The debt brake: business cycle and welfare consequences of Germany’s new fiscal policy rule

Abstract

In a New Keynesian DSGE model with non-Ricardian consumers, we show that automatic stabilization according to a countercyclical spending rule following the idea of the debt brake is well suited both to steer the economy and in terms of welfare. In particular, the adjustment account set up to record public deficits and surpluses serves well to keep the level of government debt stable. However, it is essential to design its feedback to government spending correctly, where discretionary lapses should be corrected faster than lapses due to estimation errors.

Appendices

Appendix 1: How to set the feedback of the adjustment account

Assuming that the fundamental shocks (technology shocks (TS), shocks to consumer preferences (CPS), price mark-up shocks (PMS), monetary shocks (MS) and fiscal spending shocks (FS)) are orthogonal as standard in the literature, we can decompose the welfare loss function as a linear combination of the structural shocks, i.e ${{\mathbb{W}}_0(\rho)={\mathbb{W}}_0^{TS}(\rho)+ {\mathbb{W}}_0^{CPS}(\rho) +{\mathbb{W}}_0^{PMS}(\rho) +{\mathbb{W}}_0^{MS}(\rho)+{\mathbb{W}}_0^{FS}(\rho)}$, where all parameters are fixed at their baseline value except ρ. Then, we continue by calculating ${\mathbb{W}_0(\rho)}$, where ρ is defined over the following tuple [0.00, 0.05, 0.10, 0.15, 0.20]. While conducting this exercise we find that the welfare loss metric ${\mathbb{W}_0(\rho)}$ takes its lowest value for ρ = 0.05, which we take as our baseline value. A more sophisticated approach would be to find the globally optimal value for each fundamental shock by, for example, the MATLAB routine fmincon, which finds a constrained minimum of a scalar function, starting from an initial estimate. In Fig. 12, we report the outcome of such an exercise graphically. It suggests that, if ρ could be fine-tuned towards a specific shock, the value optimally differed with the shock. If movements in the adjustment account can be traced back to technology or price mark-up shocks, fiscal authorities are well advised not to correct fiscal expenditures to sharply in the following period. For the case of fiscal, monetary and consumer preference shocks, the recommendation is somewhat reversed. If fiscal authorities are the source of economic disturbance, the welfare metric reports evidence that a sharper correction in the following period is appropriate as the relative damage imposed on the consumer can be reduced by a factor of four compared to the case in which fiscal authorities only moderately respond to past lapses in expenditures. For the case of consumer preference and monetary shocks, the welfare metric can be reduced by 10% if fiscal authorities move from a very low feedback (ρ = 0.01) to a somewhat higher feedback (ρ = 0.05). To be in line with debt brakes actually implemented in Switzerland or which are planed to be implemented in Germany, we assume that the government has no technology at hand to find-tune the response of the adjustment account towards the specific shock and thus set to ρ = 0.05 for all shocks, which is—on average—the best response to movements in the adjustment account.

Fig. 12

Optimal feedback coefficient for each shock

Appendix 2: Welfare approximation

We know that per-period utility of household j of type i is given by

$$ \left\{\underbrace{\zeta_t\left[(1-\chi)log\left(C_t^i(j)\right)+\chi log(G_t)\right]}_{=u^{i}}+\underbrace{\zeta_t\upsilon_t log\left(L_t^i(j)\right)}_{=V^{i}}\right\}, $$

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where i = o, r (see also Eq. 6). In what follows, we will derive the second-order Taylor approximation of the consumption part of this equation (indicated by u ⁱ) and the leisure part (indicated by V ⁱ) separately for convenience. For consumption, we then get

$$ \begin{aligned} u^{i}_t&\approx\bar{u}^i+\bar{u}^i_{C^i}\left(C_t^i-\bar{C}^i\right) +{{1}\over {2}}\bar{u}^i_{C^iC^i}\left(C_t^i-\bar{C}^i\right)^2+ \bar{u}^i_{G}\left(G_t-\bar{G}\right) +{{1}\over {2}}\bar{u}^i_{GG}\left(G_t-\bar{G}\right)^2 \\ &\quad+\bar{u}^i_{C^i\zeta}\left(C_t^i-\bar{C}^i\right)\left(\zeta_t-\bar{\zeta}\right)+ \bar{u}^i_{G\zeta}\left(G_t-\bar{G}\right)\left(\zeta_t-\bar{\zeta}\right) \\ &=\bar{u}^i +(1-\chi)\frac{\left(C_t^i-\bar{C}^i\right)} {\bar{C}^i}-(1-\chi)\frac{1} {2}\frac{\left(C_t^i-\bar{C}^i\right)^2}{(\bar{C}^i)^2} +\chi\frac{\left(G_t-\bar{G}\right)}{\bar{G}}-\chi\frac{1} {2}\frac{\left(G_t-\bar{G}\right)^2}{\bar{G}^2}\\ &\quad+\frac{\left(\zeta_t-\bar{\zeta}\right)}{\bar{\zeta}} \left[(1+\chi)\frac{\left(C_t^i-\bar{C}^i\right)}{\bar{C}^i} +\chi\frac{\left(G_t-\bar{G}\right)}{\bar{G}}\right] \\ &=\bar{u}^i+(1+\hat{\zeta}_t)\left\{(1-\chi)\left[\frac{\hat{C}^i_t} {\gamma_i}-\frac{1}{2}\frac{(\hat{C}^i_t)^2}{\gamma_i^2}+ \frac{1}{2}\frac{(\hat{C}^i_t)^2}{\gamma_i^2}\right]+ \chi\left[\hat{G}_t-\frac{1}{2}(\hat{G}_t)^2+ \frac{1}{2}(\hat{G}_t)^2\right]\right\} \\ &=\bar{u}^i+(1+\hat{\zeta}_t)\left\{(1-\chi)\frac{\hat{C}^i_t}{\gamma_i}+ \chi\hat{G}_t\right\}, \end{aligned} $$

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where we have used the fact that we defined $\hat{C}^i_t=\frac{\left(C_t^i-\bar{C}^i\right)}{\bar{C}}$ earlier, used the definitions for $\gamma_r=\frac{\upsilon}{1-\chi+\upsilon}\frac{1}{1-\bar{N}}$ and $\gamma_o=\frac{1-\gamma_r\lambda}{1-\lambda}$, respectively, and made use of the commonly known fact that, for any variable X, it holds that $(X_t-\bar{X})\approx\bar{X}\left[\hat{X}_t+\frac{1} {2}\hat{X}_t^2\right]$ and $(X_t-\bar{X})^2\approx\frac{1}{2}\hat{X}_t^2$ when approximating second order. Furthermore, we have neglected the individual household parameter j for notational convenience and remembered that $\bar{\zeta}=1$. In an analogous proceeding as before, for the disutility of labor (utility of leisure) this yields

$$ \begin{aligned} V^{i}_t&\approx\bar{V}^i+\bar{V}^i_{L^i} \left(L_t^i-\bar{L}^i\right)+\frac{1}{2}\bar{V}^i_{L^iL^i} \left(L_t^i-\bar{L}^i\right)^2 +\bar{V}^i_{L^i\zeta} \left(L_t^i-\bar{L}^i\right)\left(\zeta_t-\bar{\zeta}\right) \\ &=\bar{V}^i +\upsilon\frac{\left(L_t^i-\bar{L}^i\right)} {\bar{L}^i}-\upsilon\frac{1}{2}\frac{\left(L_t^i-\bar{L}^i\right)^2}{(\bar{L}^i)^2} +\upsilon\frac{\left(L_t^i-\bar{L}^i\right)\left(\zeta_t-1\right)} {\bar{L}^i}\\ &=\bar{V}^i+\upsilon\left\{ \left[\frac{\hat{L}^i_t}{\gamma_i}- \frac{1}{2}\frac{\left(\hat{L}^i_t\right)^2}{\gamma_i^2}+\frac{1}{2} \frac{\left(\hat{L}^i_t\right)^2}{\gamma_i}\right]\right\}+ \frac{\hat{L}^i_t}{\gamma_i}\left(\upsilon\hat{\zeta}_t\right) \\ &=\bar{V}^i+\left(1+\hat{\zeta}_t\right)\upsilon\frac{\hat{L}^i_t}{\gamma_i} =\bar{V}^i +(1+\hat{\zeta}_t)\upsilon\frac{\hat{L}^i_t}{\gamma_i}. \end{aligned} $$

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Combining the utility of consumption and disutility of labor, we get for household j of type i = o, r that

$$ U^i_t(j)=\underbrace{\bar{u}^i(j)+\bar{V}^i(j)}_{\bar{U}^i}+ (1+\hat{\zeta}_t)\left\{(1-\chi)\frac{\hat{C}^i_t(j)}{\gamma_i}+ \chi\hat{G}_t\right\}+(1+\hat{\zeta}_t)\upsilon\frac{\hat{L}^i_t(j)}{\gamma_i}. $$

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Noting that individuals of type r have a constant consumption pattern due to constant labor supply (see Eqs. 12 and 13), we know that $\hat{C}^r_t(j)=\hat{C}^r_t$, where the latter is given by $\hat{C}^r_t=\gamma_r\hat{C}_t+\varphi\gamma_r\hat{N}_t$. Due to the assumption of complete markets and state-contingent claims that can be purchased by households of type o, we know that $\hat{C}^o_t(j)=\hat{C}^o_t$ (see Woodford 2003, chapter 2 for more details), where the latter is given by $\hat{C}_t^o=\gamma_o\hat{C}_t- \frac{\lambda\gamma_r\varphi}{1-\lambda}\hat{N}_t.$ Unfortunately, this does not hold for the labor supply (i.e. leisure) except for households of type r. We will come back to this in a second. As we further know that a share λ of households is of type r, while the remainder, i.e. (1 − λ), is of type o, aggregate per-period utility can be expressed through the second-order Taylor approximation

$$ \begin{aligned} U_t&=\underbrace{\lambda\bar{U}^r+(1-\lambda)\bar{U}^o}_{=\bar{U}}+ (1+\hat{\zeta}_t)\left\{(1-\chi)\left[\lambda\frac{\hat{C}^r_t}{\gamma_r} +(1-\lambda)\frac{\hat{C}^o_t}{\gamma_o}\right] +\chi\hat{G}_t\right\} \\ & \quad +(1+\hat{\zeta}_t)\upsilon\left[\lambda \frac{\frac{1}{\lambda} \int_0^{\lambda}\hat{L}^r_t(j)dj}{\gamma_r} +(1-\lambda)\frac{\frac{1}{(1-\lambda)}\int_{\lambda}^1 \hat{L}^o_t(j)dj}{\gamma_o}\right] \end{aligned} $$

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We can use the definition of the consumption aggregate and the labor aggregate, where it holds that

$$ \hat{L}_t=\lambda \frac{1}{\gamma_r}\hat{L}_t^r+(1-\lambda)\frac{1}{\gamma_o} \hat{L}_t^o\quad {\rm and} \quad\hat{C}_t=\lambda\frac{1}{\gamma_r} \hat{C}_t^r+(1-\lambda)\frac{1}{\gamma_o}\hat{C}_t^o, $$

where $\hat{L}_t^i$ and $\hat{C}_t^i$ denote the per capita log-deviations in the respective household segment. By definition, we know that $C_t^r=\frac{1}{\lambda}\int_{0}^{\lambda}C_t^r(j)dj$ and, henceforth, $\frac{1}{\gamma_r}\hat{C}_t^r=\frac{1}{\gamma_r} \frac{1}{\lambda} \int_{0}^{\lambda}\hat{C}_t^r(j)dj$. In complete analogy, we get $\frac{1}{\gamma_r}\hat{L}_t^r=\frac{1}{\gamma_r} \frac{1}{\lambda} \int_{0}^{\lambda}\hat{L}_t^r(j)dj,\,\frac{1}{\gamma_o} \hat{C}_t^o= \frac{1}{\gamma_o} \frac{1}{(1-\lambda)} \int_{\lambda}^{1}\hat{C}_t^o(j)dj$ and $\frac{1}{\gamma_o} \hat{L}_t^o=\frac{1}{\gamma_o} \frac{1}{(1-\lambda)} \int_{\lambda}^{1}\hat{L}_t^o(j)dj$. Substitution into Eq. 34 and rearranging gives

$$ U_t=\bar{U}+(1+\hat{\zeta}_t)\left[(1-\chi)\hat{C}_t+\chi\hat{G}_t\right] -(1+\hat{\zeta}_t)\upsilon \varphi\hat{N}_t, $$

where we have substituted leisure for labor through $\hat{L}_t=-\frac{\bar{N}}{\bar{L}}\hat{N}_t =-\frac{\bar{N}}{1-\bar{N}}\hat{N}_t=-\varphi\hat{N}_t$. Further, we can use

$$ N_t=\int\limits_0^1N_t(j)dj=\int\limits_o^1\frac{Y_t(j)}{A_t}dj=\frac{Y_t}{A_t} \int\limits_0^1\frac{Y_t(j)}{Y_t}dj =\frac{Y_t}{A_t} \int\limits_0^1\left(\frac{P_t(j)}{P_t}\right)^{-\epsilon}dj $$

and log-linearize, which yields $\hat{N}_t=\hat{Y}_t-\hat{A}_t+\hat{q}_t$, where $\hat{q}_t={\rm log}\left(\int_0^1\left(\frac{P_t(j)}{P_t}\right)^{-\epsilon}dj\right)$. Using standard results as in Woodford (2003), we know that $q_t=\left(\epsilon/2\right)\sigma^2_t$, where $\sigma^2_t=\int_0^1\left[p_t(j)-p_t\right]^2dj$, in which the lower case letters p denote second-order log deviations. Substituting into the latest equation for the second-order Taylor approximation, we get

$$ U_t=\bar{U}+(1+\hat{\zeta}_t)\left[(1-\chi)\hat{C}_t+\chi\hat{G}_t\right] -(1+\hat{\zeta}_t)\upsilon \varphi\left[\hat{Y}_t-\hat{A}_t+\frac{\epsilon}{2}\sigma^2_t\right], $$

which can be simplified to

$$ U_t=\bar{U}+(1+\hat{\zeta}_t)\left[(1-\chi)\hat{C}_t+\chi\hat{G}_t\right] -(1+\hat{\zeta}_t)\upsilon \varphi\hat{Y}_t-\upsilon\varphi\frac{\epsilon}{2} \sigma^2_t+o\left(||a^3||\right)+t.i.p., $$

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where terms of order three (such as $\sigma_t^2\zeta_t^2$) are collected in $o\left(||a^3||\right)$, while terms independent of policy (such as $(1+\hat{\zeta}_t)\upsilon \varphi\hat{A}_t$) have been put into t.i.p. Using the income identity $\hat{Y}_t=\gamma_C\hat{C}_t+\gamma_G\hat{G}_t$ and the fact that χ = γ _G  = (1 − γ _C ) in the efficient steady state, we get

$$ U_t=\bar{U}+\left[1-\upsilon \varphi\right](1+\hat{\zeta}_t)\hat{Y}_t-\upsilon\varphi\frac{\epsilon}{2} \sigma^2_t+o\left(||a^3||\right)+t.i.p., $$

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Noting that $\hat{\zeta}_t\hat{Y}_t={{1}\over {2}}\left[\hat{\zeta}_t^2+\hat{Y}_t^2-(\hat{Y}_t-\hat{\zeta}_t)^2\right]$, we are able to rearrange this to

$$ U_t=\frac{(1-\upsilon\varphi)}{2}\left[(1+\hat{Y}_t)^2-(\hat{Y}_t- \hat{\zeta}_t)^2\right] -\upsilon\varphi\frac{\epsilon}{2} \sigma^2_t+o\left(||a^3||\right)+\overline{t.i.p.}, $$

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where

$$ \overline{t.i.p.}=t.i.p.+\hat{\zeta}_t^2\frac{(1-\upsilon\varphi)}{2}- \frac{(1-\upsilon\varphi)}{2} $$

is the full set of variables independent of policy. Noting that $\frac{1}{2}\sum\nolimits_{t=0}^{\infty}\beta^t\sigma_t^2=\frac{\epsilon}{\kappa} \sum\nolimits_{t=0}^{\infty}\beta^t\hat{\pi}_t^2$ (see Woodford 2003) and taking conditional expectations at date zero and neglecting al terms higher than second order, the discounted sum of utility streams can be written as Eq. 28.