2.1. Model

Below are a constant-returns-to-scale CES production function and the corresponding CES unit cost function for the $j$ th sector (index omitted) of $n$ sectors, with $i = 1, \cdots, n$ being an intermediate and a single primary factor of production labeled $i=0$ .

\begin{align*} x = z \left ( \sum _{i=0}^n (\alpha _{i})^{\frac{1}{\sigma }} (x_{i})^{\frac{\sigma -1}{\sigma }} \right )^{\frac{\sigma }{\sigma -1}}, && \pi =\frac{1}{z}\left ( \sum _{i=0}^n \alpha _{i} (\pi _{i})^{\gamma } \right )^{{1}/{\gamma }} \end{align*}

Here, $\sigma =1-\gamma$ denotes the elasticity of substitution, while $\alpha _{i}$ denotes the share parameter with $\sum _{i=0}^n \alpha _{i}=1$ . Quantities and prices are denoted by $x$ and $\pi$ , respectively. Note that the output price equals the unit cost due to the constancy of returns to scale. The Hicks-neutral productivity level of the sector is denoted by $z$ . The duality asserts zero profit in all sectors $j=1,\cdots,n$ , that is, $\pi _j x_j = \sum _{i=0}^n \pi _i x_{ij}$ .

By applying Shephard’s lemma to the unit cost function of the $j$ th sector, we have:

(1) \begin{align} s_{ij} = \alpha _{ij} \left ( \frac{z_j \pi _j}{\pi _i} \right )^{-\gamma _j} \end{align}

where $s_{ij}$ denotes the $i$ th factor cost share of the $j$ th sector. For later convenience, let us calibrate the share parameter at the benchmark where price and productivity are standardized, that is, $\pi _0 = \pi _1 = \cdots = \pi _n = 1$ and $z_1 = \cdots = z_n =1$ . Since we know the benchmark cost-share structure from the input-output coefficients of the benchmark period $a_{ij}$ , the benchmark-calibrated share parameter must therefore be $\alpha _{ij} = a_{ij}$ . Taking this into account, the equilibrium price $\boldsymbol{\pi }=(\pi _1, \cdots, \pi _n)$ given $\boldsymbol{z}=(z_1, \cdots, z_n)$ must be the solution to the following system of $n$ equations:

\begin{align*} \pi _1 &= (z_1)^{-1} \left ( a_{01} (\pi _0)^{\gamma _1} + a_{11} (\pi _1)^{\gamma _1} + \cdots + a_{n1} (\pi _n)^{\gamma _1} \right )^{1/\gamma _1} \\ \pi _2 &= (z_2)^{-1} \left ( a_{02} (\pi _0)^{\gamma _2} + a_{12} (\pi _1)^{\gamma _2} + \cdots + a_{n2} (\pi _n)^{\gamma _2} \right )^{1/\gamma _2} \\ &\vdots \\ \pi _n &= (z_n)^{-1} \left ( a_{0n} (\pi _0)^{\gamma _n} + a_{1n} (\pi _1)^{\gamma _n} + \cdots + a_{nn} (\pi _n)^{\gamma _n} \right )^{1/\gamma _n} \end{align*}

where we can set the price of the primary factor $\pi _0$ as the numéraire.

Note that this system of equations is equivalent to Miranda-Pinto and Young (Reference Miranda-Pinto and Young2022)’s specification of equilibrium price for a CES economy with sectoral distortions (Proposition 3), which reads as follows:

\begin{align*} P^{1-\epsilon _Q} & = a \cdot (Z \cdot \vartheta ^w )^{ \cdot (\epsilon _Q -1)} + \left ( (1-a) \cdot (Z \cdot \vartheta ^m)^{ \cdot (\epsilon _Q -1)} 1^\prime \right ) \cdot \left ( \Omega \cdot (P 1^\prime )^{ \cdot ((1-\epsilon _Q)1^\prime )^\prime } \right )^\prime 1 \end{align*}

The correspondence of symbols are $P = (\pi _1, \pi _2, \cdots, \pi _n)$ , $(1-\epsilon _{Q}) = (\gamma _1, \gamma _2, \cdots, \gamma _n)$ , $Z = (z_1, z_2, \cdots, z_n)$ , $a = (a_{01}, a_{02}, \cdots, a_{0n})$ , and

\begin{align*} (1-a) \cdot \Omega ^\prime = \left ({\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} a_{11} & a_{12} & \cdots & a_{1n}\\[4pt] a_{21} & a_{22} & \cdots & a_{2n}\\[4pt] \vdots & \vdots & \ddots & \vdots \\[4pt] a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{array} } \right ). \end{align*}

The only difference is that the sectoral wedges ( $\vartheta ^m, \vartheta ^w$ ) are all equal to 1 in our model, since the firms are assumed to be unconstrained, that is, that there is no binding constraint other than the budget. Sectoral production choices are, therefore, undistorted in our model.Footnote ²

For later convenience, we write the above system in a more concise form as follows:

(2) \begin{align} \boldsymbol{\pi }=\left \lt \boldsymbol{z} \right \gt ^{-1} \boldsymbol{c}\left (\boldsymbol{\pi };\, \pi _0 \right ) \end{align}

Here, angled brackets indicate the diagonalization of a vector. Note that $\boldsymbol{c}: \mathbb{R}^{n}_{+} \to \mathbb{R}^{n}_{+}$ is strictly concave and $\boldsymbol{z} \in \mathbb{R}^{n}_{+}$ . Consider a mapping $\mathcal{E}$ that nests the equilibrium solution (fixed point) $\boldsymbol{\pi }$ of equation (2) and maps the (exogenous) productivity $\boldsymbol{z}$ onto the equilibrium price $\boldsymbol{\pi }$ , that is,

(3) \begin{align} \boldsymbol{\pi }=\mathcal{E} (\boldsymbol{z};\, \pi _0 ) \end{align}

There is no closed-form solution to equation (3). However, one can be found for the case of uniform elasticity, that is, $\gamma _1 = \cdots = \gamma _n = \gamma$ , which is as follows:

(4) \begin{align} \boldsymbol{\pi } = \pi _0 \left ( \boldsymbol{a}_0 \left [ \left \lt \boldsymbol{z} \right \gt ^\gamma - \mathbf{A} \right ]^{-1} \right )^{1/\gamma } && \text{Uniform CES economy} \end{align}

where the $n$ row vector $\boldsymbol{a}_0 = (a_{01}, \cdots, a_{0n})$ is called the primary factor coefficient vector and the $n \times n$ matrix $\mathbf{A} = \{ a_{ij} \}$ is called the input-output coefficient matrix. The case of a Leontief economy, where $1-\gamma =0$ , equation (4), can be reduced straightforwardly as follows:

(5) \begin{align} \boldsymbol{\pi } = \pi _0 \boldsymbol{a}_0 \left [ \left \lt \boldsymbol{z} \right \gt - \mathbf{A} \right ]^{-1} && \text{Leontief economy} \end{align}

For the case of a Cobb-Douglas economy, where $\gamma =0$ , we first take the log and let $\gamma \to 0$ where L’Hôspital’s rule is applicable since $\sum _{i=0}^n a_{ij}= 1$ for $j=1,\cdots,n$ .

\begin{align*} \ln \pi _{j} z_{j} = \frac{\ln \left ( \sum _{i=0}^n a_{ij} (\pi _{i})^\gamma \right )}{\gamma } \to \frac{\sum _{i=0}^n a_{ij} (\pi _i)^\gamma \ln \pi _{i}}{\sum _{i=0}^n a_{ij} (\pi _i)^\gamma } \to \sum _{i=0}^n a_{ij} \ln \pi _{i} \end{align*}

Thus, equation (4) can be reduced in the following manner:

(6) \begin{align} \ln \boldsymbol{\pi } = \left ( \boldsymbol{a}_0 \ln \pi _0 - \ln \boldsymbol{z} \right ) \left [ \mathbf{I} - \mathbf{A} \right ]^{-1} && \text{Cobb-Douglas economy} \end{align}

It is notable that the growth of the equilibrium price $\textrm{dln}\, \boldsymbol{\pi }=(\textrm{dln}\, \pi _1, \cdots, \textrm{dln}\, \pi _n)$ is a linear combination of the growth of sectoral productivity $\textrm{dln}\, \boldsymbol{z}=(\textrm{dln}\, z_1, \cdots, \textrm{dln}\, z_n)$ in the case of a Cobb-Douglas economy.

Otherwise, the fixed point $\boldsymbol{\pi }$ given $\boldsymbol{z}$ can be searched for by using the simple recursive method applied to equation (2). Since the unit cost function $\pi _j = (z_j)^{-1} c_j(\boldsymbol{\pi };\, \pi _0)$ is monotonically increasing and strictly concave in $\boldsymbol{\pi }$ , we know by Krasnosel’skiĭ (Reference Krasnosel’skiĭ1964) and Kennan (Reference Kennan2001) that equation (2) is a contraction mapping that globally converges onto a unique fixed point, if it exists in $\mathbb{R}^{n}_{+}$ . Note that if $\pi _0=1$ and $(z_1,\cdots,z_n)=(1,\cdots,1)$ , then $(\pi _1,\cdots,\pi _n)=(1, \cdots,1)$ is an equilibrium, which must be unique. Moreover, note that obviously from equation (6), the existence of a positive fixed point $\boldsymbol{\pi } \in \mathbb{R}^n_{+}$ for any given $\boldsymbol{z} \in \mathbb{R}^n_{+}$ can be asserted for the case of a Cobb-Douglas economy. Specifically, it is possible to show from equation (6) that:

\begin{align*} \left ( \pi _1, \cdots, \pi _n \right ) = \left ( \prod _{i=1}^n \frac{e^{a_{01} \ell _{i1}\ln \pi _0}}{(z_i)^{\ell _{i1}}}, \cdots, \prod _{i=1}^n \frac{e^{a_{0n} \ell _{in}\ln \pi _0}}{(z_i)^{\ell _{in}}} \right ) \gt \left ( 0, \cdots, 0 \right ) \end{align*}

where $\ell _{ij}$ denotes the $ij$ element of the Leontief inverse $[\mathbf{I}-\mathbf{A}]^{-1}$ . Conversely, $\boldsymbol{\pi }$ can have negative elements or may not even exist in $\mathbb{R}^n$ for non-Cobb-Douglas economies. One may see this by replacing $\boldsymbol{z}$ with small (but positive) elements in equations (4) and (5).

2.2. Viability of the equilibrium structure

From another perspective, $c_j(\boldsymbol{\pi };\, \pi _0)$ is homogeneous of degree one in $(\pi _0, \cdots,{\pi _n})$ , so by Euler’s homogeneous function theorem, it follows that:

\begin{align*} \pi _j = \sum _{i=1}^n (z_j)^{-1}\frac{\partial c_j}{\partial \pi _i}{\pi _i} +(z_j)^{-1}\frac{\partial c_j}{\partial \pi _0}{\pi _0} = \sum _{i=1}^n b_{ij} \pi _{i} + b_{0j} \pi _{0} \end{align*}

Here, by Shephard’s lemma, $b_{ij}$ denotes the equilibrium physical input-output coefficient. In matrix form, this is equivalent to:

\begin{align*} \boldsymbol{\pi }= \boldsymbol{\pi } \nabla \boldsymbol{c} \left \lt \boldsymbol{z} \right \gt ^{-1} + \pi _0 \nabla _0 \boldsymbol{c} \left \lt \boldsymbol{z} \right \gt ^{-1} = \boldsymbol{\pi } \mathbf{B} + \pi _0 \boldsymbol{b}_0 \end{align*}

Let us hereafter call $(\mathbf{B}, \boldsymbol{b}_0)$ the equilibrium structure (of an economy). Note that if $\boldsymbol{\pi }$ exists in $\mathbb{R}^{n}_{+}$ , while $\boldsymbol{b}_{0} \in \mathbb{R}^{n}_{+}$ (i.e. all sectors, upon production, physically utilize the primary factor), then $[\mathbf{I}-\mathbf{B}]$ , where $\boldsymbol{\pi } [ \mathbf{I} - \mathbf{B} ] =\pi _0 \boldsymbol{b}_0$ , is said to satisfy the Hawkins-Simon (HS) condition [Theorem 4.D.4, Takayama (Reference Takayama1985); Hawkins and Simon (Reference Hawkins and Simon1949)].

The existence of a solution $\boldsymbol{y} \in \mathbb{R}^{n}_{+}$ for $[\mathbf{I}-\mathbf{B}] \boldsymbol{y} = \boldsymbol{d}$ given any $\boldsymbol{d} \in \mathbb{R}^{n}_{+}$ and the matrix $[\mathbf{I}-\mathbf{B}]$ satisfying the HS condition are two equivalent statements [Theorem 4.D.1, Takayama (Reference Takayama1985)]. Thus, a structure that $[\mathbf{I}-\mathbf{B}]$ satisfies the HS condition is said to be viable. Conversely, for an unviable structure (that $[\mathbf{I}-\mathbf{B}]$ does not satisfy the HS condition), no positive production schedule $\boldsymbol{y} \in \mathbb{R}^{n}_{+}$ can be possible for fulfilling any positive final demand $\boldsymbol{d} \in \mathbb{R}^{n}_{+}$ . For a Cobb-Douglas economy, we can assert that the equilibrium structure is always viable, since we know from equation (6) that it is always the case that $\boldsymbol{\pi } \in \mathbb{R}^n_+$ . Otherwise, $\boldsymbol{\pi }$ may have negative elements, in which case the equilibrium structure must be unviable. An unviable equilibrium structure may never appear during the recursive process, however, if the equilibrium price search is such that it is installed in the recursive process of equation (2); instead, the recursive process will not be convergent since equation (2) maps into an open set $\mathbb{R}^n_+$ .

Last, let us specify below the structural transformation (as the physical input-output coefficient matrix $\mathbf{B}$ ) and network transformation (as the cost-share structure or the monetary input-output coefficient matrix $\mathbf{S}$ ) given $\boldsymbol{z}$ in a uniform CES economy. Since an element of the gradient of the CES aggregator is:

\begin{align*} \frac{\partial c_j}{\partial \pi _i}= a_{ij}(z_j)^{1-\gamma _j} \left ( \frac{\pi _j}{\pi _i} \right )^{1-\gamma _j} \end{align*}

the gradient of the uniform CES aggregator can be written as follows:

\begin{align*} \nabla \boldsymbol{c} = \left \lt \boldsymbol{\pi } \right \gt ^{\gamma -1}\mathbf{A}\left \lt \boldsymbol{\pi } \right \gt ^{1-\gamma } \left \lt \boldsymbol{z} \right \gt ^{1-\gamma } \end{align*}

Thus, below are the transformed structure and networks, where $\boldsymbol{\pi }$ is given by equation (4):

\begin{align*} \mathbf{B} =\left \lt \boldsymbol{\pi } \right \gt ^{\gamma -1}\mathbf{A}\left \lt \boldsymbol{\pi } \right \gt ^{1-\gamma } \left \lt \boldsymbol{z} \right \gt ^{-\gamma }, && \mathbf{S} =\left \lt \boldsymbol{\pi } \right \gt ^{\gamma }\mathbf{A}\left \lt \boldsymbol{\pi } \right \gt ^{-\gamma } \left \lt \boldsymbol{z} \right \gt ^{-\gamma } \end{align*}

Observe that $\mathbf{S}=\mathbf{A}$ in a Cobb-Douglas economy ( $\gamma =0$ ) and $\mathbf{B}=\mathbf{A} \langle 1/\boldsymbol{z} \rangle$ in a Leontief economy ( $\gamma =1$ ).

4.1. Representative household

Let us now consider a representative household that maximizes the following CES utility:

\begin{align*} H(\boldsymbol{h}) = \left ( (\mu _1)^{\frac{1}{1-\kappa }} (h_1)^{\frac{\kappa }{1-\kappa }} + \cdots + (\mu _n)^{\frac{1}{1-\kappa }} (h_n)^{\frac{\kappa }{1-\kappa }} \right )^{\frac{1-\kappa }{\kappa }} \end{align*}

The household determines the consumption schedule $\boldsymbol{h}=(h_1, \cdots, h_n)^\intercal$ given the budget constraint $W = \sum _{i=1}^n \pi _i h_i$ and prices of all goods $\boldsymbol{\pi }=(\pi _1, \cdots, \pi _n)$ . The source of the budget is the renumeration for the household’s supply of the primary factor to the production sectors, so we know that $ W = \sum _{j=1}^n v_{j}$ (total value added, or GDP of the economy) is the representative household’s (or national) income. The indirect utility of the household can then be specified as follows:

\begin{align*} H(\boldsymbol{h}(\boldsymbol{\pi }, W)) = \frac{W}{\left ( \mu _1 (\pi _1)^{\kappa } + \cdots + \mu _n (\pi _n)^{\kappa } \right )^{1/\kappa }} =\frac{W}{\Pi (\boldsymbol{\pi })} \end{align*}

where $\Pi$ , as defined as above, denotes the representative household’s CES price index. Note that $H=W$ at the baseline $(z_1, \cdots, z_n)=(1, \cdots, 1)$ where $\Pi =1$ . Thus, $H$ is the utility (in terms of money) that the representative household can obtain from its income $W$ given the price change $\boldsymbol{\pi }$ (due to the productivity shock $\boldsymbol{z}$ ) while holding the primary input’s price constant at $\pi _0 =1$ . In other words, $H$ is the real GDP if $W$ is the nominal GDP.

From another perspective, we note that the household’s income $W$ can also be affected by the productivity shock. When there is a productivity gain in a production process, this process can either increase its output while holding all its inputs fixed or reduce the inputs while holding the output fixed. In the former case, the national income $W$ (nominal GDP) remains at the baseline level, which equals real GDP in the previous year $\overline{H}$ , and GDP growth ( $\Delta \ln H = \ln H - \ln \overline{H}=-\ln \Pi (\boldsymbol{\pi })$ ) is fully accounted for. In the latter case, however, the national income can be reduced by as much as $W=\overline{H}\Pi (\boldsymbol{\pi })$ , in which case we have $\Delta \ln H=0$ that is, no GDP growth will be accounted for.

Of course, the reality must be in between the two extreme cases. In this study, we conservatively evaluate national income (as nominal GDP) to the following extent:

\begin{align*} W = \overline{H} \left ( \mu _1 (1/z_1)^\kappa + \cdots +\mu _n (1/z_n)^\kappa \right )^{1/\kappa } = \overline{H}\Pi (1/\boldsymbol{z}) \end{align*}

The real GDP under the equilibrium price, which equals the household’s expenditure, can then be evaluated as follows:

(9) \begin{align} \ln H = \ln{W} - \ln{\Pi ( \boldsymbol{\pi } )} =\ln \overline{H} -\ln{\Pi ( \boldsymbol{\pi } )} + \ln{\Pi (1/\boldsymbol{z})} \end{align}

If we assume Cobb-Douglas utility ( $\kappa \to 0$ ) and normalize the initial real GDP ( $\overline{H}=1$ ), we have the following exposition:

\begin{align*} \ln H = -\ln \Pi (\boldsymbol{\pi }) +\Pi (1/\boldsymbol{z}) = - \left ( \ln \boldsymbol{\pi } + \ln \boldsymbol{z} \right) \boldsymbol{\mu } \end{align*}

where $\boldsymbol{\mu }=(\mu _1,\cdots,\mu _n)^\intercal$ denotes the column vector of expenditure share parameters. The first identity indicates that the real GDP growth is the negative price index growth of the economy less the negative price index growth of a simple economy.Footnote ⁴ Moreover, if we assume a Cobb-Douglas economy ( $\gamma _j \to 0$ ), we arrive at the following:

\begin{align*}{\ln H} = - (\!\ln \boldsymbol{z} ) \left ( \left [ \mathbf{I} - \mathbf{A} \right ]^{-1} + \mathbf{I} \right ) \boldsymbol{\mu } = \sum _{j=1}^n \lambda _j \ln z_j \end{align*}

Note that $\lambda _j$ is the Domar weight [Hulten (Reference Hulten1978)] in this particular case.

The parameters of the utility function are also subject to estimation. By applying Roy’s identity, that is, $h_i = - \frac{\partial H}{\partial p_i}/\frac{\partial H}{\partial W}$ , we have the following expansion for the household’s expenditure share on the $i$ th commodity:

(10) \begin{align} s_i = \frac{\pi _i h_i}{\sum _{i=1}^n \pi _i h_i} = \frac{\mu _i (\pi _i)^\kappa }{\sum _{i=1}^n \mu _i (\pi _i)^\kappa } = \mu _i \left ( \frac{\Pi }{\pi _i} \right )^{-\kappa } \end{align}

where $s_{i}$ denotes the expenditure share of the $i$ th commodity for the representative household. By taking the log of equation (10) and indexing observations by $t=1,\cdots,T$ , we obtain the following regression equation where the parameter $\kappa$ can be estimated via FE.

(11) \begin{align} \ln m_{it} = \ln \mu _i - \kappa \ln (\Pi _t) + \kappa \ln p_{it} + \delta _{it} \end{align}

As is typical, the error term $\delta _{it}$ is assumed to be iid normally distributed with mean zero. Here, we replace $\pi$ with $p$ to emphasize that they are observed data. For the response variable, we use the expenditure share $m_{it}$ of the final demand available from the input-output tables. The cross-sectional dimension of the data for regression equation (11) is $i=1,\cdots,n$ , whereas it is $i=0,1,\cdots,n$ for equation (7). Thus, we apply sectoral TFP available for $t=1, \cdots, T$ from the JIP (2019) database as instruments to fix the endogeneity of the explanatory variable. The estimation result for $\kappa$ using time dummy variables as in equation (8) (such that we may retrieve the estimates for $\Pi _t$ ) is presented in Table 1 (id = 101).

4.2. Tail asymmetry and robustness

For a quantitative illustration, we study the distribution of aggregate output $\ln H$ when sectoral shocks $\ln \boldsymbol{z}$ are drawn from a normal distribution. Specifically, we use 10,000 $\ln z_j \sim \mathcal{N}(0, 0.2)$ iid samples for $j=1,\cdots,n$ , where the standard deviation (i.e. annual volatility of 20%) is chosen with reference to the annual volatility of the estimated sectoral productivity growth $\ln \zeta _j$ (see Appendix A). Let us first examine the granularity of our baseline production networks (i.e. 2011 input-output linkages). Below are both Cobb-Douglas price indices in terms of productivity shocks $\ln \boldsymbol{z}$ for the Cobb-Douglas and simple economies:

\begin{align*} \ln \Pi _{\text{CD}} = - (\!\ln \boldsymbol{z}) \left [\mathbf{I} - \mathbf{A} \right ]^{-1} \boldsymbol{m}, && \ln \Pi _{\text{SE}} = - \left ( \ln \boldsymbol{z} \right )\boldsymbol{m} \end{align*}

Here, we set the share parameter $\boldsymbol{\mu }$ at the standard expenditure share of the final demand $\boldsymbol{m}=(m_1, \cdots, m_n)^\intercal$ for the year 2011. Both indices must follow normal distributions because they are both linear with respect to the normal shocks $\ln \boldsymbol{z}$ . The variances differ, however, and the left panel of Figure 2 depicts the difference. Observe the dilation of the variance in the Cobb-Douglas economy where the power-law granularity of the Leontief inverse causes the difference [Gabaix (Reference Gabaix2011); Acemoglu et al. (Reference Acemoglu, Carvalho, Ozdaglar and Tahbaz-Salehi2012)].

Figure 2. Left: Price index dispersion in Cobb-Douglas and Simple economies under Cobb-Douglas utility. Dispersion is larger in the Cobb-Douglas than in the Simple economy, due to the power-law granularity of the Leontief inverse. Right: QQ plot of the aggregate output fluctuations generated by the linear Domar aggregator for the Cobb-Douglas economy and utility.

Replacing the equilibrium price of equation (9) with the output of equation (3) yields the following Domar aggregator, where exogenous productivity shocks ( $\ln \boldsymbol{z}$ ) are aggregated into the growth of output ( $\ln H$ ) for a CES economy.

\begin{align*} \ln H = -\! \ln \Pi \left ( \mathcal{E}\left ( \boldsymbol{z}; \pi _0=1 \right ) \right ) + \ln \Pi \left ( 1/\boldsymbol{z} \right ) \end{align*}

Note that this aggregator involves recursion in $\mathcal{E}$ , as specified by equation (2), for a CES economy with non-uniform substitution elasticities. In this section, Cobb-Douglas utility is assumed for comparison with previous research, whence the Domar aggregator becomes:

(12) \begin{align} \ln H = - \left ( \ln \mathcal{E}\left ( \boldsymbol{z}\right ) + \ln \boldsymbol{z} \right ) \boldsymbol{m} && \text{CES economy} \end{align}

Again, we set the share parameter $\boldsymbol{\mu }$ at the standard expenditure share of the final demand. In the case of the Leontief economy, the closed form is available from equation (5) as follows:

(13) \begin{align} \ln H = - \left ( \ln \left ( \boldsymbol{a}_0 \left [ \left \lt \boldsymbol{z} \right \gt - \mathbf{A} \right ]^{-1} \right ) + \ln \boldsymbol{z} \right )\boldsymbol{m} && \text{Leontief economy} \end{align}

The case for the Cobb-Douglas economy is also obtainable from equation (6) as follows:

(14) \begin{align} \ln H = - (\!\ln \boldsymbol{z}) \left ( \left [ \mathbf{I} - \mathbf{A} \right ]^{-1} + \mathbf{I} \right )\boldsymbol{m} && \text{Cobb-Douglas economy} \end{align}

As is obvious from the linearity of equation (14), the aggregate fluctuations must be normally distributed in the Cobb-Douglas economy. The resulting QQ plot is depicted in the right panel of Figure 2. Note further that $\mathbb{E} [\ln H] = 0$ because of the linearity of equation (14) and $\mathbb{E}[\ln \boldsymbol{z}]=(0,\cdots,0)$ ; this is recognizable in the same figure. That is, the expected economic growth is zero against zero-mean turbulences. In other words, the unit elasticity of the Cobb-Douglas economy precisely absorbs the turbulences as if there were none to maintain zero expected growth. Such robustness is absent in a Leontief economy with zero elasticity of substitution. The left panel of Figure 3 illustrates the resulting QQ plot of the aggregate fluctuations generated by the same zero-mean normal shocks by way of equation (13).Footnote ⁵ In this case, we observe negative expected output growth, that is, $\overline{\ln H} =-1.57\%$ , whose absolute value, in turn, can be interpreted as the robustness of the unit elasticity of substitution.Footnote ⁶ Moreover, we observe that normal shocks to the Leontief economy result in aggregate fluctuations with tail asymmetry similar to that depicted in the left panel of Figure 1.

Figure 3. QQ plots of the aggregate output fluctuations generated by the Domar aggregators for Leontief (left) and elastic CES (right) economies under Cobb-Douglas utility.

In the CES economy with a sector-specific elasticity of substitution, we use the Domar aggregator of general type (12) with recursion. The right panel of Figure 3 depicts the resulting QQ plot of the aggregate fluctuations generated by zero-mean normal shocks. In this case, we observe positive expected output growth that is, $\overline{\ln H} =1.10\%$ . The value demonstrates the robustness of the elastic CES economy relative to the Cobb-Douglas economy. We also observe that normal shocks to the elastic CES economy result in aggregate fluctuations with tail asymmetry similar to that depicted in the right panel of Figure 1. For sake of credibility, we show in Figure 4 (right) the empirical aggregate output fluctuations focusing on the period 1994–2015 from which our sectoral elasticities are estimated. It is obvious that extreme observations belong to periods around the GFC (global financial crisis), which was a massive external (non-sectoral) shock to the Japanese economy. The plot otherwise appears rather positively skewed, as predicted by our empirical result ( $\bar{\hat{\sigma }}=1.54$ ).

Figure 4. Quantile-quantile plots of recent (USA: 1997I–2020IV; Japan: 1994I–2015IV) quarterly HP-detrended log GDP against the normal distribution for the USA (left) and Japan (right). The time periods correspond to those when the sectoral elasticities are estimated. Open dots indicate 2007I–2009IV (GFC) and 2019IV– (COVID-19 pandemic) observations. Source: FRED (2021); Cabinet Office (2021).

Figure 4 (left) shows the aggregate output fluctuations focusing on the period 1997–2020 from which we estimated sectoral elasticities for the USA (see Appendix B for details). In this case, the extreme observations also belong to periods around the GFC and COVID-19 pandemic. The plot, however, seems to be in a straight line, which is consistent with our empirical result ( $\bar{\hat{\sigma }}=1.08$ ). Our elasticity estimates for the US economy also coincide with the estimates of the elasticity of substitution across intermediate inputs ( $\epsilon _M\text{ all}= 1.05$ ) by Miranda-Pinto and Young (Reference Miranda-Pinto and Young2022) based on 1997–2007 input-output accounts. These elasticity estimates, however, differ from those obtained by Atalay (Reference Atalay2017) based on 1997–2013 input-output accounts that Baqaee and Farhi (Reference Baqaee and Farhi2019) employed in their simulation ( $\varepsilon = 0.001$ ). An inelastic economy as such is rather consistent with negatively skewed aggregate fluctuations spanning the postwar US economy as depicted in Figure 1 (left) than those of recent times depicted in Figure 4 (left).