3.1 Definitions

Definition 1. $v_k(t)$ is defined as the sales share or Domar weight of sector $k$ in the economy. $\tilde{\alpha }_k(t)$ is defined as the labor income share in the production of goods $k$ . $\gamma _k(t)$ is defined as the share of labor used in sector $k$ in the total labor supply $L(t)$ . $\tilde{\Lambda }_L(t)$ is defined as the share of total labor income in the economy.

\begin{align*} v_k(t) \equiv \frac{P_k(t)Q_k(t)}{P(t)C(t)}, \quad \tilde{\alpha }_k(t)\equiv \frac{W(t) L_k(t)}{P_k(t)Q_k(t)},\quad \gamma _k(t) \equiv \frac{L_k(t)}{L(t)}, \quad \tilde{\Lambda }_L(t)\equiv \frac{W(t)L(t)}{P(t)C(t)}. \end{align*}

Definition 2. Define input–output matrix $\boldsymbol{\Omega}$ ( $N\times N$ ), Leontief inverse $\boldsymbol{\Psi}$ ( $N\times N$ ), consumption share vector $\boldsymbol{\beta }$ ( $N\times 1$ ), labor income share vector $\boldsymbol{\tilde{\alpha }}(t)$ ( $N\times 1$ ) in the production, sales share (Domar weight) vector $\boldsymbol{v}(t)$ ( $N\times 1$ ), and sectoral productivity vector $\boldsymbol{A}$ ( $N\times 1$ ).

\begin{align*} \boldsymbol{\Omega} &=\left [ \begin{array}{c@{\quad}c@{\quad}c} \theta _{11} & \ldots & \theta _{1N} \\[4pt] \vdots & \ddots & \vdots \\[4pt] \theta _{N1} & \ldots & \theta _{NN}\end{array}\right ] ; \quad \boldsymbol{\Psi}=\left (\boldsymbol{ I}-\boldsymbol{\Omega}\right ) ^{-1}=\left [ \begin{array}{c@{\quad}c@{\quad}c} \psi _{11} & \ldots & \psi _{1N} \\[4pt] \vdots & \ddots & \vdots \\[4pt] \psi _{N1} & \ldots & \psi _{NN}\end{array}\right ] ; \\[4pt] \boldsymbol{\beta } &=[\beta _1,\cdots,\beta _N];\ \boldsymbol{\tilde{\alpha }}(t)=[\tilde{\alpha }_1(t),\cdots,\tilde{\alpha }_N(t)]; \\[4pt] \boldsymbol{v}(t)&=[v_{1}(t),\cdots,v_{N}(t)];\quad \boldsymbol{A}=[A_1,\cdots, A_N]. \end{align*}

As in the production network literature, $\boldsymbol{\Omega}$ represents the input–output linkages between sectors of the economy, $\boldsymbol{\Psi}$ is the Leontief inverse matrix, and $\boldsymbol{v}(t)$ is the Domar weight vector representing the importance of each sector in the economy. The competitive equilibrium is defined as usual.

Definition 3. The competitive equilibrium is defined as an allocation $(C(t),\boldsymbol{C}(t),\boldsymbol{Q}(t), \\[4pt] L(t), \boldsymbol{L}(t), \boldsymbol{M}(t))$ and a price system $(P(t), \boldsymbol{P}(t), W(t))$ such that

1. (1) given the price system, the allocation solves households’ problem;

2. (2) given the price system, the allocation solves producers’ problem;

3. (3) all markets are cleared;

4. (4) the government runs a balanced budget,

where

\begin{align*} \boldsymbol{C}(t)&=[C_1(t),\cdots,C_N(t)], \boldsymbol{Q}(t)=[Q_1(t),\cdots,Q_N(t)]; \\[4pt] \boldsymbol{L}(t)&=[L_1(t),\cdots,L_N(t)], \boldsymbol{P}(t)=[P_1(t),\cdots,P_N(t)]; \\[4pt] \boldsymbol{M}(t)&=\left [ \begin{array}{c@{\quad}c@{\quad}c} M_{11}(t) & \ldots & M_{1N}(t) \\[4pt] \vdots & \ddots & \vdots \\[4pt] M_{N1}(t) & \ldots & M_{NN}(t)\end{array}\right ] ; \end{align*}

and $P(t)$ is normalized to 1.

We use black boldface to represent matrices and vectors. We denote $\boldsymbol{X}^0$ and $\boldsymbol{X}^1$ to represent steady state vector before and after the tax cuts where $\boldsymbol{X}$ can be output $\boldsymbol{Q},\boldsymbol{C},\boldsymbol{v}, \widetilde{\Lambda }_L, L$ .

Definition 4. The TFP $A(t)$ is defined by the aggregate production function:

(5) \begin{align} C(t) = A(t)L(t), \end{align}

where $C(t)$ is the final goods consumption or GDP and $L(t)$ is the total labor supply in the economy.

As the labor market is distorted by income taxes, the TFP $A(t)$ is distorted by income taxes.

3.2 Equilibrium results

We show that the Domar weight $\boldsymbol{v}(t)$ , labor income share vector $\widetilde{\boldsymbol{\alpha }}(t)$ , total income share $\widetilde{\Lambda }_L(t)$ , and sectoral labor share in the labor supply $\gamma _k$ are functions of the primitives of the economy with production netowrks in Lemma 1.

Check Appendix A to see the proof. After the one-time income tax rate cuts at $t=0$ , the sales shares or Domar weights $\boldsymbol{v}$ do not change and the labor income shares increase shown in Lemma 2.

Refer to Appendix A to see the proof. This lemma shows that the income tax cuts do not change the sales shares or Domar weights, which is different from the results in Bigio and La’o (Reference Bigio and La’o2020) where real distortions are in the form of revenue taxes. This is because income tax cuts do not change the comparative price between intermediate goods and their intermediate inputs. The total labor income shares $\widetilde{\Lambda }(t)$ increase because the wage increases due to higher labor demand after the tax cuts.

We assume elastic labor supply in this model. Actually, elastic labor supply is an important channel to drive the effects of income tax reductions in this paper. We follow Saez (Reference Saez2001) by defining elasticities to solve the labor supply.

Lemma 3. The change of labor supply is given by:

(6) \begin{align} \ln \frac{L^1}{L^0} = -\frac{\kappa }{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}\gt 0. \end{align}

The wage increases after income taxes are reduced and thus incentivizes households to supply more labor. This lemma is intuitive, and this labor supply channel augments the effects of income tax reductions to the whole economy.

Proposition 1. The change of steady state output of sector $k$ is

\begin{align*} \Delta _{k} &=\ln \frac{Q_{k}^{1}}{Q_{k}^{0}}=\sum _{j=1}^{N}\psi _{kj}\alpha _{j}\ln z_{j} +\frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \end{align*}

Or in the matrix form,

\begin{align*} \boldsymbol{\Delta}&=\ln \boldsymbol{Q}^{1} - \ln \boldsymbol{Q}^{0}=\boldsymbol{\Psi} diag(\boldsymbol{\alpha }) \ln \boldsymbol{z}+ \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}\boldsymbol{1}. \end{align*}

And

\begin{align*} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}= \ln \frac{\sum _{k=1}^{N}\frac{\alpha _{k}v_{k}}{1+\tau _{k}}}{\sum _{k=1}^{N}\frac{\alpha _{k}v_{k}}{1+\tau _{k}}z_k}\lt 0. \end{align*}

The proof is given in Appendix C. This proposition indicates that the income tax reduction in sector $k$ influences production of sector $k$ through two channels:

1. (1) One channel is through positive downstream demand effect induced by lower comparative price due to tax cuts: $\sum _{j=1}^{N}\psi _{kj}\alpha _{j}\ln z_{j}\gt 0$ . When the income tax in sector $j$ which is the upstream sector or supplier of sector $k$ is cut by $\ln z_j$ , the price of intermediate good of sector $j$ is reduced with an augmentation of labor share $\alpha _j$ . Sector $k$ demands more intermediate goods from sector $j$ and thus produces more, which is amplified by the production network $\psi _{kj}$ . Obviously, only the downstream sectors are influenced through this channel implied by $\psi _{kj}$ .

2. (2) The other channel is through negative labor effect $\frac{1}{1+\kappa }\ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda } _{L}^{1}}$ . We further decompose this effect into negative labor demand effect and positive labor supply effect.

   1. (i) The first sub-channel is called labor demand effect which is negative. The labor cost is higher in the economy induced by more labor demand: $\ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda } _{L}^{1}}\lt 0$ . The wage is higher after the income tax cut because labor demand is higher keeping aggregate labor supply fixed. The total labor income share in the economy is higher after the income tax cut, which reduces the steady state output.

   2. (ii) The second sub-channel is through positive labor supply effect because labor supply is higher in the economy induced by the higher wage: $\ln \frac{L^1}{L^0}=-\frac{\kappa }{1+\kappa }\ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda } _{L}^{1}}\gt 0$ . The wage is higher after the income tax cut, and households are incentivized to supply more labor and produce more.

   The latter two effects are combined to the term $\frac{1}{1+\kappa }\ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda } _{L}^{1}}$ which is negative. When $\kappa =0$ is 0 or labor supply is fixed, the labor supply effect disappears. When $\kappa$ goes to infinity or labor supply is fully elastic, the two sub-channels counteract each other.

Notice that both channels are resulted from the distorting effects on the labor market instead of intermediate goods market. The first effect and the second effect are opposite and the total change of production depends on the magnitude of each one.

We are interested in the scenario when the income taxes of all sectors are cut by the same percentages as this is a probable case in the real world. This scenario is given by Corollary 1.

Corollary 1. The change of steady state output of sector $k$ after homogeneous income tax reductions across all sectors is zero that

\begin{align*} \Delta _k = \ln \frac{Q_k^1}{Q_k^0} =\frac{\kappa }{1+\kappa }\ln z,\quad if\quad z_k=z,\forall k. \end{align*}

Especially, when labor is inelastic or fixed $\kappa =0$ , the change is zero $\Delta _k=0$ .

When labor supply is fixed and income tax reductions are the same across sectors, the positive downstream demand effect counteract the negative labor demand effect completely.

Proposition 2. The TFP is given by:

(7) \begin{align} \ln A(t) = \boldsymbol{v}^T \ln \boldsymbol{A} - \boldsymbol{v}^T diag(\boldsymbol{\alpha })\ln (1+\boldsymbol{\tau }(t)) - \ln \widetilde{\Lambda }_L(t). \end{align}

When there are no income taxes $\boldsymbol{\tau }(t)=0$ , the TFP goes to the efficient one that $\ln A(t) = \boldsymbol{v}^T \ln \boldsymbol{A}$ . When income taxes are reduced, the TFP is impacted through two channels discussed in Proposition 1: the downstream demand channel and labor demand channel.

Corollary 2. The change of the TFP is given by:

\begin{align*} \ln A^1 - \ln A^0 = \boldsymbol{v}^T diag(\boldsymbol{\alpha }) \ln \boldsymbol{z}+ \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}} \end{align*}

As final goods consumption $C(t)$ is a product of the TFP $A(t)$ and labor supply $L(t)$ , we can see that the change of the final consumption $C(t)$ is the sum of the change of the TFP and labor supply $\Delta C(t) = \Delta A(t) + \Delta L(t)$ . Next, we show the effect of the income tax reductions on the GDP or the TFP in Proposition 3.

Proposition 3. The change of GDP in the economy after income tax reductions is given by:

\begin{align*} \ln C^{1} - \ln C^{0}= \ln A^{1} - \ln A^{0} + \ln L^1 - \ln L^0 =\beta ^T\boldsymbol{\Delta}=\boldsymbol{v}^T diag(\boldsymbol{\alpha }) \ln \boldsymbol{z}+ \frac{1}{1+\kappa }\ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}. \end{align*}

The marginal effect of the income tax reductions in sector $k$ on GDP is given by a sufficient statistics $\xi _k$ :

\begin{align*} \frac{d({\ln}\, C^{1} - \ln C^{0})}{d \ln z_k} = \xi _k, \end{align*}

where

\begin{align*} \xi _k=v_k \alpha _k - \frac{1}{1+\kappa } \gamma _k^0, \quad \gamma _k^0=\frac{L_k^0}{L^0}=\frac{\frac{\alpha _k v_k}{1+\tau _k}}{\sum _{k=1}^N \frac{\alpha _k v_k}{1+\tau _k}}. \end{align*}

Especially, in the efficient economy when $\tau _k=0,\forall k$ , we have

\begin{align*} \xi _k =\frac{\kappa }{1+\kappa }v_k \alpha _k, \end{align*}

and further when labor is inelastic $\kappa =0$ in the efficient economy, the sufficient statistics is zero $\xi _k=0$ .

The proof of Proposition 3 is obvious and the proof is in Appendix D. The first-order effect of income tax reductions on the GDP is just the sum of these two channels, which is represented by the sufficient statistics $v_k\alpha _k-\frac{1}{1+\kappa }\gamma _k^0$ . We can still decompose the sufficient statistics into the two channels mentioned in Proposition 1:

1. (1) The sufficient statistics for the positive downstream demand channel of the effects of income tax reductions on the final goods consumption is just the product of labor demand elasticity in the production function $\alpha _k$ and the Domar weight $v_k$ .

2. (2) The sufficient statistics for the negative labor channel of the effects of income tax reductions on the GDP is just opposite of the sectoral labor share in the total labor supply before income tax reductions adjusted for the labor supply $-\frac{1}{1+\kappa }\gamma _k^0$ . This channel can be further decomposed into two sub-channels.

   1. (i) The sufficient statistics for the negative labor demand channel of the effects of income tax reductions on the GDP is just opposite of the sectoral labor share in the total labor supply before income tax reductions $-\gamma _k^0$ .

   2. (ii) The sufficient statistics for the positive labor supply channel of the effects of income tax reductions on the GDP is $\frac{\kappa }{1+\kappa }\gamma _k^0$ .

We can see that when labor supply is inelastic, the second sub-channel in the labor effect disappears and the first-order effect is just $v_k \alpha _k - \gamma _k^0$ , which is the product of labor demand elasticity in the production function $\alpha _k$ and the Domar weight $v_k$ minus the sectoral labor share before income tax reduction $\gamma _k^0$ .

The marginal effect of income tax reductions around the efficient economy is $\frac{\kappa }{1+\kappa }v_k \alpha _k$ . When labor is inelastic $\kappa =0$ , the marginal effect is zero when the two channels counteract each other, which has the same insight as Bigio and La’o (Reference Bigio and La’o2020). This is because the labor effect is equal to $-v_k \alpha _k$ in the efficient economy, which is shown in Corollary 3.Footnote ⁵

Corollary 3. The first-order effect of the income tax reduction in sector $k$ on the labor wedge or total labor income share $\widetilde{\Lambda }_L$ is $\alpha _k v_k$ in the efficient economy. Footnote ⁶

\begin{align*} \frac{d\left({\ln}\, \widetilde{\Lambda }_L ^{1} - \ln \widetilde{\Lambda }_L ^{0}\right)}{d \ln z_k} =\alpha _k v_k. \end{align*}

The proof of this corollary is embedded in that of Proposition 3. Corollary 3 shows that the sufficient statistics of the first-order effect of income tax reductions on the labor wedge around the efficient economy is the product of labor demand elasticity $\alpha _k$ and Domar weight $v_k$ . The result is different from that in Bigio and La’o (Reference Bigio and La’o2020) who show that the first-order effect of real distortions in general is the Domar weight $v_k$ with revenue taxes.

The difference lies in the setup in their paper that real distortions are represented by revenue taxes. This difference changes the comparative price among produced goods, labor, and intermediate inputs. In their setup, revenue taxes do change the comparative price between intermediate goods and their intermediate inputs. Here in our setup, income taxes do not change the comparative price between intermediate goods and their intermediate inputs, and thus Domar weights do not change with income tax reductions. The reason why $\alpha _k$ appears in the sufficient statistics is because real distortions in terms of income taxes exert effects on sectoral output and the GDP only through the labor input in the production and the elasticity of labor demand is $\alpha _k$ .

Our result shows that revenue taxes are not general enough to include all the real distortions. The setup in this paper is one example that income taxes have different first-order effects from revenue taxes.

3.3 Direct effect and indirect effect

Production networks act as shock amplifiers in the economy. Sectoral shocks not only affect the sectors that shocks originate but also spill over to other sectors through input–output linkages. We decompose the direct and indirect/spillover effect in this subsection. This decomposition hinges on the Leontief inverse matrix $(\boldsymbol{I}-\boldsymbol{\Omega})^{-1}$ , which denotes the total effects of shocks to the economy. The direct effect is represented by the identity matrix $\boldsymbol{I}$ and the rest $(\boldsymbol{I}-\boldsymbol{\Omega})^{-1}-\boldsymbol{I}$ goes to the spillover effect:

\begin{align*} (\boldsymbol{I}-\boldsymbol{\Omega})^{-1} = \left (\boldsymbol{I}\right ) + \left ((\boldsymbol{I}-\boldsymbol{\Omega})^{-1}-\boldsymbol{I} \right ). \end{align*}

We decompose the direct effect and indirect/spillover effect in Proposition 4.

Proposition 4. The direct effect and indirect/spillover effect to sectoral production change $\boldsymbol{\Delta}$ , TFP change $\ln A^1 - \ln A^0$ , and GDP change $\ln C^1 - \ln C^0$ are represented by:

\begin{align*} \boldsymbol{\Delta}= \boldsymbol{\Delta}_d + \boldsymbol{\Delta}_s, \quad \ln A^1 - \ln A^0=\Delta _d^A + \Delta _s^A, \quad \ln C^1 - \ln C^0=\Delta _d^C + \Delta _s^C, \end{align*}

where

\begin{align*} \boldsymbol{\Delta}_d & = \boldsymbol{I}\ diag(\boldsymbol{\alpha }) \ln \boldsymbol{z} + \boldsymbol{I}\ diag(\boldsymbol{\alpha }) \boldsymbol{1} \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}},\\[4pt] \boldsymbol{\Delta}_s & = \left (\boldsymbol{\Psi}-\boldsymbol{I}\right ) diag(\boldsymbol{\alpha }) \ln \boldsymbol{z} + \left (\boldsymbol{\Psi}-\boldsymbol{I}\right ) diag(\boldsymbol{\alpha }) \boldsymbol{1} \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \\[4pt] \Delta _d^A & = \boldsymbol{\beta }^T \boldsymbol{I}\ diag(\boldsymbol{\alpha }) \ln \boldsymbol{z} + \boldsymbol{\beta }^T \boldsymbol{I}\ diag(\boldsymbol{\alpha }) \boldsymbol{1} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \\[4pt] \Delta _s^A & = \boldsymbol{\beta }^T \left (\boldsymbol{\Psi}-\boldsymbol{I}\right ) diag(\boldsymbol{\alpha }) \ln \boldsymbol{z} + \boldsymbol{\beta }^T \left (\boldsymbol{\Psi}-\boldsymbol{I}\right ) diag(\boldsymbol{\alpha }) \boldsymbol{1} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \\[4pt] \Delta _d^C & = \boldsymbol{\beta }^T \boldsymbol{I}\ diag(\boldsymbol{\alpha }) \ln \boldsymbol{z} + \boldsymbol{\beta }^T \boldsymbol{I}\ diag(\boldsymbol{\alpha }) \boldsymbol{1} \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \\[4pt] \Delta _s^C & = \boldsymbol{\beta }^T \left (\boldsymbol{\Psi}-\boldsymbol{I}\right ) diag(\boldsymbol{\alpha }) \ln \boldsymbol{z} + \boldsymbol{\beta }^T \left (\boldsymbol{\Psi}-\boldsymbol{I}\right ) diag(\boldsymbol{\alpha }) \boldsymbol{1} \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}. \end{align*}

Proof is illustrated in Appendix E. Notice that all direct effects are related to the identity matrix $\boldsymbol{I}$ which denotes the self-impacted from the income tax reduction of the same sector. All the indirect effects are related to the matrix $\boldsymbol{\Psi}-\boldsymbol{I}=(\boldsymbol{I}-\boldsymbol{\Omega})^{-1}-\boldsymbol{I}$ , which denotes the difference between total effect represented by Leontief matrix $(\boldsymbol{I}-\boldsymbol{\Omega})^{-1}$ and direct effect represented by identity matrix $\boldsymbol{I}$ .

There are two terms in all the direct effects $\boldsymbol{\Delta}_d, \Delta _d^A, \Delta _d^C$ and indirect effects $\boldsymbol{\Delta}_s, \Delta _s^A, \Delta _s^C$ . The first term is due to the downstream demand channel, and the second term is due to the labor channel. In the literature of production networks, the division of direct effect and indirect/spillover effect relies on whether there is the spillover through input–output linkages. The downstream demand channel functions absolutely through input–output linkages, so without these linkages the direct effect must be the one to the sector itself.

But the labor channel functions not only through input–output linkages but also through economy-wide labor demand and labor supply. It is because the income tax reduction of any sector results in the economy-wide wage increase even without input–output linkages. Thus, the direct effect of labor channel not only affects the sector itself but also spillovers to other sectors through the wage change. This seems not so “direct,” but this is related to the definition of what we mean by “direct effect.” We still follow the literature of production networks by defining “direct effect” as the effect without input–output linkages. Under this definition, the labor channel of income tax reductions still spill over to other sectors through labor demand and labor supply.

4.1 Horizontal economy

Example 1. Horizontal Economy is an economy that production is not organized in a network that each sector employs only labor as inputs and does not employ goods of other sectors as inputs, as depicted in Fig. 1. The production structure of the horizontal economy can be represented by $( \boldsymbol{\Omega},\boldsymbol{\Psi},\boldsymbol{\tilde{\alpha }}(t),\boldsymbol{v},\widetilde{\Lambda }_L(t))$ :

\begin{align*} \boldsymbol{\Omega}&=\left [\begin{array}{c@{\quad}c@{\quad}c} 0 & \cdots & 0 \\[4pt] \vdots & \vdots & 0 \\[4pt] 0 & \cdots & 0 \end{array} \right ], \boldsymbol{\Psi}=\left [\begin{array}{c@{\quad}c@{\quad}c} 1 & \cdots & 0 \\[4pt] \vdots & \vdots & 0 \\[4pt] 0 & \cdots & 1 \end{array} \right ], \\[4pt] \boldsymbol{\alpha }&=[1,\cdots,1],\quad, \boldsymbol{\widetilde{\alpha }}(t) =[1/(1+\tau _1(t)),\cdots,1/(1+\tau _N(t))], \boldsymbol{v}=\boldsymbol{\widetilde{\Psi }}^T \boldsymbol{\beta }= [\beta _1,\cdots,\beta _N], \\[4pt] \widetilde{\Lambda }_L(t) &= \boldsymbol{\widetilde{\alpha }}(t)^T \boldsymbol{v}=\sum _{k=1}^N \frac{ \beta _k}{1+\tau _k(t)}. \end{align*}

After one-time income tax cuts, the change of steady state output of sector $k$ is

\begin{align*} \Delta _{k} &=\ln \frac{Q_{k}^{1}}{Q_{k}^{0}}=\ln z_{k} + \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \end{align*}

where

\begin{align*} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}&= \ln \frac{\sum _{k=1}^{N}\frac{\beta _{k}}{1+\tau _{k}}}{\sum _{k=1}^{N}\frac{\beta _{k}}{1+\tau _{k}}z_k}\lt 0. \end{align*}

We can see that the positive downstream demand effect is collapsed to a self-influenced term $\ln z_{k}$ and all the network effects are gone due to no input–output structure in the horizontal economy. The production of sector $k$ increases due to higher final demands as price is lowered after the tax reduction.

The network effect is also missing in the labor effect or the income share change $\ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}$ where the Domar weight $v_k$ is no longer determined by a weighted average of consumption shares of all sectors but the consumption share $\beta _k$ of itself.

The welfare change of one-time income tax cuts of sector $k$ is given by:

\begin{align*} \ln C^{1} - \ln C^{0}=\sum _{k=1}^N \beta _k \ln z_k + \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}, \end{align*}

where

\begin{align*} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}&= \ln \frac{\sum _{k=1}^{N}\frac{\beta _{k}}{1+\tau _{k}}}{\sum _{k=1}^{N}\frac{\beta _{k}}{1+\tau _{k}}z_k}\lt 0. \end{align*}

Due to the missing network effect, the change of GDP is a weighted average of self-influenced terms $\beta _k \ln _k$ of all sectors, where the weight is just the consumption share of each sector, along with the labor demand effect.

The sufficient statistics for the marginal effect of income tax reductions in sector $k$ on GDP is given by:

\begin{align*} \xi _k = \beta _k - \frac{1}{1+\kappa } \gamma _k^0,\quad \gamma _k^0=\frac{L_k^0}{L(t)}=\frac{\frac{ \beta _k}{1+\tau _k}}{\sum _{k=1}^N \frac{ \beta _k}{1+\tau _k}}. \end{align*}

When labor supply is inelastic or $\kappa =0$ , we have $\xi _k=\beta _k - \gamma _k^0$ . The marginal effect of the income tax reduction of sector $k$ on GDP is just the consumption ratio of sector $k$ in the final goods minus the labor share of sector $k$ .

4.2 Vertical economy

Example 2. Vertical Economy is an economy that production is organized in a vertical network that each sector employs goods of the neighbor sector as inputs and does not employ labor as inputs except sector $1$ which only employs labor as its input, as depicted in Fig. 2. The production network structure in the vertical economy can be represented by $( \boldsymbol{\Omega}(t),\boldsymbol{L},\boldsymbol{\alpha },\boldsymbol{v},\widetilde{\Lambda }_L(t))$ :

Figure 2. Vertical economy.

Note: This figure denotes the production organization of the vertical economy. Each sector employs goods of the neighbor sector as inputs and does not employ labor as inputs except sector $1$ which only employs labor as its input. The household or consumer directly consume the goods of sector $N$ .

\begin{align*} \boldsymbol{\Omega}&=\left [\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & \cdots & 0 \\[4pt] 1 &0 &\cdots & 0 \\[4pt] \vdots & \cdots &\cdots & \vdots \\[4pt] 0 & \cdots & 1 & 0 \end{array} \right ], \boldsymbol{\Psi}=\left [\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & \cdots & 0 \\[4pt] 1 & 1 & \cdots & 0 \\[4pt] 1&\cdots & 1 & 0 \\[4pt] 1&\cdots & 1 &1 \end{array} \right ], \\[4pt] \boldsymbol{\alpha }&=[1,0,\cdots,0],\quad \boldsymbol{\widetilde{\alpha }}(t) =[1/(1+\tau _1(t)), 0, \cdots,0],\quad \boldsymbol{\beta }=[0,\cdots,1],\\[4pt] \boldsymbol{v}&=\boldsymbol{\Psi}^T \boldsymbol{\beta }=[1,\cdots,1],\quad \widetilde{\Lambda }_L(t) = \boldsymbol{\widetilde{\alpha }}(t)^T \boldsymbol{v}=1/(1+\tau _1(t)). \end{align*}

After one-time income tax cuts, the change of steady state output of sector $k$ is

\begin{align*} \Delta _{k} &=\ln \frac{Q_{k}^{1}}{Q_{k}^{0}}=\ln z_{1} + \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}=\frac{\kappa }{1+\kappa }\ln z_{1}, \end{align*}

where

\begin{align*} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}&=- \ln z_1. \end{align*}

When labor supply is inelastic $\kappa =0$ , the change of sectoral output is just zero.

From the example of the vertical economy, we can clearly see what each of the two impact channels is. First, only the tax reduction in the first sector matters to the two channels because only this sector hires labor as its input and other sectors only hire upstream intermediate goods as inputs. The tax reduction results the reduction of the price of sector one and transmits to every downstream sector through vertical production networks. Second, the positive downstream demand effect is collapsed to the tax reduction term of the first sector $\ln z_1$ transmitted through the production networks. Third, the negative labor demand effect $-\ln z_1$ is negative as the first sector demands more labor thus pushes up the wage. Fourth, the labor supply effect $\frac{\kappa }{1+\kappa }\ln z_1$ is positive as higher wage induces households to supply more labor and thus sectoral production is higher. Last, the positive downstream demand effect and negative labor demand effect offsets each other in the vertical economy, and it is totally due to the Cobb–Douglas production function in which the elasticity of substitution is just one. That’s why the total effect is equal to the positive labor supply effect.

The welfare change of one-time income tax cuts of sector $k$ is given by:

\begin{align*} \ln C^{1} - \ln C^{0}=\ln z_1 + \frac{1}{1+\kappa } \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}= \frac{\kappa }{1+\kappa } \ln z_1. \end{align*}

where

\begin{align*} \ln \frac{\widetilde{\Lambda }_{L}^{0}}{\vphantom{{}^{\int}}\widetilde{\Lambda }_{L}^{1}}&=- \ln z_1. \end{align*}

Especially, when labor is inelastic or $\kappa =0$ , we have

\begin{align*} \ln C^{1} - \ln C^{0}=0. \end{align*}

As only labor supply effect exists in the effect of income tax reductions on sectoral production, the effect on the GDP is also from this sub-channel. The sufficient statistics for the marginal effect of income tax reductions in sector $k$ on GDP is given by:

\begin{align*} \xi _k = \frac{\kappa }{1+\kappa }. \end{align*}

Especially, when labor is inelastic or $\kappa =0$ , we have

\begin{align*} \xi _k=0. \end{align*}

The marginal effect of the income tax reduction of sector $k$ on GDP is zero even in the inefficient economy with vertical production networks and inelastic labor supply.