Heterogeneity, monetary policy, Mirrleesian taxes, and the Friedman rule

Abstract

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.

Appendices

Appendix A

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:

$$\begin{aligned} {\mathcal {L}}&=\sum _{j=\ell ,h}\delta ^{j}v^{j}+\mu \left[ \sum _{j=\ell ,h} \pi ^{j}\left( I^{j}-z^{j}+\frac{\tau }{1+r}d^{j}\right) -{\bar{R}}\right] \nonumber \\&\quad +\eta \sum _{j=\ell ,h}\pi ^{j}\left( b^{j}-\frac{\theta }{1+g} \gamma ^{j}d^{j}\right) +\lambda \left( v^{h}-v^{h\ell }\right) , \end{aligned}$$

(A1)

with the non-negativity constraint $\theta -\frac{g-r}{1+r}\ge 0$. The first-order conditions associated with Lagrangian (A1) are:

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial I^{h}}&=\left( \delta ^{h}+\lambda \right) \left( \frac{\partial v^{h}}{\partial I^{h}}+\frac{\partial v^{h}}{ \partial q^{h}}\frac{\partial q^{h}}{\partial \gamma ^{h}}\frac{\partial \gamma ^{h}}{\partial I^{h}}\right) \nonumber \\&\quad +\mu \pi ^{h}\left[ 1+\frac{\tau }{1+r} \left( \frac{\partial d^{h}}{\partial I^{h}}+\frac{\partial d^{h}}{\partial q^{h}}\frac{\partial q^{h}}{\partial \gamma ^{h}}\frac{\partial \gamma ^{h}}{ \partial I^{h}}\right) \right] \nonumber \\&\quad -\eta \frac{\theta \pi ^{h}}{1+g}\left[ \gamma ^{h}\frac{\partial d^{h}}{\partial I^{h}}+\left( \gamma ^{h}\frac{\partial d^{h}}{\partial q^{h}}\frac{ \partial q^{h}}{\partial \gamma ^{h}}+d^{h}\right) \frac{\partial \gamma ^{h}}{ \partial I^{h}}\right] \nonumber \\&=0, \end{aligned}$$

(A2)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial I^{\ell }}&=\delta ^{\ell }\left( \frac{ \partial v^{\ell }}{\partial I^{\ell }}+\frac{\partial v^{\ell }}{\partial q^{\ell }}\frac{\partial q^{\ell }}{\partial \gamma ^{\ell }}\frac{\partial \gamma ^{\ell }}{\partial I^{\ell }}\right) -\lambda \left( \frac{\partial v^{h\ell }}{\partial I^{\ell }}+\frac{\partial v^{h\ell }}{\partial q^{h\ell }} \frac{\partial q^{h\ell }}{\partial \gamma ^{h\ell }}\frac{\partial \gamma ^{h\ell }}{\partial I^{\ell }}\right) \nonumber \\&\quad +\mu \pi ^{\ell }\left[ 1+\frac{\tau }{1+r}\left( \frac{\partial d^{\ell }}{ \partial I^{\ell }}+\frac{\partial d^{\ell }}{\partial q^{\ell }}\frac{\partial q^{\ell }}{\partial \gamma ^{\ell }}\frac{\partial \gamma ^{\ell }}{\partial I^{\ell }}\right) \right] \nonumber \\&\quad -\eta \frac{\theta \pi ^{\ell }}{1+g}\left[ \gamma ^{\ell }\frac{\partial d^{\ell }}{\partial I^{\ell }}+\left( \gamma ^{\ell }\frac{\partial d^{\ell }}{\partial q^{\ell }}\frac{\partial q^{\ell }}{\partial \gamma ^{\ell }}+d^{\ell }\right) \frac{\partial \gamma ^{\ell }}{\partial I^{\ell }}\right] \nonumber \\&=0, \end{aligned}$$

(A3)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial z^{h}}&=\left( \delta ^{h}+\lambda \right) \frac{\partial v^{h}}{\partial y^{h}}+\mu \left( -\pi ^{h}+\frac{ \tau \pi ^{h}}{1+r}\frac{\partial d^{h}}{\partial y^{h}}\right) -\eta \frac{ \gamma ^{h}\theta \pi ^{h}}{1+g}\frac{\partial d^{h}}{\partial y^{h}}=0, \end{aligned}$$

(A4)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial z^{\ell }}&=\delta ^{\ell }\frac{ \partial v^{\ell }}{\partial y^{\ell }}-\lambda \frac{\partial v^{h\ell }}{ \partial y^{\ell }}+\mu \left( -\pi ^{\ell }+\frac{\tau \pi ^{\ell }}{1+r}\frac{ \partial d^{\ell }}{\partial y^{\ell }}\right) -\eta \frac{\gamma ^{\ell }\theta \pi ^{\ell }}{1+g}\frac{\partial d^{\ell }}{\partial y^{\ell }}=0, \end{aligned}$$

(A5)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial b^{h}}&=\left( \delta ^{h}+\lambda \right) \frac{\partial v^{h}}{\partial y^{h}}+\mu \frac{\tau \pi ^{h}}{1+r} \frac{\partial d^{h}}{\partial y^{h}}+\eta \left( \pi ^{h}-\frac{ \gamma ^{h}\theta \pi ^{h}}{1+g}\frac{\partial d^{h}}{\partial y^{h}}\right) =0, \end{aligned}$$

(A6)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial \tau }&=\sum _{j}\delta ^{j}\frac{ \partial v^{j}}{\partial \tau }+\lambda \left( \frac{\partial v^{h}}{\partial \tau }-\frac{\partial v^{h\ell }}{\partial \tau }\right) +\frac{\mu }{1+r} \sum _{j}\pi ^{j}\left( d^{j}+\tau \frac{\partial d^{j}}{\partial \tau }\right) \nonumber \\&\quad -\eta \frac{\theta }{1+g}\sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{ \partial \tau }=0, \end{aligned}$$

(A7)

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial \theta }&=\sum _{j}\delta ^{j}\frac{ \partial v^{j}}{\partial \theta }+\lambda \left( \frac{\partial v^{h}}{ \partial \theta }-\frac{\partial v^{h\ell }}{\partial \theta }\right) -\eta \frac{1}{1+g}\sum _{j}\pi ^{j}\gamma ^{j}\left( d^{j}+\theta \frac{\partial d^{j}}{\partial \theta }\right) \nonumber \\&\quad +\frac{\mu \tau }{1+r}\sum _{j}\pi ^{j}\frac{ \partial d^{j}}{\partial \theta }\le 0, \end{aligned}$$

(A8)

$$\begin{aligned}&\left( \theta -\frac{g-r}{1+r}\right) \frac{\partial {\mathcal {L}}}{\partial \theta }=0, \end{aligned}$$

(A9)

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

$$\begin{aligned} \frac{\partial q^{j}}{\partial \tau }&=\frac{\partial q^{jk}}{\partial \tau }=\frac{1}{1+r}, \end{aligned}$$

(A11)

$$\begin{aligned} \frac{\partial q^{j}}{\partial \theta }&=\frac{\gamma ^{j}}{1+g}, \end{aligned}$$

(A12)

$$\begin{aligned} \frac{\partial q^{jk}}{\partial \theta }&=\frac{\gamma ^{jk}}{1+g}. \end{aligned}$$

(A13)

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

$$\begin{aligned} \frac{\partial d^{j}}{\partial \tau }&=\frac{1}{1+r}\frac{\partial d^{j}}{ \partial q^{j}}, \end{aligned}$$

(A14)

$$\begin{aligned} \frac{\partial d^{j}}{\partial \theta }&=\frac{\gamma ^{j}}{1+g}\frac{ \partial d^{j}}{\partial q^{j}} \end{aligned}$$

(A15)

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,

$$\begin{aligned} \frac{\partial v^{j}}{\partial \tau }|_{\theta ,b^{j},z^{j},I^{j}}&=\frac{ \partial v^{j}}{\partial q^{j}}|_{y^{j},I^{j}}\frac{\partial q^{j}}{\partial \tau }|_{\theta }=\frac{-\alpha ^{j}d^{j}}{1+r}, \end{aligned}$$

(A16)

$$\begin{aligned} \frac{\partial v^{jk}}{\partial \tau }|_{\theta ,b^{k},z^{k},I^{k}}&=\frac{ \partial v^{jk}}{\partial q^{jk}}|_{y^{k},I^{k}}\frac{\partial q^{jk}}{ \partial \tau }|_{\theta }=\frac{-\alpha ^{jk}d^{jk}}{1+r}, \end{aligned}$$

(A17)

$$\begin{aligned} \frac{\partial v^{j}}{\partial \theta }|_{\tau ,b^{j},z^{j},I^{j}}&=\frac{ \partial v^{j}}{\partial q^{j}}|_{y^{j},I^{j}}\frac{\partial q^{j}}{\partial \theta }|_{\tau }=\frac{-\gamma ^{j}\alpha ^{j}d^{j}}{1+g}, \end{aligned}$$

(A18)

$$\begin{aligned} \frac{\partial v^{jk}}{\partial \theta }|_{\tau ,b^{k},z^{k},I^{k}}&=\frac{ \partial v^{jk}}{\partial q^{jk}}|_{y^{k},I^{k}}\frac{\partial q^{jk}}{ \partial \theta }|_{\tau }=\frac{-\gamma ^{jk}\alpha ^{jk}d^{jk}}{1+g}. \end{aligned}$$

(A19)

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:

$$\begin{aligned} \left( \delta ^{h}+\lambda \right) \alpha ^{h}+\mu \pi ^{h}\left( \frac{ \gamma ^{h}\theta }{1+g}+\frac{\tau }{1+r}\right) \frac{\partial d^{h}}{ \partial y^{h}}-\mu \pi ^{h}&=0, \end{aligned}$$

(A20)

$$\begin{aligned} \delta ^{\ell }\alpha ^{\ell }-\lambda \alpha ^{h\ell }+\mu \pi ^{\ell }\left( \frac{ \gamma ^{\ell }\theta }{1+g}+\frac{\tau }{1+r}\right) \frac{\partial d^{\ell }}{ \partial y^{\ell }}-\mu \pi ^{\ell }&=0, \end{aligned}$$

(A21)

$$\begin{aligned}&\lambda \alpha ^{h\ell }d^{h\ell }-\delta ^{\ell }\alpha ^{\ell }d^{\ell }-\left( \delta ^{h}+\lambda \right) \alpha ^{h}d^{h}+\mu \sum _{j}\pi ^{j}d^{j} \nonumber \\&\quad +\mu \sum _{j}\pi ^{j}\left( \frac{\theta \gamma ^{j}}{1+g}+ \frac{\tau }{1+r}\right) \frac{\partial d^{j}}{\partial q^{j}}=0, \end{aligned}$$

(A22)

$$\begin{aligned}&\lambda \alpha ^{h\ell }\gamma ^{h\ell }d^{h\ell }-\delta ^{\ell }\alpha ^{\ell }\gamma ^{\ell }d^{\ell }-\left( \delta ^{h}+\lambda \right) \alpha ^{h}\gamma ^{h}d^{h}+\sum _{j}\pi ^{j}\gamma ^{j}d^{j} \nonumber \\&\quad +\mu \left( \frac{\theta }{1+g}\sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial d^{j}}{\partial q^{j}}+\frac{\tau }{1+r} \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{\partial q^{j}}\right) \le 0, \end{aligned}$$

(A23)

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

$$\begin{aligned}&\lambda \alpha ^{h\ell }\left( d^{h\ell }-d^{\ell }\right) +\mu \left[ \frac{ \tau }{1+r}\sum _{j}\left( \pi ^{j}\frac{\partial d^{j}}{\partial q^{j}} +\pi ^{j}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right. \nonumber \\&\quad \left. +\frac{\theta }{1+g }\sum _{j}\left( \pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{\partial q^{j}} +\pi ^{j}\gamma ^{j}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right] =0. \end{aligned}$$

(A24)

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

$$\begin{aligned}&\mu \left[ \frac{\tau }{1+r}\sum _{j}\left( \pi ^{j}\gamma ^{j}\frac{\partial d^{j}}{\partial q^{j}}+\pi ^{j}\gamma ^{j}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right. \nonumber \\&\quad \left. +\frac{\theta }{1+g}\sum _{j}\left( \pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial d^{j}}{\partial q^{j}}+\pi ^{j}\left( \gamma ^{j}\right) ^{2}d^{j}\frac{\partial d^{j}}{\partial y^{j}}\right) \right] \nonumber \\&\quad +\lambda \alpha ^{h\ell }\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \le 0, \end{aligned}$$

(A25)

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

$$\begin{aligned} q=\frac{1}{1+r}+\gamma \left( \frac{1}{1+g}-\frac{1}{1+r}\right) +\frac{\tau }{1+r}+\frac{\gamma \theta }{1+g}. \end{aligned}$$

(A26)

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

$$\begin{aligned} d\tau =\frac{1+r}{1+g}\left( -\gamma d\theta \right) , \end{aligned}$$

(A27)

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

$$\begin{aligned} dR=\pi ^{h}db^{h}+\pi ^{\ell }db^{\ell }-\frac{\gamma d\theta \,}{1+g} \sum _{j}\pi ^{j}d^{j}. \end{aligned}$$

(A28)

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

$$\begin{aligned} \pi ^{h}db^{h}+\pi ^{\ell }db^{\ell }=\frac{\gamma \sum _{j}\pi ^{j}d^{j}\,}{ 1+g}d\theta . \end{aligned}$$

(A29)

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

$$\begin{aligned} \left[ \begin{array}{cl} \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} &{}\quad \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} \\ \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} &{}\quad \sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j} }{\partial q^{j}} \end{array} \right] \left[ \begin{array}{c} \frac{\tau }{1+r} \\ \frac{\theta }{1+g} \end{array} \right] =\frac{-1}{\mu }\left[ \begin{array}{c} \lambda \alpha ^{h\ell }\left( d^{h\ell }-d^{\ell }\right) \\ \lambda \alpha ^{h\ell }\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \end{array} \right] . \nonumber \\ \end{aligned}$$

(A30)

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

$$\begin{aligned} \left[ \begin{array}{c} \frac{\tau }{1+r} \\ \frac{\theta }{1+g} \end{array} \right]&=\frac{-1}{\Gamma }\left[ \begin{array}{cl} \sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j} }{\partial q^{j}} &{}\quad -\sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}} ^{j}}{\partial q^{j}} \\ -\sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} &{}\quad \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}} \end{array} \right] \left[ \begin{array}{c} \lambda \alpha ^{h\ell }\left( d^{h\ell }-d^{\ell }\right) \\ \lambda \alpha ^{h\ell }\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \end{array} \right] \\&=\frac{\lambda \alpha ^{h\ell }}{\Gamma }\left[ \begin{array}{c} -\left( d^{h\ell }-d^{\ell }\right) \sum _{j}\pi ^{j}\left( \gamma ^{j}\right) ^{2}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}}+\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \sum _{j}\pi ^{j}\gamma ^{j}\frac{ \partial {\widetilde{d}}^{j}}{\partial q^{j}} \\ \left( d^{h\ell }-d^{\ell }\right) \sum _{j}\pi ^{j}\gamma ^{j}\frac{\partial {\widetilde{d}}^{j}}{\partial q^{j}}-\left( \gamma ^{h\ell }d^{h\ell }-\gamma ^{\ell }d^{\ell }\right) \sum _{j}\pi ^{j}\frac{\partial {\widetilde{d}} ^{j}}{\partial q^{j}} \end{array} \right] . \end{aligned}$$

Or

$$\begin{aligned} \tau&=\frac{\left( 1+r\right) \lambda \alpha ^{h\ell }}{\Gamma }\Phi , \\ \theta&=\frac{\left( 1+g\right) \lambda \alpha ^{h\ell }}{\Gamma }\Psi , \end{aligned}$$

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

$$\begin{aligned} u=10\left( \ln c+\frac{1}{95}d^{0.95}\right) -\frac{L}{2000}\left( L+d\right) . \end{aligned}$$

(B1)

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

$$\begin{aligned} q^{j}=\frac{1}{1+r}+\gamma ^{j}\left( \frac{1}{1+g}-\frac{1}{1+r}\right) + \frac{\tau ^{j}}{1+r}+\frac{\gamma ^{j}\theta }{1+g}. \end{aligned}$$

(B1)

It follows from this expression that if the fiscal authority changes $\tau ^{j}$ by

$$\begin{aligned} d\tau ^{j}=-\gamma ^{j}\frac{1+r}{1+g}d\theta , \end{aligned}$$

(B2)

$dq^{j}=0$ 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)

$$\begin{aligned} dR=\pi ^{h}db^{h}+\pi ^{\ell }db^{\ell }-\frac{1\,}{1+g} \sum _{j=\ell ,h}\pi ^{j}\gamma ^{j}d^{j}d\theta . \end{aligned}$$

(B3)

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

$$\begin{aligned} \sum _{j=\ell ,h}\pi ^{j}db^{j}=\frac{1}{1+g}\sum _{j=\ell ,h}\pi ^{j}\gamma ^{j}d^{j}d\theta . \end{aligned}$$

(B4)

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

$$\begin{aligned} q^{jk}=\frac{1}{1+r}+\gamma ^{jk}\left( \frac{1}{1+g}-\frac{1}{1+r}\right) + \frac{\tau ^{k}}{1+r}+\frac{\gamma ^{jk}\theta }{1+g}. \end{aligned}$$

(B5)

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.