Title

## Prudential fiscal stimulus

MMF Conference, 6  September 2022
<table width=1100>
<td width=500 style="vertical-align:top;">
#### Alfred Duncan
#### University of Kent
</td>
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#### Charles Nolan
#### University of Glasgow
</td>
</table>
<br>
<br>
<br>
This project is supported by research funding from UKRI Grant Ref: ES/V015559/1

---
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### The problem (the Greenspan put)
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Stimulus policies can increase moral hazard.
<br><br>
Anticipating to be rescued in downturns, firms
might take more risk today.
<br><br>
This paper asks:
- Can we stimulate the economy in a way that
encourages prudence?
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</td>
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<br>

---
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### Our contribution
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- We show that countercyclical wage subsidies can improve welfare,
  even in the absence of aggregate demand externalities.

- We characterise the optimal wage subsidies.

- We estimate that simple rule implementations can generate large welfare gains.
<br>
</td>
<td width=650 style="vertical-align:top;">
<img src="images/map.svg" width=650></img>
  Figure: Countries with new or existing wage subsidy schemes
during the Covid-19 pandemic (Sources: ILO, IMF, authors’ calculations)

</td>
</table>
<br>

<!-- ---
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### The contribution
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Anticipated wage subsidy stimulus policies can have an ex ante
macroprudential effect.
<br><br>
- We solve for optimal wage subsidies under log utility.

- We estimate that
the welfare gains from countercyclical wage subsidies are large.

</td>
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</td>
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<br> -->

<!-- ---
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### Conventional wisdom
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Traditionally, there is thought to be a conflict between ex post stimulus and
ex ante incentives for financial stability.

<br>
Sometimes this is referred to as the *Greenspan put*.

<br>
The idea is that firms have an incentive to take *excessive* cyclical
risk during expansions because policymakers will pursue expansionary policies
in downturns.

<br>
Lack of formal literature.
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</td>
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<br> -->

---
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### Intuition behind our result
Absent intervention

<br>

---
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### Intuition behind our result
Introduction of the wage subsidy

<br>

---
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### Intuition behind our result
Firm's risk allocation response

<br>

---
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### Intuition behind our result

<br>

---
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### Intuition behind our result
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Labour supply is a complement to firms' inside wealth.

<br>
Anticipating wage subsidies in recessions, firms will be more prudent in
expansions.

<br>
The ex-post wage subsidy replicates an ex-ante macroprudential intervention.
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</td>
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<br>

<!-- ---
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### Wage subsidies
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- Are not the only prudential fiscal stimulus.

- Illustrate the theory in a clear way (hopefully!)

- Are popular right now.

- prudential austerity programmes also exist (not presented here).
</td>
<td width=650 style="vertical-align:top;">
<img src="images/map.svg" width=650></img>
  Figure: Countries with new or existing wage subsidy schemes
during the Covid-19 pandemic (Sources: ILO, IMF, authors’ calculations)

</td>
</table>
<br> -->

<!-- ---
class: left, middle

### Wage subsidies

<img src="images/map.svg" width=1200></img>
  Figure: Countries with new or existing wage subsidy schemes
during the Covid-19 pandemic (Sources: ILO, IMF, authors’ calculations)

---
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### Related literature
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Macropru
- di Tella (2017); Duncan and Nolan (2022).

- Farhi and Werning (2016); Schmitt-Grohe and Uribe (2012);

Information economics
- Arnott and Stiglitz (1991)

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</td>
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<br>

<!-- ---
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### Rest of this talk

- General model and macroprudential externality (close to di Tella, 2017 JPE)
- Theory results
- Quantitative exercise -->

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# The macroprudential externality

---

# The macroprudential externality

## Three ingredients

1. Anonymous unrestricted trade in aggregate-state contingent securities.
2. Agency costs restricting trade in idiosyncratic-state contingent securities.
3. Risk aversion over individual specific states.

<a href='/#28'>Jump to results</a>

---

### The entrepreneur's intratemporal problem

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$$ R(\theta,s),b^e,h^e =\arg \max \mathbb{E} \left\lbrace \hat{v}^e(R(\theta,s)q^e) | \theta\right\rbrace $$

subject to

$$  R(\theta,s)q^e \leq  f(\theta,k,h) - r^b(\theta)b^e - wh^e\qquad \forall \theta$$

in addition to
$$ [\text{capital budget constraint}] $$
$$ [\text{truth telling constraint}] $$
$$ [\text{lender participation constraint}] $$
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</td>
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<br>

---

### The entrepreneur's intratemporal problem

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$$ R(\theta,s),b^e,h^e =\arg \max \mathbb{E} \left\lbrace {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\hat{v}^e}(R(\theta,s)q^e) | \theta\right\rbrace $$

subject to

$$  R(\theta,s)q^e \leq  f(\theta,k,h) - r^b(\theta)b^e - wh^e\qquad \forall \theta$$

in addition to
$$ [\text{capital budget constraint}] $$
$$ [\text{truth telling constraint}] $$
$$ [\text{lender participation constraint}] $$
</td>
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\${\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\hat{v}^e}\$ is concave.

Entrepreneurs are risk averse, but behave as risk neutral when financial markets
are perfect.
</td>
</table>
<br>

---

### The entrepreneur's intratemporal problem

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$$ R(\theta,s),b^e,h^e =\arg \max \mathbb{E} \left\lbrace \hat{v}^e(R(\theta,s)q^e) | \theta\right\rbrace $$

subject to

$$  R(\theta,s)q^e \leq  f(\theta,k,h) - r^b(\theta)b^e - wh^e\qquad \forall \theta$$

in addition to
$$ [\text{capital budget constraint}] $$
$$ {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}[\text{truth telling constraint}]} $$
$$ [\text{lender participation constraint}] $$
</td>
<td width=400 style="vertical-align:top;">

Quantitative model:<br>
Imperfect state verification (Duncan and Nolan, 2019)

Theory results can accommodate<br>
Grossman Hart (1983), Krasa and Villamil (2000), Lacker and Weinberg (1989).

</td>
</table>
<br>

---

### The entrepreneur's intratemporal problem

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$$ R(\theta,s),b^e,h^e =\arg \max \mathbb{E} \left\lbrace \hat{v}^e(R(\theta,s)q^e) | \theta\right\rbrace $$

subject to

$$  R(\theta,s)q^e \leq  f(\theta,k,h) - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}r^b(\theta)b^e} - wh^e\qquad \forall \theta$$

This is state-contingent, negative in default.

Partially insures fluctuations in
\$\theta\$.
</td>
</table>
<br>

---

### The entrepreneur's intratemporal problem

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$$ R(\theta,s),b^e,h^e =\arg \max \mathbb{E} \left\lbrace \hat{v}^e(R(\theta,s)q^e) | \theta\right\rbrace $$

subject to

$$  R(\theta,s)q^e \leq  f(\theta,k,h) - r^b(\theta)b^e - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}wh^e}\qquad \forall \theta$$

Workers are hired before the realisation of \$\theta\$.

Wages are not contingent on \$\theta\$.
</td>
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<br>

---

### The entrepreneur's intertemporal problem

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$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e  v^e({q^e}')\right\rbrace $$

subject to

$$ {q^e}' = R(\theta,s)q^e - c^e - \int_{s'\in S} p(s') x^e(s') ds + {x^e}(s')$$
</td>
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</td>
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<br>

---

### The entrepreneur's intertemporal problem

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$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e  v^e({q^e}')\right\rbrace $$

subject to

$$ {q^e}' = {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}R(\theta,s)}q^e - c^e - \int_{s'\in S} p(s') x^e(s') ds + {x^e}(s')$$
</td>
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\$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}R(\theta,s)\$ is an indirect gross return function.

It is the result of privately optimal borrowing and hiring, from the
entrepreneur's intratemporal problem.
</td>
</table>
<br>

---

### The entrepreneur's intertemporal problem

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$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e  v^e({q^e}')\right\rbrace $$

subject to

$$ {q^e}' = R(\theta,s)q^e - c^e - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\int_{s'\in S} p(s') x^e(s') ds + {x^e}(s')}$$
</td>
<td width=400 style="vertical-align:top;">

\${\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\int_{s'\in S} p(s') x^e(s') ds + {x^e}(s')}\$

captures trade in aggregate state contingent securities.

Markets for aggregate risks are complete.

</td>
</table>
<br>

---

### The entrepreneur's intertemporal problem

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$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e  v^e({q^e}')\right\rbrace $$

subject to

$$ {q^e}' = R(\theta,s)q^e - c^e - \int_{s'\in S} p(s') x^e(s') ds + {x^e}(s')$$
</td>
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**Assumption (anonymity)**

Entrepreneurs are anonymous across financial markets.

<br>

**Assumption (interior borrowing)**

Intratemporal financial allocations \$x^e(s')\$ are unconstrained.

<br>

**Assumption (regularity)**

\$R\$ invertible, continuously differentiable, convex.

</td>
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<br>

---

### The household's problem

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$$ v(q) = \max_{x,c,h,{q}'} \mathbb{E} \left\lbrace u(c,h)+ \beta  v({q}')\right\rbrace $$

subject to

$$ q' = (1+r)q + wh - c - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\int_{s'\in S} p(s') x(s') ds + x(s')}$$
</td>
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\${\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\int_{s'\in S} p(s') x(s') ds + {x}(s')}\$

captures trade in aggregate state contingent securities.

Markets for aggregate risks are complete.

</td>
</table>
<br>

<!-- ---

class: left, top
### How the intervention works
<br>
<br>
<img src="images/IncentiveImprovingStimulusTable.svg" width=1000></img> -->

---

### Factor markets
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$$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255} l = \dfrac{\mathbb{E}_\Theta f(\theta,k,h)}{(1+r)q^e} $$

$$\dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r} = 1+l\tau$$

$$w = \mathbb{E}_\Theta f_h(\theta,k,h)( 1-\tau )$$
</td>
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\$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\text{Leverage, }l\$
</td>
</table>
<br>

---

### Factor markets
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$$ l = \dfrac{\mathbb{E}_\Theta f(\theta,k,h)}{(1+r)q^e} $$

$$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r} = 1+l\tau$$

\$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\rho := \dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r}\$
</td>
</table>
<br>

---

### Factor markets
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$$ l = \dfrac{\mathbb{E}_\Theta f(\theta,k,h)}{(1+r)q^e} $$

$$\dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r} = 1+l\tau$$

$$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255} w = \mathbb{E}_\Theta f_h(\theta,k,h)( 1-\tau )$$
</td>
<td width=400 style="vertical-align:top;">
\$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\text{Wages, }w\$

</td>
</table>
<br>

<!-- ---

<br>
<br>
<br>
Markets equate innovations in expected marginal value
$$\dfrac{\beta(1+r')u'(c',h')}{u'(c,h)} =
 \mathbb{E}_{\Theta} \dfrac{\beta^e R'(θ',s'){u^e}'({c^e}'(\theta'))}{{u^e}'(c^e)}$$
<br>
To change the allocation of risk, you can either
- tax / regulate financial markets directly, or
- manipulate future marginal values.
<br> -->

---

<br>
<br>
$$  \dfrac{\beta^e\ {\mathbb{E}'_{\Theta}u^{e}}'({c^e}'(\theta'))}{{u^e}'(c^e)} =  \dfrac{\beta\ u'({c}',h')}{u'(c,h)}  $$
<br>
<br>

---

<br>
<br>
$$ (1+\omega) \dfrac{\beta^e\ {\mathbb{E}'_{\Theta}u^{e}}'({c^e}'(\theta'))}{{u^e}'(c^e)} =  \dfrac{\beta\ u'({c}',h')}{u'(c,h)}  $$
<br>
<br>

Under optimal policy
$$ \dfrac{\partial\omega}{\partial l},\dfrac{\partial\omega}{\partial \sigma}  > 0.$$

Optimal macroprudential policy leans against

- fluctuations in leverage, and
- entrepreneurs' exposure to risk shocks.

---

- Cyclical risk is a complement to downturn moral hazard.

- Entrepreneurs accept too much cyclical risk,

- amplifying the cost of moral hazard in downturns,

- Arnott-Stiglitz: Regulate cyclical risk (macroprudential)

<!-- ---

Denote the wage subsidy by \$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\varsigma\$

$$ q' = (1+r)q + wh(1+{\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\varsigma}) - c - \int_{s\in S} p(s) x(s) ds + x(s') - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}T}$$

Balanced budget constraint,

$$ \color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}T = wh\varsigma$$

<br>
**Proposition**

In the absence of macroprudential policy, the optimal wage subsidy satisfies
$$ \dfrac{\partial\varsigma}{\partial l},\dfrac{\partial\varsigma}{\partial \sigma}  > 0.$$ -->

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# Optimal wage subsidy policy

---

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**Proposition**

Let
$$u(c,h) = \log c - \dfrac{h^{1+\psi}}{1+\psi},\qquad u^e(c^e) = \log c^e. $$

Optimal wage subsidy:
$$\varsigma^* = \dfrac{\tau}{1-\tau} - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\hat{\lambda'}}(1-\beta^e) \dfrac{ 1+l\tau }{l(1-\tau)}$$

where

$$\lambda = \frac{{u^e}'(\bar{c}^e)}{u_c(c,h)},\qquad\hat{\lambda}' := \frac{\lambda'-\lambda_0}{ \lambda'}.$$
  <br>
</td>
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<br>

</td>
</table>
<br>

---

Optimal wage subsidy
$$\varsigma^* = {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\dfrac{\tau}{1-\tau}} - \hat{\lambda'}(1-\beta^e) \dfrac{ 1+l\tau }{l(1-\tau)}$$

</td>
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<br>
  <br>
    <br>
      <br>
Optimal wage subsidy
- completely offsets the financial wedge on impact,
</td>
</table>
<br>

---

Optimal wage subsidy
$$\varsigma^* = \dfrac{\tau}{1-\tau} - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\hat{\lambda'}}(1-\beta^e) \dfrac{ 1+l\tau }{l(1-\tau)}$$

where

$$\hat{\lambda}' := \frac{\lambda'-\lambda_0}{ \lambda'},$$

\$\lambda' = \lambda\dfrac{\beta^e}{\beta}(1+l\tau).\$

\$\hat{\lambda}'\$ is a summary statistic for history of past financial wedges.

<br>
</td>
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<br>
Optimal wage subsidy
- completely offsets the financial wedge on impact,
- is moderated by past financial wedges.
</td>
</table>
<br>

---

<br>

*Benefit*

Wage subsidies
- complement firms' wealth,
- encourage precaution during expansions, and
- decreases financial frictions in downturns.
- First order welfare gain.

<br>

*Cost*

Wage subsidies
- introduces a distortion between labour supply and demand,
- Second order welfare cost.

---

# Quantitative exercise

---

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### The entrepreneur

combines their own wealth with borrowed wealth and labour.

Contracts are endogenously incomplete. Entrepreneurs can hide income from
external creditors.

External creditors can audit the firm and uncover hidden income, but these
audits are noisy.

(Duncan and Nolan, 2019)

</td>
<td width=550 style="vertical-align:top;">
  <br>  <br>
<a title="linkedineditors, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons" href="https://commons.wikimedia.org/wiki/File:Carlos_Ghosn_(15766546977).jpg"><img width="512" alt="Carlos Ghosn (15766546977)" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Carlos_Ghosn_%2815766546977%29.jpg/512px-Carlos_Ghosn_%2815766546977%29.jpg"></a>
</td>
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<br>

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Leverage and the factor price wedge

<!-- ---

### Preferences

$$ u^e(c^e) = \log(c^e)$$

$$ u(c,h) = \dfrac{c^{1-γ}}{1-\gamma} + \dfrac{h^{1+\psi}}{1+\psi}$$ -->

<!-- ---

class: left, top
<table width=1100>
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  ## The model

#### Production

$$ y = zh^\alpha$$

#### Aggregate demand
  $$ y = c + c^e$$

#### Labor supply
  $$ \dfrac{h^\psi}{c^{-\gamma}} = w $$

#### Entrepreneurs' optimal consumption
  $$ c^e = (1-\beta^e)\rho n^e $$

</td>
<td width=500 style="vertical-align:top;">

#### Factor prices
  $$ \rho = 1+l\tau $$
  $$ w = \dfrac{\alpha y}{h}(1-\tau)$$

#### Intratemporal financial contracts

$$ l = \dfrac{y}{n^e}$$
  $$ \tau = \mathcal{T}(l,\sigma)$$

#### Risk sharing & the distribution
  $$ \lambda' =  \dfrac{c^{-\gamma}}{(c^e)^{-1}}  $$
  $$ \lambda' =  \lambda \dfrac{\beta^e}{\beta}\rho  $$

</td>
</table>
<br> -->

<!--
class: left, top
<table width=1100>
<td width=500 style="vertical-align:top;">
  ## The model

- Competitive equilibrium real allocations under rational expectations
can be expressed in terms of backward looking equations.

- Solving for financial allocations and asset prices requires projection.
    - Agents make forward looking portfolio allocations, but these have
    consequences for real allocations only in the next period, and only
    contingent on contemporaneous realised states.

- Solving for constrained efficient real allocations requires projection.

</td>
<td width=500 style="vertical-align:top;">

</td>
</table>
<br> -->

---

<table width=1100>
<td width=600 style="vertical-align:top;">
- We estimate the model on US business cycle data, with no macroprudential or wage subsidy policy.

- We add a wage subsidy, via a simple rule, and find the optimal
simple rule and the associated welfare gain.
</td>
<td width=400 style="vertical-align:top;">

</td>
</table>
<br>

---

<table width=1100>
<td width=700 style="vertical-align:top;">
We propose the following simple rule:

$$ \varsigma = -\phi_\varsigma (y-y_0)$$

where \$\varsigma\$ is the wage subsidy (tax if negative), and \$y_0\$ is deterministic
steady state output.

<!-- <br><br>
The policymaker balances their budget with a lump sum tax paid by households

$$ T = wh\varsigma$$
$$ q' = (1+r)q + wh(1+\varsigma) - c - \int_{s\in S} p(s) x(s) ds + x(s') - T$$ -->

</td>
<td width=400 style="vertical-align:top;">

</td>
</table>
<br>

---

Welfare gain is expressed as a share of business cycle welfare losses.<br>
Shaded area indicates 90% credible interval.

---

Welfare gain is expressed as a share of business cycle welfare losses.

---

If the margin that the policy is acting on is distorted,<br>
then the Arnott-Stiglitz logic doesn't necessarily hold.

The cost of the tax distortion could be first order.

We add
1. A static labour tax
2. Dynamic New Keynesian markups

---

Welfare gain is expressed as a share of business cycle welfare losses.<br>
Shaded area indicates 90% credible interval.

---

We derive a small-scale log-linear New Keynesian version of our model.
The terms in black are as in Gali (2007).

IS Curve

$$({\color{blue}\zeta} + \gamma{\color{blue}-1})y = ({\color{blue}\zeta} + \gamma{\color{blue}-1})\mathbb{E}[y'] - {\color{blue}\zeta}(i - \mathbb{E}[\pi'])  - {\color{blue}(\zeta-1)\varphi l   - \gamma\Delta \varphi (\rho_\sigma-\varphi) \sigma}$$

Phillips Curve

$$\pi = \beta \mathbb{E}[\pi'] + \lambda\left(\chi + \gamma - 1\right) y -\lambda\chi z {\color{blue}- \lambda \varsigma + \lambda(\zeta + \delta -1) l + \lambda(\delta_\sigma-\gamma\Delta\varphi)\sigma}$$

Leverage updating

$${\color{blue}\zeta l    =  \left(\zeta-\varphi\right) L l + \gamma\Delta\varphi \sigma - (1+\gamma\Delta)\varphi L\sigma - (\gamma-1)(y-Ly)}$$

The policy variables are \$i\$ and \${\color{blue}\varsigma}\$.

---

---

<table width=1100>
<td width=600 style="vertical-align:top;">
Proposition 2<br>
Under joint optimal monetary and wage subsidy policy, the optimal path of inflation
is zero in all periods \$\pi_t = 0 \: \forall t\$.
<br>
<br>
<br>
<br>
<br>
<br>
Proposition 3<br>
Let \$\gamma>1\$. When the convexity of monitoring costs is relatively high (low),
optimal output growth is lower (higher) when leverage is high, all else equal.
<br>
<br>
<br>
</td>
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Proposition 2<br>
Under joint optimal monetary and wage subsidy policy, the optimal path of inflation
is zero in all periods \$\pi_t = 0 \: \forall t\$.
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- Would get the same result from a standard NK model with markup shocks and the same
policy instruments.
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Proposition 3<br>
Let \$\gamma>1\$. When the convexity of monitoring costs is relatively high (low),
the optimal output growth is lower (higher) when leverage is high, all else equal.
<br>
- The convexity of monitoring costs generates a wedge between the social and
private marginal costs of monitoring (see Hillier and Worral 1994 EJ).
When high, the planner seeks a smoother path of leverage.
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United Kingdom
- Furlough scheme.
- Not conditioned on firm outcomes, complements inside wealth.
- Furloughed workers not permitted to work.
  - Not great stimulus.
  - Not much value for employers (not very prudential).
  - Seems to not have been enforced (maybe a good thing).

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We present a model where moral hazard generates a macroprudential externality.

In lieu of aggregate demand externalities, there is still a role for fiscal
stimulus.

If the stimulus programme complements inside wealth, like a labour subsidy,
then it will
- encourage firms' prudence during the preceding expansion, and
- reduce the costs of the moral hazard friction,
- increasing welfare.

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