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): q i = quantity produced by partner i (= office visits per week) h i = hours t i = nonpartner hours used k i = capital services e i = effort i = individual characteristics Well-behaved production relationship (1). q i = f(h i, t i, k i, e i, i )(1) Partner wants to maximize utility with function: u i = y i - v i (e i, h i )(2) u i = utility, y i = net income, v i = nonmonetary costs of effort and hours.
Compensation Structure Compensation structure determines y i for each partner as (3): y i = a (P - C)q i + (1/n)(1 - a)(P - C) q i. (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): u i = y i - v i (e i, h i )(2) [a + (1/n)(1 - a)](P - C) f e - v e = 0. (4) First term = marginal net income product of effort Second term = marginal utility cost. Effort, e i $/e i [a + (1/n)(1 - a)](P - C) f e e* i q i = f(h i, t i, k i, e i, i ) a decreases
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: q i eff = f(h i, t i, k i, e i max, i ), leading to R i = q i /q i eff = f (h i, t i, k i, e i, i )/f(h i, t i, k i, e i max, i ) (related to 6) R i < 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 R i 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.
Inputs Output White line is best feasible. But we can’t use plain old OLS
Frontier Methods – saw these before Efficient is efficient We want to identify the difference between “bad luck” and inefficiency. q i = f(X i ) u i v i, (6') v i = random shock, and u i = multiplicative efficiency term. u i = q i /f(X i )v i u i [0, 1]. Equation (7): q i = X ji b j + e j, then: (8)e i = u i + v i, where u i are one-sided, non- negative, and v i ~ N (0, 2 v ). Normal v i $ Disturbance Truncated normal u i Efficiency Q
What they estimate To estimate this they use function (10) - write out: which yields: that is, the ratio of the observed output to the stochastic frontier. Allows 0 values for the Y’s.
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 q i = office visits per week h i = 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.