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