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Examples

• promotion from being an associate in a law firm to the partner of the firm brings about a substantial pay raise, but the productivity of the lawyer does not increase overnight
• ditto for promotion from assistant professorship to associate professorship, though the pay raise is not substantial
• salesperson of the year award
• promotion from vice-president to president of the company--usually associated with substantial changes in job duties

The tournament model

Suppose there are two workers, Joe and Katie. Let

q_j = m_j + e_j where q is output, m is Joe's effort, and e is a luck component. The cost of effort is C = C(m). Similarly, let q_k = m_k + e_k The value of each unit of output is p. Assume workers are risk neutral.

Efficiency can be induced by paying workers a piece rate of p dollars per unit of output. Workers who choose the piece rate will choose the optimal effort level m* such that p = C'(m*).

But piece rate is not the only compensation system that can lead to efficiency. A suitably designed tournament can also produce the same results. In particular, suppose there is an implicit contest between Joe and Katie. If one worker produces more output than the other, the first worker gets W_h while the second gets W_l. One can think of W_h - W_l as the "prize" of winning. The prize of winning can take the form of a bonus or a promotion. Also note that we assume that the outcome of the contest depends only on the ranking of output but not on the actual output levels of the two workers--hence the name "rank-order tournaments."

Consider the choice problem for Joe. Let

P = probability that Joe wins Joe chooses m_j to maximize P W_h + (1-P) W_l - C(m_j) The first-order condition is (W_h - W_l)∂ P/∂ m_j - C'(m₁) = 0 Now, Joe wins the tournament if and only if q_j > q_k. This can be written as m_j + e_j > m_k + e_k. Let η = e_k - e_j, and let G() and g() represent the c.d.f. and p.d.f. of η. The probability that Joe wins the contest is P = Pr[ m_j - m_k > η ] = G(m_j - m_k) Hence, ∂ P/∂ m_j = g(m_j - m_k) If Joe and Katie are identical, then m_j = m_k. This allows us to simplify the first-order condition into (W_h - W_l) g(0) - C'(m_j) = 0 From this, we can conclude

• The higher is the prize spread B = W_h - W_l, the higher is the effort level. (It is the prize spread rather than the level of the prize that motivate workers.)
• Effort is higher the higher is g(0). The magnitude of g(0) depends on the variance of the luck component. If the luck component has a large variance, then a marginal increase in effort may only increase the chance of winning slightly (g(0) is small), and therefore the tournament may not be very effective in encouraging effort.
  • pure lottery
  • promotion in US firms versus Japanese firms
  • promotion in new industries versus established industries

Optimal wage structure

What determines the optimal prize structure? Let m(B) be the effort chosen by the workers when the prize spread is B. The value of the firm is

p m(B) - C(m(B)) The optimal B must satisfy (p - C'(m(B))) dm(B)/dB = 0 Since dm(B)/dB > 0, this equation implies that we must have p - C'(m(B)) = 0. Comparing this equation with the first-order condition for the workers' choice, we see that optimality can be achieved by setting (W_h - W_l)g(0) = p When the prizes are chosen in this way, we have m(B) = m*. Furthermore, the zero profit condition implies that the value of output must be equal to the value of wage payments. So 2 p m* = W_h + W_l These two equations allow us to solve for the equilibrium prize structure. In particular, we have W_h = p m* + p/(2g(0))
W_l = p m* - p/(2g(0)) Several observations follow:

• The optimal spread is B* = p/g(0). Thus, the optimal spread is larger if p is larger or g(0) is smaller.
• The outputs of the workers are not equal to their wages. Yet the equilibrium is fully efficient.
• The same outcome can also be achieved by giving a bonus to workers whose output exceed a certain fixed standard. However, firms would have an incentive to claim that the standard has not been met.

Absolute versus relative performance

• One advantage of the tournament over piece rates is that it requires only ordinal but not cardinal measure of output. In management level jobs, a cardinal measure of output is sometimes simply impossible.
• Another advantage of relative performance evaluation is that it eliminates that component of luck which is common to all contestants (e.g., weather, the state of the economy, overly generous or mean evaluators).
• One disadvantage of relative performance evaluation is that it could invite collusion among contestants.
• Alternatively, workers may also become too competitive among themselves.

Empirical evidence

• The same golfer achieve better scores on the same course when prize spread is greater.
• Absenteeism falls when firms give larger raises upon promotion.
• Do professors slack off once they obtain tenure?

Implications for organizational design

Consider the following organization chart of a firm:

CEO:                  o                   $350000
EVP:   o     o     o     o     o     o    $300000
VP:   o o   o o   o o   o o   o o   o o   $150000

What is wrong with this organizational structure?

Heterogeneous work force

• When worker ability is not symmetric, the effort level of both may fall. (Why?)
• If workers can change their effort during the contest, workers may slack off when they find that the distance between the contestants is too far.
• The firm may want to group workers in such a way that they may have similar chances of winning within their group.
• The firm may want to provide some reward at every level of promotion to prevent workers from giving up.
• External recruitment may dilute the incentives of internal contestants.
• Promotion is often associated with a different kind of tasks to be performed. To the extent that tournaments select a more able worker (instead of just a worker who puts in more effort), it performs the selection function as well.
• But workers may also be promoted to the level of their incompetence--the Peter principle.

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Lecture Notes