Lecture 17
Gaynor and Pauly - JPE 1990
© Allen C. Goodman, 2012
Effects of Compensation
Arrangements - Review
- Productivity-based arrangements (e.g. piece
rates) are best when production is non-joint
across agents.
- Jointness calls for some kind of revenue
sharing, and others.
Formal Model
They want to look at "effort" that is exerted by partners. Effort
is defined as a variable input supplied by an individual
partner that determines the efficiency of production.
Production function (1):
qi = quantity produced by partner i (= office visits per week)
hi = hours
ti = nonpartner hours used
ki = capital services
ei = effort
i = individual characteristics
Well-behaved production relationship (1).
qi = f(hi, ti, ki, ei, i)
(1)
Partner wants to maximize utility with function:
ui = yi - vi (ei, hi)
(2)
ui = utility, yi = net income, vi = nonmonetary costs of effort
and hours.
qi= f(hi, ti, ki, ei, i)
Compensation Structure
Compensation structure determines yi
for each partner as (3):
yi = a (P - C)qi + (1/n)(1 - a)(P - C)  qi. (3)
P = price, C = average cost.
First term is portion a of net income
generated by i that she keeps, and
second term is her share from net
operating pool.
Optimizing (2), and using (3), we get (4):
ui = yi - vi (ei, hi)
(2)
[a + (1/n)(1 - a)](P - C) fe - ve = 0.
(4)
First term = marginal net income product
of effort
Second term = marginal utility cost.
[a + (1/n)(1 - a)](P - C) fe
$/ei
vi ()
ei
a decreases
e*i
Effort, ei
Compensation makes a difference
Clearly, in this case the compensation structure makes a
difference  behavioral production function. (Toggle back
to shift yellow curve). It shifts down as a .
In contrast (1) is a "technical" production function.
Formulation (1') [with technical changes] should work
better. We'll compare them.
They then want to look at a measure of efficiency. Define:
qieff = f(hi, ti, ki, eimax,  i), leading to
Ri= qi/qieff= f (hi, ti, ki, ei,  i)/f(hi, ti, ki, eimax, i)
(related
to 6)
Ri < 1 doesn't necessarily mean inefficiency in a welfare sense,
although it is inefficient in a technical sense. Why?
Why?
Since deviations of effort from the maximum are due in
part to the agents' taste for effort, the technical
efficiency measure captures these preferences.
In fact, the distribution of the Ri over the agents reflects
the distribution of the agents' tastes for effort (jointly
with technical inefficiency). Consequently, we can
think of this measure as measuring agents'
preferences for effort, all else equal.
Those with a greater preference for effort will be
measured as more technically efficient and vice versa.
Big leap here is the notion that all inefficiency is
chosen. (I'm not sure about this.)
They also point out that the incentives may be
endogenous. After some discussion, they note that
multi-stage estimates must be used.
White line is best feasible.
But we can’t use plain old OLS
Output
We’ve seen
this before.
Inputs
… and this!
Frontier Methods – saw these before
Efficient is efficient
We want to identify the
difference between “bad
luck” and inefficiency.
qi = f(Xi) ui vi, (6')
vi = random shock, and ui =
multiplicative efficiency
term.
ui = qi/f(Xi)vi ui  [0, 1].
Equation (7):
qi =  Xji bj + ej, then:
(8) ei = ui + vi,
where ui are one-sided,
non-negative, and vi ~ N
(0, 2v).
$
Normal vi
Truncated
normal ui
Disturbance
Efficiency
Q
What they estimate
Allows 0
values for
To estimate this they use function (10) - write out: the Y’s.
qi  A ( X ji  j ) exp[  kYki   k (Yki ) 2 ] exp( i ).
k
j
k
which yields:
qi
exp( ui ) 
A ( X ji  j ) exp[   k Yki    k (Yki ) 2 ] exp( vi ).
j
k
k
that is, the ratio of the observed output to the stochastic
Estimation
Estimate “traditional” and “behavioral” cost functions.
“Behavioral“ should work better because it includes
behavioral relationship between inputs and outputs
as determined by internal organization of the firm.
They end up looking pretty similar, since the behavioral
parameters are related to the “traditional” variables.
Variables
qi = office visits per week
hi = physician hours
capital = examining rooms per physician
Incentives
Two types of variables for incentives
average price, wage rate, number of full-time physicians in the
group.
 comes from a 1-10 scale, where a value of 1 indicates that
the physician regarded his compensation as completely
unrelated to productivity, and 10 indicated a perfect
relationship.
I believe 10, and I believe that 10 implies more control than 1,
but I don't buy that this is the best .
Also come up with a joint incentive variable as:
[ + (1/n )(1 - )] (P - C).
Estimates are on Table 2. Basically, the frontier methods give
similar parameters as do regular two stage methods.
Findings
As  increases, you get more office visits per week. As
 goes from 1 to 10, output increases by 28 percent.
Consistent with observation of output without ability
to observe effort.
Average level of technical efficiency is approx. 0.66.
Comes out about the same with traditional or
behavioral model.
Re behavioral v. traditional:
For behavioral measures, we get increased marginal
products and elasticities of scale as compensation
scale increases, in contrast, presumably, to the
traditional.
Last words
Incentives affect quantity produced but not measured
technical efficiency, because increased incentives
call forth a greater supply of effort from all agents
but do not change tastes for effort.
Consistent with theoretical work that predicts that
productivity-based compensation schemes will work
well for firms with non-joint production and
observable output.
Last words
Noelle Molinari did a dissertation here a couple of
years ago looking at compensation schemes.
She tested the hypothesis that medical partnerships
can effectively respond to moral hazard by
monitoring their members. A partnership model was
developed to describe the decision-making process
in medical groups.
Empirical analysis was performed using data collected
by the Medical Group Management Association.
Research was funded by Agency for Healthcare
Research and Quality (AHRQ) at approx. $30,000.