Herrick and Squires Measuring fishing fleet productivity 



87 



proportions with which all inputs are used remain con- 

 stant for any given level of resource stock abundance. 

 The productivity growth-accounting measure is fully 

 developed in the Appendix, while only its final form 

 is presented here: 



A _ Y •'^' ^ X, B 

 A Y h ' X, B' 



(2) 



where the dots over variables represent time deriva- 

 tives, S,(t) = P,(t)X,{t)IP{t)Y(t); i.e., the income or 

 cost share of input X, , where P,{t ) is the price of input 

 X, in time t, and P(t ) is the price of output Y in time 

 t. Productivity growth AIA is the residual of output 

 growth YIY after accounting for aggregate input 

 growth Z,S,(Z,/Z,) and changes in resource abun- 

 dance BIB. This index of total-factor productivity 

 growth is also called a Divisia index. 



Tornqvist economic index numbers 



A convenient discrete-time approximation to the con- 

 tinuous Divisia index is provided by Tornqvist (1936). 

 The Tornqvist discrete approximation to the Divisia 

 index of total-factor productivity growth (TFP) is 

 expressed as: 



TFPt 

 TFP, 



= n [i'y/i',,/-.]"-'^.'' 



■«,,,-i) 



^-1 



j=i 



A'+l 



n [x„/x.,_,r'2(s..s„, 



1 = 1 



(3) 



where the Yj are outputs, the .Y, are the A^ -i- 1 inputs, 

 B is the index of resource abundance, the R^ = P^YJ 

 "^jPjYj are output revenue shares, and the S, areN + 1 

 input cost shares. 6 As the interval between time- 

 periods approaches zero, discrete-time Equation (3) 

 approaches continuous-time Equation (2). 



«In practice, the logarithmic form of Equation (3) is computed, which 

 gives the productivity growth between two periods. The exponent 

 of this computation is then taken to obtain TFP, ,^ , . These period- 

 to-period changes are then typically chained as in Equation (4) to 

 form the Tornqvist bilateral chain index. The logarithmic form of 

 the Tornqvist index is: 



Chain indices 



We use the method of chain indices to calculate the 

 Tornqvist total-factor productivity index. Chain indices 

 directly compare adjacent observations in a sequence 

 of economic index numbers.' Nonadjacent observations 

 are only indirectly compared, using the intervening 

 observations as intermediaries, a practice resulting in 

 transitive comparisons. The general form of the chain 

 index can be written: 



rFP,,,,vh"" = TFP,.,.:  TFP,,,j,. 



TFP, 



t +N -l.l +N' 



(4) 



where each individual term of Equation (4), TFP, , ^ i , 

 is computed by the bilateral Tornqvist formula given 

 in Equation (3), and represents the change from time 

 period t to time-period ^ -i-l; i.e., TFP, , + i = TFP, ^^1 

 TFP,. 



Productivity and capacity utilization 



If firms are in l(jng-run equilibrium, quasi-fixed inputs 

 are optimally utilized in that the total cost of produc- 

 tion per unit of output is minimized. This long-run op- 

 timal utilization is called full economic capacity utiliza- 

 tion. Under long-run equilibrium, the flow of services 

 from a quasi-fixed input is assumed proportional to the 

 stock of that input, so that the available services from 

 each of the quasi-fixed inputs are fully utilized: the 

 observed stocks replace unobserved service flows in 

 Equation (2) (Berndt and Fuss 1986). 



When quasi-fixed inputs are not optimally utilized, 

 i.e., the firm is not in long-run equilibrium, the observed 

 productivity growth is composed of both the true tech- 

 nical progress impact, captured by AIA in Equation 

 (2), and the rate of change in capacity utilization. An 

 additional source of output variation is added to Equa- 

 tion (3): variations in capacity utilization (CU). To 

 develop this argument, suppose that the TV -t- 1"' input 

 is now a quasi-fixed input capital (A'), while the first 

 N inputs are variable inputs. Then growth in produc- 

 tivity is written as (Hulten 1986): 



A Y r, ' X, 



Sk 



B 



b' 



(5) 



\n{TFP,ITFP,^,) = 0.5 ^j (R^, + R^ 



,)l"(>Vi;,-, 



where S^ is capital's cost share. 



0.5 2 (S„ ■^ S,,_,)ln(A'„/A', 



ln(B,/B, ,). 



'The alternative approach is that of fixed-base indices in which TFP 

 in any time / is directly compared with TFP in the initial or base 

 period. See Squires (1988) for an extensive discussion within 

 fisheries. 



