The Measurement of Nonmarket Sector Outputs and Inputs Using Cost Weights 2008 World Congress on National Accounts and Economic Performance Measures for Nations, May by W. Erwin Diewert, Department of Economics, University of British Columbia

Introduction Can we obtain meaningful measures of Total Factor Productivity growth in the nonmarket sector, where real output growth is generally measured as real input growth? Answer: Yes, under two sets of conditions: (1)We can measure the output of an activity, the corresponding inputs and input prices or (2)We can measure a vector of outputs, the corresponding inputs and input prices and we are willing to undertake an econometric exercise. See the Appendix for an elaboration of this approach.

The simple cost based output prices model The model: (1)x i t = a i t y i t ; i = 1,2 (two activities) ; t = 0,1. x i t is a vector of inputs used by activity i in period t a i t is a vector of input output coefficients that describe the constant returns to scale, no substitution technology for activity i in period t; note that a i 0  a i 1. y i t is the (scalar) amount of output that is produced by the establishment in period t. Note that given x i t and y i t, we can determine a i t. We have no output price so we will use unit cost as the price of output in each period.

(2) C i t  w i t  x i t   n=1 N(i) w in t x in t ; i = 1,2 ; t = 0,1 C i t is the total cost in period t of producing y i t w i t is the vector of period t input prices The above framework allows for technical progress going from period 0 to 1; i.e., a i 0  a i 1. (3) w i 1  a i 1  w i 1  a i 0 ; i = 1,2; No technological regress (4) a i 0 = a i 1 ; i = 1,2; No technical change (5) w i 1  a i 1 < w i 1  a i 0 ; i = 1,2; Technical progress

Define the Laspeyres industry input quantity index Q L * as follows: (6) Q L *  [w 1 0  x w 2 0  x 2 1 ]/[w 1 0  x w 2 0  x 2 0 ] = [w 1 0  a 1 1 y w 2 0  a 2 1 y 2 1 ]/[w 1 0  a 1 0 y w 2 0  a 2 0 y 2 0 ] Similarly, the Paasche and Fisher industry input quantity indexes, Q P * and Q F *, can be defined as follows: (7) Q P *  [w 1 1  x w 2 1  x 2 1 ]/[w 1 1  x w 2 1  x 2 0 ] = [w 1 1  a 1 1 y w 2 1  a 2 1 y 2 1 ]/[w 1 1  a 1 0 y w 2 1  a 2 0 y 2 0 ] (8) Q F *  [Q L * Q P * ] 1/2.

Constructing industry input indexes is easy. But how are we to construct output indexes? The “best” approach would be to estimate user prices; i.e., how do users of the outputs value these nonmarket outputs? Unfortunately, such final demand prices are either not available or are not reproducible: so what to do? SNA 1993 recommends valuing publicly provided services at their costs of production. We follow this advice. (See also the Atkinson Review 2005). Thus in our present simple model, we set the price of each activity’s output, p i t, equal to average cost. In our more general Appendix model, we use econometrically estimated marginal costs.

(9) p i t  w i t  a i t ; i = 1,2 ; t = 0,1. Using the above cost based procedure i output prices p i t along with the corresponding period t output quantities y i t, we can readily define the Laspeyres, Paasche and Fisher output quantity indexes, Q L, Q P and Q F respectively: (10) Q L  [p 1 0 y p 2 0 y 2 1 ]/[p 1 0 y p 2 0 y 2 0 ] = [w 1 0  a 1 0 y w 2 0  a 2 0 y 2 1 ]/[w 1 0  a 1 0 y w 2 0  a 2 0 y 2 0 ] (11) Q P  [p 1 1 y p 2 1 y 2 1 ]/[p 1 1 y p 2 1 y 2 0 ] = [w 1 1  a 1 1 y w 2 1  a 2 1 y 2 1 ]/[w 1 1  a 1 1 y w 2 1  a 2 1 y 2 0 ] (12) Q F  [Q L Q P ] 1/2.

Proposition 1: If there is no technical progress going from period 0 to 1 in each procedure, then the Laspeyres, Paasche and Fisher output indexes are exactly equal to the corresponding Laspeyres, Paasche and Fisher input indexes; i.e., we have: (13) Q L = Q L * ; (14) Q P = Q P * ; (15) Q F = Q F *. The above Proposition justifies current SNA practice, which frequently estimates nonmarket output growth by the corresponding input growth. But what if there is technical progress?

Proposition 2: If there is technical progress going from period 0 to 1 in either activity or procedure, then the Laspeyres, Paasche and Fisher output indexes are strictly greater than the corresponding Laspeyres, Paasche and Fisher input indexes; i.e., we have: (16) Q L > Q L * ; (17) Q P > Q P * ; (18) Q F > Q F *. Thus if there is technical progress in either procedure, then the above Proposition tells us that there will be Laspeyres, Paasche or Fisher productivity growth in the industry if we use the same index number formula (Laspeyres, Paasche or Fisher) for measuring both input and output growth.

The Introduction of a New More Efficient Technology The methodology developed in the previous section does not give reasonable results when, for example, a new medical technology is developed that is more efficient than an existing technology but the new technology does not immediately displace the old technology.

Procedure 1 is a well established “incumbent” procedure which has input-output coefficient vectors a 1 0 and a 1 1 in periods 0 and 1. Procedure 2 is a “new” improved procedure that accomplishes exactly what procedure 1 accomplishes but it is only available in period 1. The vector of input-output coefficients for procedure 2 in period 1 is a 2 1. We assume that one unit of the new procedure has a lower unit cost than one unit of the old procedure in period 1 so that (19) w 2 1  a 2 1 < w 1 1  a 1 1.

We assume that the incumbent technology does not suffer from technological regress; i.e., we assume that (3) w 1 1  a 1 1  w 1 1  a 1 0. We can use the period 1 input-output coefficients for the new technology as imputed input-output coefficients for period 0 but of course, the corresponding period 0 output and input for the new procedure, y 2 0 and x 2 0, should be set equal to zero; i.e., we make the following assumptions: (20) a 2 0  a 2 1 ; y 2 0  0 ; x 2 0  0.

We assume that the new procedure is also more efficient than the old procedure using the input prices of period 0; i.e., we assume that: (21) w 2 0  a 2 0 < w 1 0  a 1 0. We now get to the point of all this algebra. The “correct” output growth index should be the following unit value quantity index, Q UV : (22) Q UV  (y y 2 1 )/y 1 0. Thus output in period 1 is the sum of the two procedure outputs, y y 2 1, and this is divided by the procedure 1 output in period 0, y 1 0, to obtain the quantity output index.

But if we compute the Laspeyres output quantity index Q L defined by (10) under our current assumptions, we find that: (23) Q L  [p 1 0 y p 2 0 y 2 1 ]/[p 1 0 y p 2 0 y 2 0 ] = [w 1 0  a 1 0 y w 2 0  a 2 0 y 2 1 ]/[w 1 0  a 1 0 y w 2 0  a 2 0 y 2 0 ] = [w 1 0  a 1 0 y w 2 0  a 2 0 y 2 1 ]/w 1 0  a 1 0 y 1 0 using y 2 0 = 0 = [y 1 1 /y 1 0 ] + [w 2 0  a 2 0 /w 1 0  a 1 0 ][y 2 1 /y 1 0 ] < [y 1 1 /y 1 0 ] + [y 2 1 /y 1 0 ] = Q UV using definition (22). Thus treating the improved procedure as a totally new procedure that produces a different output and using the methodology in the previous section leads to a Laspeyres rate of growth that is below the “true” output growth, Q UV.

Similarly, if we compute the Paasche output quantity index Q P defined by (11) under our current assumptions, we find that: (24) Q P < Q UV and of course, (23) and (24) imply that (25) Q F < Q UV. Thus under the assumptions of this section, the methodology developed in the previous section will lead to estimates of output growth and productivity growth that are biased downwards.

The bias that results from the incorrect measurement of the effects of the introduction of a new and more efficient procedure is similar in many respects to new outlets bias; i.e., a new output is linked into an index in such a way that the index shows no change when in fact, a change should be recorded. The methodology for dealing with a new procedure developed above is relatively straightforward and is reproducible and objective under the assumptions of the model. However, in the real world, a new procedure is unlikely to have an output that is exactly equivalent to the output of an existing procedure and hence in real life, it will not be so straightforward to deal with new medical procedures.

Conclusion The valuation of outputs produced by the nonmarket sector is a complicated task. A first best solution would be to have unambiguous, objective and reproducible final demand prices but this solution is generally not available to statistical agencies. A second best solution is to value outputs at their average costs as is recommended in SNA We have developed the algebra associated with this second approach and found (not surprisingly) that it is possible to have Total Factor Productivity growth using this methodology, provided that sufficient information is available.