# Appendix A: Technical Typology

## Health Care Efficiency Measures: Identification, Categorization, and Evaluation

### Intellectual History of Efficiency in Economics

In the first half of the 20th century, microeconomic theory approached the efficiency concept from a Pareto perspective. The Pareto criterion is satisfied if no person can be made better off without making someone else worse off. The classic first welfare theorem holds that Pareto efficiency obtains if and only if:

- Markets exist for all possible goods.
- Markets are perfectly competitive.
- Transaction costs are negligible.
- There are no externalities.

The implicit assumption was that firms always make optimal decisions on the use of inputs, and that any inefficiencies in an economy have their origin in the way resources are allocated across firms, rather than within firms. Two main threats to efficiency in this paradigm were monopolies and (international) trade restrictions.^{1}

In the second half of the 20th century, the assumption that firms always make optimal input decisions was challenged. It became accepted that besides the original "social" or "allocative" efficiency, the efficiency within firms was worthy to be analyzed as well. This had traditionally been an operations research (OR) field, concerned with "activity analysis," where the manager was the subject of interest; hence the term "managerial efficiency."

During the 50's, several scholars^{2-4} tried to formalize both types of efficiency. These are sometimes referred to as the neo-Walrasian school. Within the neo-Walrasian school the seminal paper on the measurement of efficiency is Farrell.^{4} Farrell's definition of productive efficiency was inspired by Koopmans' work on "activity analysis,"^{3} and his measure of technical efficiency is similar to Debreu's "coefficient of resource utilization."^{2} The novelty of Farrell's approach is that his efficiency measure explicitly allows the inclusion of multiple inputs and outputs, whereas previous work (e.g., index numbers) was often limited to single inputs or outputs (e.g., the average productivity of labor).

Farrell's definition of the efficient firm is "its success in producing as large as possible an output from a given set of inputs." Farrell introduces the efficient production function as a special case of the traditional (Paretian) production function, defined as "the output that a perfectly efficient firm could obtain from any given combination of inputs."^{4}

Farrell distinguishes between technical-, price-, and overall efficiency. Technical efficiency is defined as a firm's success in producing maximum output from a given set of inputs, i.e., producing on the "technical frontier." Price efficiency is defined as the firm's success in choosing an optimal set of inputs, i.e., the set that would minimize cost if the firm were producing on the technical frontier. Overall efficiency (commonly known as productive efficiency) is the product of price and technical efficiency. Technical and price inefficiency each imply overall inefficiency (as Farrell defines the term).

Many economists define technical efficiency like Farrell but define productive efficiency as minimizing costs, i.e., subsuming technical efficiency. Under this approach technical inefficiency implies productive inefficiency, which in turn implies Pareto inefficiency.

Figure 1 shows the classic framework by Farrell which makes it possible to decompose overall efficiency into technical and allocative (price) efficiency. Consider the case of a simple output (Y) that is produced by using two inputs (X1, X2). Under the assumption that the production function Y=f(X1, X2) is linearly homogeneous, the efficient unit isoquant, Y=1, shows all technically efficient combinations. In Figure 1, P represents a firm, country, individual, etc., that also produces at Y=1, but uses higher levels of inputs, and is therefore less efficient in a technical sense. The magnitude of the efficiency can be expressed as the ratio between optimal and actual resource use (OR/OP). By taking into account the iso cost line (representing relative factor prices), we can identify allocative efficiency. Any point on the line Y=1 has technical efficiency, but only Q receives technical efficiency at minimum cost. Allocative (price) efficiency can be expressed as the ratio between minimum and actual cost (OS/OR), and overall efficiency is the product of technical and allocative efficiency.

#### Figure 1: Technical, allocative, and overall efficiency

Leibenstein^{1} makes a similar distinction, albeit less formal than Farrell, and proposes the term X-efficiency, which is essentially the same as Farrell's technical efficiency. Aigner and Chu^{5} show that from an empirical perspective (in)efficiency can be modeled through either linear or quadratic programming, and that Farrell's original assumptions on returns to scale for the industry production function are then no longer necessary.

Starting in the 70's the first empirical papers appear that estimate technical efficiency within a regression framework or using Data Envelopment Analysis (DEA).

Efficiency, particularly technical efficiency, is most commonly associated with measurements taken at a single point in time. Changes over time in the technical frontier are usually studied within the framework of productivity, which in its modern form has its origin in the 50's as well.

**References**

1. Leibenstein H. Allocative efficiency vs "X-efficiency." *The American Economic Review* 1966;53(3):392-415.

2. Debreu G. The coefficient of resource utilization. *Econometrica* 1951;19(3):273-92.

3. Koopmans TC. Efficient allocation of resources. *Econometrica* 1951;19(4):455-65.

4. Farrell MJ. The measurment of productive efficiency. *Journal of the Royal Statistical Society Series* A 1957;120(3):253-90.

5. Aigner DJ, Chu SF. On estimating the Industry Production Function. *The American Economic Review* 1968;58(4):826-39.