这是一份liverpool利物浦大学ECON335的成功案例

Spreadsheet simulation (more advanced): Suppose that stock growth is given by $F(X)=a X-b X^{2}$, where $(a=r, b=r / k)$. Note that steady-state harvest $H=E[a / b-E / b]$. If for simplicity we assume that $P=1$ and total costs are given by $T C=c E$, then the open-access equilibrium occurs where $T R=T C$, implying that $E[a / b-E / b]=c E$, or $E^{0}=a-b c$. The group optinum equilibrium occurs where $M R=M C$. With $P=1$, then $M R=a / b-2 E / b$. Thus the group-optimum equilibrium occurs where $a / b-2 E / b=c$, or $E^{*}=a-b c$ /2. Assuming that $a=1000, b=1, \mathrm{p}=\$ 1, c=\$ 100$ :
a. Solve for the “group-optimum” and “open-access” equilibrium values for $E, X$, and $H$.
b. Create a table that shows $F(X)$ values for 50 -unit increments of $X$ (starting at $X=0$ up to $X=1000$ ). Plot the $F(X)$ values from the table in a fully labeled diagram and show the “group optimum” and “open access” equilibrium values for $X$ on the horizontal axis, and for $H$ on the vertical axis.

ECON335 COURSE NOTES ：

$$
P V_{N B}=\left(B_{0}-C_{0}\right) /(1+r)^{0}+\left(B_{1}-C_{1}\right) /(1+r)^{1}+\ldots+\left(B_{n}-C_{n}\right) /(1+r)^{n}
$$
Note that $C=$ cost in a given time period, $B=$ benefit in a given time period, $r=$ discount rate, and $n$ is the end period of the project in years from the present. $\left(B_{1}-C_{1}\right)$, for example, refers to net benefits received one year from the present. The expression $(1+r)^{n}$ means that the sum $(1+r)$ is taken to the $n^{\text {th }}$ power.