94 Fishery Bulletin 88(1). 1990 



or after rearranging: 



A Y TTx X, B 



where the dots over the variables represent time 

 derivatives. 



Equation (A.5) is the fundamental equation of growth 

 accounting in its continuous time form. Thus full, long- 

 run equilibrium total-factor productivity growth, A I A, 

 identified with technical progress, is a residual after 

 the sources of output growth have been allocated 

 among intertemporal changes in inputs and resource 

 abundance for a constant-returns-to-scale production 

 technology. 



Equation (A.5) allocates the growth rate of Y{t ) 

 among A (t ), X{t ), and B{t ) as required, but because 

 the E, are not observable, two additional steps are re- 

 quired for empirical analysis. Assuming that all inputs 

 are paid the value of their marginal product, then 

 6FI6XXt ) = P,{t )IP(t )■ where P(t ) and P,(t ) are the 

 full equilibrium prices of output and inputs, respec- 

 tively. This implies that: S,(t) = E,(t) = PXt)X,(t)l 

 P{t)Y(t), where S,(t) is the income or cost share of 

 input Z,. Under constant returns to scale, total costs 

 equal total revenue; i.e., S,P,(/ )X,(t ) = P(t )Y{t ), and 

 1,5,(0 = 1. 



The final step is to substitute E,{t ) = S,{t) into equa- 

 tion (A.5), which gives an equation in which all variables 

 are measurable except A/A, which is calculated as a 

 residual: 



A Y ^4-\ X, B 



= - 1 S, ' - -, (A.6) 



A Y f~i X, B 



where the notation for time-period t is again suppressed 

 and 'Z,S,(X,IX,) represents aggregate input growth. 

 Productivity growth equals the rate of change of out- 

 put flow minus a share-weighted index of rates of 

 change of inputs minus the rate of change of the re- 

 source stock. The index of productivity growth in Equa- 

 tion (A.6) is also called a Divisia index (Hulten 1974). 



