11 to allow for quality and not just focus on „outputs‟ such as enrollments or cases treated. We can allow for quality by graphing average cost per person, _C_ / _y_ , against quality, _q_ . For example, _y_ might be students enrolled, _C_ / _y_ cost per student, and _q_ the average test score.<sup>12</sup> In the first stage of our analysis, we identify a group of efficient service delivery units (or, in the case of panel data, efficient service delivery units _at a point in time_ ), defined as those that have the smallest (total or average) expenditure for each level of output, or the smallest expenditure per student or patient treated for each level of quality. Because there will be relatively few units that have _exactly_ the same output (or quality), we work with output (or quality) _ranges_ . We define a caliper of size _c_ , and move the caliper along the _y_ (or _q_ ) axis in steps of size _s_ ≤ _c_ . In this case where _y_ (or _q_ ) is a scalar, the caliper is a line of length _c_ which gets moved up the _y_ (or _q_ ) axis up to the maximum value of the outcome in steps of _s_ . In each step, the unit with the smallest expenditure within the caliper is identified and labeled an „efficient unit‟. Next we create a (stochastic) frontier by running a nonparametric Lowess smoother through the datapoints of these efficient units, and defining the frontier as the predicted cost for each efficient unit. The grid-search process thus identifies efficient units, and the smoothing process produces the frontier, with all efficient units being moved to the frontier.

In the second stage, we compute the inefficiency of inefficient delivery units by matching each unit off the frontier with the closest unit on the frontier in terms on the outcome _y_ ; the unit‟s inefficiency is the difference between its expenditure and the expenditure of the closest match on the frontier.<sup>13</sup> In contrast to both DEA and SFA where units are compared with hypothetical units, the matched unit for each inefficient unit in our approach is a real-life service delivery unit, not a hypothetical point on the frontier. We see this as a strength of our approach; all are units that have actually managed to produce (close to) output _y_ at a cost _C_ , not ones that ought to have been capable of doing so.

Two points are worth clarifying at this stage. First, the first stage may leave some inefficient units below the smoothed frontier. These are units that emerge with somewhat higher

> 12 For such a graph to be justified, the underlying two-product cost function would have to have the form _C_ ( _y_ , _q_ ) = _y_ ∙ _c_ ( _q_ ), giving _C_ ( _y_ , _q_ )/ _y_ = _c_ ( _q_ ). Crampes and Hollander (1995) use such a cost function, but do not explore its properties. For the most part, the properties are fairly innocuous. The extent of ray economies (the effect on cost of doubling both _y_ and _q_ ) depends on the shape of _c_ ( _q_ ), with _c_ '( _q_ )>0 implying ray diseconomies and _c_ '( _q_ )<0 implying ray diseconomies. There are economies of scale with respect to quality if _c_ '( _q_ )< _c_ ( _q_ ). The only odd feature of the cost function is that it implies that average incremental costs for quantity always exceeds the marginal cost for quantity.

> 13 We use the Stata module PSMATCH2 (Leuven and Sianesi 2003) to do the matching.