2.1 Static neoclassical model with variable labor

Before analyzing the neoclassical model, we briefly review the main predictions of the traditional Keynesian model regarding the macroeconomic effects of changes in government purchases. In the traditional Keynesian model, a one-unit increase in government purchases $G$ can raise output $Y$ by more than one unit because of positive consumption responses. This mechanism has several implications. First, fiscal expansions that raise $G$ are expected to stimulate output significantly. Second, the government spending multiplier, $\partial Y/\partial G$ , falls as the income tax rate rises because income tax payments weaken consumption responses by reducing disposable income. Thus, consumption and output are expected to move in the same direction in response to changes in government purchases because output is primarily driven by changes in consumption. As discussed, neoclassical models provide different predictions of the effects of fiscal expansion. Hence, in Sections 3 and 4, we evaluate the predictions of the two types of models in our empirical analysis.

Our theoretical analysis is based on static neoclassical models developed by Baxter and King (Reference Baxter and King1993) and Woodford (Reference Woodford2011). In this model, homogeneous households choose consumption $C$ and labor supply $L$ to maximize utility $u\left ( C\right ) -v\left ( L\right )$ , that satisfies $u^{\prime }\gt 0\gt u^{\prime \prime }$ and $v^{\prime }\gt 0$ , $\ v^{\prime \prime }\gt 0$ , subject to the budget constraint:

\begin{equation*} C=\left ( 1-\tau \right ) \left ( wL+\Pi \right ) +TR, \end{equation*}

where $\tau \in \left ( 0,1\right ), \ w,\ \Pi, $ and $TR$ denote the income tax rate, wages, dividends from firm profits, and lump-sum transfers from the government, respectively. Then, labor supply is determined by the following condition:

\begin{equation*} \left ( 1-\tau \right ) w=\frac {v^{\prime }\left ( L\right ) }{u^{\prime }\left ( C\right ) }\end{equation*}

On the supply side, identical firms produce output using labor. Their problem is to maximize profits by optimally choosing labor.

\begin{equation*} \Pi =\max _{L}\left [ f\left ( L\right ) -wL\right ], \end{equation*}

where $Y=f\left ( L\right )$ is a production function satisfying $f^{\prime }\gt 0\gt f^{\prime \prime }.$ Note that profit $\Pi$ is distributed among households as dividends. The firms’ labor demand is determined by

\begin{equation*} w=f^{\prime }\left ( L\right ) . \end{equation*}

The government receives income tax from households, makes purchases $G$ , and provides lump-sum transfers $TR$ to households. Thus, the government budget constraint is given as

\begin{equation*} G+TR=\tau \left ( wL+\Pi \right ) . \end{equation*}

In this model, we assume that the government chooses $G$ exogenously but maintains $\tau$ at a given value. Then, the budget surplus, $TR=\tau \left ( wL+\Pi \right ) -G,$ is rebated to households as lump-sum transfers. This type of government behavior is suitable for our analysis because the government spending multiplier represents the endogenous response of output $Y$ to an exogenous change in  $G$ .

Alternatively, the current model could be combined with a Ramsey government that optimizes the policy instruments $\left ( \tau, G,TR\right ) .$ Such a model can be useful to analyze how $G$ responds to various shocks. For example, Vegh and Vuletin (Reference Vegh and Vuletin2016) consider a Ramsey local government problem in a model analogous to the current one to account for the flypaper effect, which refers to the phenomenon that local government spending responds more significantly to unconditional grants from the national government than to local income. They show that the flypaper effect arises in their model only under distortionary taxation.Footnote ⁸ Indeed, if we assumed a Ramsey government in the current model, we could replicate their results: there is a flypaper effect with $TR=0$ but no flypaper effect with $\tau =0.$ However, since our main focus is on the government spending multiplier in this analysis, we assume that the government changes $G$ exogenously, rather than optimizing it, in what follows.

In equilibrium, the resource constraint $Y=C+G$ and the conditions for labor supply and demand should be satisfied. Combining these results yields the following equation:

(1) \begin{equation} \left ( 1-\tau \right ) u^{\prime }\left ( Y-G\right ) =\frac{v^{\prime }\left ( L\right ) }{f^{\prime }\left ( L\right ) }\equiv \tilde{v}^{\prime }\left ( Y\right ), \end{equation}

where $\tilde{v}\left ( Y\right ) \equiv v\left ( f^{-1}\left ( Y\right ) \right )$ represents the (utility) cost of producing output $Y.$ It is trivial to verify $\tilde{v}^{\prime }\left ( Y\right ) \gt 0$ and $\tilde{v}^{\prime \prime }\left ( Y\right ) \lt 0$ using the properties of $v\left ( L\right )$ and $f\left ( L\right )$ . Equation (1) has only one endogenous variable $Y$ with policy variables $\tau$ and $G.$ Thus, we can use the equation to characterize the effects of $\tau$ and $G$ on $Y.$ First, total differentials of equation (1) with respect to $\tau$ and $Y$ yields

(2) \begin{equation} \frac{\partial Y}{\partial \tau }=-\frac{u^{\prime }}{\tilde{v}^{\prime \prime }-\left ( 1-\tau \right ) u^{\prime \prime }}\lt 0. \end{equation}

This equation implies that the income tax rate $\tau$ has a negative impact on equilibrium output $Y.$ This appears to be intuitive because a large $\tau$ discourages labor supply, thereby reducing output. We can also derive the government spending multiplier, $m$ , from the total differentials of equation (1) with respect to $G$ and $Y$ .

\begin{equation*} m\equiv \frac {\partial Y}{\partial G}=\frac {-\left ( 1-\tau \right ) u^{\prime \prime }}{\tilde {v}^{\prime \prime }-\left ( 1-\tau \right ) u^{\prime \prime }}\end{equation*}

Moreover, because $1-\tau =\tilde{v}^{\prime }/u^{\prime }$ from equation (1), we can rewrite the above equation as

(3) \begin{equation} m=\frac{\eta _{u}}{\eta _{v}+\eta _{u}}=\frac{1}{\eta _{v}/\eta _{u}+1}, \end{equation}

where $\eta _{u}\equiv -Yu^{\prime \prime }/u^{\prime }\gt 0$ and $\eta _{v}\equiv Y\tilde{v}^{\prime \prime }/\tilde{v}^{\prime }\gt 0$ denote the elasticities of marginal utility $u^{\prime }$ and marginal cost $\tilde{v}^{\prime }$ with respect to $Y$ .

Equation (3) implies that the government spending multiplier is positive but less than one. Hence, the static neoclassical model predicts a smaller multiplier effect than the traditional Keynesian model. This difference arises because government purchases influence output through different channels in the neoclassical model. Specifically, an increase in $G$ has a negative wealth effect on households because, given $C=Y-G$ , such fiscal expansion effectively removes resources from households. Hence, they reduce consumption but raise labor supply in response to fiscal expansion. In other words, $dG\gt 0$ leads to $dY\gt 0$ (because $dL\gt 0$ ) but $dY\lt dG$ because $dC=dY-dG\lt 0$ . These results imply $0\lt m\lt 1$ , as suggested by equation (3). Interestingly, while consumption decreases under the static neoclassical model, it increases under the traditional Keynesian model.

We also examine the impact of the income tax rate on the government spending multiplier using $\partial m/\partial \tau$ . Therefore, we decompose the derivative as follows.

\begin{equation*} \frac {\partial m}{\partial \tau }=\frac {\partial Y}{\partial \tau }\times \frac {\partial \left ( \eta _{v}/\eta _{u}\right ) }{\partial Y}\times \frac {dm}{d\left ( \eta _{v}/\eta _{u}\right ) }\end{equation*}

In this equation, $\partial Y/\partial \tau \lt 0$ from equation (2) and $dm/d\left ( \eta _{v}/\eta _{u}\right ) \lt 0$ from equation (3). Thus, $\partial m/\partial \tau$ should have the same sign as $\partial \left ( \eta _{v}/\eta _{u}\right ) /\partial Y$ in the model. Intuitively, the sign of $\partial \left ( \eta _{v}/\eta _{u}\right ) /\partial Y$ determines the magnitude of labor supply and consumption responses to changes in government purchases. To observe this, we suppose that $\eta _{v}/\eta _{u}$ is small. In other words, the marginal cost $\tilde{v}^{\prime }$ is relatively inelastic whereas the marginal utility $u^{\prime }$ is relatively elastic. In this case, when government purchases rise, output increases significantly because the marginal cost $\tilde{v}^{\prime }$ increases relatively slowly, whereas consumption does not decrease significantly because the marginal utility $u^{\prime }$ increases relatively rapidly. Consequently, the government spending multiplier is high when $\eta _{v}/\eta _{u}$ is low. Then, we can conclude that $\partial m/\partial \tau \gt 0$ is associated with $\partial \left ( \eta _{v}/\eta _{u}\right ) /\partial Y\gt 0$ because $\eta _{v}/\eta _{u}$ decreases as $\tau$ increases through a decrease in $Y$ by equation (2). By contrast, $\partial m/\partial \tau \lt 0$ if $\partial \left ( \eta _{v}/\eta _{u}\right ) /\partial Y\lt 0$ because a large value of $\tau$ results in a large value of $\eta _{v}/\eta _{u}.$

The sign of $\partial m/\partial \tau$ is generally ambiguous because $\partial \left ( \eta _{v}/\eta _{u}\right ) /\partial Y$ can take any sign, depending on the form of the utility and production functions. However, $\partial m/\partial \tau \gt 0$ may arise realistically in the static neoclassical model, even though it is the opposite of the prediction of the traditional Keynesian model. To demonstrate this, we present the following group of widely used functional forms of $u,\ v,$ and $f$ that give rise to $\partial m/\partial \tau \gt 0$ .

(4) \begin{equation} \left . \begin{array} [c]{l}u\left ( C\right ) =\frac{C^{1-\sigma }-1}{1-\sigma },\ \ \sigma \gt 0\\[5pt] v_{1}\left ( L\right ) =-\frac{\xi }{1-\phi }\left ( 1-L\right ) ^{1-\phi },\ \ \phi \gt 0,\ \xi \gt 0,\ L\in \left [ 0,1\right ] \\[5pt] v_{2}\left ( L\right ) =\frac{\xi }{1+\phi }L^{1+\phi },\ \ \phi \gt 0,\ \xi \gt 0,\ L\geq 0\\[5pt] f\left ( L\right ) =L^{\theta },\ 0\lt \theta \lt 1 \end{array} \right . \end{equation}

Each of these functions has been widely used in economics as a standard form of the utility and production functions. The consumption utility function $u\left ( C\right )$ and one of the labor disutility functions, $v_{1}\left ( L\right )$ , exhibit constant relative risk aversion. The other labor disutility function $v_{2}\left ( L\right )$ is characterized by the constant Frisch elasticity of labor supply. The production function is a standard concave power function.

Using algebra, we can calculate $\eta _{v}/\eta _{u}$ as implied by the functions in equation (4) as follows.

\begin{equation*} \frac {\eta _{v}}{\eta _{u}}=\left \{ \begin {array} [c]{cc}\frac {1}{\sigma }\left ( \frac {\phi }{\alpha }\frac {Y^{1/\theta }}{1-Y^{1/\theta }}+\frac {1}{\theta }-1\right ) \left ( 1-\frac {G}{Y}\right ) \text { } & \text {if }v\left ( L\right ) =v_{1}\left ( L\right ) \\[5pt] \left ( \frac {1-\theta +\phi }{\theta \sigma }\right ) \left ( 1-\frac {G}{Y}\right ) & \text {if }v\left ( L\right ) =v_{2}\left ( L\right ) \end {array} \right . \end{equation*}

In both cases, $\eta _{v}/\eta _{u}$ increases with $Y$ . Consequently, $\partial m/\partial \tau \gt 0$ with functions in equation (4). In other words, the government spending multiplier increases with the income tax rate if the economy is well represented by these functions. Admittedly, this result does not prove that the income tax rate has a positive impact on the multiplier effect. However, it may suggest that the positive impact of the income tax rate on the government spending multiplier can be a realistic possibility because it is based on the standard utility and production functions that can account for numerous macroeconomic phenomena. Indeed, in our empirical analysis, we find that an unanticipated increase in government purchases tends to have more significant and positive effects on output when the tax-GDP ratio is high. This finding may corroborate our theoretical result in the static neoclassical model with the functions in equation (4).

2.2 Dynamic neoclassical model with labor and capital

In this subsection, we consider a dynamic neoclassical model with capital and labor following Baxter and King (Reference Baxter and King1993). This is more general than the static neoclassical model because it has more channels through which government purchases can influence output. Even in this general dynamic model, we quantitatively show that the income tax rate tends to have a positive impact on output responses to a rise in government purchases.

2.2.1 Model setup

The economy is composed of households, firms, and a government. First, homogeneous households choose consumption $C_{t}$ , labor $L_{t}\in \left [ 0,1\right ]$ , and capital $K_{t+1}$ in period $t$ to maximize the following sum of utilities:

\begin{equation*} \mathbb {E}\left [ \sum _{t=0}^{\infty }\beta ^{t}\left [ \ln C_{t}+\xi \ln \left ( 1-L_{t}\right ) \right ] \right ], \ \ \beta \in \left ( 0,1\right ), \ \xi \gt 0, \end{equation*}

where $\ln C_{t}$ and $\ln \left ( 1-L_{t}\right )$ represent the utilities from consumption $C_{t}$ and leisure $\left ( 1-L_{t}\right ), $ respectively. Notice that both functions have the form given in equation (4) for $dm/d\tau \gt 0$ in a static neoclassical model. Household choices $\left \{ C_{t},L_{t},K_{t+1}\right \}$ should satisfy the following budget constraint.

\begin{equation*} C_{t}+K_{t+1}=\left ( 1-\tau \right ) \left ( W_{t}L_{t}+R_{t}^{k}K_{t}\right ) +\left ( 1-\delta \right ) K_{t}+TR_{t}, \end{equation*}

where $W_{t}$ and $R_{t}^{k}$ denote the wage and rental price of capital in period $t$ . As before, $\tau \in \left ( 0,1\right )$ and $\ TR_{t}$ are the income tax rate and lump-sum government transfers in period $t$ . Finally, $\delta \in \left [ 0,1\right ]$ is the depreciation rate of capital.

On the supply side, homogeneous firms choose $L_{t}$ and $K_{t}$ to solve the following problem:

\begin{equation*} \max _{\left \{ K_{t},L_{t}\right \} }\left [ K_{t}^{\alpha }L_{t}^{1-\alpha }-W_{t}L_{t}-R_{t}^{k}K_{t}\right ], \ \ \alpha \in \left ( 0,1\right ), \end{equation*}

where $Y_{t}=K_{t}^{\alpha }L_{t}^{1-\alpha }$ denotes the production function. Finally, the government collects income tax, purchases goods $G_{t},$ and provides lump-sum transfers $TR_{t}$ to households in period $t$ . Hence, the government budget constraint can be written as

\begin{equation*} G_{t}+TR_{t}=\tau \left ( W_{t}L_{t}+R_{t}^{k}K_{t}\right ) . \end{equation*}

Note that $TR_{t}$ can be interpreted as the government budget surplus.

The competitive equilibrium of this economy is characterized by the following equations.

(5) \begin{align} \xi \frac{C_{t}}{1-L_{t}}=\left ( 1-\tau \right ) \left ( 1-\alpha \right ) K_{t}^{\alpha }L_{t}^{-\alpha } \end{align}

(6) \begin{align} \frac{1}{C_{t}}=\beta \mathbb{E}_{t}\left [ \frac{1}{C_{t+1}}\left \{ 1-\delta +\left ( 1-\tau \right ) \alpha K_{t+1}^{\alpha -1}L_{t+1}^{1-\alpha }\right \} \right ] \end{align}

(7) \begin{align} C_{t}+G_{t}+K_{t+1}=K_{t}^{\alpha }L_{t}^{1-\alpha }+\left ( 1-\delta \right ) K_{t} \end{align}

We derive equation (5) by combining the first-order conditions for labor supply and demand. Similarly, we obtain equation (6) from the households’ intertemporal Euler equation and condition for capital demand. In addition, equation (7) is the resource constraint or goods-market-clearing condition that should hold in competitive equilibrium. In the next subsections, we characterize the economy using equilibrium conditions (5)–(7) given government policy $\left \{ \tau, G_{t}\right \}$ .

2.2.2 The effects of a permanent fiscal expansion in the steady state

We begin by analyzing the effects of a permanent rise in government purchases on output and other steady-state variables. This exercise could be interpreted as a generalization of the static neoclassical model, as the dynamic model incorporates both the intratemporal and intertemporal responses of economic agents. In the steady state, equilibrium conditions (5)–(7) are simplified as follows:

(8) \begin{align} \xi \frac{C^{\ast }}{1-L^{\ast }}=\left ( 1-\tau \right ) \left ( 1-\alpha \right ) \left ( K^{\ast }\right ) ^{\alpha }\left ( L^{\ast }\right ) ^{-\alpha }, \end{align}

(9) \begin{align} 1=&\; \beta \left [ 1-\delta +\alpha \left ( 1-\tau \right ) \left ( K^{\ast }\right ) ^{\alpha -1}\left ( L^{\ast }\right ) ^{1-\alpha }\right ], \\& C^{\ast }+G^{\ast }+\delta K^{\ast }=\left ( K^{\ast }\right ) ^{\alpha }\left ( L^{\ast }\right ) ^{1-\alpha },\nonumber \end{align}

where $X^{\ast }$ denotes the steady-state value of a variable $X_{t}$ . For any policy mix $\left ( \tau, G^{\ast }\right ), $ we find $\left ( C^{\ast },K^{\ast },L^{\ast },Y^{\ast }\right )$ from those conditions and the production function $Y^{\ast }=\left ( K^{\ast }\right ) ^{\alpha }\left ( L^{\ast }\right ) ^{1-\alpha }$ .

We can also obtain the derivatives of $\left ( C^{\ast },K^{\ast },L^{\ast },Y^{\ast }\right )$ with respect to $G^{\ast }$ from the total differentials of steady-state equilibrium conditions. Our primary interest is in $\partial Y^{\ast }/\partial G^{\ast }$ , denoted by $m^{\ast }.$ This can be interpreted as a long-run government spending multiplier, because it represents the long-run effect of a permanent rise in government purchases on steady-state output. Appendix A shows, $m^{\ast }$ is expressed in terms of the income tax rate $\tau$ and other model parameters as follows:

(10) \begin{equation} m^{\ast }=\frac{1}{1+\psi \left ( 1-\tau \right ) }, \ \text{with }\,\psi \equiv \frac{1-\alpha }{\xi }-\frac{\delta \alpha }{\beta ^{-1}-1+\delta }. \end{equation}

The properties of $m^{\ast }$ depend crucially on $\psi$ , that is, the coefficient on $\left ( 1-\tau \right )$ . If $\psi \gt 0$ , then $\partial m^{\ast }/\partial \tau \gt 0$ and $0\lt m^{\ast }\lt 1$ . In this case, $m^{\ast }$ exhibits qualitative properties similar to those of $m$ in equation (3) into a static neoclassical model.Footnote ⁹ By contrast, if $\psi \lt 0$ , then $\partial m^{\ast }/\partial \tau \lt 0$ and $m^{\ast }\gt 1$ . In this case, $m^{\ast }$ has properties similar to those of the government spending multiplier in the traditional Keynesian model.

In equation (10), the sign of $\psi$ is primarily determined by $\xi$ because there are standard values of $\alpha$ (capital share in GDP), $\beta$ (discount factor), and $\delta$ (depreciation rate of capital) in the literature. Specifically, let us define $\bar{\xi }$ as the value of $\xi$ corresponding to $\psi =0$ , given the values of $\left ( \alpha, \beta, \delta \right )$ . From equation (10), we obtain $\bar{\xi }$ as follows:

(11) \begin{equation} \bar{\xi }\equiv \frac{\left ( 1-\alpha \right ) (\beta ^{-1}-1+\delta )}{\alpha \delta }. \end{equation}

Then, $\xi \lt \bar{\xi }$ is required for $\psi \gt 0$ because $\psi$ decreases in $\xi$ . This condition is likely to be satisfied by realistic parameter values. It should be noted that the equilibrium labor $L^{\ast }$ tends to decrease with $\xi$ because the parameter represents the importance of leisure in the utility function. However, if $\xi \geq \bar{\xi }$ with standard values of $\left ( \alpha, \beta, \delta \right )$ , $L^{\ast }$ tends to be unrealistically low. Thus, the value of $\xi$ is likely to be smaller than $\bar{\xi }$ to generate an empirically plausible $L^{\ast }$ . In other words, the condition $\psi \gt 0$ tends to be satisfied in the steady state.

To illustrate these results, we simulate the model using realistic parameter values. To this end, we choose $\alpha =0.33,\ \beta =0.96$ , and $\delta =0.1$ because we interpret one model period as one year. These parameter values are widely used in the literature. We set $\xi$ depending on fiscal policy $\left ( \tau, G^{\ast }\right )$ so that $L^{\ast }=0.33$ may be attained in the steady state under any given combination of $\tau$ and $G^{\ast }.$ Then, we compare $\xi$ with the threshold $\bar{\xi }=2.876,$ which can be calculated from equation (11). We also examine the behavior of the long-run multiplier $m^{\ast }$ and related variables from the simulated model.

To assess the plausibility of $\xi \lt \bar{\xi }$ through this simulation, we need to consider a wide range of fiscal policy (e.g., from a small government to a big one). Therefore, we choose 20%, 40%, and 60%, as values of $\tau .$ These values can represent government size observed in the real world. For each value of $\tau$ , we choose $G^{\ast }$ ranging from 95% of total tax revenue $\tau Y^{\ast }$ to 120%. As a result, the government deficit to GDP ratio falls on the interval $[{-}5\%,20\%]$ in the simulation. As such, the combinations of $\tau$ and $G^{\ast }$ can capture the various types of fiscal policy observed in the real world.

The simulation results are shown in Fig. 1. In Panel (a), we display $\xi$ that can yield $L^{\ast }=0.33$ for each combination of $\tau$ and $G^{\ast }.$ The figures in all other panels are drawn for the various values of $\tau$ and $G^{\ast }$ and corresponding values of $\xi .$ In Panel (a), we can clearly see that $\xi \lt \bar{\xi }$ , which is the condition for $\psi \gt 0$ , is satisfied for most values of $\tau$ and $G^{\ast }$ . We observe $\xi \gt \bar{\xi }$ only when both $\tau$ and $G^{\ast }$ are unrealistically large. By equation (10), the result implies $\psi \gt 0$ , $\ 0\lt m^{\ast }\lt 1$ , and $\partial m^{\ast }/\partial \tau \gt 0$ in cases where $\xi \lt \bar{\xi }$ , as shown in Panels (b)–(d) of Fig. 1. These results are consistent with the properties of the multiplier $m$ in equation (3) in the static neoclassical model. They suggest that $m^{\ast }$ tends to be smaller than one, but increases in $\tau$ in the steady states under the realistic types of fiscal policy. Moreover, we observe a negative wealth effect of an increase in government purchases in the steady states of the dynamic neoclassical model. As discussed previously, such fiscal expansion reduces consumption, but raises labor supply in the static neoclassical model through the negative wealth effect. Indeed, Fig. 1 shows that $\partial L^{\ast }/\partial G^{\ast }\gt 0$ in Panel (e) and $\partial C^{\ast }/\partial G^{\ast }\lt 0$ in Panel (f) regardless of $\tau$ and $G^{\ast }$ in the steady states of the dynamic model.Footnote ¹⁰

Figure 1. Long-run effects of a permanent rise in government purchases. This figure displays the steady-state outcomes of the simulated dynamic neoclassical model with labor and capital as discussed in Subsection 2.2. In all panels, the horizontal axis represents government purchases $G^{\ast }$ as a percentage of total tax revenue $ \tau Y^{\ast }.$ Also, in all panels, for each combination of $ \tau$ and $G^{\ast },$ we choose a different $ \xi$ so that $L^{\ast }=0.33$ may be attained. Such $ \xi$ is displayed for all combinations of $ \tau$ and $G^{\ast }$ in panel (a). In other panels, we display $ \psi, $ defined in equation (10), in panel (b); $m^{\ast }=\partial Y^{\ast }/\partial G^{\ast },$ also defined in equation (10), in panel (c); $\partial m^{\ast }/\partial \tau$ in panel (d); $\partial L^{\ast }/\partial G^{\ast }$ in panel (e); and $\partial C^{\ast }/\partial G^{\ast }$ in panel (f). The black dashed line in panel (a) represents a threshold value $\bar{ \xi }$ defined in equation (11). If $ \xi \lt \bar{ \xi },$ $ \psi \gt 0$ in panel (b), $m^{\ast }\lt 1$ in panel (c), and $\partial m^{\ast }/\partial \tau \gt 0$ in panel (d). See the equations and related discussion in Subsection 2.2.2 for details.

Overall, the simulation results from the steady state of the dynamic neoclassical model tend to confirm the analytical results from the static neoclassical model. In addition, the main simulation results, such as $m^{\ast }\in \left ( 0,1\right )$ and $\partial m^{\ast }/\partial \tau \gt 0$ are quite robust, because they are shown in Fig. 1 for almost all values of $\tau$ and $G^{\ast }$ . These results suggest that a relatively small multiplier effect and the positive effect of the income tax rate on the multiplier can be realistic outcomes of an increase in government purchases and not just theoretical possibilities.

2.2.3 Impulse responses to a one-time fiscal expansion

In the second exercise using the dynamic neoclassical model, we quantify the effects of an unanticipated one-time increase in government purchases. To this end, we characterize the impulse responses of macroeconomic variables to one-time fiscal expansion. In particular, we focus on the magnitude of the impulse responses of output and the impact of the income tax rate on these impulse responses. The results in this subsection can be compared with the empirical results for the government spending multipliers in Section 3.

We log-linearize the equilibrium conditions (5)–(7) and production function $Y_{t}=K_{t}^{\alpha }L_{t}^{1-\alpha }$ around the steady state:

\begin{align*} c_{t} & =\alpha k_{t}-\left ( \alpha +\frac{L^{\ast }}{1-L^{\ast }}\right ) l_{t},\\ c_{t} & =\mathbb{E}_{t}\left [ c_{t+1}+\left \{ 1-\beta \left ( 1-\delta \right ) \right \} \left ( 1-\alpha \right ) \left ( k_{t+1}-l_{t+1}\right ) \right ], \\ s_{K}k_{t+1} & =\left [ \alpha +\left ( 1-\delta \right ) s_{K}\right ] k_{t}+\left ( 1-\alpha \right ) l_{t}-s_{C}c_{t}-s_{G}g_{t},\\ y_{t} & =\alpha k_{t}+\left ( 1-\alpha \right ) l_{t}, \end{align*}

where $x_{t}\equiv \ln X_{t}-\ln X^{\ast }$ and $s_{X}\equiv X^{\ast }/Y^{\ast }$ for each variable $X_{t}$ with steady-state output $Y^{\ast }$ . Government purchases are assumed to evolve as an AR(1) process as follows:

(12) \begin{equation} g_{t}=\rho g_{t-1}+\nu _{t},\ \ \rho \in \left ( 0,1\right ), \end{equation}

where $g_{t}\equiv \ln \left ( G_{t}/G^{\ast }\right )$ and $\nu _{t}$ is an i.i.d. shock. For the quantitative analysis, we assume $\rho =0.7$ . In addition, we choose $\alpha =0.33,\ \beta =0.96,$ and $\delta =0.1$ as in the steady-state analysis. For fiscal policy, we consider the two values of $\tau$ , 0.2 and 0.4, to examine the role of the income tax rate. For each $\tau$ , we choose $G^{\ast }$ such that the government budget is balanced in the steady state (i.e., $G^{\ast }=\tau Y^{\ast }$ or $TR^{\ast }=0$ ). Subsequently, we set $\xi$ such that $L^{\ast }=0.33$ can be obtained in the steady state for each combination of $\tau$ and $G^{\ast }$ .

To obtain impulse responses, we assume that each economy has been in a balanced-budget steady state up to period $-1$ . However, a one-time positive shock occurs to government purchases in period 0, such that $\nu _{t}=1$ for $t=0$ and $\nu _{t}=0$ for $t\geq 1$ in equation (12). In other words, government purchases increase unexpectedly by 1% in period 0 and change in subsequent periods according to equation (12). Given the process for $g_{t}$ and parameter values, we can calculate the impulse responses $\left ( c_{t},l_{t},k_{t},y_{t}\right )$ using log-linearized equilibrium conditions. However, the impulse responses of output are not government spending multipliers because both $g_{t}$ and $y_{t}$ are logarithmic. Hence, we calculate the cumulative output responses as follows:

\begin{equation*} M^{h}\equiv \frac {\sum _{t=0}^{h}\left ( Y_{t}-Y^{\ast }\right ) }{\sum _{t=0}^{h}\left ( G_{t}-G^{\ast }\right ) }\end{equation*}

Intuitively, $M^{h}$ represents the total change in output over $h$ periods, due to the total change in government purchases. In this sense, the cumulative responses can be interpreted as cumulative government spending multipliers.

Figure 2 presents the simulated output, labor, and consumption responses. In Panels (a) and (b), the characteristics of the impulse and cumulative responses of output are consistent with the main theoretical results. First, a positive shock to government purchases can stimulate output. However, all impulse responses are smaller than 0.2 in Panel (a), and the cumulative responses are far below one in Panel (b). These results indicate that government spending multipliers are smaller than one. Another interesting finding is that the negative wealth effect of fiscal expansion is also observed in impulse responses. In Panels (c) and (d) of Fig. 2, labor supply increases, whereas consumption decreases in response to a positive shock to government purchases. As previously discussed, these are characteristics of the negative wealth effect of fiscal expansion.

Figure 2. Simulated impulse responses to a one-time fiscal expansion. Panels (a), (c), and (d) display the impulse responses of log of output, labor, and consumption to an unexpected one-time 1% rise in government purchases in period 0. Panel (b) shows the cumulative responses of output. All panels are based on the calibrated dynamic neoclassical model. Refer to discussion in Subsection 2.2 for details regarding the model specification and calibration.

Regarding the role of the income tax rate, a positive shock to government purchases has a more significant stimulating impact on output when $\tau$ is higher. In Panel (a) of Fig. 2, output exhibits stronger impulse responses when $\tau =0.4$ for up to six periods after the shock. Similarly, the cumulative responses of output are larger for $\tau =0.4$ than for $\tau =0.2$ in Panel (b). Such differences in impulse and cumulative responses indicate that the higher the income tax rate, the stronger the multiplier effect of a rise in government purchases. Obviously, this relationship is not generalizable. However, this has been proven analytically in the static neoclassical model with standard forms of the utility and production functions. It is also verified quantitatively using a dynamic neoclassical model calibrated with standard functional forms and parameter values. With a high income tax rate, a permanent increase in government purchases raises the steady-state output more significantly, and a temporary increase in government purchases leads to stronger positive output responses. Overall, the positive impact of the income tax rate on the government spending multiplier is quite robust in our theoretical analysis using both static and dynamic neoclassical models.

2.3 Economies with monopolistic competition

So far, we have analyzed both static and dynamic neoclassical models. In these models, we showed that high tax rates can amplify the multiplier effect of increased government purchases with realistic functional forms and parameter values. However, this result might hold less relevance for other types of models if they have a quite different structure from the neoclassical models. In particular, the New Keynesian models feature monopolistic competition and various types of frictions such as nominal rigidities. Hence, our theoretical results from the neoclassical models may not necessarily apply to the New Keynesian models. Given the importance of New Keynesian models in macroeconomics, we discuss in this subsection how our theoretical findings can be extended to the New Keynesian models.

First, we show that our theoretical results still hold even if monopolistic competition is introduced to the model. For this purpose, we consider the Dixit–Stiglitz model of monopolistic competition. In the economy, there is a continuum of differentiated goods on the interval $\left [ 0,1\right ] .$ Each good is produced by a producer with some degree of market power. Households consume differentiated goods and supply labor for the production of those goods. Let $c\left ( i\right )$ and $l\left ( i\right )$ denote consumption and labor supply for good $i.$ The utility function for households is assumed as follows:

\begin{equation*} u\left ( C\right ) -\int _{0}^{1}v\left ( l\left ( i\right ) \right ) di, \end{equation*}

where $C$ is composite consumption defined as

\begin{equation*} C\equiv \left [ \int _{0}^{1}c\left ( i\right ) ^{\frac {\theta -1}{\theta }}di\right ] ^{\frac {\theta }{\theta -1}},\ \ \theta \gt 1. \end{equation*}

As in the neoclassical models, we assume $u^{\prime }\gt 0\gt u^{\prime \prime }$ and $v^{\prime }\gt 0,\ v^{\prime \prime }\gt 0.$ With differentiated goods, the households’ budget constraint is modified as follows:

\begin{equation*} \int _{0}^{1}p\left ( i\right ) c\left ( i\right ) di=\left ( 1-\tau \right ) \left [ w\int _{0}^{1}l\left ( i\right ) di+\Pi \right ] +TR, \end{equation*}

where $p\left ( i\right )$ is the price of good $i.$ As defined earlier, $w,\ \tau, \ \Pi, $ and $TR$ denote the wage, income tax rate, dividends from firms and lump-sum transfers from the government, respectively.

Each household maximizes its utility subject to the budget constraint. The first-order conditions of the households’ problem imply that $l\left ( i\right )$ is the same for all $i.$ Hence, we denote common labor supply by $L.$ Then, labor disutility can be simplified to $v\left ( L\right ), $ which coincides with labor disutility in the static neoclassical model. Also, the demand for good $i$ is obtained as

\begin{equation*} c\left ( i\right ) =\left ( \frac {p\left ( i\right ) }{P}\right ) ^{-\theta }C\ \ \text {for all }i\in \left [ 0,1\right ], \end{equation*}

where $P$ is the price index defined as

\begin{equation*} P=\left [ \int _{0}^{1}p\left ( i\right ) ^{1-\theta }di\right ] ^{\frac {1}{1-\theta }}. \end{equation*}

Finally, labor supply and consumption should satisfy the following condition:

\begin{equation*} v^{\prime }\left ( L\right ) =\left ( 1-\tau \right ) \frac {w}{P}u^{\prime }\left ( C\right ) \end{equation*}

On the production side, firm $i$ produces output $y\left ( i\right )$ using labor $l\left ( i\right )$ according to a production function $y\left ( i\right ) =f\left ( l\left ( i\right ) \right )$ . The production function $f$ is assumed to be identical across firms. Also, it satisfies the standard assumption, $f^{\prime }\gt 0\gt f^{\prime \prime }.$ As firms have market power, firm $i$ chooses $p\left ( i\right )$ to maximize its profit $p\left ( i\right ) c\left ( i\right ) -wf^{-1}\left ( c\left ( i\right ) \right )$ . Solving the firm’s problem provides the following condition for labor demand.

\begin{equation*} p\left ( i\right ) =\mu \frac {w}{f^{\prime }\left ( l\left ( i\right ) \right ) }, \end{equation*}

where $\mu \equiv \frac{\theta }{\theta -1}$ represents the degree of the markup. From the household’s problem, $l\left ( i\right ) =L$ for all $i.$ Then, the labor demand condition implies $p\left ( i\right ) =P$ for all $i.$

In equilibrium, both labor supply and demand conditions are satisfied. Also, $Y=C+G$ for goods market clearing. Combining all those conditions, we obtain the following equation:

(13) \begin{equation} \left ( 1-\tau \right ) u^{\prime }\left ( Y-G\right ) =\mu \tilde{v}^{\prime }\left ( Y\right ), \end{equation}

where $\tilde{v}\left ( Y\right ) =v\left ( f^{-1}\left ( Y\right ) \right )$ is defined in the static neoclassical model. This equation is almost identical to equation (1), except for the markup $\mu .$ Moreover, all results in the static neoclassical model still hold in the current model with monopolistic competition. More specifically, (i) the income tax rate has a negative impact on the equilibrium output (i.e., $\partial Y/\partial \tau \lt 0$ ); (ii) the government spending multiplier $m$ takes exactly the same form as in equation (3); and (iii) $\partial m/\partial \tau$ will be positive for the functions in equation (4). As such, the main theoretical results in the static neoclassical model can be generalized to the economies with monopolistic competition. In addition, the negative wealth effect channel is still relevant even if the economy is characterized with monopolistic competition.

Our interest lies in whether it is feasible to derive analogous theoretical outcomes in New Keynesian dynamic models. Although New Keynesian models are more complicated than the simple model of monopolistic competition. However, even in the New Keynesian models, an equilibrium condition analogous to equation (13) should be satisfied for the labor market. In addition, households’ labor supply and firms’ labor demand in the New Keynesian models are based on those in the simple model of monopolistic competition. Given these similarities, it is reasonable to anticipate that in the New Keynesian models, an increase in government purchases may still induce a negative wealth effect, reducing consumption but raising labor supply. Moreover, the New Keynesian models are often calibrated with the functions presented in equation (4). Consequently, the negative wealth effect of fiscal expansion can also generate a positive impact of the income tax rate on the government spending multiplier in the New Keynesian models.

On the other hand, there are many other channels through which the income tax rate affects the government spending multiplier. Consequently, the overall effect of the income tax rate on the government spending multiplier may often be negative in the New Keynesian models. However, the negative wealth effect is expected to exist and interact with the income tax rate in these models. Therefore, if the New Keynesian models are calibrated with the functions specified in equation (4), the income tax rate can positively affect the government spending multiplier through the negative wealth effect. Taken together, the assertion that an increase in government purchases stimulates output more effectively when the income tax rate is high is not generally applicable in the New Keynesian models. Nevertheless, the positive effect of the income tax rate on the government spending multiplier is likely to coexist with other channels in the New Keynesian models.

3.1 Data and the identification of shocks to government purchases

For our empirical analysis, we construct a dataset of both macroeconomic and fiscal variables for 12 OECD countries over the period 1985–2019. All variables in our dataset are obtained from the OECD Economic Outlook and OECD Statistics and Projections databases. Our dataset includes Belgium, Denmark, France, Germany, Italy, the Netherlands, Spain, Sweden, Portugal, Switzerland, the United Kingdom, and the United States.Footnote ¹¹ However, our dataset is an unbalanced panel, due to differences in data availability across the sample countries.

In our empirical analysis, we estimate the macroeconomic effects of unanticipated changes in government purchases using the state-dependent local projection method (Jordà (Reference Jordà2005)). To implement this method, the shocks to government purchases must be identified. Following Auerbach and Gorodnichenko (Reference Auerbach and Gorodnichenko2012, Reference Auerbach, Gorodnichenko, Auerbach and Gorodnichenko2013), we use forecast errors of government purchases as “government spending shocks” because they represent the component of government purchases unexpected by economic agents. Specifically, let $G_{it}$ and $G_{it}^{e}$ denote actual and forecasted government purchases, respectively, for country $i$ in year $t$ .Footnote ¹² The forecast $G_{it}^{e}$ is predetermined in year $t-1.$ We define the government spending shock $\varepsilon _{it}$ for country $i$ in year $t$ as follows:

(14) \begin{equation} \varepsilon _{it}\equiv \frac{G_{it}-G_{it}^{e}}{Y_{i,t-1}}, \end{equation}

where $Y_{i,t-1}$ is GDP of country $i$ in year $t-1$ . In other words, the government spending shock in a year is the forecast error of government purchases scaled by GDP in the previous year.Footnote ¹³ $^{,}$ Footnote ¹⁴ Note that $\varepsilon _{it}$ is analogous to the shock $\nu _{t}$ in equation (12) for the calibrated dynamic neoclassical model in Section 2 because both can be interpreted as forecast errors of government purchases.

Ideally, $G_{it}^{e}$ should be determined at the end of year $t-1$ . In this case, $\varepsilon _{it}$ could be a pure shock to $G_{it}$ because it would only include the component of $G_{it}$ that was totally unanticipated until the end of the previous year. Moreover, a regression of $Y_{i,t+h}$ on $\varepsilon _{it}$ yields an unbiased estimate of the output response in period $t+h$ to an unexpected change in $G_{it}$ in period $t$ . In other words, we can avoid the well-known issues of fiscal foresight (Leeper et al. (Reference Leeper, Walker and Yang2013)) and potential feedback from the state of the economy to fiscal policy. In contrast, if $G_{it}^{e}$ is determined relatively early in year $t-1$ , $\varepsilon _{it}$ may include the component of $G_{it}$ that is fully expected in year $t-1$ based on the newly available information after $G_{it}^{e}$ is determined. In this case, a regression of $Y_{it}$ on $\varepsilon _{it}$ may also include part of output that is not caused by shocks to government purchases in period $t$ .

Considering the importance of the forecasting timing, we draw $G_{it}^{e}$ from forecasts of government purchases published in the fall issue of the previous year in the OECD Economic Outlook and Statistics and Projections databases. Hence, there is only a two- or three-month gap between the publication of forecast $G_{it}^{e}$ and the beginning of year $t$ . Consequently, $\varepsilon _{it}$ can identify the shock component of $G_{it}$ reasonably well as significant events or policy changes are not as frequent after the publication of OECD forecasts of government purchases. If this is the case, then the regression of the macroeconomic variables on $\varepsilon _{it}$ can properly capture their responses to unanticipated changes in government purchases. Based on these advantages, we interpret $\varepsilon _{it}$ as a government spending shock, which is the main explanatory variable in our empirical analysis.

3.2 Estimation equations

To evaluate the dynamic responses of the macroeconomic variables to unexpected changes in government purchases, we employ a state-dependent local projection method. We estimate the following equation for each $h=0,1,\cdots, 4$ :

(15) \begin{equation} \ln Z_{i,t+h}-\ln Z_{i,t-1}=\alpha _{i}^{h}+\theta _{t}^{h}+\left [ \beta _{LT}^{h}H\left ( T_{i,t-1}\right ) +\beta _{HT}^{h}\left \{ 1-H\left ( T_{i,t-1}\right ) \right \} \right ] \varepsilon _{it}+\omega ^{h}X_{it}+u_{i,t+h}^{h}, \end{equation}

with the transition function $H\left ( \cdot \right )$ defined as

(16) \begin{equation} H\left ( T_{i,t-1}\right ) =\frac{1}{1+\exp \left ( \gamma T_{i,t-1}\right ) },\text{ }\,\gamma \gt 0. \end{equation}

In the estimation equation, $Z$ is a variable of interest, and $i$ and $t$ are the indices for the country and year, respectively. $\alpha _{i}^{h}$ and $\ \theta _{t}^{h}$ are the country and year dummies, respectively, and $X_{it}$ is a vector of the control variables.Footnote ¹⁵ $\varepsilon _{it}$ denotes the government spending shock defined in equation (14). Note that the coefficient on $\varepsilon _{it}$ consists of two parameters, $\beta _{LT}^{h}$ and $\beta _{HT}^{h}$ , which correspond to the “low-tax (LT)” and “high-tax (HT)” states, as explained below. The error term $u_{i,t+h}^{h}$ may not be spherical. Hence, we calculate the Driscoll–Kraay (1998) standard errors, as they are robust to heteroskedasticity, autocorrelation, and cross-sectional dependence. We estimate equation (15) for various $Z$ , such as output, employment, and consumption, over the horizon $h=0,1,\cdots, 4$ , so that we can characterize their dynamic responses over four years after a government spending shock occurs.

In the transition function $H$ in equation (16), we use the lagged tax-GDP ratio, denoted by $T_{i,t-1},$ as a transition variable. The tax-GDP ratio is a useful indicator of the overall tax burden or government size. Also, it is directly related to the income tax rate $\tau$ in the theoretical analysis in Section 2 because it coincides with the tax-GDP ratio in equilibrium. The parameter $\gamma$ in equation (16) determines the speed of transition. We assume $\gamma =1.5$ following Auerbach and Gorodnichenko (Reference Auerbach, Gorodnichenko, Auerbach and Gorodnichenko2013) and Honda and Miyamoto (Reference Honda and Miyamoto2021). However, the main results are robust to changes in $\gamma$ . We also normalize $T_{it}$ in equation (16) so that $\mathbb{E}\left ( T_{it}\right ) =0$ and $var\left ( T_{it}\right ) =1$ .

We use the lagged tax-GDP as the transition variable to avoid potential issues caused by contemporaneous effects of government purchases and the dependent variables on tax revenue. For example, there can be potential reverse causality from output to tax revenue. Also, if government purchases can influence tax revenue, a positive $\varepsilon _{it}$ , or equivalently, an unanticipated increase in government purchases, may also affect tax revenue. In other words, the government spending shock may influence the transition variable. These issues can create some bias in our regression analysis. Therefore, we use the lagged tax-GDP ratio $T_{i,t-1}$ as the transition variable because it is unlikely to be affected by the government spending shock $\varepsilon _{it}$ or output $Y_{it}$ in period $t.$ Footnote ¹⁶

With the transition function discussed so far, we can interpret the regression equation (15). In particular, the coefficient on $\varepsilon _{it}$ is expressed as a weighted average of $\beta _{LT}^{h}$ and $\beta _{HT}^{h}$ with the weights, $H$ and $\left ( 1-H\right )$ . With this specification, the tax-GDP ratio controls the dynamic responses of the macroeconomic variables to government spending shocks because weights are fully determined by $T_{i,t-1}$ . Specifically, as the lagged tax-GDP ratio increases, the weight on $\beta _{HT}^{h}$ increases, but the weight on $\beta _{LT}^{h}$ decreases in the coefficient on $\varepsilon _{it}$ because $H$ is a decreasing function of $T_{i,t-1}$ . Thus, $\beta _{HT}^{h}$ is more relevant for large governments with high tax rates, whereas $\beta _{LT}^{h}$ is more relevant for small governments with low tax rates. We compare $\beta _{LT}^{h}$ and $\beta _{HT}^{h}$ to evaluate the quantitative effects of the tax-GDP ratio or government size on the dynamic responses of the macroeconomic variables to government spending shocks. Such an analysis is closely related to the theoretical analysis in Section 2 because the tax-GDP ratio is represented by the income tax rate $\tau$ in theoretical models. We focus on the comparison between $\beta _{LT}^{h}$ and $\beta _{HT}^{h}$ when interpreting the estimation results in the next section.

4.1 Responses of output

First, we discuss the estimated output responses to positive government spending shocks. Noticeably, the estimated output responses are almost opposite in the high- and low-tax economies shown in the top panels of Fig. 3. In the left panel, the estimated $\beta _{HT}^{h}$ is positive for all $h=0,1,\cdots, 4$ and even statistically significant for $h=1,2$ . By contrast, the estimated $\beta _{LT}^{h}$ in the right panel is negative regardless of $h$ , although all estimates are statistically insignificant. These contrasting results clearly suggest that a positive government spending shock can stimulate output only when the tax-GDP ratio is relatively high. In an economy with a relatively low tax-GDP ratio, such fiscal expansion may fail to increase output.

These empirical results are consistent with the corresponding theoretical results. In Section 2, using the standard neoclassical models, we find that the stimulus effect of an increase in government purchases can increase with the income tax rate. This result is proven analytically in a static neoclassical model with standard functional forms and confirmed quantitatively through steady-state and impulse-response analyses in dynamic neoclassical models calibrated with standard functional forms and parameter values. The theoretical result is similar to the empirical result $\beta _{HT}^{h}\gt \beta _{LT}^{h}$ for output because both indicate that increases in government purchases have stronger stimulating effects as tax rates rise or government size grows.

Despite these similarities, we also observe differences between the estimated impulse responses of output in Fig. 3 and the simulated impulse responses in Panel (a) of Fig. 2 from the dynamic neoclassical model. First, the estimated impulse responses in Fig. 3 are positive only with a high tax-GDP ratio, whereas the simulated impulse responses in Panel (a) of Fig. 2 are always positive for $h=0,1,\cdots, 4$ whether the income tax rate $\tau$ is high or low. In addition, the estimated responses are U-shaped or inverted U-shaped, whereas the simulated impulse responses are monotonically decreasing up to $h=4$ regardless of the value of $\tau$ .

However, these differences appear plausible because the dynamic neoclassical model is intended to clearly highlight how the income tax rate influences the macroeconomic effects of a government spending shock. Thus, we set up a dynamic neoclassical model as simply as possible, abstracting from various factors and frictions that could help the model better match the macroeconomy of the countries in the sample. Due to its simplicity, the dynamic neoclassical model may not generate impulse responses that match their empirical counterparts, which could explain the differences between the theoretical and empirical impulse responses. From this perspective, our primary focus is on the qualitative properties of $\left ( \beta _{HT}^{h}-\beta _{LT}^{h}\right )$ when we compare the theoretical results in Fig. 2 with the empirical results in Fig. 3.

4.2 Responses of employment

Next, we analyze the impulse responses of employment, as presented in the second row of Fig. 3. In the left panel, the estimates of $\beta _{HT}^{h}$ are positive for all $h=0,1,\cdots, 4$ , but statistically significant only for $h=1,2$ . By contrast, in the right panel, the estimates of $\beta _{LT}^{h}$ are negative and statistically insignificant, regardless of $h$ . In other words, a positive government spending shock can boost employment when the tax-GDP ratio is high but it fails to do so when the tax-GDP ratio is low. These results are qualitatively identical to those of the output responses shown in the top panels of Fig. 3. Such similarities suggest that employment responses could be one of the key driving forces for output responses.

Neoclassical models appear to explain the important characteristics of employment impulse responses. As discussed in Section 2, an increase in government purchases can increase output because it induces people to work more through a negative wealth effect. In this sense, whether and how much such fiscal expansion stimulates output depends crucially on the magnitude of the positive labor response. Hence, output and labor are expected to exhibit qualitatively similar impulse responses, as theoretically confirmed in Fig. 2. Thus, the similarity in the estimated responses between output and labor in Fig. 3 is consistent with the theoretical prediction.

Moreover, the empirical result $\beta _{HT}^{h}\gt \beta _{LT}^{h}$ for employment can be explained using the neoclassical models with the standard functional forms considered in Section 2. In this case, the negative wealth effect of an unanticipated increase in government purchases tends to have a stronger positive impact on labor supply as the income tax rate $\tau$ rises. Therefore, output can be further stimulated by a high value of $\tau .$ For this reason, the theoretical impulse responses of both output and labor are more significant with a high value of $\tau$ , especially for $h=0,1,\cdots, 4$ , as shown in Panels (a) and (c) of Fig. 2. As such, the income tax rate can influence the characteristics of the negative wealth effect, thereby affecting both output and labor responses. This effect could also contribute to the empirical result $\beta _{HT}^{h}\gt \beta _{LT}^{h}$ as shown in Fig. 3.

4.3 Responses of consumption

We characterize the estimated impulse responses of consumption to a positive government spending shock. In the third row of Fig. 3, all estimates of $\beta _{HT}^{h}$ and $\beta _{LT}^{h}$ are statistically insignificant for consumption. This result indicates that consumption does not respond strongly to changes in government purchases regardless of the tax-GDP ratio. Focusing on the point estimates, both $\beta _{HT}^{h}$ and $\beta _{LT}^{h}$ remain close to zero for $h\leq 2$ , but diverge for $h=3,4$ . Consequently, $\left ( \beta _{HT}^{h}-\beta _{LT}^{h}\right )$ for consumption is essentially zero for $h\leq 2$ but becomes negative for $h=3,4$ . This result could be related to the theoretical counterpart in Panel (d) of Fig. 2, in which the simulated impulse responses of consumption are larger for the low-income tax rate. However, we do not observe a significant drop in consumption in Fig. 3, unlike the impulse responses obtained from the dynamic neoclassical model through a negative wealth effect. As previously discussed, this inconsistency could be caused by factors that were not considered in our theoretical analysis.

Moreover, the estimated responses of output and consumption tend to support the predictions of the neoclassical models rather than those of the traditional Keynesian model. In response to a positive shock to government purchases, output rises but consumption drops, according to neoclassical models, through the negative wealth effect of fiscal expansion. If this prediction is correct, then the output and consumption responses should exhibit a negative correlation. In contrast, according to the traditional Keynesian model, both output and consumption rise in response to fiscal expansion because output is stimulated through the positive reaction of consumption. In this case, the output and consumption responses exhibit a positive correlation. However, the estimated output and consumption responses tend to be negatively correlated, as shown in Fig. 3, because $\left ( \beta _{HT}^{h}-\beta _{LT}^{h}\right ) \gt 0$ for output and $\left ( \beta _{HT}^{h}-\beta _{LT}^{h}\right ) \lt 0$ for consumption. Overall, these empirical results tend to be more consistent with neoclassical models than traditional Keynesian models.

4.4 Discussion

Our empirical results can be summarized as follows. First, a positive shock to government purchases can significantly stimulate output only when the tax-GDP ratio is sufficiently high. This result is inconsistent with the predictions of the traditional Keynesian model. However, this can be better explained by static and dynamic neoclassical models, as discussed in Section 2. We find that government spending multipliers tend to increase in the income tax rate in neoclassical models with standard functional forms and parameter values. Taken together, these theoretical and empirical results suggest that fiscal expansion through rises in government purchases tends to be more effective in large governments with relatively high tax rates.

Second, we provide evidence of a negative wealth effect when governments make more purchases. In Fig. 3, output tends to exhibit a strong positive correlation with employment, but a weak negative correlation with consumption. Again, such differences in the signs of the correlations are better explained by the negative wealth effect in the neoclassical models than by the multiplier effect in the traditional Keynesian model. Thus, the empirical results corroborate the theoretical analysis.

4.5 Robustness checks

In this subsection, we confirm the robustness of our empirical results from state-dependent local projections. Specifically, we obtain state-dependent impulse responses of output to a government spending shock using various specifications as robustness checks in Fig. 4: (a) using an alternative government spending shock based on the forecast of government spending in the fall issue of the same year rather than that in the fall issue of the previous year as in the baseline case, (b) including control variables, such as current and lagged output growth shocks defined as the forecast error of GDP growth to deal with the endogeneity issue, which is caused by unexpected business cycle conditions, (c) including country-specific time trends, (d) excluding control variables such as government spending (over GDP), consumer price inflation, and short-term interest rate, and (e) controlling for tax response.

Figure 4. Robustness checks. Estimated impulse responses of output to a positive shock to government purchases using various specifications, with 90% confidence bands.

The estimated fiscal multipliers, using alternative specifications in Panels (a) through (e), remain similar to those from the baseline case as reported in Panel (a) of Fig. 3. That is, it suggests that a positive government spending shock can boost output when the tax-GDP ratio is relatively high. However, in an economy with a relatively low tax-GDP ratio, such fiscal expansion yields statistically insignificant responses of output. Overall, our main findings in Fig. 3 remain qualitatively unaltered across alternative specifications.