Promotion as Tournament


The tournament model

Suppose there are two workers, Joe and Katie. Let

qj = mj + ej
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
qk = mk + ek
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 Wh while the second gets Wl. One can think of Wh - Wl 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 mj to maximize
P Wh + (1-P) Wl - C(mj)
The first-order condition is
(Wh - Wl)∂ P/∂ mj - C'(m1) = 0
Now, Joe wins the tournament if and only if qj > qk. This can be written as mj + ej > mk + ek. Let η = ek - ej, 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[ mj - mk > η ] = G(mj - mk)
∂ P/∂ mj = g(mj - mk)
If Joe and Katie are identical, then mj = mk. This allows us to simplify the first-order condition into
(Wh - Wl) g(0) - C'(mj) = 0
From this, we can conclude

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
(Wh - Wl)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* = Wh + Wl
These two equations allow us to solve for the equilibrium prize structure. In particular, we have
Wh = p m* + p/(2g(0))
Wl = p m* - p/(2g(0))
Several observations follow:

Absolute versus relative performance

Empirical evidence

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

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