Economics 145Economics of Ocean ResourcesGordon-Schaefer ModelMaximum Sustainable Yield (MSY)The yield-effort curve with the Schaefer model employs a logistic growth equation and assumes steady-state equilibrium.Start from the simple differential equation for the resource stock: € dXtdt= F(Xt) − Yt (1)where Xt denotes the resource stock in time t, F(Xt) denotes growth of the resource stock in time t, and Yt denotes yield from the resource stock in time t. Substitute into the Graham-Schaefer production function: € Yt= qEtXt (2)which is the simple version of a Cobb-Douglas production function (in which the production elasticities are set equal to unity), but which this time includes the catchabilitycoefficient, q, and where Et denotes fishing effort in time t and Xt denotes the resource stock biomass in time t. Next, substitute in the logistic growth function,€ F(Xt) = rXt[1−XtK]= rXt−rKXt2 (3) where r denotes the intrinsic growth rate of the resource stock and K denotes the environmental carrying capacity. This substitution gives:€ dXtdt= rXt[1−XtK] − qEtXt (4)To obtain steady-state equilibrium, € dXtdt= 0(so that€ F(X) = Yor the addition to the resource stock F(X), is counterbalanced by the removals from the resource stock, Y), and note that€ Et= E and € Xt= X in steady-state. Solving for the steady-state level of X gives:1€ X* = K[1−qEr] = K −qEKr. (5)The yield-effort curve for the logistic growth function, or Schaefer model, and simple form of the production function chosen is obtained by substituting in X* in Y = EX and solving for Y solely in terms of E, r, and K.€ Y = qEX= qKE[1−qEr] = qKE −q2KE2r (6)This equation is quadratic in E. For sufficiently high levels of effort (E > r/q) the yield is zero, i.e. beyond the critical level of effort the yield is zero. Therefore, if the effort level Eexceeds r/q, the population will be driven towards extinction.A common objective of renewable resource management has been to maintain standing stock Xt ≡ X so as to give a maximum sustainable yield or MSY. Mathematically, in terms of the continuous time model, € dXtdt= F(Xt) − Yt, one seeks to maximize sustainable yield Y = F(X), which requires F’(X) = 0. For the logistic growth function:€ F'(X) =dF(X)dE= 0 at € XMSY=K2 and € YMSY=rK4. (7)Note that steady-state equilibrium means that corresponding to each biomass level X is a certain rate of harvest Y = F(X) that just balances the natural rate of growth and thus maintains an equilibrium. This rate of harvest, at which Y = F(X), is sustainable yield, and MSY is simply the maximum sustainable yield.The resource stock level corresponding to maximum sustainable yield or MSY, € XMSY, isobtained from the logistic growth function as follows:€ F(X) = rX[1−XK] = rX −rKX2.From the first-order conditions for a maximum:2€ F'(X) = r − 2rKX = 0,so that:€ r = 2rKX or € 1 =2KXor€ XMSY=K2. (8)In words, the resource stock at MSY is one-half of the environmental carrying capacity, K. (Note that this result is due to the use of the logistic growth function, which is quadratic in the resource stock, X.)Since the logistic growth function is quadratic in X, the second-order conditions for a maximum are met, and we will ignore their explicit consideration.€ YMSY is obtained by substituting€ XMSY=K2in F(X) for the logistic growth function and solving as follows: € F(XMSY) = YMSY= rXMSY[1−XMSYK]= rK2[1−K2K] = rK2[1−12]=rK4. (9)The level of effort corresponding to maximum sustainable yield, € EMSY, is obtained by taking the first derivative of the Schaefer yield-effort curve, Equation (6), with respect to E and solving for the corresponding level of E, € EMSY. Because the Schaefer model is quadratic in E, the second-order conditions will automatically hold and we will not checkthem. From the first-order conditions with respect to E for a maximum in Y and solving:€ ∂Y∂E= qK −2q2EKr= 0.Solving for € EMSY gives:3€ EMSY=r2q. (10)Note that the critical level of effort – the effort level which is sufficient to drive the population to extinction – is € rq from above, which equals 2€ EMSY.Substituting€ EMSY=r2qinto the yield-effort curve also gives€ YMSY=rK4:€ YMSY= qKEMSY[1−qEMSYr]= qK[r2q][1−qr2qr] =rK2[1−12]=rK4. (11) Again, we do not check the second-order conditions for a maximum, since the yield-effort curve with the logistic growth function – the Schaefer model – is quadratic in E.Bionomic Equilibrium under Open-AccessTo examine the bionomic equilibrium under open access, two basic equations are used: (1) the steady-state equilibrium in the biomass with the logistic growth equation and (2) the steady-state equilibrium condition for all sustainable rents π to be dissipated (equal to zero):€ dXtdt= F(Xt) − Yt= 0π = T R − TC= PYt(Et) − cEt= 0,where c denotes the constant cost per unit of effort, P is the constant price per unit of output, and Y(E) denotes sustainable yield (output) for the Schaefer yield-effort curve.Substituting in the logistic growth function for the resource stock and the simple Cobb-Douglas production function and allowing for steady-state equilibrium (thereby dropping the time subscripts) gives:€ rX[1−XK] − cEX = 0π = PqEX − cE = 0.4All subsequent derivations start with the above two basic equations for steady-state / sustainable resource stock and rents. From these two basic equations, we find the two basic property right conditions of open-access and the social welfare optimum (under ideal property rights, either common property with effective management or individual private property, or the sole owner case), which gives us our two conditions for comparative statics. That is, the comparative statics compare the socially sub-optimal open-access levels of effort, resource stock, yield, and rents with the social optimum levels of effort, resource stock, yield, and rents.The open-access bionomic equilibrium level of the resource stock, € X∞,