Growth of the Service Sector and Economic Fluctuations: A Production Network Perspective

2.1 Firm

Assume there are n sectors in the economy, and each sector i has a representative firm that produces according to CES technology^[1], namely:

In Equation (1), the intermediate input bundle M_i is equal to:

Among them, y_i, z_i, L_i represent the output, total factor productivity, and initial input (labor) of the representative firm in sector i, respectively. M_ij denotes the intermediate input that sector i purchases from sector j. a_i signifies the importance of the initial input (labor) to total output, while (1 − a_i) indicates the importance of intermediate inputs to total output. ω_ij represents the importance of sector j as a provider of intermediate inputs to sector i, hence the square matrix Ω (i.e., n×n matrix containing elements ω_ij) represents the input-output structure (network structure) of the economy. ε_y denotes the elasticity of substitution between the initial input (labor) and intermediate inputs, and ε_M represents the elasticity of substitution among different intermediate inputs.

2.2 Household

The representative household maximizes utility:

The budget constraint is:

Among them, c is the vector of consumption quantities, where c_i represents the consumption of product or service i, and p_i denotes the price of product or service i. c_i indicates the exogenously given basic subsistence consumption level for product or service i, below which consumption does not “generate utility. This setting ensures that the fitting line between the “income (I)” and “product consumption expenditure” has a positive intercept, thereby approximately reflecting the Engel effect. β_i represents the marginal expenditure share (marginal consumption amount) for different products, i.e., βi=dpicidI . Let the share of consumption of different products in the household utility function be γi=piciI , and ∑i=1nγi=1 , then the income elasticity of demand εd,i=βiγi for that product or service is εd,i=βiγi . This implies that when ε_d,i = 1, the marginal expenditure share of the product or service equals the average expenditure share; when ε_d,i > 1, the marginal expenditure share of the product or service exceeds the average expenditure share. L¯ represents the total labor supply of the household, w denotes the price of the initial factor (labor), and π_i represents the profit of sector i.

2.3 Market Equilibrium

The competitive equilibrium in the economy is constituted by the set of price levels w,piin the set of resource allocations ci,Li,yi,Miin,Mijijn . Given sectoral productivity shocks and price levels, the representative household maximizes utility under the budget constraint; firms maximize profits; and the product and labor markets clear (yi=ci+∑j=1nMji,L¯=∑i=1nLi) .^[1]

2.4 Production Network Structure and Economic Fluctuations

We follow the method of Carvalho and Gabaix (2013) to express the overall economic fluctuation σ as:^[1]

Among them, σ_i represents the idiosyncratic fluctuation of sector i.^[2] λ_i is the Domar weight of sector i, equal to the sales value of sector i (Sales_i) divided by GDP. According to Hulten’s theorem, in efficient economy, the impact of (productivity) shock (z_i) on sector i on total output (Y) is equal to the Domar weight (λ_i) of that sector, i.e., dlogYdlogzi=λi=SalesiGDP . In other words, the Domar weight of a sector reflects the sufficient statistic of how shocks to the sector affect GDP. The higher the Domar weight of sector, the greater the amplification of shocks propagated through the production network.

Based on the market clearing conditions and the general equilibrium solution, the sector Domar weight matrix can be obtained as:

Among them, ° denotes the Hadamard product. Ω (n×n matrix with elements ω_ij) represents the economy’s input-output structure (production network structure). p^M, p, z, γ and a denote the intermediate goods price vector, price vector, sectoral productivity vector, consumption share vector, and the vector of the importance of primary factors (labor) in the production process, respectively. The exponent ε_M of Ω can also reflect the cost of complexity. In summary, the production network structure, consumption expenditure shares, primary factor input shares, and the two substitution elasticities ε_y and ε_M all influence the Domar weights, thereby affecting aggregate economic fluctuations. Clearly, depending on the specific assumptions about z, ε_y and ε_M, Equation (6) can take different forms.

Case 1: Let z_i = 1 and L̅ = 1. When ε_y ≠ ε_M, Equation (6) becomes:^[1]

Among them, S is the sparsity vector of the production network, with its elements Sj=(∑i=1nωjiεMi)1εM−1 . The sparsity of the production network can measure the distribution of connections (edges) in the input-output network. The lower the sparsity of the production network, the higher the degree of diversification in intermediate inputs and the lower the degree of specialization, leading to more uniform distribution of input-output linkages across sectors.^[2]

Specifically, when ε_y ≠ ε_M = 1, Equation (7) simplifies to: λ=I−Sεy−1∘(1−a)εy1′′∘Ω−1γ Here, S is the sparsity vector of the production network, with its elements Sj=∏i=1nωjiωji , representing the sparsity of sector j.

Taking the partial derivative of Equation (7) with respect to S_j yields:

Among them, ω_j,i∈n is 1 × n dimensional vector representing the importance of intermediate inputs from other sectors i to sector j; 0 is 1 × n dimensional 0 vector.

If and only if ε_y > ε_M, then dλ/dS_j > 0, which implies that the lower the sparsity of the production network, the smaller the Domar weight, and consequently, the smaller the magnitude of economic fluctuations. Existing research, such as Atalay (2017), estimates that the substitution elasticity between primary inputs (labor) and intermediate inputs, ε_y, is generally greater than the substitution elasticity between different intermediate inputs, ε_M. In other words, the condition ε_y > ε_M in Equation (8) generally holds. Appendix 2 of this paper provides further proofs.

Case 2: Let z_i = 1 and L̅ = 1. When ε_y = ε_M, Equation (6) becomes:^[1]

Specifically, when ε_y =ε_M = 1, Equation (9) simplifies to: λ = [I − ((1 − a)1′)′ ∘ Ω]⁻¹γ. Here, L = [ℓ_ji] = [I − ((1− a)1′)′ ∘ Ω]⁻¹ represents the Leontief inverse matrix of the economy. In this case, although the sparsity metric (S) mentioned above no longer appears, the distribution of elements ℓ_ij in the Domar weights and the Leontief inverse matrix can still reflect the sparsity of the production network. Specifically, the sparsity of production network connections can influence the transmission path and distribution of upstream sector heterogeneity shocks to downstream sectors, thereby affecting macroeconomic fluctuations. The higher the sparsity of the production network, the greater the impact of upstream sector heterogeneity shocks on individual downstream sectors, leading to larger overall economic fluctuations. Further proof of this is provided in Appendix 3 of this paper.

Integrating the previous two scenarios, we can derive the following hypothesis:

Hypothesis 1: Given other conditions (especially when ε_y is not less than ε_M), the lower the sparsity of the production network, the smaller the magnitude of economic fluctuations.

Furthermore, the structural characteristics and evolution of the production network are also influenced by sectoral structure, with the rise of the service sector being a fundamental feature of sectoral structural changes. Evidence shows that, compared to other sectors, the service sector uses lower share of intermediate inputs and higher share of initial inputs (value-added); longitudinally, as the proportion of the service sector increases, the share of intermediate inputs in the service sector tends to decrease, while the share of initial inputs (value-added) tends to rise.^[1] This implies that the rise of the service sector leads to more equalized centrality of nodes within the production network, thereby making the overall production network connections more uniform. We discuss this mechanism further in Appendix 4. Consequently, we get the following hypothesis:

Hypothesis 2: A rising proportion of the service sector leads to a decreasing sparsity of production network. The mechanisms include two aspects: firstly, the increase in the share of initial inputs (the role of the service sector as demander of initial inputs increases); secondly, the increase in the share of final consumption (the role of the service sector as provider of final goods rises).

3.1 Econometric Model Setting

Since Hypothesis 1 concerns how changes in the structure of the production network affect economic fluctuations, and Hypothesis 2 focuses on the impact of the rising proportion of the service sector on the structure of the production network, the variable representing the structure of the production network appears both as explanatory variable and as explained variable. In this case, a system of simultaneous equations model is required, and the iterative three-stage least squares (3SLS) method can be used for estimation. The specific econometric model is specified as:

Among them, ln(Sparsity_ct) represents the sparsity of network connections for economy c in year t (logarithmically transformed). σ_ct denotes the actual GDP fluctuation of economy c in year t. The explanatory variables ln(Share_ct) in Equation (11) include ln(PI_ct) and ln(FC_ct), which represent the proportion of initial inputs and final consumption in the service sector of economy c in year t, respectively (logarithmically transformed). X_ct contain control variables, including economic growth rate, openness to trade, government expenditure share, inflation rate, etc. μ_c and η_t represent country fixed effects and year fixed effects, respectively. ϑ_ct and ν_ct are random error terms. If the coefficient φ₁ before ln(Sparsity_ct) is positive, Hypothesis 1 holds, indicating that the higher the sparsity of network connections, the greater the magnitude of economic fluctuations. If the coefficient φ₂ before ln(Share_ct) is negative, Hypothesis 2 holds, suggesting that the higher the proportion of initial inputs and final consumption in the service sector, the lower the sparsity of production network connections.

3.2 Data and Indicator

The main variables involved in the econometric analysis include the production network structure indicators, economic fluctuation rates, economic growth rates, service sector shares, and the proportions of initial inputs and final consumption in the service sector for various economies. We primarily utilize the input-output data in basic price from the Eora database for the years 1991–2016 and the PWT10.0 database. The former covers 188 economies and 26 sectors/items worldwide, and after matching with the latter, the number of economies included is 170.^[1]

3.3 Stylized Facts^[1]

First, observe the relationship between the structure of the production network and economic fluctuations. Figure 2 shows that the sparsity of network connections is positively correlated with economic fluctuations, but network density is negatively correlated with economic fluctuations. This indicates that the higher the sparsity of network connections and the lower the network density, the greater the magnitude of economic fluctuations. This is consistent with the prediction of Hypothesis 1, but the causal relationship between them requires further identification and confirmation.

Figure 2

The Relationship Between the Structure of the Production Network and Economic Fluctuations

Note: The authors estimate and produce the data based on PWT data and Eora global input-output data.

Next, we observe the relationship between the structural characteristics of the production network and the proportion of initial inputs and final consumption in the service sector, respectively. As can be seen from Figure 3, the sparsity of network connections is negatively correlated with both the proportion of initial inputs and the proportion of final consumption in the service sector. That is, the higher the proportion of initial inputs and final consumption in the service sector, the lower the sparsity of network connections, which is consistent with the expectation of Hypothesis 2.

Figure 3

The Relationship Between the Proportion of Initial Input and Final Consumption in the Service Sector and the Structure of Production Networks: Cross-Country Evidence

Note: The authors estimate and produce the data based on Eora global input-output data.

4.1 Benchmark Regression

Columns (1) to (4) in Table 1 only control for country and year fixed effects without including other control variables. We consider two simultaneous equation setting. Firstly, with the proportion of initial inputs in the service sector ln(PI) as the explanatory variable (columns (1) and (2)), the sparsity of the production network has a significant positive impact on economic fluctuations, with coefficient of 16.606, meaning that 1% increase in the sparsity of the production network leads to 0.166 unit rise in the economic fluctuation rate; the increase in the proportion of initial inputs in the service sector results in the decrease in the sparsity of the production network, with 1% increase in the proportion of initial inputs leading to 0.047% decrease in sparsity. Secondly, with the proportion of final consumption in the service sector ln(FC) as the explanatory variable (columns (3) and (4)), the impact of the sparsity of the production network on economic fluctuations remains positive, with coefficient of 8.818, indicating that 1% increase in sparsity leads to 0.088 unit rise in the fluctuation rate; whereas the increase in the proportion of final consumption in the service sector leads to the decrease in the sparsity of the production network, with 1% increase in the proportion of final consumption leading to 0.142% decrease in sparsity.

Columns (5) to (8) in Table 1 further introduce additional country-level control variables on the basis of the first four columns. With the proportion of initial inputs in the service sector as the explanatory variable (columns (5) and (6)), the coefficient for ln(Sparsity) is 15.632, meaning that 1% increase in the sparsity of the production network leads to 0.156 unit rise in the economic fluctuation rate; and 1% increase in the proportion of initial inputs in the service sector results in 0.042% decrease in sparsity. When the proportion of final consumption in the service sector is used as the explanatory variable (columns (7) and (8)), 1% increase in sparsity leads to 0.079 unit rise in the fluctuation rate; and 1% increase in the proportion of final consumption in the service sector leads to 0.134% decrease in sparsity.

The results validate the two hypotheses of the theoretical model presented earlier. In other words, the increase in the proportion of initial inputs and final consumption associated with the growth of the service sector leads to the decrease in the sparsity of production network connections in the economy, thereby reducing economic fluctuations.^[1]

4.2 Endogenous Issue

The endogeneity issues mainly include omitted variables and reverse causality. We have already controlled for fixed effects and multiple control variables in the baseline regression, so we primarily employ the instrumental variable method to address the reverse causality problem caused by the mutual influence among the proportion of the service sector, economic fluctuations, and the structure of the production network.^[2]

The instrumental variable method requires estimation based on single equation and utilizes the two-stage least squares (2SLS) approach. Drawing on the approach of Autor et al. (2013), we use the shift-share method to construct Bartik instrumental variables (Bartik, 1991) for the core explanatory variables. The construction method of this instrumental variable is to use the initial share composition at the national level and the overall growth rate of the corresponding variables at the sectoral level to simulate the estimated values for each year. Specifically, this paper uses the initial share of network sparsity for each sector in each country in the initial year and the inner product of the average growth rate of network sparsity across global sectors as the instrumental variable for production network sparsity, with the formula as follows:

Among them, ϕ_ci,1990 represents the output share of sector i in country c at the beginning of the year 1990; Sparsity_ci,1990 represents the network sparsity of sector i in country c at the beginning of the year 1990; the product of ϕ_ci,1990 and Sparsity_ci,1990 is the initial share of network sparsity for sector i in country c; g_it represents the growth rate of the average network sparsity of global sector i in year t relative to the beginning of the year.

Similarly, the instrumental variables for the proportion of initial inputs and final consumption in the service sector are the inner product of the initial share of initial inputs or final consumption in the service sector for each country in the initial year and the average growth rate of initial inputs or final consumption in the global service sector.^[1] The instrumental variable regression results are shown in Table 2, which are consistent with the baseline scenario.^[2]

4.3 Robustness Test

This section further examines the robustness of the baseline regression results by altering the calculation method of economic fluctuations, replacing the network connection sparsity indicators, and changing the division of the sample period.^[3]

4.4 Counterfactual Analysis

We first decompose the sources of economic fluctuations into three parts based on the above mentioned theoretical model and Carvalho and Gabaix (2013), namely σt=∑i=1n(SalesitGDPitGDPitGDPt)2σi2 . In other words, σ_t depends on changes in three components: (1) the ratio of the total output (total sales value) of sector i to its value-added SalesitGDPit , (2) the proportion of the value-added of sector i to the total value-added of the entire economy GDPitGDPt , and (3) the productivity fluctuation σ_i of sector i. For example, even if GDPitGDPt and σ_i remain unchanged, the decrease in SalesitGDPit can reduce economic fluctuations. Economic fluctuations may also arise from the diversification effect, meaning that, given other conditions, if the value-added shares of sectors GDPitGDPt shift from being concentrated in a few sectors to being evenly distributed across different sectors, economic fluctuations will also decrease. This effect reflects changes in the sectoral structure. Additionally, economic fluctuations may stem from the compositional effect, that is, if the Domar weights of sectors with lower fluctuation rates increase while those of sectors with higher fluctuation rates decrease, economic fluctuations will consequently diminish.

To analyze the relative importance of the three sources of economic fluctuations and to assess the contribution of the rise of the service sector to smoothing economic fluctuations, we take China as an example and conduct counterfactual analysis using the WIOD SEA data from 1995–2011.^[1] Firstly, we assume that the output-to-value-added ratio of each sector SalesitGDPit in China does not change over time, thereby isolating the contribution of SalesitGDPit to China’s economic fluctuations. The economic fluctuation of China at this time is σt=∑i=1n(Salesi¯GDP¯GDPitGDPt)2σi2 , where Salesi¯GGP¯ is the cross-period average of SalesitGDPit .

Secondly, we assume that the proportion of each sector in China GDPitGDPt does not change over time to isolate the contribution of the diversification effect to China’s economic fluctuations. The economic fluctuation at this time is σt=∑i=1n(SalesitGDPitGDP¯iGDP¯)2σi2 . Finally, we examine the role of the compositional effect in China’s economic fluctuations. The economic fluctuation at this time is σt=∑i=1n(SalesitGDPitGDPitGDPt)2σ2 , where σ is the average productivity fluctuation of sectors in the Chinese economy. The final results are shown in Figure 4.

Figure 4

Counterfactual Analysis of the Sources of Economic Fluctuation: the Case of China

Note: The solid line in the graph represents the baseline economic fluctuations calculated by the economic fluctuation decomposition formula, and the diamond line represents the counterfactual economic fluctuations. Among them, (a) shows the counterfactual economic fluctuations after isolating changes SalesitGDPit ; (b) shows the counterfactual economic fluctuations after isolating changes in GDPitGDPt ; (c) shows the counterfactual economic fluctuations after isolating changes in σ_i.

From Figure 4(a), it can be seen that after isolating the impact of changes in SalesitGDPit , the downward trend of China’s economic fluctuations from 1995 to 2002 was unaffected. This means that during this sample period, changes in SalesitGDPit had a minor influence on the trend of China’s economic fluctuations. After 2002, when controlling for changes in SalesitGDPit , China’s economic fluctuations did not exhibit an upward trend. This indicates that at this time, changes in GDPitGDPt had a significant contribution to China’s economic fluctuations. In contrast, Figure 4(b) shows that after isolating changes in GDPitGDPt , the changes in GDPitGDPt around 2002 had impact on China’s economic fluctuations. Meanwhile, Figure 4(c) reveals that after isolating changes in σ_i, the changes in σ_i had greater contribution to China’s economic fluctuations before 2002, but smaller impact after 2002.

The results of the above mentioned counterfactual analysis indicate that the primary reasons for the decline in China’s economic fluctuations between 1995 and 2002 are the changes in the proportion of value-added across different sectors in the economy ( GDPitGDPt ) and the shift in Domar weights towards sectors with lower productivity fluctuations (σ_i), that is, the transformation of the sectoral structure largely contributed to the decrease in China’s economic fluctuations. This finding coincides with the fact that China’s service sector experienced significant growth from the late 1990s to the early 2000s. This implies that the stability of China’s economic fluctuations during this period was mainly due to the rise of the service sector. After 2000, China’s economic fluctuations increased, which is primarily attributed to changes in the ratio of total output to value-added in sector i (SalesitGDPit) and changes in value-added across different sectors (GDPitGDPt) , that is, changes in sectoral Domar weights. During this period, the proportion of China’s service sector stabilized or even declined, while the proportion of the manufacturing sector increased, leading to the increase in Domar weights and greater economic fluctuations.

We further compare China with the United States, and the results are shown in Table 3. It can be seen that China’s GDP fluctuation rate is 18.1%, while that of the United States is 5.2%. This difference mainly stems from three aspects: firstly, the proportion of China’s service sector in the overall economy (38%) is much lower than that of the United States (78%); secondly, the ratio of total output to value-added in China’s service sector (SalessGDPs) and the ratio of total output to value-added in China’s non-service sectors (SalesmGDPm) are both higher than those in the United States; thirdly, the productivity fluctuation in China’s service sector (4.5%) is higher than that in the United States (3.8%).

Based on this, we conduct counterfactual analysis of China’s economic fluctuations by replacing China’s service sector share, industry intermediate input share, and service sector productivity fluctuations with the corresponding data from the United States while keeping other components unchanged. The results are shown in Table 3. It can be seen that after doing so, China’s economic fluctuations decrease to varying degrees. Among them, changes in the service sector share and the ratio of total output to value-added lead to larger changes in overall economic fluctuations, with contribution rates of 69.8% and 73.6%, respectively; whereas the contribution of service sector productivity fluctuations to the overall economic fluctuation changes is smaller, at only 20.2%. This means that the growth of the service sector and the increase in the share of initial inputs help to smooth out economic fluctuations. This further validates the results of the previous econometric analysis.