# 高级微观经济学 Advanced Microeconomics ECON3010

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with complementary slackness between $\lambda_{t}$ and $\left(A_{t}+y_{t}-c_{y}\right)$. By the envelope property, $x_{t-1}$ will satisfy
$$E_{t-1} \frac{d V_{t}(\cdot)}{d x_{t-1}}=E_{t-1} U^{\prime}\left(c_{t}\right) \frac{\partial y}{\partial x_{t-1}}=0$$
Substituting for $u^{\prime}(c)$, we have
$$E_{t-1}\left[\beta(1+r) V_{t+1}^{\prime}(\cdot)+\lambda_{t}\right] \frac{\partial y}{\partial x_{t-1}}=0$$
So if $\lambda_{t}=0$ in all states of period $t$, so that the individual knows that the liquidity constraint will not bind in period $t$, then $x_{p-1}$ is chosen so that
$$E_{t-1} V_{t+1}^{\prime}(\cdot) \frac{\partial y}{\partial x_{t-1}}=0$$

## ECON3010 COURSE NOTES ：

$$\frac{U_{1}(c, W-R)}{U_{2}(c, W-R)}=1+r$$
(10)
This suggests a further interpretation of the equilibrium implicit contract. Suppose that the landlord charges peasants an interest rate on consumption loans of $r$ plus an ‘entry fee’ of $f$ for the privilege of borrowing at this rate. A peasant will optimally choose a consumption credit, $c$, that satisfies the optimal intertemporal consumption condition,
$$\frac{U_{1}(c, W-(1+r) c-f)}{U_{2}(c, W-(1+r) c-f)}=1+r$$
If $f=\mathrm{R}^{\prime}-(1+r) i$, then the peasant will optimally choose a credit of $\iota=i$. Thus, the equilibrium implicit contract effects an outcome equivalent to a two-part

# 高级微观经济学 Advanced Microeconomics ECON342

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First, we assume that there are a large number of competitive lenders. We initially assume that the lenders can observe the borrowers’ choice of $a$. Therefore, they can write contracts that specify both the interest factor $i$ and the effort level a. We define an equilibrium to be a pair $\left(i_{1}, a_{1}\right)$ such that: (a) $U\left(i_{1}, a_{1}\right) \geq W ;(b) \Pi\left(i_{1}, a_{1}\right) \geq \varrho$; and $(c)$ there is no other pair $(i, a)$ that yields a return greater than or equal to $\varrho$ to a lender and which a borrower would prefer to $\left.\left(i_{1}, a_{1}\right)\right)^{38}$ If there is an equilibrium with lending, it is characterized by the solution to
$$\underset{i, a}{\operatorname{Max}} \pi(a)(R-i)-D(a)$$
subject to
$$\pi(a) i \geq \rho$$
and
$$\pi(a)(R-i)-D(a) \geq W .$$
There may be no solution to this problem. If there is no $(i, a)$ such that both constraints are satisfied, then there is no lending in equilibrium. For any fixed $\varrho$, there is a Wlow enough so that both constraints may be satisfied.

## ECON342 COURSE NOTES ：

Suppose that a village has a single resident with enough wealth to act as a moneylender. (His wealth is larger than $N$, the number of residents in the village.) The moneylender lives in the village and has the opportunity costlessly to monitor the activities of anyone who borrows from him. He can deposit his wealth at the risk-free rate of $\varrho$, so this is the opportunity cost of his funds. The moneylender will set the interest rate and level of effort to solve:
$\underset{a, i}{\operatorname{Max}} i \pi(a)$
subject to
$$\pi(a)(R-i)-D(a) \geq W$$
and
$$i \pi(a) \geq \rho$$