(18) X = (l -(1-/C)' )W 

 or, writing W in terms of N 



That is, whenever we find qj' in equation (16) 

 we replace it with equation (17). 



Restricting ourselves to equilibrium values 

 (that is, where catch is equal to growth), 

 differentiation of (18) with respect to N yields 

 the marginal physical product of effort in long 

 run equilibrium: 



(20) dX/dN = 



N 

 (l-k) 



{1-k) lx\(l-k)W + |1 - 

 \dW/dN 



The first expression on the right of (20) is the 

 short run marginal physical product of another 

 unit of effort, and is always positive for any 

 positive W, and declines as TV increases, thus 

 illustrating short run diminishing returns. The 

 second expression on the right of (20) shows 

 the effect on long i-un equilibrium catch of 

 another pound of fish stock, and is equal to the 

 percentage of the stock caught by N units of 

 effort multiplied by the change in equilibrium 

 stock resulting from a marginal change in 

 equilibrium effort. Thus, (20) includes both 

 congestion and growth externalities. 



Solving explicitly ior dW I DN and rearrang- 

 ing terms can also yield the expression for 

 dX I dN in terms of W, N, and the parameters 

 k, m, and H: 



(21) dX/dN=-[(l-/e) In(l-fe)] 

 ,l-(l-fe)'' 



{m-l)H 



W 



+ W]. 



If we assume that the cost per unit of effort 

 is some constant A, then the marginal cost per 

 pound of fish under biological equilibrium con- 

 ditions, and including both congestion and 

 growth externalities (the long run marginal 

 co.st, LRMC) is 



(22) LRMC = A HdX/dN). 



Estimation of the parameters H. K, m, and 

 k, together with data on demand and the cost 

 of effort can be used to estimate long i-un 

 equilibrium catch and the welfare losses in any 

 year associated with MSY regulation or other 

 nonoptimal output. 



LITERATURE CITED 



BELL, FREDERICK W. 1969. Estimation of the Econ- 

 nomic Benefits to Fishermen, Vessels, and Society from 

 Limited Entry to the Inshore U. S. Northern Lobster 

 Fishery. National Marine Fisheries Service, Working 

 Paper Number 36. 



CARLSON, ERNEST W. 1969. Bio-Economic Model of 

 a Fishei-y. National Marine Fisheries Service, Working 

 Paper Number 12. 



CRUTCHFIELD. JAMES and ARNOLD ZELLNER. 

 1962. Economic .Aspects of the Pacific Halibut P^ishery, 

 Fishery Industrial Research, 1(1). 



PELLA. J. J. and P. K. TOMLINSON. 1969. A Gen- 

 eralized Stock Production Model, Inter-.-\merican Tropical 

 Tuna Commission, Bulletin 13(3). 



SMITH. VERNON. 1968. Economics of Production from 

 Natural Resources, American Economic Review, 58(3): 

 409-431. 



SOUTHWARD, G. MORRIS. 1968. A Simulation of 

 Management Strategies in the Pacific Halibut Fishei-y. 

 International Pacific Halibut Commission, Report 

 Number 47. 



WORCESTER, DEAN A., JR. 1969. Pecuniai-y and 

 Technological Externality, Factor Rents, and Social 

 Cost, American Economic Review, 59(5):873-885. 



71 



