Open Access Fisheries

Q1.Open access fishery with c (y) = 5y, p = 10-20y, Logistic growth with γ = 0.04 and K=100.

(a)

In the case of stock independent costs : \(p_t=MC_t\)

Harvest rule is: \(min(x, [a - MC_t]/b)\) \[ 10-2y=\frac{dC(y)}{dy}=5\] \[y=min(x,1/4)\]

Logistic Growth equation is given by : \(F(x)= \gamma * x_t(1-x_t/100)\)

(b)

The equation that determines the steady states for this open access fishery:

\[F(x) -y = \gamma * x_t(1-x_t/100)-min(x,1/4)=0\]

(c)

for, \(y=min(x,1/4)=x\)

f1_1 <- function(x) {0.04*x*(1-(x/100)) - x}
print(paste("For harvest,y(x) = min[x, 1/4] = x, the steady state stock is: ",uniroot(f1_1, lower = 0, upper = 100)[1]))
## [1] "For harvest,y(x) = min[x, 1/4] = x, the steady state stock is:  0"

for, \(y=min(x,1/4)= 1/4\)

f1_2 <- function(x) {0.04*x*(1-(x/100)) - 1/4}
print(paste("For harvest,y(x) = min[x, 1/4] = 1/4, the steady state stock is: ",uniroot.all(f1_2, lower = 0, upper = 100)[1], "& ",uniroot.all(f1_2, lower = 0, upper = 100)[2]))
## [1] "For harvest,y(x) = min[x, 1/4] = 1/4, the steady state stock is:  6.69872967587282 &  93.3012703241272"

(d)

f1 <- function(x) {0.04*x*(1-(x/100))}
print(ggplot(data.frame(x=c(0, 100)), aes(x=x)) +
        labs(x = "x", y = "F(x)", title = "Open acess growth rate") +
        theme(panel.grid = element_blank()) +
        geom_hline(yintercept = 0.25, color = "firebrick", linetype = "dotted", size=1)+
        geom_vline(xintercept = 6.69, color = "orange", linetype = "dashed", size=1)+
        geom_vline(xintercept = 93.30, color = "orange", linetype = "dashed", size=1)+
        theme_classic()+
        stat_function(fun=f1))

Stable at x={0, 93.30}

(e)

Producer Surplus is zero & consumer surplus is \(\int_{0}^{y} p(y) dy-p^*y^*\)

CS is 0 at y=0 and is 0.625 at y=0.25

Q2.Open access fishery with y = 1.5xE, p = 10-8y, Logistic growth with γ = 0.08 and K=60.

(a)

Given the cost of a unit effort is w = $3.

The cost of harvesting y, given the stock x, is given by: harvest cost: \(c(x,y) = w*E = w*y/(q*x) = C*y/x\) where, \(C=w/q = 3/1.5=2\) so, harvest cost= \(2*y/x\)

(b)

\[y=1/b(a-C/x) for 1/b(a-C/x)>=0\] \[y=0 for 1/b(a-C/x)<0\]

The equation that determines the steady states for this open access fishery:

\[F(x) -y = 0.08 * x(1-x/60)-1/8(10-2/x)=0\]

(c)

f2 <- function(x) {0.08*x*(1-(x/60)) - 1/8*(10-2/x)}
print(paste("For harvest,y(x) = 1/8(10-2/x), the steady state stock is: ",uniroot(f2, lower = 0, upper = 60)[1]))
## [1] "For harvest,y(x) = 1/8(10-2/x), the steady state stock is:  0.202619746550559"

And the steady state harvest is 1/8(10-2/2.02)= 1.13

(d)

print(ggplot(data.frame(x=c(0, 100)), aes(x=x)) +
        labs(x = "x", y = "F(x)-y(x)", title = "Graph the steady states") +
        theme(panel.grid = element_blank()) +
        geom_hline(yintercept = 1.12625, color = "black", linetype = "dotted", size=1)+
        geom_hline(yintercept = 0, color = "firebrick", linetype = "dotted", size=1)+
        geom_vline(xintercept = 0, color = "orange", linetype = "dashed", size=1)+
        theme_classic()+
        stat_function(fun=f2))

Stable at x=0

(e)

Producer Surplus is zero & consumer surplus is \(\int_{0}^{y} p(y) dy-p^*y^*\)

CS is 0 at y=0

Sole Owner Fisheries

Q3.F(x) = 0.08x(1-x/100), r = 0.04, inverse demand is p = 60-y, and harvest costs are c(x,y) = Cy

(a)

Find candidate interior stead state with positive profits. Does it satisfy x_> 0? \[F(x) = 0.08x(1-x/100)\] \[r=\frac{dF(x)}{dx}\] \[0.04=0.08-2*0.08*x/100\] \[x_\infty = 25 > 0\]

(b)

Find a condition on C such that the candidate steady state satisfied R(y_) > 0. \[y_\infty=F(x)\] \[y_\infty=F(x) = 0.08x(1-x/100)=1.5;x=25\]

\[R>0 => R(y_\infty)>0\]

\[P-C>0\] \[60-y-C>0\] \[60-1.5>C\] \[C<58.5\]

(c)

Assuming the fishery is open access and the costs of harvest are given by c(x,y) = C*y 10y. \[Tax=Rent=P-C\] \[Tax=58.5-10=48.5\]