Macro Study Notes
Xinyu Zhou
Module 8
How do financial market imperfections amplify shocks? The financial amplification mechanism shows how collateral constraints create a feedback loop between asset prices and borrowing, turning small shocks into large recessions.
Households maximize $u(c_1) + \beta u(c_2)$ subject to:
$c_1 = y_1 + d_1, \quad c_2 + d_1(1+r^*) = y_2, \quad d_1 \leq \kappa y_1$
where $d_1$ is debt and $\kappa$ is the borrowing limit (fraction of income).
Under $\beta(1+r^*) = 1$, unconstrained consumption is flat: $c_1^u = c_2^u = \frac{1+r^*}{2+r^*} y^{PDV}$.
When $d_1^u > \kappa y_1$, the constraint binds: $d_1 = \kappa y_1$.
The household can no longer perfectly smooth consumption. A credit crunch (decline in $\kappa$) forces $c_1$ down and $c_2$ up, reducing welfare.
Household buys an asset $a_1$ at price $p_1$ that pays dividend $D$ in period 2:
$\max u(c_1) + \beta u(c_2)$ subject to $c_1 + p_1 a_1 = y_1 + p_1 a_0$, $c_2 = y_2 + D a_1$.
FOC:
Equivalently: $p_1 = \beta D \frac{u'(y_2 + D)}{u'(y_1)}$ in equilibrium (with $a_0 = a_1 = 1$).
In infinite horizon with risk:
$p_t = \mathbb{E}_t\!\left[\beta\frac{u'(c_{t+1})}{u'(c_t)}(D_{t+1} + p_{t+1})\right]$
The stochastic discount factor $m_{t+1} \equiv \beta\frac{u'(c_{t+1})}{u'(c_t)}$ prices all assets. In bad times, $c_{t+1}$ is low, $u'(c_{t+1})$ is high, and $m_{t+1}$ is high — assets that pay off in bad times are more valuable.
The key mechanism (Bernanke-Gertler 1989, Kiyotaki-Moore 1990):
This creates a "diabolic loop" where small initial shocks are amplified through the financial system.
$d_1 \leq \kappa p_1 a_0$
Borrowing is constrained by the value of existing assets (collateral).
$d_1 \leq \kappa p_1 a_1$
Borrowing is constrained by the value of assets being purchased (like a mortgage requiring a down payment).
With $u(c_1) = \ln c_1$ and $u(c_2) = c_2$ (log-linear preferences):
Asset pricing: $p_1 = \beta D c_1$
Budget: $c_1 = y_1 + \kappa p_1$
Solving:
We need $\beta D \kappa < 1$ for a finite, positive equilibrium (otherwise the amplification "blows up").
The derivative of consumption with respect to income:
Consumption is more volatile than income! Without financial frictions ($\kappa = 0$ or no collateral), consumption moves one-for-one with income. With amplification, the impact is magnified.
Under the mortgage formulation $d_1 \leq \kappa p_1 a_1$:
The household's budget constraint becomes $c_1 + (1-\kappa)p_1 a_1 = y_1 + p_1 a_0$, where $(1-\kappa)p_1 a_1$ is the down payment. The analysis is similar but yields slightly different amplification dynamics.
Mian and Sufi (2011, 2014) argue that a financial amplification mechanism played a central role in the 2007-2009 crisis.
Counties with the largest declines in housing net worth experienced the most severe employment losses from 2007-2009, consistent with the amplification mechanism.