Technical efficiency is achieved when a producer is able to select the prices of inputs and outputs in such a way as to make a zero profit as all other producers make a zero profit or a loss.  Taking the frontier and all points behind the frontier gives a production possibilities set – the union of the technically efficient DMUs and the technically inefficient DMUs.

Farrell’s “Diagram 1” is reproduced as Figure 1, with the modification that the axes representing the quantities of the primary factors have been labeled as x₁ and x₂ to conform with the notation used in my dissertation.

Primary factors are the raw materials: inputs to the production process.

Figure 1 – Farrell's Diagram 1.

SS’ is an isoquant representing combinations of the primary factors x₁ and x₂ required to produce a single output y₁ at a given level, q, with 100 percent technical efficiency.

AA’ is a market price line whose slope is the ratio of the market prices of the primary factors. Production of quantity, q, of the output at Q’ (where AA’ is tangent to SS’ ) is both technically efficient and price efficient. Q’ represents the combination of the primary factors which results in the production of quantity, q, at the lowest cost.

A producer operating at P is both technically inefficient and price inefficient at the market prices implicit in AA’.  P can improve its technical efficiency by reducing its usage of either or both of the primary factors.  Assuming proportionate reductions in each primary factor, P would operate technically efficiently at Q.   However to be as cost efficient as Q’ at the market prices implied by AA’ would require P to be operating at R if it were possible for P to further proportionately reduce its use of the primary factors.

Alternatively, P, operating at Q would be price efficient if AA’ were tangent to SS’ at Q.  If a producer on SS’ is allowed to select its own “market prices” such that AA’ is tangent to SS’ where the producer sits on SS’ then the producer will be both 100 percent Technically and 100 percent Price Efficient.  Put in another way, if a producer is allowed to choose its own market prices and it can achieve 100 percent Price Efficiency at those prices, then it is 100 percent Technically Efficient.

Farrell takes the ratio OR/OQ as the measure of Price Efficiency of P and OQ/OP as the measure of Technical Efficiency of P.  The Overall Efficiency of P is the Price Efficiency * Technical Efficiency = OR/OQ * OQ/OP = OR/OP.  Both measures are known as “radial” measures since they are derived from the ratio of the lengths of radii – lines from the geometric origin.

Reduction from P’s usage to Q’s usage involves reductions in x₁ and x₂ proportional to their actual usage, since PQO is a straight line.  This quality is known as “equi-proportional reduction”.

Actual examples of the production of output y₁ from primary factors x₁ and x₂ would be plotted as a series of points, rather than a continuous curve.  Drawing a set of lines between a subset of the points, such that no point was closer to the origin than the lines, would give an estimation of SS’.  Inefficient producers can then derive an estimated technical efficiency score by reference to the estimate of SS’.

Koopmans (1951) defined efficiency as arising when the sum of the Priced Outputs equals the sum of the Priced Inputs. The ratio is one for an efficient DMU. Using the notation described in the Sidebar and substituting Q for the subscript j we can express the efficiency of Q, an efficient DMU, as:

This expression of efficiency allows for up to s outputs: only one output is assumed in Farrell’s model.

Evaluation of P at the market prices for Q gives P an efficiency of:

The Technical Efficiency and Price Efficiency of P, OP /OQ , is equivalent to h_P /h_Q. So, if both P and Q are evaluated at the “best” prices for them: P will be judged inefficient in relation to Q to the extent OP /OQ.

In the discussion of Farrell's Model, P was evaluated at prices (AA’) that allowed DMU Q’ on SS’ to be both Technically and Price Efficient. Prices that would allow P to operate as efficiently as Q – if P managed to reduce input usage of x₁ and x₂ proportionately to be operating at the same point as Q – are given by BB’ the line tangent to SS’ at Q – See Figure 2. Q, the Technically Efficient point is also R, from Figure 1, the Price Efficient point at which P would operate: if it could.

Figure 2 – The Evaluation of Point P.

Each DMU, in turn, is evaluated against its own Isoquant, which is to say at prices that allow the sum of the priced inputs to be equal to 1. The DMU being evaluated then compares itself to the other DMUs at this Isoquant.

If we let the sum of the Priced Outputs for any given DMU, 0, be equal to 1:

, then DMU 0 is efficient if:

but DMU, 0, is inefficient if : .

For any given DMU, 0, minimizing , while at the same time ensuring that:

1. no other DMU can do better than DMU, 0, at the prices for DMU, 0, and
2. 

will yield the prices for DMU, 0, that allow it to achieve its highest level of efficiency relative to the other DMUs and will allow calculation of an efficiency score. This gives an algorithm that can be applied to the n DMUs as follows:

For 0 from 1 to n, solve:

The result is an efficiency score h₀ for each DMU and a set of input prices and a set of output prices for each DMU.

Consider P and Q in this framework. There are two DMUs: so n = 2. For Q one first evaluates Q against P at Q’s prices:

Expanding the constraints as (1Q) and (1P) make it explicit that Q is being evaluated against P at Q’s prices.

Then one evaluates P against Q at P’s prices: