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2017 | OriginalPaper | Chapter

11. Can Consumption Taxes Stabilize the Economy in the Presence of Consumption Externalities?

Authors : Teresa Lloyd-Braga, Leonor Modesto

Published in: Sunspots and Non-Linear Dynamics

Publisher: Springer International Publishing

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Abstract

We discuss the stabilization role of consumption taxes under a balanced-budget rule in the presence of consumption externalities of the “keeping up with the Joneses” type. We consider a finance constrained economy and depart from a situation where sufficiently strong externalities make the steady state indeterminate, if government intervention is absent. Sufficiently procyclical consumption tax rates are able to ensure local saddle path stability. However, this procyclicality leads to the appearance of another steady state with lower levels of output which is a source or indeterminate. Therefore, government intervention with stabilization purposes may not be successful.

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previous chapter The Stabilizing Virtues of Monetary Policy on Endogenous Bubble Fluctuations

next chapter Uncertainty and Sentiment-Driven Equilibria

See also in a Woodford (1986) framework, Gokan (2006) and Pintus (2004) with income tax rates and Lloyd-Braga et al. (2008) with consumption taxation.

One exception is Guo and Lansing (1998), who studied the stabilization role of (variable) tax rates on income in a one sector Ramsey model with infinitely-lived households, where indeterminacy is created by increasing returns in production.

On this issue see also Gokan (2008), Guo and Harrison (2008) and Kamiguchi and Tamai (2011).

Since the tax rate policy is only used for stabilization purposes, it makes sense to assume that the consumption tax rate depends on income and not on the tax base.

See Alonso-Carrera et al. (2008), Gali (1994), Ljungqvist and Uhlig (2000), Wendner (2010).

As in Lloyd-Braga et al. (2014), we denote by effective consumption the argument \(c_{t+1}^{w}\varphi (\overline{c}_{t+1}^{w})/B\) of the function U(.), which includes consumption externalities.

The desire to ‘keep up with the Joneses’ is supported by empirical studies, see for example Carlsson et al. (2003), Ferrer-i-Carbonell (2005) and Maurer and Meier (2008).

Since in our framework the tax rate depends on aggregate income, individuals, being atomistic, take tax rates as given.

Note that our assumptions on U and V imply that consumption and leisure are gross substitutes.

We do not introduce consumption externalities into capitalists’ preferences because, since they have a log-linear utility function, such externalities would not affect the dynamics.

See (11.6) and the budget constraint, according to which current employment depends on future values of consumption and expectations of the future price of output.

In Sect. 11.4, we will show that when \(\varepsilon _{\gamma }(l)\) crosses the critical value \(\varepsilon _{\gamma }^{*}\left( l\right) \) a transcritical bifurcation generically occurs. Note that we disregard the case where \(\varepsilon _{\gamma }(l)=\varepsilon _{\gamma }^{*}\left( l\right) \) for all l. In this case Z(l) becomes constant describing an horizontal line in the space \(\left( l,Z(l)\right) \). Hence, either \( Z(l)\ne B\) for all l and there would be no steady state, or \(Z(l)=B\) for all l and we would obtain a continuum of steady states.

The same result was obtained by Alonso-Carrera et al. (2008), in a growth model with endogenous labor supply.

This result, which links constant elasticities with steady state uniqueness is not specific to our framework. See, for example, Guo and Lansing (1998).

See also Schmitt-Grohe and Uribe (1997) for a seminal contribution on the relation between the Laffer curve and equilibrium indeterminacy.

Note that \(M\left( \tau \right) \equiv \tau \left[ c^{w}\left( \tau \right) +c^{k}\left( \tau \right) \right] \) and \(H\left( \tau \right) \equiv h\left( Al(\tau )f(x_{ss})\right) \left[ c^{w}\left( \tau \right) +c^{k}\left( \tau \right) \right] -\left[ c^{w}\left( \tau \right) +c^{k}\left( \tau \right) \right] \) with \(c^{w}\left( \tau \right) =\frac{1}{1+\tau }Al\left[ f(x_{ss})-f^{\prime }(x_{ss})x_{ss}\right] \) and \(c^{k}\left( \tau \right) = \frac{1}{1+\tau }\left\{ k\left[ 1-\delta +Af^{\prime }(x_{ss})\right] -k\right\} \).

Note that \(\epsilon _{M,\tau }-\epsilon _{H,\tau }=\epsilon _{Z,l}/[ 1+\chi ( c^{w}) -\varepsilon _{\gamma }( l) ] \), with \(c^{w}=Al\left[ f(x_{ss})-f^{\prime }(x_{ss})x\right] /h(lf(x_{ss}))\) and l being given by \(l(\tau )\). Hence, given that \(1+\chi _{\left( \tau \right) }-\varepsilon _{\gamma (\tau )}\ne 0\) for any \(\tau \ge 0\), a change in the sign of \((\epsilon _{M,\tau }-\epsilon _{H,\tau })\) is equivalent to a change in the sign of \(\epsilon _{Z,l}\), a condition already shown in Proposition 3 to be necessary for steady state multiplicity.

Remark that our model is nested in the general framework of analysis provided in Lloyd-Braga et al. (2014) with effective consumption per unit of labor given by \(\Omega \left( k,l\right) =\frac{\omega (k,l)}{h\left( Alf\left( k/l\right) \right) }\varphi \left( \frac{\omega (k,l)l}{h\left( Alf\left( k/l\right) \right) }\right) \), the real interest rate relevant to capitalists given by \(R\left( k,l\right) =Af^{^{\prime }}(x_{t})\), and the (generalized) offer curve given by \(\Gamma \left( k,l\right) =\gamma (l)\). Therefore, the trace and the determinant can be obtained using their expressions (11.7)–(11.8), where, from their expression (9) we have: \(\varepsilon _{Rk}=-\frac{1-s}{\sigma }\), \(\varepsilon _{Rl}=\frac{1-s}{\sigma }\), \( \varepsilon _{\Gamma k}=0\), \(\varepsilon _{\Gamma l}=\varepsilon _{\gamma }\), \(\varepsilon _{\Omega k}=\left( 1+\chi \right) \left[ \frac{s}{\sigma } -\eta s\right] \) and \(\varepsilon _{\Omega l}=\chi -\left( 1+\chi \right) \left[ \frac{s}{\sigma }+\eta \left( 1-s\right) \right] \).

Indeterminacy occurs when the number of eigenvalues strictly lower than one in absolute value is larger than the number of predetermined variables.

See for instance Grandmont et al. (1998).

See also Alonso-Carrera et al. (2008) for a similar result.

This result could be immediately obtained from Lloyd-Braga et al. (2014). Indeed our parameterization falls into their configuration (i). Therefore, from their table 1, we can immediately see that the steady state is always a saddle when \(\varepsilon _{\gamma }\) exceeds the critical value for which a transcritical bifurcation occurs, i.e., \(\varepsilon _{\gamma }^{*}\) in the current paper.

We disregard the case of a saddle node bifurcation since in Sect. 11.5 we apply our local dynamics analysis to a normalized steady state whose persistence is ensured. We also disregard Pitchfork bifurcations since they are non generic. When \(\varepsilon \) crosses the value \(\varepsilon _{\gamma 1}\), we expect that a Hopf bifurcation occurs, through which a pair of complex conjugate eigenvalues cross the unit root.

Note that with \(\tau _{c}\left( y_{t}\right) \) given as in (11.21), we have that at the normalized steady state \(h(1)=1+\alpha \), i.e., the tax wedge does not depend on \(\phi \). Hence, existence of the normalized steady state is persistent for all values of \(\phi ,\) including \( \phi =\eta =0\).

To compute the \(\lim _{l\rightarrow \infty }Z(l)\) we apply l’hopital rule, obtaining \(\lim _{l\rightarrow \infty }Z(l)= \lim _{l\rightarrow \infty }\frac{\left( 1+\chi -\varepsilon _{\gamma }\right) \left[ A\left( 1-a\right) \left[ ax_{_{ss}}^{\frac{\sigma -1}{ \sigma }}+1-a\right] ^{\frac{1}{\sigma -1}}\right] ^{1+\chi }}{\left( 1+\chi \right) \phi \alpha \left( A\left[ ax_{_{ss}}^{\frac{\sigma -1}{\sigma }}+1-a \right] ^{\frac{\sigma }{\sigma -1}}\right) ^{\phi }\left[ l^{\frac{\phi -1-\chi +\varepsilon _{\gamma }}{\chi }}+\alpha \left( A\left[ ax_{_{ss}}^{ \frac{\sigma -1}{\sigma }}+1-a\right] ^{\frac{\sigma }{\sigma -1}}\right) ^{\phi }l^{\frac{\phi \left( 1+\chi \right) -1-\chi +\varepsilon _{\gamma }}{ \chi }}\right] }\). It is easy to see that this limit takes the value 0 or \( \infty \) depending on the parameter’s values. Therefore \(\lim _{l\rightarrow \infty }Z(l)<B_{nss}\) can only be satisfied when \(\lim _{l\rightarrow \infty }Z(l)=0\). This will happen if and only if \(\phi >\frac{1+\chi -\varepsilon _{\gamma }}{1+\chi }\).

They report robust estimates contained in \(\left[ 1.24,3.24\right] \).

Since \(x_{ss}=1=f(x_{ss})\), this implies \(k_{nss}/y_{nss}=\) \(l_{nss}=2.5\), i.e., the capital-output ratio takes a value that is empirically plausible at annual data.

Concerning the value of \(\chi \) Maurer and Meier (2008) estimate significant peer effects, but in any case lower than 0.44.

Remark that \(\sigma ^{**}\) must be evaluated at the lower steady state, i.e., \(\sigma ^{**}\equiv \frac{s-\theta \left( 1-s\right) (1-\eta (y_{a}))}{\eta (y_{a})s}=2.88\).

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Title: Can Consumption Taxes Stabilize the Economy in the Presence of Consumption Externalities?
Authors: Teresa Lloyd-Braga
Leonor Modesto
Publisher: Springer International Publishing
Book: Sunspots and Non-Linear Dynamics
Print ISBN: 978-3-319-44074-3

Electronic ISBN: 978-3-319-44076-7

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-44076-7_11