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23-BioEcon2 - BIOECONOMIC MODEL OF A FISHERY(continued...

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BIOECONOMIC MODEL OF A FISHERY (continued) Dynamic Maximum Economic Yield In our derivation of maximum economic yield (MEY) we examined a system at equilibrium and our analysis made no distinction between profits in the present versus profits in the future. Both types of profits were treated as being equally valuable. This MEY is sometimes described as the static MEY . In contrast, the dynamic MEY considers the fact that we place greater value on goods and services that we receive today than on those that we might receive in the future. For example, if I offer you the choice of receiving $100 today or $100 in a month, chances are you will choose to take the money now. But, you might be indifferent between $100 today versus $101 in a month. The extra dollar reflects the time-value of the money. Suppose you have a bank account that earns interest at rate r, and you have made an initial deposit of X 0 . Your balance: after 1 time period would be X 1 X 0 1 r + ( ) = and after 2 periods it would be X 2 X 1 1 r + ( ) = X 0 1 r + ( ) 2 = and after n periods it would be X n X 0 1 r + ( ) n = We can manipulate this last equation to derive the relationship between money in the present and money in the future. X 0 X n 1 r + ( ) n = ==> Present_Value Future_Value Discount_Factor = We can write the discount factor in an equivalent exponential form as 1 r + ( ) n - exp δ - n ( 29 = where δ ln 1 r + ( ) = is the discount rate . For small values of r, the discount rate and interest rate are almost equal, δ 2245 r. r 0.010 = δ r ( ) ln 1 r + ( ) = δ r ( ) 0.009950 = r 0.050 = δ r ( ) 0.048790 = This is analogous to the relationship between the mortality fraction μ and the instantaneous mortality rate M. r 0.100 = δ r ( ) 0.095310 = Rather than limiting our analysis of maximum economic yield to a fishery at equilibrium, we will allow the fish stock and fishing fleet to change over time. The objective of dynamic MEY is to find the conditions that will maximize the discounted present value of the stream of profits flowing from the fishery as the fishery changes over time. In mathematical terms the problem is to maximize 0 t e δ - t p Y t ( ) c f t ( ) - ( ) d with Y t ( ) q f t ( ) B t ( ) = FW431/531 Copyright 2008 by David B. Sampson BioEcon2 - Page 149
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subject to the conditions dB dt G B t ( ) ( ) Y t ( ) - = and B t ( ) 0 and 0 f t ( ) f max This type of problem is sometimes described as an optimal control or dynamic optimization problem. Clark (1985), on the Supplemental Reading list, shows that a necessary condition for obtaining the maximum net present value from our model fishery is that the fishing effort f should be controlled according to the following policy f max ... if B > B opt f opt = G(B opt )/(q·B opt ) ... if B = B opt 0 ... if B < B opt where G(B) is the latent growth function for the fish stock and B opt , which is a target biomass level towards which we should force the stock, satisfies the following equation.
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