- 1 What?
- 2 Why?
- 3 How?
- 3.1 Firms
- 3.2 Households
- 3.3 Banks
- 3.4 Evolution of Bank networths
- 3.5 Equilibrium
- 4 And?
1 What?
Study three credit policies:
- Liquidity facilities: Permitting discount window lending to banks secured by private credits
- Lending facilities: lend directly in relatively high grade credits markets (including commercial paper, agency debt and mortgage-backed securities)
- Equity injections: Direct assistance to large financial institutions (equity injections and debt guarantees)
2 Why?
2008 Financial crisis. Sorta important.
3 How?
3.1 Firms
Cobb-Douglas production function \( Y_{t} = A_{t} K_{t}^{\alpha} L_{t}^{1-\alpha} \), where \( A \) follows a Markov process.
Idiosyncratic risk with investment oppotunity: \( \pi^{i} = 1 - \pi^{n} \).
Law of motion for capital:
\begin{align*} K_{t+1} &= \psi_{t+1} [I_{t} + \pi^{i}(1 - \delta)K_{t}] + \psi_{t+1}\pi^{n}(1 - \delta)K_{t} \\ &= \psi_{t+1} [I_{t} + (1 - \delta) K_{t}] \end{align*}
where \( \psi_{t+1} \) is the capital quality shock (cheap exogeneous shock to capital quality, see Merton 1973 ecma).
Output allocation: \[ Y_{t} = C_{t} + G_{t} + \left[1 + f \left(\frac{I_{t}}{I_{t-1}} \right) \right]I_{t} \] where \( f\left( \frac{I_{t}}{I_{t-1}} \right)I_{t} \) is the physical adjustment costs. This create an inertia, preventing for investment to response instantly to any shocks. Gleaned from the literature, the inclusion of this condition most likely stems from a macroeconomic need to match data, rather than microeconomic evidence.
3.1.1 Non-financial and financial firms
3.2 Households
Preferences \[ \mathbb{E} \sum\limits_{i=0}^{\infty} \beta^{i} \left[ \log(C_{t+i} - \gamma C_{t + i - 1}) - \frac{\chi}{1 + \varepsilon} L^{1 + \varepsilon}_{1 - i} \right] \]
- Uses habut instead of other nk frictions cuz it’s
cheaptractable and doesn’t matter
Type: \( 1 - f \) “workers” and \( f \) “bankers”. Bankers dies with probability \( (1 - \sigma) \) and are replaced by new workers, so the sum remains unchanged.
Flow of funds constraints: \[ C_{t} = W_{t} L_{t} + \Pi_{t} - T_{t} + R_{t} D_{ht} - D_{h t - 1} \]
FoCs:
\begin{align*} L_{t} &:\qquad \mathbb{E} u_{c_{t}} W_{t} = \chi L_{t}^{\varepsilon} \\ C_{t} &:\qquad \mathbb{E} \Lambda_{t,t+1} R_{t+1} = 1 \end{align*}
where \( u_{c_{t}} \equiv (C_{t} - \gamma C_{t-1})^{-1} - \beta \gamma (C_{t+1} - C_{t})^{-1} \), \( \lambda_{t,t+1} \equiv \beta \frac{u_{c_{t+1}}}{u_{c_{t}}} \).
3.3 Banks
Flow of funds: \[ Q_{t} s_{t}^{h} = n_{t}^{h} + b_{t}^{h} + d_{t} \]
where networth is defined as \[ n_{t}^{h} = \left[ Z_{t} + (1 - \delta)Q_{t}^{h} \right]\psi_{t} s_{t-1} - R_{bt}b_{t-1} - R_{t} d_{t-1} \]
Objective: expected present value of future dividends \[ V_{t} = \mathbb{E} \sum\limits_{i = 1}^{\infty} (1 - \sigma)\sigma^{i - 1} \Lambda_{t, t+i} n_{t+i}^h \] However, there an agency problem: After raising funds, bankers can transfer \( \theta \) of “divertable” assets (\( Q_{t}^{h}s_{t}^{h} - \omega b_{t}^{h} \)) to their family. Creditors reclaim \( 1 - \theta \) of funds.
- \( \omega = 1 \): interbank market operates frictionlessly
- \( \omega = 0 \): interbank is simlar to the deposit market
Incentive constraints \[ V_{t}(s_{t-1}, b_{t-1}, d_{t-1}) \geq \theta(Q_{t}^{h}, s_{t}^{h} - \omega b_{t}^{h}) \] The value of bank at the end of period \( t-1 \) solves the Bellman equation:
\begin{multline*} V_{t-1}(s_{t-1}, b_{t-1}, d_{t-1}) = \\ \mathbb{E}_{t-1} \Lambda_{t-1,t} \sum\limits_{h=i,n} \pi^{h} \left\{ (1 - \sigma)n_{t}^{h} + \sigma \max_{d_{t}} \left[ \max_{s_{t}^{h}, b_{t}^{h}} V_{t} \left( s_{t}^{h}, b_{t}^{h}, d_{t} \right) \right] \right\} \end{multline*}
Guess and verify: \[ V_{t}(s_{t}^{h}, b_{t}^{h}m d_{t}) = \nu_{st}s_{t}^{h} - \nu_{bt}b_{t}^{h} - \nu_{t}d_{t} \]
Let \( \lambda_{t}^h \) be the Lagrangian multiplier for the incentive constraint faced by bank of type \( h \) and let \( \overline{\lambda}_{t} \equiv \sum\limits_{h=i,n}\pi^{h}\lambda_{t}^{h} \) be the average of this multiplier across states. With the conjectures we have the focs for \( d_{t}, s_{t}^h \) and \( \lambda_{t}^{h} \):
\begin{align*} d_{t} &:\qquad (\nu_{bt} - \nu_{t})(1 + \overline{\lambda}_{t}) = \theta\omega \overline{\lambda}_{t} \\ s_{t}^{h} &:\qquad \left( \frac{\nu_{st}}{Q_{t}^{h}} - \nu_{bt} \right) \left( 1 + \lambda_{t}^h \right) = \lambda_{t}^{h}\theta(1-\omega) \\ \lambda_{t}^{h} &:\qquad \left[ \theta - \left( \frac{\nu_{st}}{Q_{t}^{h}} - \nu_{t} \right) \right]Q_{t}^{h}s_{t}^{h} - [\theta\omega - (\nu_{bt} - \nu_{t})] b_{t}^{h} \leq \nu_{t} n_{t}^{h} \end{align*}
- We can only have \( \nu_{bt} > \nu_{t} \) if:
- \( \overline{\lambda}_{t} > 0 \) i.e. the incentive constrant is bind for some state and
- \( \omega > 0 \) i.e. inter-bank operates more efficiently than retail deposit market.
- The marginal value of asses in terms of goods \( \frac{\nu_{st}}{Q_{t}^{h}} \) exceeds the marginal cost of interbank \( v_{bt} \) if \( \lambda_{t}^{h} > 0 \) (incentive constraints binding) and \( \omega < 1 \)(there exists frictions)
3.4 Evolution of Bank networths
Total net workth for type \( h \) banks equal the sum of networth for existing bankers and entering bankers \[ N_{t}^{h} = N_{ot}^{h} + N_{yt}^{h} \] where
\begin{align*} N_{ot}^{h} &= \sigma \pi^{h} \left\{ \left[ Z_{t} + (1 - \delta)Q_{t}^{h} \right]\psi_{t} S_{t-1} - R_{t} D_{t-1} \right\} \\ N_{yt}^{h} &= \xi\pi^{h}\left[ Z_{t} + (1 - \delta)Q_{t}^{h} \right]\psi_{t} S_{t-1} \end{align*}
The aggregated balance sheet is written as: \[ D_{t} = \sum\limits_{h = i,n} \left( Q_{t}^{h}S_{t}^{h} - N_{t}^{h} \right) \]
3.5 Equilibrium
4 And?
4.1 Economy under different assumption on \( \omega \)
4.1.1 Special case: Frictionless wholesale market \( \omega = 1 \)
The incentive constraints for bankers in this case is just \[ Q_{t} s_{t} - b_{t} = \phi_{t} n_{t} \,, \] where
- \( \phi_{t} = \frac{\nu_{t}}{\theta - \mu_{t}} \) is the leverage ratio net of interbank borrowing
- \( \mu_{t} \equiv \frac{\nu_{st}}{Q_{t}} - \nu_{t} > 0 \) represents the excess value of bank assets
4.1.2 Special case: Symmetric friction between wholesale and retail market \( \omega = 0 \)
The FoC for \( d_{t} \) implies the marginal costs of interbank equals to the marginal costs of deposits \[ \nu_{bt} = \nu_{t} \]
4.1.3 General case: \( 0 < \omega < 1 \)
- The interbank rate is lower than that of the deposit rate, higher than the return on loans. Intuitively, because a dollar interabank credit tighten the incentive constraint by less than a dollar of deposit.
- Because lending banks are not able to perfectly recover asset \( \omega < 1 \), there is still imperfect arbitrage which keeps the expected discounted interbank rate below the expected discount rate of return to loans.
4.2 Credit policies
4.2.1 Lending facilities (direct lending)
4.2.2 Liquidity facilities (discount window lending)
4.2.3 Equity Injections
This post is in the collection of my public reading notes.