class: center, middle ## Prudential fiscal stimulus
### Alfred Duncan #### University of Kent
### Charles Nolan #### University of Glasgow
--- class: left, middle ### 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?
--- class: left, middle
### Framework Similar to di Tella (2017, JPE). ### 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.
--- class: left, middle ### Intuition behind our result
Labour supply is a complement to firms' inside wealth.
Subsidised labour in downturns increases the marginal value of 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.
--- class: left, middle ### Wage subsidies
- 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).
Figure: Countries with new or existing wage subsidy schemes during the Covid-19 pandemic (Sources: ILO, IMF, authors’ calculations)
--- class: left, middle ### Related literature
Macropru - di Tella (2017); Duncan and Nolan (2022). - Farhi and Werning (2016); Schmitt-Grohe and Uribe (2012); - Gertler, Kiyotaki, and Prestipino (2020); Bianchi and Mendoza (2018). Information economics - Arnott and Stiglitz (1991)
--- class: left, top
### 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)
Main results also hold under - Hidden action (Grossman Hart, 1983) - Costly state verification with limited commitment (Krasa and Villamil, 2000) - Costly state falsification (Lacker and Weinberg, 1989).
--- class: left, middle ### The entrepreneur's within period problem
$$ R(\theta,s),b^e,h^e =\arg \max \mathbb{E} \left\lbrace \hat{v}^e(R(\theta,s)q^e) | \theta\right\rbrace $$ 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 $$ [\text{capital budget constraint}] $$ $$ [\text{truth telling constraint}] $$ $$ [\text{lender participation constraint}] $$
--- class: left, middle ### The entrepreneur's between periods problem
$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e v^e({q^e}')\right\rbrace $$ subject to $$ {q^e}' = R(\theta,s)q^e - c^e - \int_{s\in S} p(s) x^e(s) ds + {x^e}(s')$$
--- class: left, middle ### The entrepreneur's between periods problem
$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e v^e({q^e}')\right\rbrace $$ subject to $$ {q^e}' = {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}R(\theta,s)}q^e - c^e - \int_{s\in S} p(s) x^e(s) ds + {x^e}(s')$$
\\(\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 within period problem.
--- class: left, middle ### The entrepreneur's between periods problem
$$ v^e(q^e) = \max_{x^e,c^e,{q^e}'} \mathbb{E} \left\lbrace u^e(c^e) + \beta^e v^e({q^e}')\right\rbrace $$ subject to $$ {q^e}' = R(\theta,s)q^e - c^e - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\int_{s\in S} p(s) x^e(s) ds + {x^e}(s')}$$
\\({\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.
--- class: left, middle ### The household's problem
$$ v(q) = \max_{x,c,h,{q}'} \mathbb{E} \left\lbrace u(c,h)+ \beta v({q}')\right\rbrace $$ subject to $$ q' = (1+r)q + wh - c - {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\int_{s\in S} p(s) x(s) ds + x(s')}$$
\\({\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.
--- class: left, middle ### Resulting equilibrium conditions
$$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255} l = \dfrac{\mathbb{E}_\Theta f(\theta,k,h)}{(1+r)q^e} $$ $$\dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r} = 1+l\tau$$ $$w = \mathbb{E}_\Theta f_h(\theta,k,h)( 1-\tau )$$
\\(\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\text{Leverage, }l\\)
--- class: left, middle ### Resulting equilibrium conditions
$$ l = \dfrac{\mathbb{E}_\Theta f(\theta,k,h)}{(1+r)q^e} $$ $$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r} = 1+l\tau$$ $$w = \mathbb{E}_\Theta f_h(\theta,k,h)( 1-\tau )$$
\\(\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\text{The equity risk premium, }\\) \\(\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\rho := \dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r}\\)
--- class: left, middle ### Resulting equilibrium conditions
$$ l = \dfrac{\mathbb{E}_\Theta f(\theta,k,h)}{(1+r)q^e} $$ $$\dfrac{\mathbb{E}_\Theta [R(\theta,s)]}{1+r} = 1+l\tau$$ $$\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255} w = \mathbb{E}_\Theta f_h(\theta,k,h)( 1-\tau )$$
\\(\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\text{Wages, }w\\)
--- class: left, top ### The evolution of leverage Log-linear example for illustration $$ l' =\underbrace{\left(1-\frac{\psi}{\zeta}\right) l}_{\text{persistent}} + \dfrac{\gamma\delta\phi}{\zeta} \sigma - \dfrac{(1+\gamma\delta)\phi}{\zeta} L\sigma - \dfrac{\gamma-1}{\zeta}(y - Ly) $$ --- class: left, top ### The evolution of leverage Log-linear example for illustration $$ l' =\left(1-\frac{\psi}{\zeta}\right) l + \underbrace{\dfrac{\gamma\delta\phi}{\zeta} \sigma - \dfrac{(1+\gamma\delta)\phi}{\zeta} L\sigma}_{\text{increases in risk shocks}} - \dfrac{\gamma-1}{\zeta}(y - Ly) $$ --- class: left, top ### The evolution of leverage Log-linear example for illustration $$ l' =\left(1-\frac{\psi}{\zeta}\right) l + \dfrac{\gamma\delta\phi}{\zeta} \sigma - \dfrac{(1+\gamma\delta)\phi}{\zeta} L\sigma - \underbrace{\dfrac{\gamma-1}{\zeta}(y - Ly)}_{\text{financial accelerator}} $$ #### The macroprudential externality - Financial accelerator: \\(\downarrow\\) output \\(\rightarrow\\) \\(\uparrow\\) costs of moral hazard. - \\(\uparrow\\) Cyclical firm risk \\(\rightarrow\\) \\(\uparrow/\downarrow\\) costs of moral hazard in recessions / expansions. --- class: left, top ### Optimal macroprudential policy
From related work, the macroprudential wedge \\(\omega\\)
acts on the allocation of cyclical risk.
$$ \lambda' = \dfrac{\beta^e}{\beta} \lambda \rho (1+\omega)$$ where $$ \lambda' = \dfrac{u'(c,h)}{{u^e}'(c^e)} , $$ and \\(\rho\\) is the equity risk premium. Under optimal policy $$ \dfrac{\partial\omega}{\partial l},\dfrac{\partial\omega}{\partial \sigma} > 0.$$ --- class: left, top ### Optimal wage subsidy policy
Wage subsidies \\(\varsigma\\) also distort the
competitive allocation of cyclical risk.
$$ \lambda' = \dfrac{\beta^e}{\beta} \lambda \left[1+l\left(1-\dfrac{-u\_h}{u\_c}\dfrac{1}{\mathbb{E}_\Theta f\_h(\theta,h)(1+\varsigma)}\right)\right]$$
**Proposition 1** In the absence of macroprudential policy,
the optimal wage subsidy satisfies $$ \dfrac{\partial\varsigma}{\partial l},\dfrac{\partial\varsigma}{\partial \sigma} > 0.$$ --- class: left, top ### Optimal wage subsidy policy - example with log utility
**Proposition 2** 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 $$\hat{\lambda}' := \frac{\lambda'-\lambda_0}{ \lambda'}.$$
--- class: left, top ### Optimal wage subsidy policy - example with log utility
Optimal wage subsidy $$\varsigma^* = {\color[rgb]{0.7372549019607844,0.5882352941176471, 0.9019607843137255}\dfrac{\tau}{1-\tau}} - \hat{\lambda'}(1-\beta^e) \dfrac{ 1+l\tau }{l(1-\tau)}$$
Optimal wage subsidy - completely offsets the financial wedge on impact,
--- class: left, top ### Optimal wage subsidy policy - example with log utility
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.
Optimal wage subsidy - completely offsets the financial wedge on impact, - is moderated by past financial wedges.
--- class: left, middle # Quantitative exercise --- class: left, top
## The model #### Production $$ y = zh^\alpha$$ #### Aggregate demand $$ y = c + c^e$$ #### Labor supply $$ \dfrac{h^\psi}{c^{-\gamma}} = w $$ #### Entrepreneurs' optimal consumption $$ c^e = (1-\beta^e)\rho n^e $$
#### Factor prices $$ \rho = 1+l\tau $$ $$ w = \dfrac{\alpha y}{h}(1-\tau)$$ #### within period financial contracts $$ l = \dfrac{y}{n^e}$$ $$ \tau = \mathcal{T}(l,\sigma)$$ #### Risk sharing & the distribution $$ \lambda' = \dfrac{c^{-\gamma}}{(c^e)^{-1}} $$ $$ \lambda' = \lambda \dfrac{\beta^e}{\beta}\rho $$
--- class: left, top ### Wage subsidy simple rule
We propose the following simple rule: $$ \varsigma = -\phi_\varsigma (y-y_0)$$
Balanced budget constraint: $$ T = wh\varsigma$$ $$ q' = (1+r)q + wh(1+\varsigma) - c - \int_{s\in S} p(s) x(s) ds + x(s') - T$$
--- class: center, top #### 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. --- class: center, top #### With 40% constant labour tax
Welfare gain is expressed as a share of business cycle welfare losses.
Shaded area indicates 90% credible interval. --- class: center, top #### Persistence of TFP shocks Samples are drawn from the posterior density. Wage subsidy coefficient \\(\phi_\varsigma = 1\\).
Welfare gain is expressed as a share of business cycle welfare losses.
Shaded area indicates 90% credible interval. --- class: left, top ### 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.