这是一份anu澳大利亚国立大学ECON1101作业代写的成功案例

Let us return to the case of a village with a monopolistic, omniscient moneylender. He knows which villagers have access to which type of land. As before, we assume that his opportunity cost of funds is $\varrho$. His problem is to set an interest rate $i_{3}(t)$ for each type of borrower to solve:
$\underset{i(t)}{\operatorname{Max}} \pi(t) i(t)$
subject to
$$
\pi(t)(R(t)-i(t)) \geq W
$$
and
$$
\pi(t) i(t) \geq \rho
$$
As long as $\mathrm{R}-\varrho \geq W$, the equilibrium will involve lending to each type of borrower at interest rates $i_{3}(t)=(\mathrm{R}-W) /$ $\pi(t)=i(t)$. Each type of borrower achieves an expected utility of $W$, and the lender earns an expected return of $\mathrm{R}-\varrho \geq$ $W$ on each loan.

ECON1101 COURSE NOTES ：

$$
U_{i}=\sum_{t=1}^{T} \beta^{t} \sum_{s=1}^{S} \pi_{s} u_{i}\left(c_{\text {ist }}\right)
$$
where $u()$ is twice continuously differentiable with $u^{\prime}>0, u^{\prime \prime}<0$ and $\operatorname{Lim}{x \rightarrow e^{\prime}} u^{\prime}(x)=+\infty .^{49} \mathrm{~A}$ Pareto-efficient allocation of risk within the village can be found by maximizing the weighted sum of the utilities of each of the $N$ households, where the weight of household $i$ in the Pareto programme is $\lambda, 0<\lambda{i}<1, \Sigma \lambda_{i}=1$ :
$$
\operatorname{Max}{C{\text {iht }}} \sum_{i=1}^{N} \lambda_{i} U_{i}
$$subject to the resources available in the village at each point in time in each state of nature:
$$
\sum_{i=1}^{N} c_{\text {ist }}=\sum_{i=1}^{N} y_{\text {ist }} \forall s, t
$$
$$
c_{\mathrm{ist}} \geq 0 \forall i, s, t
$$