Prudential fiscal stimulus

MMF Conference, 6 September 2022

Alfred Duncan

University of Kent

Charles Nolan

University of Glasgow

This project is supported by research funding from UKRI Grant Ref: ES/V015559/1

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The problem (the Greenspan put)

Stimulus policies can increase moral hazard.

Anticipating to be rescued in downturns, firms
might take more risk today.

This paper asks:
Can we stimulate the economy in a way that
encourages prudence?

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Our contribution

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.

Figure: Countries with new or existing wage subsidy schemes during the Covid-19 pandemic (Sources: ILO, IMF, authors’ calculations)

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Intuition behind our result

Absent intervention

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Intuition behind our result

Introduction of the wage subsidy

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Intuition behind our result

Firm's risk allocation response

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Intuition behind our result

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Intuition behind our result

Labour supply is a complement to firms' inside wealth.

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

The ex-post wage subsidy replicates an ex-ante macroprudential intervention.

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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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The macroprudential externality10 / 46

The macroprudential externality

Three ingredients

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

Jump to results

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The entrepreneur's intratemporal problem

R (θ, s), b^{e}, h^{e} = \arg max E {{\hat{v}}^{e} (R (θ, s) q^{e}) | θ}

subject to

$R (θ, s) q^{e} \leq f (θ, k, h) - r^{b} (θ) b^{e} - w h^{e} \forall θ$

in addition to $[capital budget constraint]$ $[truth telling constraint]$ $[lender participation constraint]$

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The entrepreneur's intratemporal problem

R (θ, s), b^{e}, h^{e} = \arg max E {{\hat{v}}^{e} (R (θ, s) q^{e}) | θ}

subject to

$R (θ, s) q^{e} \leq f (θ, k, h) - r^{b} (θ) b^{e} - w h^{e} \forall θ$

in addition to $[capital budget constraint]$ $[truth telling constraint]$ $[lender participation constraint]$

{\hat{v}}^{e}

is concave.

Entrepreneurs are risk averse, but behave as risk neutral when financial markets are perfect.

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The entrepreneur's intratemporal problem

R (θ, s), b^{e}, h^{e} = \arg max E {{\hat{v}}^{e} (R (θ, s) q^{e}) | θ}

subject to

$R (θ, s) q^{e} \leq f (θ, k, h) - r^{b} (θ) b^{e} - w h^{e} \forall θ$

in addition to $[capital budget constraint]$ $[truth telling constraint]$ $[lender participation constraint]$

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

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

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The entrepreneur's intratemporal problem

R (θ, s), b^{e}, h^{e} = \arg max E {{\hat{v}}^{e} (R (θ, s) q^{e}) | θ}

subject to

$R (θ, s) q^{e} \leq f (θ, k, h) - r^{b} (θ) b^{e} - w h^{e} \forall θ$

in addition to $[capital budget constraint]$ $[truth telling constraint]$ $[lender participation constraint]$

r^{b} (θ) b^{e}

is the net interest paid on loans.

This is state-contingent, negative in default.

Partially insures fluctuations in $θ$ .

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The entrepreneur's intratemporal problem

R (θ, s), b^{e}, h^{e} = \arg max E {{\hat{v}}^{e} (R (θ, s) q^{e}) | θ}

subject to

$R (θ, s) q^{e} \leq f (θ, k, h) - r^{b} (θ) b^{e} - w h^{e} \forall θ$

in addition to $[capital budget constraint]$ $[truth telling constraint]$ $[lender participation constraint]$

w h^{e}

is the total wage bill.

Workers are hired before the realisation of $θ$ .

Wages are not contingent on $θ$ .

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The entrepreneur's intertemporal problem

v^{e} (q^{e}) = max_{x^{e}, c^{e}, {q^{e}}^{'}} E {u^{e} (c^{e}) + β^{e} v^{e} ({q^{e}}^{'})}

subject to

${q^{e}}^{'} = R (θ, s) q^{e} - c^{e} - \int_{s^{'} \in S} p (s^{'}) x^{e} (s^{'}) d s + x^{e} (s^{'})$

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The entrepreneur's intertemporal problem

v^{e} (q^{e}) = max_{x^{e}, c^{e}, {q^{e}}^{'}} E {u^{e} (c^{e}) + β^{e} v^{e} ({q^{e}}^{'})}

subject to

${q^{e}}^{'} = R (θ, s) q^{e} - c^{e} - \int_{s^{'} \in S} p (s^{'}) x^{e} (s^{'}) d s + x^{e} (s^{'})$

R (θ, s)

is an indirect gross return function.

It is the result of privately optimal borrowing and hiring, from the entrepreneur's intratemporal problem.

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The entrepreneur's intertemporal problem

v^{e} (q^{e}) = max_{x^{e}, c^{e}, {q^{e}}^{'}} E {u^{e} (c^{e}) + β^{e} v^{e} ({q^{e}}^{'})}

subject to

${q^{e}}^{'} = R (θ, s) q^{e} - c^{e} - \int_{s^{'} \in S} p (s^{'}) x^{e} (s^{'}) d s + x^{e} (s^{'})$

$\int_{s^{'} \in S} p (s^{'}) x^{e} (s^{'}) d s + x^{e} (s^{'})$

captures trade in aggregate state contingent securities.

Markets for aggregate risks are complete.

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The entrepreneur's intertemporal problem

v^{e} (q^{e}) = max_{x^{e}, c^{e}, {q^{e}}^{'}} E {u^{e} (c^{e}) + β^{e} v^{e} ({q^{e}}^{'})}

subject to

${q^{e}}^{'} = R (θ, s) q^{e} - c^{e} - \int_{s^{'} \in S} p (s^{'}) x^{e} (s^{'}) d s + x^{e} (s^{'})$

Assumption (anonymity)

Entrepreneurs are anonymous across financial markets.

Assumption (interior borrowing)

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

Assumption (regularity)

$R$ invertible, continuously differentiable, convex.

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The household's problem

v (q) = max_{x, c, h, q^{'}} E {u (c, h) + β v (q^{'})}

subject to

$q^{'} = (1 + r) q + w h - c - \int_{s^{'} \in S} p (s^{'}) x (s^{'}) d s + x (s^{'})$

\int_{s^{'} \in S} p (s^{'}) x (s^{'}) d s + x (s^{'})

captures trade in aggregate state contingent securities.

Markets for aggregate risks are complete.

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Factor markets

$l = \frac{E_{Θ} f (θ, k, h)}{(1 + r) q^{e}}$

$\frac{E_{Θ} [R (θ, s)]}{1 + r} = 1 + l τ$

$w = E_{Θ} f_{h} (θ, k, h) (1 - τ)$

Leverage, l

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Factor markets

$l = \frac{E_{Θ} f (θ, k, h)}{(1 + r) q^{e}}$

$\frac{E_{Θ} [R (θ, s)]}{1 + r} = 1 + l τ$

$w = E_{Θ} f_{h} (θ, k, h) (1 - τ)$

The equity risk premium,

$ρ := \frac{E_{Θ} [R (θ, s)]}{1 + r}$

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Factor markets

$l = \frac{E_{Θ} f (θ, k, h)}{(1 + r) q^{e}}$

$\frac{E_{Θ} [R (θ, s)]}{1 + r} = 1 + l τ$

$w = E_{Θ} f_{h} (θ, k, h) (1 - τ)$

Wages, w

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The competitive allocation of aggregate risk

$\frac{β^{e} {E_{Θ}^{'} u^{e}}^{'} ({c^{e}}^{'} (θ^{'}))}{{u^{e}}^{'} (c^{e})} = \frac{β u^{'} (c^{'}, h^{'})}{u^{'} (c, h)}$

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Optimal macroprudential policy

$(1 + ω) \frac{β^{e} {E_{Θ}^{'} u^{e}}^{'} ({c^{e}}^{'} (θ^{'}))}{{u^{e}}^{'} (c^{e})} = \frac{β u^{'} (c^{'}, h^{'})}{u^{'} (c, h)}$

Under optimal policy $\frac{\partial ω}{\partial l}, \frac{\partial ω}{\partial σ} > 0.$

Optimal macroprudential policy leans against

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

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The macroprudential externality

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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Optimal wage subsidy policy28 / 46

Optimal wage subsidy policy - example with log utility

Proposition

Let $u (c, h) = \log c - \frac{h^{1 + ψ}}{1 + ψ}, u^{e} (c^{e}) = \log c^{e} .$

Optimal wage subsidy: $ς^{*} = \frac{τ}{1 - τ} - \hat{λ^{'}} (1 - β^{e}) \frac{1 + l τ}{l (1 - τ)}$

where

$λ = \frac{{u^{e}}^{'} ({\bar{c}}^{e})}{u_{c} (c, h)}, {\hat{λ}}^{'} := \frac{λ^{'} - λ_{0}}{λ^{'}} .$

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Optimal wage subsidy policy - example with log utility

Optimal wage subsidy $ς^{*} = \frac{τ}{1 - τ} - \hat{λ^{'}} (1 - β^{e}) \frac{1 + l τ}{l (1 - τ)}$

Optimal wage subsidy

completely offsets the financial wedge on impact,

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Optimal wage subsidy policy - example with log utility

Optimal wage subsidy $ς^{*} = \frac{τ}{1 - τ} - \hat{λ^{'}} (1 - β^{e}) \frac{1 + l τ}{l (1 - τ)}$

where

${\hat{λ}}^{'} := \frac{λ^{'} - λ_{0}}{λ^{'}},$

$λ^{'} = λ \frac{β^{e}}{β} (1 + l τ) .$

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

Optimal wage subsidy

completely offsets the financial wedge on impact,
is moderated by past financial wedges.

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How the intervention works

Benefit

Wage subsidies

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

Cost

Wage subsidies

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

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Quantitative exercise33 / 46

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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Exercise

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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Wage subsidy simple rule

We propose the following simple rule:

$ς = - ϕ_{ς} (y - y_{0})$

where $ς$ is the wage subsidy (tax if negative), and $y_{0}$ is deterministic steady state output.

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Expected welfare effects of wage subsidy simple rules

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

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Persistence of TFP shocks

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

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Dixit's critique

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

The cost of the tax distortion could be first order.

We add

A static labour tax
Dynamic New Keynesian markups

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With 40% constant labour tax

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

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Interactions with monetary policy

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

IS Curve

$(ζ + γ - 1) y = (ζ + γ - 1) E [y^{'}] - ζ (i - E [π^{'}]) - (ζ - 1) φ l - γ Δ φ (ρ_{σ} - φ) σ$

Phillips Curve

$π = β E [π^{'}] + λ (χ + γ - 1) y - λ χ z - λ ς + λ (ζ + δ - 1) l + λ (δ_{σ} - γ Δ φ) σ$

Leverage updating

$ζ l = (ζ - φ) L l + γ Δ φ σ - (1 + γ Δ) φ L σ - (γ - 1) (y - L y)$

The policy variables are $i$ and $ς$ .

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Optimal policy

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Interactions with monetary policy

Proposition 2

Under joint optimal monetary and wage subsidy policy, the optimal path of inflation
is zero in all periods πt=0∀t.

Proposition 3

Let γ>1. When the convexity of monitoring costs is relatively high (low),
optimal output growth is lower (higher) when leverage is high, all else equal.

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Interactions with monetary policy

Proposition 2
Under joint optimal monetary and wage subsidy policy, the optimal path of inflation is zero in all periods

π_{t} = 0 \forall t

Would get the same result from a standard NK model with markup shocks and the same policy instruments.

Proposition 3
Let $γ > 1$ . When the convexity of monitoring costs is relatively high (low), the optimal output growth is lower (higher) when leverage is high, all else equal.

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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Covid-19 wage subsidies

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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Summary

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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Prudential fiscal stimulus

Alfred Duncan

University of Kent

Charles Nolan

University of Glasgow

The problem (the Greenspan put)

Our contribution

Intuition behind our result

Intuition behind our result

Intuition behind our result

Intuition behind our result

Intuition behind our result

Related literature

The macroprudential externality

The macroprudential externality

Three ingredients

The entrepreneur's intratemporal problem

The entrepreneur's intratemporal problem

The entrepreneur's intratemporal problem

The entrepreneur's intratemporal problem

The entrepreneur's intratemporal problem

The entrepreneur's intertemporal problem

The entrepreneur's intertemporal problem

The entrepreneur's intertemporal problem

The entrepreneur's intertemporal problem

The household's problem

Factor markets

Factor markets

Factor markets

The competitive allocation of aggregate risk

Optimal macroprudential policy

The macroprudential externality

Optimal wage subsidy policy

Optimal wage subsidy policy - example with log utility

Optimal wage subsidy policy - example with log utility

Optimal wage subsidy policy - example with log utility

How the intervention works

Quantitative exercise

The entrepreneur

Exercise

Wage subsidy simple rule

Expected welfare effects of wage subsidy simple rules

Persistence of TFP shocks

Dixit's critique

With 40% constant labour tax

Interactions with monetary policy

Optimal policy

Interactions with monetary policy

Interactions with monetary policy

Covid-19 wage subsidies

Summary

The problem (the Greenspan put)

Help