June 2019 , Volume 67, Issue 4, pp 983–1018 | Cite as
Heterogeneity, monetary policy, Mirrleesian taxes, and the Friedman rule
Firouz Gahvari
Luca Micheletto
First Online: 06 March 2018
We consider an overlapping-generations economy with money rationalized through a cash-in-advance constraint and heterogeneous agents subject to nonlinear taxation of labor income and linear taxation of commodity purchases. Some agents are more productive and more financially connected than others leading to their earning more income and requiring a proportionately smaller cash reserve to mediate their expenditures. We show that a nonlinear income tax can always fully neutralize the redistributive implications of who gets the extra money. On the other hand, with differences in financial connectedness, the tax policy cannot neutralize the redistributive implications of the monetary growth rate. Nevertheless the Friedman rule is found to be often desirable as a corner solution without having to impose arbitrary restrictions on the structure of agents' preferences. At the same time, the differences in connectedness may result in the violation of the Friedman rule.
Monetary policy Fiscal policy Redistribution The Friedman rule Second best
We would like to thank three anonymous referees for their many constructive and helpful comments.
JEL Classification
D82 E52 H21
Proof of Proposition2: Using \(v^{j}\) to denote \(v\left( q^{j},y^{j},I^{j}/w^{j}\right) \) and \(v^{jk}\) to denote \( v\left( q^{jk},y^{k},I^{k}/w^{j}\right) \), the mechanism designer's problem can be summarized by means of the Kuhn–Tucker Lagrangian:
$$\begin{aligned} {\mathcal {L}}&=\sum _{j=\ell ,h}\delta ^{j}v^{j}+\mu \left[ \sum _{j=\ell ,h} \pi ^{j}\left( I^{j}-z^{j}+\frac{\tau }{1+r}d^{j}\right) -{\bar{R}}\right] \nonumber \\&\quad +\eta \sum _{j=\ell ,h}\pi ^{j}\left( b^{j}-\frac{\theta }{1+g} \gamma ^{j}d^{j}\right) +\lambda \left( v^{h}-v^{h\ell }\right) , \end{aligned}$$
with the non-negativity constraint \(\theta -\frac{g-r}{1+r}\ge 0\). The first-order conditions associated with Lagrangian (A1) are:
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial I^{h}}&=\left( \delta ^{h}+\lambda \right) \left( \frac{\partial v^{h}}{\partial I^{h}}+\frac{\partial v^{h}}{ \partial q^{h}}\frac{\partial q^{h}}{\partial \gamma ^{h}}\frac{\partial \gamma ^{h}}{\partial I^{h}}\right) \nonumber \\&\quad +\mu \pi ^{h}\left[ 1+\frac{\tau }{1+r} \left( \frac{\partial d^{h}}{\partial I^{h}}+\frac{\partial d^{h}}{\partial q^{h}}\frac{\partial q^{h}}{\partial \gamma ^{h}}\frac{\partial \gamma ^{h}}{ \partial I^{h}}\right) \right] \nonumber \\&\quad -\eta \frac{\theta \pi ^{h}}{1+g}\left[ \gamma ^{h}\frac{\partial d^{h}}{\partial I^{h}}+\left( \gamma ^{h}\frac{\partial d^{h}}{\partial q^{h}}\frac{ \partial q^{h}}{\partial \gamma ^{h}}+d^{h}\right) \frac{\partial \gamma ^{h}}{ \partial I^{h}}\right] \nonumber \\&=0, \end{aligned}$$
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial I^{\ell }}&=\delta ^{\ell }\left( \frac{ \partial v^{\ell }}{\partial I^{\ell }}+\frac{\partial v^{\ell }}{\partial q^{\ell }}\frac{\partial q^{\ell }}{\partial \gamma ^{\ell }}\frac{\partial \gamma ^{\ell }}{\partial I^{\ell }}\right) -\lambda \left( \frac{\partial v^{h\ell }}{\partial I^{\ell }}+\frac{\partial v^{h\ell }}{\partial q^{h\ell }} \frac{\partial q^{h\ell }}{\partial \gamma ^{h\ell }}\frac{\partial \gamma ^{h\ell }}{\partial I^{\ell }}\right) \nonumber \\&\quad +\mu \pi ^{\ell }\left[ 1+\frac{\tau }{1+r}\left( \frac{\partial d^{\ell }}{ \partial I^{\ell }}+\frac{\partial d^{\ell }}{\partial q^{\ell }}\frac{\partial q^{\ell }}{\partial \gamma ^{\ell }}\frac{\partial \gamma ^{\ell }}{\partial I^{\ell }}\right) \right] \nonumber \\&\quad -\eta \frac{\theta \pi ^{\ell }}{1+g}\left[ \gamma ^{\ell }\frac{\partial d^{\ell }}{\partial I^{\ell }}+\left( \gamma ^{\ell }\frac{\partial d^{\ell }}{\partial q^{\ell }}\frac{\partial q^{\ell }}{\partial \gamma ^{\ell }}+d^{\ell }\right) \frac{\partial \gamma ^{\ell }}{\partial I^{\ell }}\right] \nonumber \\&=0, \end{aligned}$$
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial z^{h}}&=\left( \delta ^{h}+\lambda \right) \frac{\partial v^{h}}{\partial y^{h}}+\mu \left( -\pi ^{h}+\frac{ \tau \pi ^{h}}{1+r}\frac{\partial d^{h}}{\partial y^{h}}\right) -\eta \frac{ \gamma ^{h}\theta \pi ^{h}}{1+g}\frac{\partial d^{h}}{\partial y^{h}}=0, \end{aligned}$$
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial z^{\ell }}&=\delta ^{\ell }\frac{ \partial v^{\ell }}{\partial y^{\ell }}-\lambda \frac{\partial v^{h\ell }}{ \partial y^{\ell }}+\mu \left( -\pi ^{\ell }+\frac{\tau \pi ^{\ell }}{1+r}\frac{ \partial d^{\ell }}{\partial y^{\ell }}\right) -\eta \frac{\gamma ^{\ell }\theta \pi ^{\ell }}{1+g}\frac{\partial d^{\ell }}{\partial y^{\ell }}=0, \end{aligned}$$
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial b^{h}}&=\left( \delta ^{h}+\lambda \right) \frac{\partial v^{h}}{\partial y^{h}}+\mu \frac{\tau \pi ^{h}}{1+r} \frac{\partial d^{h}}{\partial y^{h}}+\eta \left( \pi ^{h}-\frac{ \gamma ^{h}\theta \pi ^{h}}{1+g}\frac{\partial d^{h}}{\partial y^{h}}\right) =0, \end{aligned}$$
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial \tau }&=\sum _{j}\delta ^{j}\frac{ \partial v^{j}}{\partial \tau }+\lambda \left( \frac{\partial v^{h}}{\partial \tau }-\frac{\partial v^{h\ell }}{\partial \tau }\right) +\frac{\mu }{1+r} \sum _{j}\pi ^{j}\left( d^{j}+\tau \frac{\partial d^{j}}{\partial \tau }\right) \nonumber \\&\quad -\eta \frac{\theta }{1+g}\sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{ \partial \tau }=0, \end{aligned}$$
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial \theta }&=\sum _{j}\delta ^{j}\frac{ \partial v^{j}}{\partial \theta }+\lambda \left( \frac{\partial v^{h}}{ \partial \theta }-\frac{\partial v^{h\ell }}{\partial \theta }\right) -\eta \frac{1}{1+g}\sum _{j}\pi ^{j}\gamma ^{j}\left( d^{j}+\theta \frac{\partial d^{j}}{\partial \theta }\right) \nonumber \\&\quad +\frac{\mu \tau }{1+r}\sum _{j}\pi ^{j}\frac{ \partial d^{j}}{\partial \theta }\le 0, \end{aligned}$$
$$\begin{aligned}&\left( \theta -\frac{g-r}{1+r}\right) \frac{\partial {\mathcal {L}}}{\partial \theta }=0, \end{aligned}$$
where comparing Eq. (A4) with (A6) reveals that \(\mu =-\eta .\)
Now substitute for i from (24) in (29) to get
$$\begin{aligned} q^{jk}=\frac{1}{1+r}+\gamma ^{jk}\left( \frac{1}{1+g}-\frac{1}{1+r}\right) + \frac{\tau }{1+r}+\frac{\gamma ^{jk}\theta }{1+g}. \end{aligned}$$
(A10)
Differentiate Eqs. (31) and (A10) with respect to \(\tau \) and \(\theta \) to get
$$\begin{aligned} \frac{\partial q^{j}}{\partial \tau }&=\frac{\partial q^{jk}}{\partial \tau }=\frac{1}{1+r}, \end{aligned}$$
$$\begin{aligned} \frac{\partial q^{j}}{\partial \theta }&=\frac{\gamma ^{j}}{1+g}, \end{aligned}$$
$$\begin{aligned} \frac{\partial q^{jk}}{\partial \theta }&=\frac{\gamma ^{jk}}{1+g}. \end{aligned}$$
Using \(\partial d^{j}/\partial \tau =\left( \partial d^{j}/\partial q^{j}\right) \left( \partial q^{j}/\partial \tau \right) \) and \(\partial d^{j}/\partial \theta =\left( \partial d^{j}/\partial q^{j}\right) \left( \partial q^{j}/\partial \theta \right) \), one finds
$$\begin{aligned} \frac{\partial d^{j}}{\partial \tau }&=\frac{1}{1+r}\frac{\partial d^{j}}{ \partial q^{j}}, \end{aligned}$$
$$\begin{aligned} \frac{\partial d^{j}}{\partial \theta }&=\frac{\gamma ^{j}}{1+g}\frac{ \partial d^{j}}{\partial q^{j}} \end{aligned}$$
Let \(\alpha ^{j}\) and \(\alpha ^{jk}\) denote the j- and jk-type agents' marginal utility of income:
$$\begin{aligned} \frac{\partial v^{j}}{\partial z^{j}}|_{\tau ,\theta ,b^{j},I^{j}}&=\frac{ \partial v^{j}}{\partial b^{j}}|_{\tau ,\theta ,z^{j},I^{j}}=\frac{\partial v^{j}}{\partial y^{j}}|_{q^{j},I^{j}}\equiv \alpha ^{j}, \\ \frac{\partial v^{jk}}{\partial z^{k}}|_{\tau ,\theta ,b^{k},I^{k}}&=\frac{ \partial v^{jk}}{\partial b^{k}}|_{\tau ,\theta ,z^{k},I^{k}}=\frac{\partial v^{jk}}{\partial y^{k}}|_{q^{jk},I^{k}}\equiv \alpha ^{jk}. \end{aligned}$$
Differentiate \(v^{j}\) and \(v^{jk}\) with respect to \(\tau \) and \(\theta \). Using Eqs. (A11)–(A13) and Roy's identity to simplify these derivatives yields,
$$\begin{aligned} \frac{\partial v^{j}}{\partial \tau }|_{\theta ,b^{j},z^{j},I^{j}}&=\frac{ \partial v^{j}}{\partial q^{j}}|_{y^{j},I^{j}}\frac{\partial q^{j}}{\partial \tau }|_{\theta }=\frac{-\alpha ^{j}d^{j}}{1+r}, \end{aligned}$$
$$\begin{aligned} \frac{\partial v^{jk}}{\partial \tau }|_{\theta ,b^{k},z^{k},I^{k}}&=\frac{ \partial v^{jk}}{\partial q^{jk}}|_{y^{k},I^{k}}\frac{\partial q^{jk}}{ \partial \tau }|_{\theta }=\frac{-\alpha ^{jk}d^{jk}}{1+r}, \end{aligned}$$
$$\begin{aligned} \frac{\partial v^{j}}{\partial \theta }|_{\tau ,b^{j},z^{j},I^{j}}&=\frac{ \partial v^{j}}{\partial q^{j}}|_{y^{j},I^{j}}\frac{\partial q^{j}}{\partial \theta }|_{\tau }=\frac{-\gamma ^{j}\alpha ^{j}d^{j}}{1+g}, \end{aligned}$$
$$\begin{aligned} \frac{\partial v^{jk}}{\partial \theta }|_{\tau ,b^{k},z^{k},I^{k}}&=\frac{ \partial v^{jk}}{\partial q^{jk}}|_{y^{k},I^{k}}\frac{\partial q^{jk}}{ \partial \theta }|_{\tau }=\frac{-\gamma ^{jk}\alpha ^{jk}d^{jk}}{1+g}. \end{aligned}$$
Finally, use the result that \(\mu =-\eta \) and Eqs. (A14)–(A19) to simplify and reduce the first-order conditions (A4)–(A5) and (A7)–(A8) into the following equations:
$$\begin{aligned} \left( \delta ^{h}+\lambda \right) \alpha ^{h}+\mu \pi ^{h}\left( \frac{ \gamma ^{h}\theta }{1+g}+\frac{\tau }{1+r}\right) \frac{\partial d^{h}}{ \partial y^{h}}-\mu \pi ^{h}&=0, \end{aligned}$$
$$\begin{aligned} \delta ^{\ell }\alpha ^{\ell }-\lambda \alpha ^{h\ell }+\mu \pi ^{\ell }\left( \frac{ \gamma ^{\ell }\theta }{1+g}+\frac{\tau }{1+r}\right) \frac{\partial d^{\ell }}{ \partial y^{\ell }}-\mu \pi ^{\ell }&=0, \end{aligned}$$
$$\begin{aligned}&\lambda \alpha ^{h\ell }d^{h\ell }-\delta ^{\ell }\alpha ^{\ell }d^{\ell }-\left( \delta ^{h}+\lambda \right) \alpha ^{h}d^{h}+\mu \sum _{j}\pi ^{j}d^{j} \nonumber \\&\quad +\mu \sum _{j}\pi ^{j}\left( \frac{\theta \gamma ^{j}}{1+g}+ \frac{\tau }{1+r}\right) \frac{\partial d^{j}}{\partial q^{j}}=0, \end{aligned}$$
$$\begin{aligned}&\lambda \alpha ^{h\ell }\gamma ^{h\ell }d^{h\ell }-\delta ^{\ell }\alpha ^{\ell }\gamma ^{\ell }d^{\ell }-\left( \delta ^{h}+\lambda \right) \alpha ^{h}\gamma ^{h}d^{h}+\sum _{j}\pi ^{j}\gamma ^{j}d^{j} \nonumber \\&\quad +\mu \left( \frac{\theta }{1+g}\sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial d^{j}}{\partial q^{j}}+\frac{\tau }{1+r} \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{\partial q^{j}}\right) \le 0, \end{aligned}$$
where (A23) is satisfied as an equality if \( \theta >\left( g-r\right) /\left( 1+r\right) \).
Now multiply Eq. (A20) by \(d^{h}\) and equation by (A21) \(d^{\ell }\), then add the resulting two equations to (A22) to get
$$\begin{aligned}&\lambda \alpha ^{h\ell }\left( d^{h\ell }-d^{\ell }\right) +\mu \left[ \frac{ \tau }{1+r}\sum _{j}\left( \pi ^{j}\frac{\partial d^{j}}{\partial q^{j}} +\pi ^{j}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right. \nonumber \\&\quad \left. +\frac{\theta }{1+g }\sum _{j}\left( \pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{\partial q^{j}} +\pi ^{j}\gamma ^{j}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right] =0. \end{aligned}$$
Let \({\widetilde{d}}^{j}\) denote the compensated version of \(d^{j}\). Then use the Slutsky equation to rewrite the above equation as (32) in the text.
Then multiply Eq. (A20) by \(\gamma ^{h}d^{h},\) and Eq. (A21) by \(\gamma ^{\ell }d^{\ell }\), and add the resulting two equations to (A23) to get
$$\begin{aligned}&\mu \left[ \frac{\tau }{1+r}\sum _{j}\left( \pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{\partial q^{j}}+\pi ^{j}\gamma ^{j}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right. \nonumber \\&\quad \left. +\frac{\theta }{1+g}\sum _{j}\left( \pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial d^{j}}{\partial q^{j}}+\pi ^{j}\left( \gamma ^{j}\right) ^{2}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right] \nonumber \\&\quad +\lambda \alpha ^{h\ell }\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \le 0, \end{aligned}$$
where (A25) is satisfied as an equality if \(\theta >\left( g-r\right) /\left( 1+r\right) \). Using the Slutsky equation, one can rewrite (A25) as (33) in the text.
Policy over-determination when there are no differences in financial connectedness Observe first that with \(\gamma ^{j}=\gamma \), from (31), \(q^{j}\) simplifies to
$$\begin{aligned} q=\frac{1}{1+r}+\gamma \left( \frac{1}{1+g}-\frac{1}{1+r}\right) +\frac{\tau }{1+r}+\frac{\gamma \theta }{1+g}. \end{aligned}$$
Consider now, starting from any initial values for \(\tau \) and \(\theta \), a change in the growth rate of money equal to \(d\theta \) while offsetting it with a corresponding change in \(\tau \) that keeps q constant. It follows from (A26) that one has to set
$$\begin{aligned} d\tau =\frac{1+r}{1+g}\left( -\gamma d\theta \right) , \end{aligned}$$
in order to have \(dq=0.\)
Next observe that the change in \(\theta \) induces a change in \(b^{j}\) as well. As in the proof of Proposition 1, let the fiscal authority also change \(z^{j}\) according to \(dz^{j}=-db^{j}\). This change ensures that \( dy^{j}=dz^{j}+db^{j}=0.\) With \(dy^{j}=dq^{j}=0\) and no change in \(I^{j},\) the instituted changes leave the utility of the h-types and the \(\ell \)-types intact. Observe also that the utility of potential mimickers, the jk-agents, remain unaffected as they continue to face the same price and income vector \(\left( q,y^{k},I^{k}\right) \). Consequently, the IC constraints continue to be satisfied. Thus, if the considered changes do not violate the government's budget constraint, they constitute a feasible change that leaves every agent just as well off as initially.
To check that the government's budget constraint is not violated, note that with \(\left( q,y^{j},I^{j}\right) \) remaining unchanged, the j-type's demand for d does not change either. With \(dd^{j}=0\), the change in the government's net tax revenue is, from the steady-state version of (20),
$$\begin{aligned} dR=-\left( \pi ^{h}dz^{h}+\pi ^{\ell }dz^{\ell }\right) +\frac{d\tau }{1+r} \sum _{j}\pi ^{j}d^{j}. \end{aligned}$$
Substituting \(-db^{j}\) for \(dz^{j}\) and the value of \(d\tau \) from (A27 ) in above, we get
$$\begin{aligned} dR=\pi ^{h}db^{h}+\pi ^{\ell }db^{\ell }-\frac{\gamma d\theta \,}{1+g} \sum _{j}\pi ^{j}d^{j}. \end{aligned}$$
Now note that the changes in \(\theta \) and \(b^{j}\) must satisfy the money injection constraint Eq. (27). Given that \(dd^{j}=0,\) we have
$$\begin{aligned} \pi ^{h}db^{h}+\pi ^{\ell }db^{\ell }=\frac{\gamma \sum _{j}\pi ^{j}d^{j}\,}{ 1+g}d\theta . \end{aligned}$$
Substituting from (A29) into (A28) results in \(dR=0\).
Proof of Lemma1: Write Eq. (32) and the equality version of (33) in matrix form as
$$\begin{aligned} \left[ \begin{array}{cl} \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} &{}\quad \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} \\ \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} &{}\quad \sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j} }{\partial q^{j}} \end{array} \right] \left[ \begin{array}{c} \frac{\tau }{1+r} \\ \frac{\theta }{1+g} \end{array} \right] =\frac{-1}{\mu }\left[ \begin{array}{c} \lambda \alpha ^{h\ell }\left( d^{h\ell }-d^{\ell }\right) \\ \lambda \alpha ^{h\ell }\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \end{array} \right] . \nonumber \\ \end{aligned}$$
The determinant of the \(2\times 2\) matrix in the left-hand side of (A30) is
$$\begin{aligned} \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}}\sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}}-\left( \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{ \partial q^{j}}\right) ^{2}=\pi ^{\ell }\pi ^{h}\frac{\partial {\widetilde{d}} ^{\ell }}{\partial q^{\ell }}\frac{\partial {\widetilde{d}}^{h}}{\partial q^{h}} \left( \gamma ^{\ell }-\gamma ^{h}\right) ^{2}, \end{aligned}$$
which is positive since \({\widetilde{d}}^{j}\) denote the j-type's compensated (Hicksian) demand for second-period consumption, so that \( \partial {\widetilde{d}}^{j}/\partial q^{j}\) represents the own-price substitution effect and is therefore negative. Premultiplying (A30) by the inverse of the \(2\times 2\) matrix, and using the notation \(\Gamma \equiv \mu \pi ^{\ell }\pi ^{h}\frac{\partial {\widetilde{d}}^{\ell }}{\partial q^{\ell }}\frac{\partial {\widetilde{d}}^{h}}{\partial q^{h}}\left( \gamma ^{\ell }-\gamma ^{h}\right) ^{2}\), yields
$$\begin{aligned} \left[ \begin{array}{c} \frac{\tau }{1+r} \\ \frac{\theta }{1+g} \end{array} \right]&=\frac{-1}{\Gamma }\left[ \begin{array}{cl} \sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j} }{\partial q^{j}} &{}\quad -\sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}} ^{j}}{\partial q^{j}} \\ -\sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} &{}\quad \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} \end{array} \right] \left[ \begin{array}{c} \lambda \alpha ^{h\ell }\left( d^{h\ell }-d^{\ell }\right) \\ \lambda \alpha ^{h\ell }\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \end{array} \right] \\&=\frac{\lambda \alpha ^{h\ell }}{\Gamma }\left[ \begin{array}{c} -\left( d^{h\ell }-d^{\ell }\right) \sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}}+\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \sum _{j}\pi ^{j}\gamma ^{j}\frac{ \partial {\widetilde{d}}^{j}}{\partial q^{j}} \\ \left( d^{h\ell }-d^{\ell }\right) \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}}-\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}} ^{j}}{\partial q^{j}} \end{array} \right] . \end{aligned}$$
$$\begin{aligned} \tau&=\frac{\left( 1+r\right) \lambda \alpha ^{h\ell }}{\Gamma }\Phi , \\ \theta&=\frac{\left( 1+g\right) \lambda \alpha ^{h\ell }}{\Gamma }\Psi , \end{aligned}$$
which lead to (39)–(38) when \(r=g\).
Appendix B: Numerical examples
Assume we are at a steady state and that skilled and unskilled workers have identical preferences represented by
$$\begin{aligned} u=10\left( \ln c+\frac{1}{95}d^{0.95}\right) -\frac{L}{2000}\left( L+d\right) . \end{aligned}$$
Observe that in this example \(u_{Ld}<0\) so that labor supply and future goods are (Edgeworth) substitutes. Further, regarding their cash-in-advance constraints, assume that \(\gamma ^{j}\left( I\right) \), \(j=\ell ,h\), is decreasing in I with the following structure:
$$\begin{aligned} \gamma ^{j}\left( I\right) =1-\beta ^{j}I^{\rho ^{j}}/\rho ^{j},\,\, \rho ^{j}\ge 1,\,\,\beta ^{j}>0, \end{aligned}$$
with \(\beta ^{\ell }<\beta ^{h}\) so that \(\gamma ^{\ell }\left( I\right) >\gamma ^{h}\left( I\right) \). The government has a (weighted) utilitarian objective function \(\sum _{j=\ell ,h}\delta ^{j}v^{j}\), where \(\delta ^{j}\) denotes the welfare weights, with \(\delta ^{\ell }>\delta ^{h}\). Set \(\pi ^{h}=0.6\) and \(\pi ^{\ell }=0.4\) so that sixty percent of workers are skilled and forty percent unskilled. Their real wage rates, reflecting their productivities, are set equal to \(w^{h}=4\) and \(w^{\ell }=2\). Assume further that \(r=g=0.4\) and that \(\delta ^{h}=0.4\) and \(\delta ^{\ell }=0.6\). As far as the government's external revenue is concerned, we set \({\bar{R}}=0\) so that optimal taxes are purely redistributive. Finally, let \(\beta ^{\ell }=0.00005\), while \(\beta ^{h}\) and \(\rho ^{j}\) are allowed to vary.
(i) The FR is violated:
Set \(\rho ^{h}=\rho ^{\ell }=1.2\) and \(\beta ^{h}=0.0003\). This yields the following solution for the tax instruments and the rate of monetary growth:
$$\begin{aligned} \tau =-0.1610,\,\,\theta =i=0.3163,\,\,T^{\prime }\left( I^{h}\right) =-0.0228, \quad T^{\prime }\left( I^{\ell }\right) =0.4772. \end{aligned}$$
Given the money injection rate of \(31.63\%\) and the population growth rate of \(40\%\), one calculates \(\varphi =-0.0598\). That is, the price level is falling at a rate of \(5.98\%\) per period. Observe also that the marginal income tax rate faced by skilled workers is nonzero, a result that is due to the presence of other policy instruments besides income taxation.50 The policy instruments result in the following values for the arguments of the utility function and real money balances:
$$\begin{aligned} c^{h}&=259.035,\,d^{h}=107.654,{\, }L^{h}=100.377,\, \\ y^{h}&=339.770,\,x^{h}=61.095,\,{\ }b^{h}=6.421, \\ c^{\ell }&=137.996,\,d^{\ell }=64.980,\,{\ }L^{\ell }=43.021,\,\\ y^{\ell }&=191.486,\,x^{\ell }=31.306,\,b^{\ell }=29.253. \end{aligned}$$
Implementing the optimal allocation by the government implies \(\gamma ^{\ell }=0.9913,\gamma ^{h\ell }=0.9476\), and \(\gamma ^{h}=0.6670\).51
(ii) The FR holds:
Set \(\rho ^{h}=\rho ^{\ell }=1\) and \(\beta ^{h}=0.0006\). Under this circumstance, we get,
$$\begin{aligned} \tau =0.0751,\,\theta =i=0,\,T^{\prime }\left( I^{h}\right) =0.0062,\,T^{\prime }\left( I^{\ell }\right) =0.4806. \end{aligned}$$
In this example, the FR is this time satisfied as a corner solution with \( \tau \) being the only instrument used to affect the price of d. With no increase in money supply and a population growth rate of \(40\%\), the price level is falling at a rate of \(28.57\%\) per period. The policy instruments result in the following values for the arguments of the utility function and real money balances:
$$\begin{aligned} c^{h}&=262.656,\,d^{h}=102.254,\,{\ }L^{h}=100.214,\,\\ y^{h}&=341.180,\,x^{h}=57.566,\,b^{h}=-2.094, \\ c^{\ell }&=129.626,\,d^{\ell }=74.902,\,{\ }L^{\ell }=42.693,\\ \,y^{\ell }&=187.147,\,x^{\ell }=50.132,\,b^{\ell }=3.142. \end{aligned}$$
Implementing the optimal allocation by the government in this case implies \( \gamma ^{\ell }=0.9957,\gamma ^{h\ell }=0.9488\), and \(\gamma ^{h}=0.7595\).
Appendix C: Observability of individual consumption levels
Let \(\tau ^{j}\) denote the tax rate levied on the second-period consumption of individuals of type j. This changes the expression for \(q^{j}\) in (31) to
$$\begin{aligned} q^{j}=\frac{1}{1+r}+\gamma ^{j}\left( \frac{1}{1+g}-\frac{1}{1+r}\right) + \frac{\tau ^{j}}{1+r}+\frac{\gamma ^{j}\theta }{1+g}. \end{aligned}$$
It follows from this expression that if the fiscal authority changes \(\tau ^{j}\) by
$$\begin{aligned} d\tau ^{j}=-\gamma ^{j}\frac{1+r}{1+g}d\theta , \end{aligned}$$
\(dq^{j}=0\) whenever the monetary authority changes \(\theta \) by \(d\theta .\) Moreover, observe again that the change in \(\theta \) induces a change in \( b^{j}\) as well. As in the proof of Proposition 1, let the fiscal authority also change \(z^{j}\) according to \(dz^{j}=-db^{j}.\) This change ensures that \( dy^{j}=dz^{j}+db^{j}=0.\) With \(dy^{j}=dq^{j}=0\) and no change in \(I^{j},\) the instituted changes leave the utility of the h-types and the \(\ell \)-types intact.
To check resource feasibility, observe first that with \(\left( q^{j},y^{j},I^{j}\right) \) remaining unchanged, the j-type's demand for d does not change either. With \(dd^{j}=0\), the change in the government's net tax revenue is, from the steady-state version of (20), while substituting \(\tau ^{j}\) for \(\tau ,\)\(-db^{j}\) for \(dz^{j},\) and the value of \(d\tau ^{j}\) from (B2)
$$\begin{aligned} dR=\pi ^{h}db^{h}+\pi ^{\ell }db^{\ell }-\frac{1\,}{1+g} \sum _{j=\ell ,h}\pi ^{j}\gamma ^{j}d^{j}d\theta . \end{aligned}$$
As in the exercises in the text, the changes in \(\theta \) and \(b^{j}\) must satisfy the money injection constraint Eq. (27). Given that \(dd^{j}=0,\) we have
$$\begin{aligned} \sum _{j=\ell ,h}\pi ^{j}db^{j}=\frac{1}{1+g}\sum _{j=\ell ,h}\pi ^{j}\gamma ^{j}d^{j}d\theta . \end{aligned}$$
Substituting from (B4) into (B3) results in \(dR=0.\)
It remains for us to check the IC constraints. To that end, consider the expression that one gets for \(q^{jk}\) when substitutes \(\tau ^{k}\) for \(\tau \) in (A10). We have
$$\begin{aligned} q^{jk}=\frac{1}{1+r}+\gamma ^{jk}\left( \frac{1}{1+g}-\frac{1}{1+r}\right) + \frac{\tau ^{k}}{1+r}+\frac{\gamma ^{jk}\theta }{1+g}. \end{aligned}$$
It then follows from (B5) and (B2) that a change in \(\theta \) accompanied by a change in \(\tau ^{k}\) that keeps \(q^{k}\) constant, changes \( q^{jk}\) by
$$\begin{aligned} dq^{jk}=\frac{d\tau ^{k}}{1+r}+\frac{\gamma ^{jk}d\theta }{1+g}=\frac{\left( \gamma ^{jk}-\gamma ^{k}\right) d\theta }{1+g}. \end{aligned}$$
As a result, the utility of a jk-mimicker will change according to
$$\begin{aligned} dv^{jk}=\frac{\partial v^{jk}}{\partial q^{jk}}dq^{jk}=-\alpha ^{jk}d^{k} \frac{\left( \gamma ^{jk}-\gamma ^{k}\right) d\theta }{1+g}. \end{aligned}$$
where \(\alpha ^{jk}\) denotes the jk-mimicker's marginal utility of income. Now if \(\gamma ^{jk}-\gamma ^{k}>0\) setting \(d\theta >0\) implies that \(dv^{jk}<0 \) and if \(\gamma ^{jk}-\gamma ^{k}<0\) setting \(d\theta <0\) implies that \( dv^{jk}<0.\)
Either way, the jk-mimicker can be made worse off allowing a Pareto-improving move. The upshot of this discussion is that, given the assumed information structure, fiscal policy becomes overarching and one would want to either keep inflating the economy or deflating it. Now, given the pattern of binding IC constraint, the relevant sign for us is that of \(\gamma ^{h\ell }-\gamma ^{\ell }\) which we know is negative. Consequently, a deflationary reform of the type described always increases welfare, resulting in the optimality of the Friedman rule as a limit solution due to the constraint on the non-negativity of the nominal interest rate.
Finally, observe that the indeterminacy problem we have mentioned in the text does not arise here despite the fact that we are enabling the fiscal authority to neutralize the redistributive effects of the monetary policy. The reason for this is that, this informational structure allows fiscal authority to achieve even more. It can even determine the "virtual" price \(q^{jk}\) thus being able to play with IC constraints.
Abel, A.: Optimal monetary growth. J. Monet. Econ. 19, 437–450 (1987)CrossRefGoogle Scholar
Albanesi, S.: Redistribution and optimal monetary policy: results and open questions. Riv. Polit. Econ. 97, 3–47 (2007)Google Scholar
Atkinson, A.B., Stiglitz, J.E.: The structure of indirect taxation and economic efficiency. J. Public Econ. 1, 97–119 (1972)CrossRefGoogle Scholar
Atkinson, A.B., Stiglitz, J.E.: The design of tax structure: direct versus indirect taxation. J. Public Econ. 6, 55–75 (1976)CrossRefGoogle Scholar
Atkinson, A.B., Stiglitz, J.E.: Lectures on Public Economics. McGraw-Hill, New York (1980)Google Scholar
Boise, G., Polemarchakis, H.: Theory and practice of monetary policy. Econ. Theor. 27, 1–23 (2006)CrossRefGoogle Scholar
Chari, V.V., Christiano, L., Kehoe, P.: Optimal fiscal and monetary policy: some new results. J. Money Credit Bank. 23, 519–539 (1991)CrossRefGoogle Scholar
Chari, V.V., Christiano, L., Kehoe, P.: Optimality of the Friedman rule in economies with distorting taxes. J. Monet. Econ. 37, 203–233 (1996)CrossRefGoogle Scholar
Correia, I., Teles, P.: Is the Friedman rule optimal when money is an intermediate good? J. Monet. Econ. 38, 223–244 (1996)CrossRefGoogle Scholar
Correia, I., Teles, T.: The optimal inflation tax. Rev. Econ. Dyn. 2, 325–346 (1999)CrossRefGoogle Scholar
Correia, I., Nicolini, J.P., Teles, P.: Optimal fiscal and monetary policy: equivalence results. J. Polit. Econ. 116, 141–170 (2008)CrossRefGoogle Scholar
Crettez, B., Michel, P., Wigniolle, B.: Cash-in-advance constraints in the Diamond overlapping generations model: neutrality and optimality of monetary policies. Oxf. Econ. Pap. 51, 431–452 (1999)CrossRefGoogle Scholar
Crettez, B., Michel, P., Wigniolle, B.: Optimal monetary policy, taxes, and public debt in an intertemporal equilibrium. J. Public Econ. Theory 4, 299–316 (2002)CrossRefGoogle Scholar
Cunha, A.B.: The optimality of the Friedman rule when some distorting taxes are exogenous. Econ. Theor. 35, 267–291 (2008)CrossRefGoogle Scholar
da Costa, C.E., Werning, I.: On the optimality of the Friedman rule with heterogeneous agents and nonlinear income taxation. J. Polit. Econ. 116, 82–112 (2008)CrossRefGoogle Scholar
De Fiore, F., Teles, P.: The optimal mix of taxes on money, consumption and income. J. Monet. Econ. 50, 871–887 (2003)CrossRefGoogle Scholar
Diamond, P.A.: National debt in a neoclassical growth model. Am. Econ. Rev. 55, 1126–1150 (1965)Google Scholar
Edwards, J., Keen, M., Tuomala, M.: Income tax, commodity taxes and public good provision: a brief guide. Finanzarchiv 51, 472–487 (1994)Google Scholar
Erceg, C.J., Henderson, D.W., Levin, A.T.: Optimal monetary policy with staggered wage and price contracts. J. Monet. Econ. 46, 281–313 (2000)CrossRefGoogle Scholar
Erosa, A., Ventura, G.: On inflation as a regressive consumption tax. J. Monet. Econ. 49, 761–795 (2002)CrossRefGoogle Scholar
Friedman, M.: The optimum quantity of money. In: Friedman, M. (ed.) The Optimum Quantity of Money and Other Essays. Aldine Publishing Company, Chicago (1969)Google Scholar
Gahvari, F.: Lump-sum taxation and the superneutrality and optimum quantity of money in life cycle growth models. J. Public Econ. 36, 339–367 (1988)CrossRefGoogle Scholar
Gahvari, F.: The Friedman rule: old and new. J. Monet. Econ. 54, 581–589 (2007)CrossRefGoogle Scholar
Gahvari, F.: Friedman rule in a model with endogenous growth and cash-in-advance constraint. J. Money Credit Bank. 44, 787–823 (2012)CrossRefGoogle Scholar
Guidotti, P.E., Vegh, C.A.: The optimal inflation tax when money reduces transaction costs. J. Monet. Econ. 31, 189–205 (1993)CrossRefGoogle Scholar
Guiso, L., Haliassos, M., Jappelli, T.: Household Portfolios. MIT Press, Boston (2001)Google Scholar
Hahn, F., Solow, R.: A Critical Essay on Modern Macroeconomic Theory. Basil Blackwell, Oxford (1995)Google Scholar
Hamada, K.: Lifetime equity and dynamic efficiency on the balanced growth path. J. Public Econ. 1, 373–396 (1972)CrossRefGoogle Scholar
Ireland, P.N.: The role of countercyclical monetary policy. J. Polit. Econ. 104, 704–723 (1996)CrossRefGoogle Scholar
Ireland, P.N.: Implementing the Friedman rule. Rev. Econ. Dyn. 6, 120–134 (2003)CrossRefGoogle Scholar
Johnson, E., Sherraden, M.S.: From financial literacy to financial capability among youth. J. Sociol. Soc. Welf. 34, 119–145 (2007)Google Scholar
Khan, A., King, R.G., Wolman, A.L.: Optimal monetary policy. Rev. Econ. Stud. 70, 825–860 (2003)CrossRefGoogle Scholar
Kotlikoff, L.J., Bernheim, B.D.: Household financial planning and financial literacy. In: Kotlikoff, L.J. (ed.) Essays on Saving, Bequests, Altruism, and Life-Cycle Planning, pp. 427–478. MIT Press, Cambridge (2001)CrossRefGoogle Scholar
Lai, C., Chin, C.: (In)determinacy, increasing returns, and the optimality of the Friedman rule in an endogenously growing open economy. Econ. Theor. 44, 69–100 (2010)CrossRefGoogle Scholar
Michel, P., Wigniolle, B.: Temporary bubbles. J. Econ. Theory 112, 173–183 (2003)CrossRefGoogle Scholar
Michel, P., Wigniolle, B.: Cash-in-advance constraints, bubbles, and monetary policy. Macroecon. Dyn. 9, 28–56 (2005)CrossRefGoogle Scholar
Mulligan, C.B., Sala-i-Martin, X.: The optimum quantity of money: theory and evidence. J. Money Credit Bank. 29, 687–715 (1997)CrossRefGoogle Scholar
Mulligan, C.B., Sala-i-Martin, X.: Extensive margins and the demand for money at low interest rates. J. Polit. Econ. 108, 961–991 (2000)CrossRefGoogle Scholar
OECD Observer: Policy brief. http://www.oecd.org/finance/financial-education/37087833.pdf (2006)
Phelps, E.: The golden rule of accumulation: a fable for growthmen. Am. Econ. Rev. 51(4), 638–643 (1961)Google Scholar
Phelps, E.S.: Inflation in the theory of public finance. Swed. J. Econ. 75, 67–82 (1973)CrossRefGoogle Scholar
Rochon, C., Polemarchakis, H.: Debt, liquidity and dynamics. Econ. Theor. 27, 179–211 (2006)CrossRefGoogle Scholar
Samuelson, P.A.: An exact consumption-loan model of interest with or without the social contrivance of money. J. Polit. Econ. 66, 467–482 (1958)CrossRefGoogle Scholar
Sandmo, A.: A note on the structure of optimal taxation. Am. Econ. Rev. 64, 701–706 (1974)Google Scholar
Schmitt-Grohé, S., Uribe, M.: Optimal fiscal and monetary policy under sticky prices. J. Econ. Theory 114, 198–230 (2004a)CrossRefGoogle Scholar
Schmitt-Grohé, S., Uribe, M.: Optimal fiscal and monetary policy under imperfect completion. J. Macroecon. 26, 183–209 (2004b)CrossRefGoogle Scholar
Shaw, M., Chang, J., Lai, C.: (Non)optimality of the Friedman rule and optimal taxation in a growing economy with imperfect competition. Econ. Lett. 90, 412–420 (2006)CrossRefGoogle Scholar
Stiglitz, J.E.: Self-selection and Pareto efficient taxation. J. Public Econ. 17, 213–240 (1982)CrossRefGoogle Scholar
Stiglitz, J.E.: In praise of Frank Ramsey's contribution to the theory of taxation. Econ. J. 125, 235–268 (2015)CrossRefGoogle Scholar
van der Ploeg, F., Alogoskoufis, G.S.: Money and endogenous growth. J. Money Credit Bank. 26, 771–791 (1994)CrossRefGoogle Scholar
Weiss, L.: The effects of money supply on economic welfare in the steady state. Econometrica 48, 565–576 (1980)CrossRefGoogle Scholar
Williamson, S.D.: Monetary policy and distribution. J. Monet. Econ. 55, 1038–1053 (2008)CrossRefGoogle Scholar
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
1.Department of EconomicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
2.Department of LawUniversity of MilanMilanItaly
3.Dondena Centre for Research on Social Dynamics and Public PolicyBocconi UniversityMilanItaly
4.UCFSUppsalaSweden
5.CESifoMunichGermany
Gahvari, F. & Micheletto, L. Econ Theory (2019) 67: 983. https://doi.org/10.1007/s00199-018-1108-x
Received 06 May 2016
First Online 06 March 2018
DOI https://doi.org/10.1007/s00199-018-1108-x
Publisher Name Springer Berlin Heidelberg