2.1. Firms

There is a single representative firm that employs a standard neoclassical production function to produce an aggregate good that can be either consumed or accumulated as capital:

(1) \begin{equation}Y=F(K,L),\quad F_{K}\gt 0, F_{KK}\lt 0, F_{L}\gt 0, F_{LL}\lt 0,\ \text{and}\ F_{KL}\gt 0,\end{equation}

where $Y$ , $K$ , and $L$ denote per capita output, capital, and labor. Profit maximization implies that the return to capital, $r,$ and the real wage rate, $w,$ are determined by the marginal product conditions:

(2) \begin{equation}r=F_{K}\kern-2pt\left(K,L\right)\kern-2pt\left[\equiv r\kern-2pt\left(K,L\right)\right] \mathrm{\ and\ } w=F_{L}\kern-2pt\left(K,L\right)\kern-2pt\left[\equiv w\kern-2pt\left(K,L\right)\right].\end{equation}

2.2. Heterogeneous households

The population comprises N heterogeneous households, indexed by i, and remains constant over time. Individual (household) $i$ owns $A_{i}(t)$ units of real wealth at time $t$ and the economy-wide average of total wealth is $A\!\left(t\right)=\frac{1}{\mathcal{N}}\int _{0}^{\mathcal{N}}A_{i}\!\left(t\right)\!di$ . The relative share of total wealth owned by individual $i$ is $a_{i}(t)\equiv A_{i}(t)/A(t)$ , the mean of which across agents is one. There are three sources of private real wealth in the economy: private capital, $K_{i}(t)$ , real money balances, $M_{i}(t),$ and real government bond, $B_{i}(t)$ . Agent $i^{\prime}s$ holding of each component of private wealth is denoted by the subscript $i$ , with his relative share of each being defined analogously to $a_{i}(t)$ .

Each household has one unit of time that can be allocated either to leisure, $l_{i}(t),$ or to labor, $1-l_{i}(t)=L_{i}(t)$ , so that summing across individuals, the average economy-wide labor supply and leisure are $L=1-l=1-\frac{1}{N}\int _{0}^{N}l_{i}di$ . The agent maximizes lifetime utility, specified as a constant elasticity function of consumption, $C_{i}(t)$ , leisure, $l_{i}(t),$ real money balances, $M_{i}(t)$ , and a multiplicatively separable function, $h(G)$ , of government expenditure, G, taken as given:

(3) \begin{equation} \Omega _{i}\equiv \frac{1}{\gamma }\int _{0}^{\infty }\left[\left(C_{i}\kern-2pt\left(t\right)l_{i}\kern-2pt\left(t\right)^{\beta }M_{i}\kern-2pt\left(t\right)^{\eta }\right)^{\gamma }\cdot h\kern-2pt\left(G\right)\right]{\kern-2pt} e^{-\rho t}dt \quad -\!\infty \lt \gamma \lt 1,\beta \,,\eta \gt 0,1\gt \gamma\! \left(1+\beta +\eta \right) \end{equation}

where $\kappa \equiv (1-\gamma )^{-1}\gt 0$ denotes the intertemporal elasticity of substitution (IES). Empirical evidence overwhelmingly supports $0\lt \kappa \lt 1$ , in which case $\gamma \lt 0$ , ensuring that the utility function is concave in the three variables, $C_{i}(t), l_{i}(t), \mathrm{and}{\ } M_{i}(t).$ Footnote ⁶ In the less common case $\kappa \gt 1$ , the utility function is concave with respect to these variables, if and only if $1\gt \gamma (1+\beta +\eta )$ .

The optimization is performed subject to the agent’s wealth accumulation constraint

(4) \begin{align} \dot{A}_{i}\!\left(t\right)&\equiv \dot{K}_{i}\!\left(t\right)+\dot{B}_{i}\!\left(t\right)+\dot{M}_{i}\!\left(t\right)=\left(1-\tau \right)\!\left(r\left(t\right)-\delta \right)\! K_{i}\!\left(t\right)+\left(1-\tau \right)\! w\!\left(t\right)\!\left[1-l_{i}\!\left(t\right)\right]-C_{i}\!\left(t\right)\nonumber\\ & \quad +\left[R\!\left(t\right)-\pi\! \left(t\right)\right]B_{i}\!\left(t\right)-\pi\! \left(t\right)M_{i}\!\left(t\right)+T_{i}\!\left(t\right) \end{align}

where $\delta$ denotes the (constant) rate of capital depreciation, assumed to be tax-deductible, $R(t)$ denotes the nominal interest rate, $\pi (t)$ denotes the rate of inflation, $\tau$ is the constant income tax rate (common to both capital and labor income), $\mathrm{and}{\ } T_{i}$ is lump-sum transfers/taxes which the government sets to maintain its intertemporal solvency and that the agent takes as given. For notational convenience, we omit the time index $t$ whenever possible.Footnote ⁷ Carrying out the optimization yields the following relationships:

(5a) \begin{equation} C_{i}^{\gamma -1}l_{i}^{\beta \gamma }M_{i}^{\eta \gamma }h\!\left(G\right)=\lambda _{i} \end{equation}

(5b) \begin{equation} \beta \frac{C_{i}}{l_{i}}=\left(1-\tau \right)\! w\!\left(K,l\right) \end{equation}

(5c) \begin{equation} \eta \frac{C_{i}}{M_{i}}=R \end{equation}

(5d) \begin{equation} \left(1-\tau \right)\!\left(r\!\left(K,l\right)-\delta \right)=R-\pi \end{equation}

(5e) \begin{equation} \left(1-\tau \right)\!\left(r\!\left(K,l\right)-\delta \right)=\rho -\frac{\dot{\lambda }_{i}}{\lambda _{i}} \end{equation}

(5f) \begin{equation}\lim _{t\rightarrow \infty } e^{-\rho t}\lambda _{i}K_{i}=0, \lim _{t\rightarrow \infty } e^{-\rho t}\lambda _{i}M_{i}=0\ {\rm and}\ \lim _{t\rightarrow \infty } e^{-\rho t}\lambda _{i}B_{i}=0.\end{equation}

where $\lambda _{i}$ is the agent’s shadow value of wealth. Using $l=1-L$ enables us to express w and r as functions of l rather than L.

Equations (5a)–(5e) are standard optimality conditions that determine allocations at a given time, while (5f) are transversality conditions that ensure intertemporal viability of the household’s consumption and savings decisions. We define the economy-wide averages $X=N^{-1}\int _{0}^{N}X_{i}di$ for all relevant variables, $X=K,B,M,C,l$ . Since all agents face the same wage rate and same rates of return on assets, the optimality conditions imply:

(6a) \begin{equation} \frac{C_{i}\!\left(t\right)}{C\!\left(t\right)}=\frac{M_{i}\!\left(t\right)}{M\!\left(t\right)}=\frac{l_{i}\!\left(t\right)}{l\!\left(t\right)}\equiv \zeta _{i} \end{equation}

so that agent $i^{\prime}$ s share of the three averages maintains the same constant value, $\zeta _{i},$ over time.Footnote ⁸ Equation (6a) further implies that all agents choose the same growth rates for consumption, leisure, and real money balances (although their levels differ):

(6b) \begin{equation} \frac{\dot{C}_{i}\!\left(t\right)}{C_{i}\!\left(t\right)}=\frac{\dot{C}\!\left(t\right)}{C\!\left(t\right)};\,\,\frac{\dot{M}_{i}\!\left(t\right)}{M_{i}\!\left(t\right)}=\frac{\dot{M}\!\left(t\right)}{M\!\left(t\right)};\,\,\frac{\dot{l}_{i}\!\left(t\right)}{l_{i}\!\left(t\right)}=\frac{\dot{l}\!\left(t\right)}{l\!\left(t\right)} \end{equation}

which, in turn, implies that the aggregates grow at the same respective rates. These relationships which are a key implication of RCTD play an important role in facilitating the aggregation.

2.3. Government

Government expenditures include its spending on a publicly provided consumption good and the interest payments on its outstanding debt, while earning revenues from the taxes imposed on labor income and capital income. It finances the deficit by issuing a mixture of government bonds and nominal money. With the constraint imposed by its monetary targeting rule, it also imposes lump-sum transfers/taxes in order to ensure its intertemporal viability. Therefore, the consolidated government budget constraint expressed in real terms is

(7) \begin{equation} \dot{M}+\dot{B}=G+\left(R-\pi \right)\! B-\pi M-\tau \!\left[\left(r-\delta \right)\! K+wL\right]-T \end{equation}

Our focus is on the specification of monetary policy, and in particular comparing the distributional consequences of:

(i) Maintaining a constant growth rate of nominal money supply, as initially proposed by Friedman (Reference Friedman1960). For this to be sustained at the constant rate, $\overline{\theta }$ , requires that the real money growth rate is $\dot{M}(t)/M(t)=\overline{\theta }-\pi (t)$ .

(ii) Pegging the nominal interest rate at $R(t)=\overline{R}$ , as initially analyzed by Poole (Reference Poole1970), with the real money supply being determined by (9c) below.

3.1. Maintaining constant nominal monetary growth rate

To derive the macroeconomic equilibrium, we proceed as follows. First, summing (5b) over individual agents and using (2) we obtain $C=(1-\tau )\beta ^{-1}F_{L}(K,L)l\equiv C(K,L)=C(K,l)$ .Footnote ⁹ The macroeconomic equilibrium can then be described by the following dynamic equations:

(8a) \begin{equation} \frac{\dot{l}}{l}=\frac{1}{\Lambda _{\theta }}\left[\left(1-\tau \right)\!\left[F_{K}\!\left(K,L\right)-\delta \right]-\!\rho +\Pi _{\theta }\frac{\dot{K}}{K}+\eta \gamma \frac{\dot{M}}{M}\right] \end{equation}

(8b) \begin{equation}\dot{K}=F(K,L)-C(K,L)-\delta K-G,\end{equation}

(8c) \begin{equation} \frac{\dot{M}}{M}=\overline{\theta }-\eta \frac{C\!\left(K,L\right)}{M}+\left(1-\tau \right)\!\left[F_{K}\!\left(K,L\right)-\delta \right] \end{equation}

together with $L=1-l,\,\,\dot{L}=-\dot{l}$ , and where $\Lambda _{\theta }\equiv \left(1-\gamma \right)\left(1+\frac{s}{\varepsilon }\frac{l}{1-l}\right)-\beta \gamma \gt 0$ , $\Pi _{\theta }\equiv -\left(1-\gamma \right)\frac{s}{\varepsilon }\lt 0, \mathrm{s}\equiv KF_{K}/Y$ is capital’s share of output, and $\varepsilon \equiv F_{K}F_{L}/FF_{KL}$ is the elasticity of substitution in production $.$ As long as the production function $F(K,L)$ is a general constant returns to scale function, both $s$ and $\boldsymbol{\varepsilon}$ are functions of the capital-labor ratio $K/L$ .Footnote ¹⁰

3.2. Pegging the nominal interest rate

If monetary policy takes the form of pegging the nominal interest rate at $R(t)=\overline{R}$ , the macroeconomic equilibrium is specified by:

(9a) \begin{equation} \frac{\dot{l}}{l}=\frac{1}{\Lambda _{R}}\left[\left(1-\tau \right)\!\left[F_{K}\!\left(K,L\right)-\delta \right]-\rho -\Pi _{R}\frac{\dot{K}}{K}\right] \end{equation}

(9b) \begin{equation}\dot{K}=F\!\left(K,L\right)-C\!\left(K,L\right)-\delta K-G\end{equation}

where $\Lambda _{R}\equiv \!\left(1-\gamma \left(1+\eta \right)\right)\left(1+\frac{s}{\varepsilon }\frac{l}{1-l}\right)-\beta \gamma \gt 0$ , $\Pi _{R}\equiv -\left(1-\gamma\! \left(1+\eta \right)\right)\frac{s}{\varepsilon }\lt 0$ . In this case, the real money stock is obtained by aggregating (5c), and having obtained $K,L$ from (9a) and (9b) is determined by

(9c) \begin{equation} M\!\left(t\right)=\left(\eta /\overline{R}\right)C\!\left(K\!\left(t\right),L\!\left(t\right)\right). \end{equation}

3.3. Steady state

Regardless of the form of monetary policy, equations (8a), (8b), (9a), and (9b) indicate that the following hold in steady state, denoted by the superscript^*

(10a) \begin{equation} \left(1-\tau \right)\!\left[F_{K}\!\left(K^{*},L^{*}\right)-\delta \right]=\rho \end{equation}

(10b) \begin{equation} F\!\left(K^{*},L^{*}\right)-C\!\left(K^{*},L^{*}\right)-\delta K^{*}-\!G=0 \end{equation}

Therefore, the steady-state values of capital, labor (leisure), output, the wage rate, return to capital, and consumption are all determined independently of the nominal money growth rate and how it is being determined. That is, in the long run monetary policy is super-neutral insofar as these real quantities are concerned.Footnote ¹¹

For the long-run value of inflation to be independent of how monetary policy is implemented, the steady-state values of real money balances obtained under the two monetary regimes must be equal. This will be met if the monetary authority adopts policies that satisfy $\overline{\theta }=\overline{R}-\rho$ , in which case

(10c) \begin{equation} M^{*}=\left(\eta /\!\left(\,\overline{\theta }+\rho \right)\right)C\!\left(K^{*},L^{*}\right)=\left(\eta /\overline{R}\,\right)C\!\left(K^{*},L^{*}\right). \end{equation}

Since our objective is to compare the dynamic consequences of the two forms of monetary policy, we assume that they satisfy this condition and are thus compatible with the same steady state.Footnote ¹²

3.4. Local aggregate dynamics

We begin by setting out the aggregate dynamics corresponding to the two forms of monetary policy.

3.4.1 Constant nominal money growth rate

To obtain the local aggregate dynamics when the monetary authority maintains the growth rate of nominal money supply constant, that is, $\theta =\overline{\theta,}$ we linearize (8) around steady state:

(11) \begin{equation}\left[\begin{array}{c} \dot{l}\\ \dot{K}\\ \dot{M} \end{array}\right]=\left[\begin{array}{ccc} \frac{\frac{l^{*}}{K^{*}}\Pi _{\theta }^{*}\cdot a_{21}+\frac{l^{*}}{M^{*}}\eta \gamma \cdot a_{31}-l^{*}\left(1-\tau \right)r_{L}^{*}}{\Lambda _{\theta }^{*}} & \frac{\frac{l^{*}}{K^{*}}\Pi _{\theta }^{*}\cdot a_{22}+\frac{l^{*}}{M^{*}}\eta \gamma \cdot a_{32}+l^{*}\left(1-\tau \right)r_{K}^{*}}{\Lambda _{\theta }^{*}} & \frac{\frac{l^{*}}{M^{*}}\eta \gamma \cdot (\theta +\rho )}{\Lambda _{\theta }^{*}}\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & \overline{\theta}+\rho \end{array}\right]\left[\begin{array}{c} l-l^{*}\\ K-K^{*}\\ M-M^{*} \end{array}\right]\!.\end{equation}

where the elements of the matrix in (11) are defined in Appendix A.1. There we suggest that under plausible conditions, convincingly met by our calibrated model, the dynamic system has one negative eigenvalue $\mu _{\theta }(\!\lt 0),$ so that all aggregate variables converge to their respective steady states at the common rate $| \mu _{\theta }|$ .

With $l(0)$ and $M(0)$ free to jump instantaneously in response to new information (the latter through a jump in the price level $P(0)$ ), while $K(t)$ is constrained to evolve gradually from its initial value, $K_{0}$ , the stable equilibrium paths for $K, l$ , and $M$ can be expressed as:

(12a) \begin{equation} K\!\left(t\right)=K^{*}+\left(K_{0}-K^{*}\right)e^{{\mu _{\theta }}t} \end{equation}

(12b) \begin{equation} l\!\left(t\right)=l^{*}+\left(\frac{\mu _{\theta }-a_{22}}{a_{21}}\right)\left(K\!\left(t\right)-K^{*}\right) \end{equation}

(12c) \begin{equation} M\!\left(t\right)=M^{*}-\left(\frac{\left(\mu _{\theta }-a_{22}\right)\! a_{31}+a_{21}a_{32}}{a_{21}\!\left(\overline{\theta }+\rho -\mu _{\theta }\right)}\right)\left(K\!\left(t\right)-K^{*}\right) \end{equation}

In Appendix A.1, it is noted that while sgn ( $\mu _{\theta }-a_{22}$ ) is potentially ambiguous, it is most likely negative, as is certainly the case in our simulations. In that case, if the economy experiences an expansion in the aggregate capital stock $(K_{0}\lt K^{*})$ , following initial jumps, average leisure, $l(t)$ , and real balances, $M(t)$ , will both increase monotonically to their respective steady-state values, $l^{*}$ and $M^{*}$ , during the subsequent transition.Footnote ¹³

Using the government budget constraint, (7), and the transversality condition, (5f), the time path for government bonds, $B(t),$ can be solved in the neighborhood of the steady state as:

(13) \begin{equation}B\!\left(t\right)=B^{*}+\frac{1}{\mu _{\theta }-\rho +\chi }\left[\Delta _{\theta }\!\left(l\!\left(t\right)-l^{*}\right)+\Gamma _{\theta }\!\left(K\!\left(t\right)-K^{*}\right)-\left(\theta +\chi \right)\left(M\!\left(t\right)-M^{*}\right)\right],\end{equation}

where $\Delta _{\theta }$ and $\Gamma _{\theta }$ are defined in Appendix A.1, and the derivation of (13) is also provided. There we point out that to satisfy the transversality condition, and with the nominal stock of bonds being sluggish, $B^{*}$ and $\chi$ are set so as to satisfy the appropriate initial and steady-state conditions for (13), given by (A5a) and (A5b). In equilibrium, government bonds also converge at the rate $| \mu _{\theta }| .$

3.4.2 Pegging the nominal interest rate

If monetary policy is specified in terms of pegging the nominal interest rate, that is, $R(t)=\overline{R}$ , the local transitional dynamics are described by:

(14) \begin{equation}\left[\begin{array}{c} \dot{l}\\ \dot{K} \end{array}\right]=\left[\begin{array}{cc} \frac{\left(l^{*}/K^{*}\right)\Pi _{R}^{*}\cdot a_{21}-l^{*}\left(1-\tau \right)r_{L}^{*}}{\Lambda _{R}^{*}} & \frac{\left(l^{*}/K^{*}\right)\Pi _{R}^{*}\cdot a_{22}+l^{*}\left(1-\tau \right)r_{K}^{*}}{\Lambda _{R}^{*}}\\ a_{21} & a_{22} \end{array}\right]\left[\begin{array}{c} l-l^{*}\\ K-K^{*} \end{array}\right].\end{equation}

where the elements of the matrix are defined in Appendix A.2. In this case, one of the two eigenvalues is negative, $\mu _{R}(\!\lt 0),$ so that all aggregate variables converge to their steady state at the common rate $| \mu _{R}| .$ From (14), the saddle path describing the transitional dynamics of capital and leisure is described by (12a) and (12b), with $\mu _{R}$ replacing $\mu _{\theta }$ . The transitional path of real money supply tracks that of consumption in accordance with equation (9c), while the dynamics of government bonds are now

(15) \begin{equation}B(t)=B^{*}+\frac{1}{\mu _{R}-\rho +\chi }\left[\Delta _{R}\!\left(l-l^{*}\right)+\Gamma _{R}(K-K^{*})\right].\end{equation}

subject to the constraints (A8a) and (A8b), and where $\Delta _{R}$ and $\Gamma _{R}$ are defined in Appendix A.2.

These results depend in part upon the form of the underlying utility function (3) and may be summarized by the following observations:

1. (i) The steady states of the real variables (output, employment, capital, consumption) are independent of how monetary policy is conducted.

2. (ii) The steady state of the monetary variables (inflation and real money) is independent of how monetary policy is conducted, provided $\overline{\theta }=\overline{R}-\rho$ .

3. (iii) The differential effects of targeting the monetary growth rate or pegging the nominal interest rate on the transitional paths of the real variables operate entirely through its impact on their respective adjustment speeds, $| \mu _{\theta }|$ , $| \mu _{R}|$ .

4. (iv) The differential effects on transitional paths of the monetary variables reflect both the differential adjustment speeds, $| \mu _{\theta }| | \mu _{R}|,$ and the initial jump in the price level necessary for the real money supply to follow the respective stable transitional paths (12c), (9c).

4.1. Relative wealth and wealth inequality

The evolution of the relative wealth of individual i, described by equations (B3) and (B4), reflects two underlying sources of dynamics. First are those of the aggregate variables, $K(t),\,\,l(t)$ , described by (11) and (14), that generate the returns to capital and labor for the two forms of monetary policy. Second are the internal dynamics of relative wealth, $a_{i}(t)$ , which depend upon the coefficient of $a_{i}$ in (B1) evaluated at steady state, and under weak restrictions can be shown to be positive. Specifically, if private consumption exceeds after-tax labor income, then aggregating the optimality condition, (5b), over individuals implies $l^{*}\gt \beta /(1+\beta )$ which in turn ensuresFootnote ¹⁴

(16) \begin{equation} l^{*}\gt \frac{\beta }{1+\beta +\eta }\equiv \upsilon \end{equation}

Imposing (16), (B2) then implies that the greater is an agent’s steady-state relative wealth, the more leisure he enjoys and the less labor he supplies. This reflects the fact that wealthier agents have a lower marginal utility of wealth.Footnote ¹⁵

In general, the individual’s initial relative wealth, $a_{i}(0)$ , is endogenous. This is because money and bonds, being denominated in nominal terms, are subject to real shocks due to the jump in the price level, $P(0)$ , at the time a policy change or structural change occurs. Equation (C2) sets out the relationship between the jump in $P(0)$ and initial relative wealth, showing that the impact of $P(0)$ on $a_{i}(0)$ depends upon the difference between the individual’s allocation between the real asset (capital) and the nominal assets (money plus bonds, $N_{0}$ ) and the economy-wide allocation.

Because of the linearity of (C3) and (C4), we can readily transform these equations into a relationship describing the evolution of the standard deviation of wealth across agents, $\sigma _{a}(t)$ , which serves as a convenient and tractable summary measure of wealth inequality:

(17) \begin{equation}\sigma _{a}\!\left(t\right)=\frac{\alpha (t)}{\alpha (0)}\sigma _{a}(0)\end{equation}

and which converges in steady state to $\sigma _{a}^{*}=\frac{1}{\alpha (0)}\cdot \sigma _{a}(0)$ . Equation (17) highlights how the evolution of $\sigma _{a}(t)$ is driven by two factors, (i) initial wealth inequality, $\sigma _{a}(0)$ , and (ii) the evolution of relative wealth, $a_{i}(t)$ , driven by $\alpha (t)$ , defined in (B3).

Appendix C shows that the initial post-shock wealth inequality across individuals is

(18) \begin{equation} \sigma _{a}\!\left(0\right)=\left[\left(1-\varpi \right)^{2}\sigma _{k,0}^{2}+2\!\left(1-\varpi \right)\varpi \sigma _{k,0}\sigma _{n,0}\nu +\varpi ^{2}\sigma _{n,0}^{2}\right]^{1/2} \end{equation}

where $\varpi \equiv N_{0}/(P(0)K_{0}+N_{0})$ is the ratio of nominal assets to total wealth, $\sigma _{k,0},\sigma _{n,0}$ are standard deviations of the initial endowments, and $\nu$ is the correlation of endowments across agents. It is evident from (18) that one channel through which any shock impinges on wealth inequality is via its impact on the initial price level, $P(0)$ , and how this interacts with the pattern of initial endowments and their distribution, see equation (C5). There we show that if: (i) the economy-wide endowment of capital exceeds that of financial assets ( $\varpi \lt 1/2$ ), and (ii) nominal assets are less unequally endowed across agents than is capital ( $\sigma _{n,0}\leq \sigma _{k,0}$ ), an increase in the initial price level will increase initial wealth inequality. This is because an increase in $P(0)$ will increase the relative quantity of capital in agents’ portfolios, and since wealthier agents hold relatively more capital, wealth inequality will increase; the opposite applies to a decrease in $P(0)$ . We view the conditions (i) and (ii) as entirely plausible and shall refer to this channel as “the initial price response” channel.

This impact on initial wealth inequality persists throughout the subsequent evolution of wealth inequality, as implied by (17). But during the transition, the other channel, $\alpha (t)$ , defined by (B3), comes into effect. From (B3) we see that $\alpha (t)$ declines as leisure increases over time. This is because the positive relationship between leisure and capital described by the aggregate dynamics, (12b) and (14), implies that if capital is accumulated over time, labor supply falls, causing the capital-labor ratio to increase. This in turn implies that during the transition an increasing wage rate is accompanied by a declining return to capital, which favors the less affluent, leading to a steady decline in wealth inequality over time. This effect, which reflects the changing productivity of capital and labor, can be termed the “wage-return to capital” channel.

4.2. Relative income and income inequality

From equation (B8a), we see that relative income is driven in part by relative wealth. To describe this channel, we need to determine the agent’s position in real money balances, M, and income-yielding assets, $V\equiv K+B$ , relative to his overall wealth. These are given by equations (B5) and (B6), respectively. In the case of the latter, with capital and bonds yielding the same rate of return, and in the absence of risk, investors view these two assets as perfect substitutes, enabling us to characterize only their composite distribution ( $v_{i}\equiv V_{i}/V$ ). Comparing (B5) and (B6), we see that the relative asset holdings of an agent who has above average wealth, $a_{i}(t)\gt 1$ are: $v_{i}(t)-1\gt a_{i}(t)-1\gt M_{i}/M-1,$ meaning that his portfolio comprises disproportionately more of the income-earning assets, and less money than his overall relative wealth position represents. For a relatively poor agent, $a_{i}(t)\lt 1$ , the opposite applies.

Of the several potential measures of income inequality, we shall focus on before-tax personal income, defined as income from the income-earning assets, (capital and bonds) plus income from labor. Thus, the relative before-tax personal income of agent i is $y_{i}(t)\equiv Y_{i}(t)/Y(t)=[r(t)V_{i}(t)+w(t)L_{i}(t)][r(t)V(t)+w(t)L(t)]^{-1}$ , which is related to the agent’s relative wealth by (B8a) and (B8b). This can then be transformed to the following relationship between income inequality and wealth inequality:

(19a) \begin{equation} \sigma _{y}\!\left(t\right)=\varphi\! \left(t\right)\sigma _{a}\!\left(t\right) \end{equation}

where

(19b) \begin{equation} \varphi\! \left(t\right)\equiv \frac{1}{Y\!\left(t\right)+r\!\left(t\right)B\!\left(t\right)}\left\{r\!\left(t\right)A\!\left(t\right)-\left(r\!\left(t\right)M\!\left(t\right)+w\!\left(t\right)l\!\left(t\right)\right)\left(1-\frac{\upsilon }{l^{*}}\right)\frac{1}{\alpha\! \left(t\right)}\right\} \end{equation}

The term $\varphi (t)$ measures the income generated by the individual’s portfolio of assets, as reflected in his relative wealth. Because of the homogeneity of the underlying utility function, it is driven by economy-wide averages and is therefore common to all individuals. It comprises two components: (i) the share of personal income from the income-earning assets, bonds plus capital, and (ii) relative labor income, reflected in the second term, which reflects the fact that more (less) wealthy agents supply less (more) labor. In our benchmark calibration, the second term, which is negative, is dominated by the first term, implying $0\lt \varphi (t)\lt 1$ . In steady state, (19) converges to:

(20) \begin{equation} \sigma _{y}^{*}=\varphi ^{*}\sigma _{a}^{*}=\left[r^{*}A^{*}-\left(r^{*}M^{*}+w^{*}l^{*}\right)\!\left(1-\upsilon /l^{*}\right)\right]\!\left[Y^{*}+r^{*}B^{*}\right]^{-1}\sigma _{a}^{*} \end{equation}

We conclude with two observations. First, it is apparent from (20) that the two channels through which the aggregate variables impinge on wealth inequality, $\sigma _{a}(t)$ , also indirectly impact income inequality. But by influencing $\varphi (t)$ , they will exert an additional direct impact on income inequality. In the short run, the discrete change in the price level, $P(0)$ , will impact the real income generated by government bonds, which in addition to any initial jumps in $l(0)$ and its impact on $r(0)$ and $w(0)$ will impact $\varphi (0)$ . Over time, $\varphi (t)$ will continue to reflect the evolving returns to capital and wage rate embodied in the wage-return to capital channel. Second, the effects of the channels we have identified are highly sensitive to the source of the structural change and specifically whether they reflect demand or supply shocks. As we demonstrate below, both analytically and in our simulations, this is because they lead to opposite responses in $P(0)$ , which have opposite effects on the initial wealth inequality, which then persist over time.

7.1. Increase in government expenditure (demand increase)

We assume that the economy is initially in steady state and at time 0 government expenditure increases from its benchmark level $G=0.115 (G/Y^{*}=15.13\% )$ to $0.135 (G/Y^{*}=17.34\% )$ .Footnote ²⁵ The resulting changes in key variables are reported in Table 4, with the relevant transitional paths illustrated in Fig. 1.

Table 4. Increase in government consumption from 0.115 to 0.135

Figure 1. Increase in government expenditure.

The responses of the aggregate real variables are reported in the first panel and are similar to those obtained by García-Peñalosa and Turnovsky (Reference García-Peñalosa and Turnovsky2011) under lump-sum tax financing and require little further comment. The long-run government expenditure-output multiplier is 0.93, which is consistent with recent empirical estimates that consistently obtain expenditure multipliers below unity,see, for example, Ramey and Zubairy (Reference Ramey and Zubairy2018). The transitional paths followed by $L(t)$ and $K(t)$ are virtually identical for the alternative forms of monetary policy, with the differential rate of convergence (5.48% vs 5.60%) having a negligible impact, see Table 3, Fig. 1a, b.

By contrast, the initial jumps in the price level necessary for the real money stock to attain the stable path (12c), (9c) are 3.0% and 2.1%, respectively (Table 4B, Fig. 1c). These are significant differences and are accounted for by the fact that on impact, the price level must satisfy $P(0)=M_{0}R(0)/\eta C(0)$ , so that $d\hat{P}(0)=d\hat{R}(0)-d\hat{C}(0)$ , (where ^ denotes proportional changes) and where $d\hat{C}(0)=-2.1\%$ is driven by the government expenditure increase and is independent of monetary policy. The difference is due to the modest increase in the nominal interest rate (from 6.00 to 6.055%) associated with the money growth policy (Table 4C, Fig. 1e).Footnote ²⁶

In both cases, the instantaneous increase in the price level, $P(0)$ , leads to an immediate increase in wealth inequality, $\sigma _{a}(0)$ , reflecting the assumption of endowments in (C5). The larger price increase associated with the fixed monetary growth policy, due to the increase in the nominal interest rate, leads to a larger initial increase in wealth inequality (0.43% versus 0.30%, Table 4D). Over time, as capital is accumulated the wage rate increases, while the rate of return to capital declines. This favors the less affluent who are more dependent on income from labor, causing wealth inequality to fall over time. For $\overline{\theta }=2\%$ , the increase in wealth inequality is reduced to 0.056%, while for $\overline{R}=6\%$ policy, wealth inequality ends up 0.083% below its pre-shock level (Table 4D). In effect, the initial increase in wealth inequality due to the initial price response channel is gradually offset over time by the decline due to the wage-return to capital channel.

With bonds being denominated in nominal terms, in the short run real bonds drop in the same proportion as do real money balances, namely 3.0% and 2.1%, respectively (Fig. 1d). But with the growth of money supply constrained by policy (2% nominal growth rate in the first case and tracking consumption in the second case) the increase in government expenditure is primarily financed by borrowing, which grows rapidly during the transition (3.63% versus 4.26%, Table 4B). The difference is because pegging the nominal interest rate at $\vec{R}=0.06$ is more constraining on monetary growth and this has consequences for the differences in income inequality.

The immediate impact of the increase in government expenditure is to increase employment by 2.66%, leading to an immediate reduction in the wage rate of 0.93% coupled with an increase in the return to capital of 0.20 p.p. (Table 4B). By favoring capital relative to labor, this decline in the wage-return to capital channel favors the more affluent individuals, increasing the “income generating” element, $\varphi$ . In fact, this effect dominates the weak wealth effect, with the result that for the fixed monetary growth policy, $\theta (t)=\overline{\theta }$ , their combined impact is to raise income inequality by 4.54%. Over time, as capital is accumulated, the wage rate increases, the return to capital falls, the initial responses are partially reversed, and the long-run increase in income inequality is reduced to 4.07%. In the case where the monetary authority pegs the nominal interest rate at $R(\mathrm{t})=\overline{R}$ , the impact on income inequality is slightly larger, namely 4.70% in the short run and 4.15% in steady state, see Table 4D.

It is clear that most of the increase in income inequality is due to the real structural change resulting from the expansion in government spending. The contribution to inequality due to the chosen monetary policy is modest, the difference being about 0.16 p.p. (4.54% versus 4.70%) in the immediate increase and 0.08 p.p. (4.07% versus 4.15%) over time. The other point to note from Table 4 is that there is a conflict with respect to the two forms of monetary policy and their respective impacts on wealth and income inequality. While pegging the nominal interest rate reduces the increase in wealth inequality relative to that of targeting the nominal money growth rate, it exacerbates the increase in income inequality. This is true both in the short run and in steady state. But again the differences are small.

7.2. Increase in productivity (supply increase)

Table 5 sets out the analogous effects in the case where TFP increases by 2% from $E=1.2$ to $1.224.$ In this case, the effects on the real aggregate variables reported in the first panel of Table 5 are similar to those reported by Turnovsky and García-Peñalosa (Reference Turnovsky and García-Peñalosa2008). In contrast to those reported in Table 4, where the increase in government consumption crowds out private consumption, now the increased output can accommodate an increase in private consumption, as well as an increase in leisure (decrease in labor). Over time, as capital is accumulated this growth continues to the new steady state.

Table 5. Increase in productivity from 1.20 to 1.224

With consumption now increasing instantaneously, the price level must now fall by 1.19% and 2.19%, respectively, under the two forms of monetary policy for the real money stocks to reach their respective stable paths, (12c) and (9c) (Table 5B, Fig. 2c). Again, the difference is due to the modest increase in the nominal interest rate associated with the money growth policy (Fig. 2e).

Figure 2. Increase in productivity.

In both cases, the instantaneous decline in the price level leads to an immediate drop in wealth inequality, with the larger price decline associated with the pegged interest rate policy leading to a larger decline in wealth inequality. Over time, as capital is accumulated, the wage rate gradually rises, and the return to capital gradually falls. The same mechanism as before causes wealth inequality to fall gradually over time, adding to the initial decline, with the long run leading to long-run declines in wealth inequality of around 0.51% and 0.65%, respectively (Table 5D).

Analogously to the response to the increase in government spending, the short-run drop in the price level causes bonds to increase immediately in real terms in the same proportion as do real money balances, 1.19% and 2.19%, respectively. Over time, the constraints on the growth of money supply mean that the deficit is financed primarily by bonds which increase by further 3.05% and 3.79%, respectively, during the transition, with long-run increases of 4.24% and 5.98% (Table 5B).

In contrast to the demand shock, the immediate effect of the increase in consumption due to the productivity increase is to increase leisure (an Edgeworth complement), thereby reducing employment by 0.23%, leading to an increase in the wage rate of 2.1%, together with a 0.22 p.p. increase in the return to capital. With both labor and capital experiencing increased rates of return, the impact on the income generating term, $\varphi$ , is weaker and the short-run decline in wealth inequality has a bigger impact on income inequality. As a result, the immediate impact is to reduce income inequality by 0.45% for the fixed monetary growth policy, $\theta (t)=\overline{\theta }$ , and by 0.25% for the $R(t)=\overline{R}$ policy. As capital is accumulated, the wage rate increases, while the return to capital falls and in the long run income inequality declines by 1.06% and 0.92%, respectively (Table 5D).

Finally, we may note that the conflict between the policies with respect to wealth and income inequality observed with respect to the increase in government expenditure applies here as well, although not as severe. A productivity increase will reduce both wealth and income inequality both in the short run and over time. But the reduction in wealth inequality will be larger if the monetary authority pegs the nominal interest rate, while income inequality will be reduced more if it chooses to maintain the nominal money growth rate.