2.3  Selecting an index number formula: The economic and axiomatic (test) approaches

As discussed in the previous subsection, there are numerous index formulae that can be used to form aggregate productivity measures. This raises the question: are there any criteria that can be used to decide on the choice of index formula? The index number literature offers two main approaches: the economic approach and the axiomatic (test) approach.[3]

The economic approach bases the choice of index number formula on a producer’s underlying production technology (that is, the production, cost, revenue or profit function). This approach assumes competitive optimising behaviour by producers. In other words producers are assumed to maximise profit (minimise costs) for a given production technology.[4]

Consider the following production technology:

(6)    

where is the level of multifactor productivity, aggregate physical capital services, and the aggregate labour input (in this case the total number of hours worked).

An index expressed in terms of the above technology is known as a theoretic index. In continuous time the theoretic index is a Divisia index.[5] As data are available in discrete time, rather than as continuous functions of time, it is necessary to use an index formula to approximate the Divisia index.[6]

A particular index is defined as an exact index when it corresponds directly to the theoretic index derived from the production technology (Diewert, 1976). For example, if production technology takes the translog functional form the Törnqvist quantity index is the corresponding index for the underlying production technology. Thus, the Törnqvist index is “exact” for translog production technology.

When an exact index corresponds to a production technology that has a flexible functional form, a functional form that is able to approximate a range of other functional forms, the index is defined as a superlative index (Diewert, 1976).[7] The Törnqvist index is a superlative index because the translog functional form can approximate a range of other functional forms. A superlative index must also be an exact index, however it is possible for an index to be an exact index but not a superlative index.

The axiomatic approach bases the choice of index number formula on properties that an index should exhibit, with these properties being embodied in axioms. One of the appealing features of this approach is that it does not make any assumptions about competitive optimising behaviour. The following four axiomatic tests are often used (see, for example, Diewert and Lawrence, 1999): the constant quantities test; the constant basket test; the proportionality test; and the time reversal test.[8]

The constant quantities test states that if quantities are identical in two periods, then the quantity index should be the same regardless of what prices are in both periods. The constant basket test states that if prices remain unchanged between two periods then the ratio of the quantity indexes between the two periods should be equal to the ratio of values between the two periods. The proportionality test requires that when all quantities increase or decrease by a fixed proportion between two periods, then the index should increase or decrease by the same fixed proportion. The time reversal test requires the index going from period 0 to period 1 to be the inverse of the index going from period 1 to period 0. In other words, if prices and quantities in period and are interchanged, the resulting index should be the inverse of the original index. The Fisher index passes all four of the above tests. The Törnqvist index does not pass the constant basket test, while the Laspeyres and Paasche indexes fail the time reversal test. These results are summarised in Table 1.

In practice it is common to use both the economic and axiomatic approaches when choosing an index number formula, while also bearing in mind the end use of the index number series. Data availability will also influence the choice of index formula. Using both the economic and axiomatic approaches a strong case can be made in favour of using the Fisher index.[9]

2.4  Numerical example using different index number formulae[10]

This subsection illustrates the use of the index formulae in equations (2) to (5) to form productivity measures using hypothetical price and quantity data.

Consider a situation in which an economy produces two outputs, and , using two inputs, and , where both output prices ( and ) and input prices ( and ) are exogenously determined. Furthermore, suppose information on the prices and quantities of outputs and inputs is available for three periods . This information is presented in Table 2.

Table 2 – Prices and quantities of outputs and inputs

	Prices and quantities of outputs				Prices and quantities of inputs
	3	6	3	5	2	6	3	7
	3	7	4	6	3	5	3	10
	4	8	6	8	6	4	4	14

To construct output quantity indexes using the formulae (2) to (5) we first construct nominal output shares. Nominal output shares are calculated by dividing nominal output (revenue) for each good by total nominal output for all goods. The nominal output shares for goods Y and Z are displayed in columns (1) and (2) in panel I of Table 3.

It is also necessary to calculate the ratio of period quantity to period quantity for both goods. This quantity ratio is known as a quantity relative. Quantity relatives for output Y and Z are presented in columns (3) and (4) in panel I of Table 3.

The Laspeyres output quantity index is constructed by first multiplying the nominal output shares in period by the corresponding quantity ratios in each period (that is, the first entries in columns (1) and (2) are multiplied by the corresponding quantity ratios in columns (3) and (4), to yield the share-weighted quantity relatives contained in columns (5) and (6)). The share-weighted quantity ratios are then added together for each period to yield the Laspeyres quantity index (that is, values in columns (5) and (6) are added for each period to yield the values in column (7)).

The Paasche output quantity index is constructed by multiplying the nominal output shares for each period by the corresponding reciprocal of the quantity ratios (that is, nominal output shares in columns (1) and (2) and multiplied by the reciprocal of the corresponding quantity ratios in columns (3) and (4)). Finally, the Paasche output quantity index is obtained by adding the share-weighted reciprocal of the quantity ratios and then taking the reciprocal (that is, adding the values in columns (8) and (9) and then taking the reciprocal to yield the values in column (10)).

The Fisher output quantity index is found by taking the geometric average of the Laspeyres and Paasche output quantity index. The Fisher output quantity index is shown in column (11) of Table 3.

Table 3 – Laspeyres, Paasche, Fisher and Törnqvist output quantity index

Panel I					Panel II						Panel III
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
	0.545	0.455	1.000	1.000	0.545	0.455	1.000	0.545	0.455	1.000	1.000	1.000	1.000	1.000
	0.467	0.533	1.167	1.200	0.636	0.545	1.182	0.400	0.444	1.184	1.183	1.081	1.094	1.183
	0.400	0.600	1.333	1.600	0.727	0.727	1.455	0.300	0.375	1.481	1.468	1.468	1.281	1.468

The first step in calculating the Törnqvist output quantity index is to take the quantity ratios to the power of the arithmetic average of the nominal output shares in period and period . The resulting exponential weighted quantity ratios are reported in columns (12) and (13) of Table 3. The final step in calculating the Törnqvist output quantity index is found by taking the product of the exponential weighted quantity ratios (that is, multiplying column (12) by column (13) for each period to yield the values in column (14).

Input quantity indices can be constructed in a similar manner using the input quantity and price data displayed in Table 1. Productivity indices are found by taking the ratio of the output quantity index to the respective input quantity index. Both the input quantity and productivity indices are reported in Table 4.

Table 4 – Input and productivity indices

	Input indices				Productivity indices
	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
	1.212	1.154	1.183	1.184	0.975	1.026	1.000	0.999
	1.515	1.250	1.376	1.389	0.960	1.185	1.067	1.057

The multifactor productivity indices displayed in Table 4 show marked differences in productivity growth for the hypothetical economy depending on which index number formulae is used. For example, the Laspeyres productivity index () suggests multifactor productivity decreased by 4% between period and period 2, while the Paasche multifactor productivity index () suggests productivity increased by 18.5%. The increase in the Fisher productivity index () lies between the increases in the Laspeyres () and Paasche () productivity indices (as it is their geometric mean). The Törnqvist productivity index () shows a similar increase to the Fisher productivity index, as the Törnqvist index usually approximates the Fisher index quite closely.