and external shocks
• Like insurance for workers.
• Example CEOs paid based on performance relative
  to industry
• Removes shocks to industry

31
Tournaments vs. Piece Rates, contd

• If hard to measure absolute output but easy to
  rank workers, then
• Use tournaments.

32
Tournaments vs. Seniority Pay

• Usually in real world
• Promotions and raises go hand in hand.

Wt
t
33
Problems with Tournaments

• Collusion among workers.
• Grading on a curve
• Ostracize workers who work too hard
• Competition between workers
• Sabotage
• Pre-meds
• Mac vs. Non-Mac team at Apple
• Enron
• Solution for collusion
• Increase wage spread and number of workers
  competing for promotion
• Harder to collude with larger number

34
Should Firm Promote Workers From Within?

• Reasons to promote from within
• Firm-specific human capital
• Opening up competition to outsiders reduces
  chance of being promoted, reducing effort
• Intuition The more workers, the greater the
  chance someone gets lucky and my effort doesnt
  matter.
• External competition reduces incentive to work
  because g(0) down as tournament becomes riskier
• Example Angels vs. Dodgers, farm team.
• Solution Increase W1-W2
• But risk averse workers prefer less competition
  and smaller wage hike
• Reasons to promote from outside
• If outside candidates better (or undervalued)
• To reduce collusion

35
Tournaments Recap

• Noise or luck has adverse effect on effort
• Too many workers competing for too few slots will
  lead to low effort (High luck factor)
• Given extreme heterogeneity of ability, need
  large number of levels
• Raises should be larger for high-level promotions
  than for low-level promotions because option
  value vanishes for final promotion
• Tournaments allow pay to depend on relative
  rather than absolute comparisons
• Sometimes relative performance easier to measure
• Pay based on relative performance reduces the
  effect of common shocks like adverse business
  conditions and offers valued insurance for
  workers, causing them to be willing to work for
  less
• One problem with tournaments May induce workers
  to collude
• Another problem Tournaments may induce harmful
  competition between workers
• Hiring from outside is an option if workers
  collude, but reduces effort by eliminating
  promotion incentive