Title

## Prudential fiscal stimulus

ZEW Public Finance Conference, 4  May 2023
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#### Alfred Duncan
#### University of Kent
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#### Charles Nolan
#### University of Glasgow
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This project is supported by research funding from UKRI Grant Ref: ES/V015559/1

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### The conventional view (the Greenspan put)
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Stimulus policies increase moral hazard.
<br><br>
Anticipating to be rescued in downturns, firms
might take more risk today.
<br><br>
<img src="images/Haldane.png" width=650></img>
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### The theorem (Arnott-Greenwald-Stiglitz)
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<i>Policymakers can reduce moral hazard, and increase efficiency,
by taxing or regulating the complements of moral hazard and/or
subsidising the substitutes for moral hazard.</i>
<br><br><br>
Doesn't align with concerns about the Greenspan put.
<br><br>
We ask, when does fiscal stimulus reduce the moral hazard
problems that concern prudential policymakers?
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### Our contribution
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- We show that countercyclical wage subsidies can improve welfare,
by reducing the cost of moral hazard frictions in firm financing.

- Result holds 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>
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<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)

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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.

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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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### 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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### 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).
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<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)

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<br> -->

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### 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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### 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

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# 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>

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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)

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  <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>
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### The entrepreneur's intratemporal problem

Extension<br>
We present an extension using an alternative financial friction applied to
a financial intermediary sector. The main results of the paper go through.
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### The entrepreneur's intertemporal problem

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<img src="programmes/Prog1.png" width=700></img>
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  \${\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.
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### 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

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

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

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

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

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

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

$$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255} w = \mathbb{E}_\Theta f_h(\theta,k,h)( 1-\tau )$$
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\$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\text{Wages, }w\$

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<br>
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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> -->

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

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- 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)

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

a. Optimal wage subsidy:
$$\varsigma^* =
\dfrac{\tau}{1-\tau}\left(1-\left(\frac{l_0 \rho -  l\rho_0}{\rho-1}\right)\frac{ (1-\beta^e)}{l_0-(1-\beta^e)\rho_0} \right).$$

b. Output is completely stabilised in response to uncertainty shocks,
and is proportional to total factor productivity.

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Under the competitive equilibrium in the absence of wage subsidy
  policy, real output follows

$$  y  = z \left(\frac{\alpha l(1-\tau)}{l-(1-\beta^e)\rho}\right)^{\frac{\alpha}{1+\psi}}.$$

Under the optimal wage subsidy, real output follows

$$    y  =
  z \left(\frac{\alpha l_0 }{l_0-(1-\beta^e)\rho_0}\right)^{\frac{\alpha}{1+\psi}}$$

Real output is proportional to
  total factor productivity in both regimes, but only responds to fluctuations
  in leverage and risk under the competitive equilibrium.

Under the optimal wage subsidy regime, output does not respond
  to uncertainty shocks.

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<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.

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# Quantitative exercise

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Leverage and the factor price wedge

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### Preferences

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

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

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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
  $$ c^e = (1-\beta^e)\rho n^e $$

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#### 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
  $$ \lambda' =  \dfrac{c^{-\gamma}}{(c^e)^{-1}}  $$
  $$ \lambda' =  \lambda \dfrac{\beta^e}{\beta}\rho  $$

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  ## 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.

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- 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.
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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

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

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

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

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### Commitment vs discretion

Optimal wage subsidy policy generates an ex post distortion, that reduces
efficiency. The benefit of the policy derives from changes in agents' actions
ex ante.

In the flexible price model, a policymaker optimising under discretion
does not use wage subsidy policies.

### Sticky prices

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

New Keynesian markups act on same margin as policy. We verify that New Keynesian
frictions do not overturn the result.

In fact, in the NK extension, the commitment problem is less severe, as there
is an ex post incentive to stimulate the economy.

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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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