3.1. Demand-side: households

Households are denoted by $z$ and are distributed uniformly along the interval between 0 and 1. Ricardian households, constituting $1-\lambda$ of the households, optimize their consumption over time. Non-Ricardian households, making up a proportion $\lambda$ , are liquidity-constrained and consume solely from their current income and endowments. All households share an identical utility function:

\begin{equation*} U_{t}\left ( z\right ) =E_{t}\sum _{s=t}^{\infty }\beta ^{s-t}\epsilon _{s}^{TP} \left [ \log C_{s}-\frac {(N_{s}(z))^{1+1/\varphi }}{1+1/\varphi }\right ] , \end{equation*}

where $E$ represents the expectation operator, $\beta$ is the discount factor, $\epsilon _{t}^{TP}$ denotes time preference, $C_{t}$ is a composite private consumption index, defined as $C_{t}=\left [ \int _{0}^{1}C_{t}^{{}}(z)^{\frac {\theta -1}{\theta }}dz\right ] ^{\frac {\theta }{\theta -1}}$ , where $C_{t}(z)$ is consumption of good $z$ and $\theta$ the elasticity of substitution between goods. $N_{t}(z)$ signifies the labor supplied in hours and $\varphi$ denotes the Frisch elasticity of the labor supply. We posit that a time preference shock adheres to a a log-linear AR(1) process, expressed as $\hat {\epsilon }_{t}^{TP}=\rho ^{TP}\hat {\epsilon }_{t-1}+\hat {\varepsilon }_{t}^{TP}$ , where $\rho ^{TP}$ denotes the persistence parameter, $\hat {\varepsilon }_{t}^{TP}$ is an i.i.d. normal error term with mean zero. The hat notation indicates the percentage deviation of a variable from its steady state in a log-linearized framework. Ricardian households, which earn dividends from firms and are subject to income and consumption taxes imposed by the government, have the following resource constraint:

\begin{eqnarray*} \frac {R_{t}^{-1}B_{t+1}}{1-\lambda }& = &\frac {B_{t}}{1-\lambda }+\big(1-\tau _{t}^{y}\big)w_{t}N_{R,t}+\frac {\big(1-\tau _{t}^{y}\big)}{{}}\big(r_{t}^{K}K_{t}+v_{t}\big)\\[3pt]&& \big(1+\tau _{t}^{c}\big)P_{t}C_{R,t}+\frac {1}{1-\lambda }P_{t}I+\frac {\phi }{2}\Big(\frac {I_{t}}{K_{t}}-\delta \Big)^{2}, \end{eqnarray*}

where $N_{R,t}$ and $C_{R,t}$ represent the labor supply and consumption of Ricardian households, the model includes $B_{t}$ the nominal price of a government bond with a payoff of $1 maturing at $t+1$ and $r_{t}$ its nominal yield. The nominal wage is denoted as $w_{t}$ , $v_{t}$ reflects the financial returns from fully dividend-imputed firms, taxed similarly to household income. Tax rates on income and consumption are represented by $\tau ^{y}$ and $\tau ^{c}$ , respectively. The return on private capital is $r_t^k$ , and $P_t$ is the price index, defined as $P_t = \left [ \int _{0}^{1} P_t(z)^{1-\theta }\, dz \right ]^{\frac {1}{1-\theta }}$ , where $P_t(z)$ is the price of good $z$ . Private investment is $I_{t}$ with quadratic adjustment costs expressed as $\phi (\cdot )=$ $\phi /2(I_{t}/K_{t}-\delta )^{2}$ , and $\delta$ is the depreciation rate of private capital. The accumulation of private capital is $K_{t+1}=(1-\delta )K_{t}+I_{t}$ .

The conditions for optimal behavior of Ricardian households are as follows:

(2) \begin{equation} \beta R_{t}E_{t}\bigg (\frac {\epsilon _{t+1}^{TP}(1+\tau _{t}^{c})P_{t}C_{R,t}}{\epsilon _{t}^{TP}(1+\tau _{t+1}^{c})P_{t+1}C_{R,t+1}}\bigg )=1, \end{equation}

(3) \begin{equation} N_{R,t}(z)=\bigg (\frac {(1-\tau _{t}^{y})w_{t}}{C_{R,t}(1+\tau _{t}^{c})P_{t}}\bigg )^{\varphi }. \end{equation}

(4) \begin{equation} q_{t}=1+\phi \bigg (\frac {I_{t}}{K_{t}}-\delta \bigg ), \end{equation}

(5) \begin{equation} q_{t}=E_{t}\bigg \{\Lambda _{t,t+1}\bigg [\big(1-\tau _{t+1}^{y}\big)r_{t+1}^{K}+q_{t+1}(1-\delta )-\phi _{t+1}+\bigg (\frac {I_{t+1}}{K_{t+1}}\bigg )\phi _{t+1}^{^{\prime }}\bigg ]\bigg \}, \end{equation}

where $\Lambda _{t,t+1}=\beta \bigg (\frac {C_{R,t}}{C_{R,t+1}}\bigg )$ . Furthermore, we assume that the log-linearized version of the investment equation incorporates investment shocks (IS), which follow a similar AR(1) process as the other shocks in the model.

Non-Ricardian households, receiving income from working in firms and public transfers while paying taxes, have these optimal conditions:

(6) \begin{equation} \big(1+\tau _{t}^{c}\big)P_{t}C_{N,t}=\big(1-\tau _{t}^{y}\big)w_{t}N_{N,t}+\omega \frac {G_{t}^{T}}{\lambda }, \end{equation}

(7) \begin{equation} N_{N,t}(z)=\bigg (\frac {(1-\tau _{t}^{y})w_{t}}{C_{N,t}(1+\tau _{t}^{c})P_{t}}\bigg )^{\varphi }, \end{equation}

Aggregate consumption and labor supply are then defined as $C_{t}=\lambda C_{N,t}+(1-\lambda )C_{R,t}$ and $N_{t}=\lambda N_{N,t}+(1-\lambda )N_{R,t}$ .

3.2. Supply-side: firms

China’s rapid growth from the 1990s to the 2010s was driven largely by industrialization and a massive reallocation of labor from agriculture to manufacturing, rather than by high-tech innovation. Empirical studies (Brandt et al. Reference Brandt, Hsieh, Zhu, Brandt, Hsieh and Zhu2008; Rodrik, Reference Rodrik2014) highlight labor reallocation, scale effects, and industrial learning as key drivers of productivity growth. A LBD mechanism therefore provides a natural way to model endogenous TFP: higher employment generates more experience and skills, which in turn raise TFP. By contrast, knowledge-capital models are more suitable for economies where R&D spending is the primary engine of growth, while variety-expansion models better capture innovation-driven growth in high-tech, open economies. Moreover, TFP strongly correlates with GDP, as shown in studies such as Fernald and Wang (Reference Fernald and Wang2016) and Hsu et al. (Reference Hsu, Lee and Zhao2020). To introduce endogenous and procyclical TFP in a DSGE model, a LBD equation can be used, as exemplified by Chang et al. (Reference Chang, Gomes and Schorfheide2002), Engler and Tervala (Reference Engler and Tervala2018), and Watson and Tervala (Reference Watson and Tervala2022):

(8) \begin{equation} Y_{t}(z)=K_{t}(z)^{\alpha }(N_{t}(z)X_{t})^{1-\alpha }K_{G,t}^{\phi _{kg}}. \end{equation}

$Y_{t}(z)$ denotes the output of firm $z$ , $K_{G,t}$ represents public capital, $\phi _{kg}$ indicating the output elasticity of this public capital. $X_{t}$ is defined as TFP, encapsulating the overall productivity level that includes the skill level of the average worker and other efficiency factors. $X_{t}$ is modeled to depend on past working hours, reflecting a LBD process, with its dynamics governed by the following law of motion:

(9) \begin{equation} X_{t}=X_{t-1}^{\rho _{x}}N_{t-1}^{\mu _{l}}(z), \end{equation}

where $\rho _{x}$ denotes the persistence of accumulated TFP stock, while $\mu _{l}$ denotes the elasticity of TFP relative to the hours of employment in the preceding period.

Cost minimization leads to a ratio of capital to labor

(10) \begin{equation} \frac {K_{t}(z)}{N_{t}(z)}=\frac {\alpha }{1-\alpha }\frac {w_{t}}{r_{t}^{K}}. \end{equation}

Firms aim to maximize the present value of their profits, denoted as $v_{t}(z)$

(11) \begin{equation} \max _{p_{t}(z)}{v_{t}(z)}=E_{t}\sum _{s=t}^{\infty }\gamma ^{s-t}Q_{t,s}\frac {v_{s}(z)}{P_{s}}, \end{equation}

where $1-\gamma$ represents the likelihood of a firm being able to adjust its price in any given period. Given the stochastic discount factor $\xi _{t,s}$ between periods $t$ and $s$ , the solution for $p_{t}(z)$ can be expressed as

(12) \begin{equation} p_{t}(z)=\frac {\theta }{\theta -1}\frac {E_{t}\sum _{s=t}^{\infty }\gamma ^{s-t}\xi _{t,s}Q_{s}MC_{s}(z)}{E_{t}\sum _{s=t}^{\infty }\gamma ^{s-t}\xi _{t,s}Q_{s}}, \end{equation}

where

\begin{equation*} Q=\left ( \frac {C_{s}+I_{s}+\phi \big(\frac {I_{s}}{K_{s}}\big)K_{s}+G_{s}^{C}+I_{s}^{G}}{P_{s}}\right )\!. \end{equation*}

Log-linearizing equation (12) yields the pricing equation $\hat {p}_{t}(z)=\beta \gamma E_{t}(\hat {p}_{t+1}(z))$ $+(1-\beta \gamma )(\hat {mc}_{t}(z))+\epsilon _{t}^{CP}$ , where $\epsilon _{t}^{CP}$ represents a cost-push shock. It follows a similar AR(1) process as the other shocks within the framework. The overall price level is determined by $\hat {p}_{t}=\gamma \hat {p}_{t-1}+(1-\gamma )\hat {p}_{t}(z)$ .

3.3. Economic policy

The equation depicting the accumulation of public capital, Equation (1), is presented and elaborated upon in the introductory section. It is assumed that public consumption indices mirror those of private consumption, with public demand functions for domestic goods formulated similarly to private demand functions. The budget constraint for the government is as follows:

(13) \begin{equation} \tau _{t}^{y}\big(w_{t}N_{t}+r_{t}^{K}K_{t}+v_{t}\big)+\tau _{t}^{c}P_{t}C_{t}=B_{t}-R_{t}^{-1}B_{t+1}+P_{t}\big(G_{t}^{C}+I_{t}^{G}+G_{t}^{T}\big). \end{equation}

The Government responds to variations in the previous period’s public debt relative to its initial steady state level by adjusting the income tax burden:

(14) \begin{equation} \tau _{t}^{y}=\tau _{0}^{y}\bigg (B_{t-1}-B_{0}\bigg )^{\Phi _{d}}. \end{equation}

The tax elasticity in relation to public debt is denoted by is $\Phi _{d}.$ Each category of government expenditure, denoted as $\hat {g}$ (where $g$ represents $G^{C}$ , $I^{G}$ , and $G^{T}$ ) follows an AR(1) process: $\hat {g}_{g,t}=\rho ^{g}\hat {g}_{g,t-1}+\varepsilon _{g,t}^{g}$ . Here $\rho ^{g}$ ranges from zero to one, $\hat {g}_{g,t}$ is defined as $(G_{g,t}-G_{g,0})/Y_{0}$ , $\epsilon _{g,t}^{g}$ is an independent and identically distributed spending shock variable with a mean of zero.

Monetary policy conforms to a Taylor (Reference Taylor1993) rule, incorporating interest rate smoothing. The log-linearized version of this monetary policy rule is as follows:

\begin{equation*} \hat {\imath }_{t}=\mu _{s}\hat {\imath }_{t-1}+(1-\mu _{s})(\mu _{p}\Delta \hat {P}_{t}+\mu _{y}\hat {Y}_{t})+\hat {\varepsilon }_{t}^{r}, \end{equation*}

where $\Delta$ is the first difference operator and $\hat {\varepsilon }_{t}^{r}$ indicates a monetary policy shock characterized by a zero mean.

4.1. Parameterization

Consistent with conventional parameters in DSGE literature, the discount factor ( $\beta$ ) is established at 0.995, and the elasticity of substitution between goods ( $\theta$ ) is determined to be 6. The Frisch elasticity of labor supply ( $\varphi$ ) is established at 2, reflecting the perspective that the macro Frisch elasticity is high (Peterman, Reference Peterman2016). The proportion of Non-Ricardian households ( $\lambda$ ) is 35%, based on the estimates for the Chinese economy by Liu and Ou (Reference Liu and Ou2020). The consumption tax rate ( $\tau ^{c}$ ) is established at 0.1, aligned with China’s standard value-added tax rates (6%, 9%, or 12%) and the effective consumption tax rate estimated by Miao et al. (Reference Miao and Bai2023). The tax rate ( $\tau _{t}^{y}$ ) is set at 0.4, exceeding the OECD’s (2021) estimate of the tax wedge. China’s public finances have seen chronic deficits, with a fiscal surplus occurring in only one year between 1995 and 2007. The model’s considerable public expenditures and the requirement for fiscal balance to ensure stability necessitate a tax rate higher than the OECD’s estimate.

In macroeconomic models focused on China, the output elasticity of private capital ( $\alpha$ ) is typically higher than in advanced economies, reflecting China’s more capital-intensive production and a labor share of GDP that has been just over 50% (United Nations, 2023). Li and Liu (Reference Li and Liu2017) use a figure of 0.4, while Fernandez-Villaverde et al. (Reference Fernandez-Villaverde, Ohanian and Yao2023) opt for 0.5. We set the share of private capital at 0.45, in line with Lei et al. (Reference Lei, Yao and Peng2023). The efficiency of public investment ( $\upsilon$ ) in China is set at 43%.

The analysis employs data from “China’s Macroeconomy: Time Series Data” by Chang et al. (Reference Chang, Chen, Waggoner and Zha2015, Reference Chang, Chen, Waggoner and Zha2023), covering all relevant variables from 1995:Q1 to 2017:Q4. The ratio of public debt-to-annual GDP has averaged 32% over the entire period (Mbaye et al. Reference Mbaye, Moreno-Badia and Chae2018). In our model, the debt ratio is set to 128% relative to quarterly GDP. Public investments, including state-owned and holding enterprises, averaged 22% over the entire period, but reduced to 18% in the last decade after excluding net exports’ contribution to GDP. Public consumption expenditures remained steady at 16% throughout the entire period and the last ten years. We set the share of public investments at 18% and public consumption at 16%. In our model, the GDP shares of private consumption and investment are determined endogenously through the model’s parameters and are 47 and 19%, respectively. According to the data (Chang et al. Reference Chang, Chen, Waggoner and Zha2015, Reference Chang, Chen, Waggoner and Zha2023), the GDP share of private consumption (investment) has been 38% (27%) over the past decade and 43% (19%) over the entire period.

4.2. Prior distribution of the parameters

The remaining model parameters are estimated utilizing Bayesian techniques. The dataset for estimation encompasses quarterly real GDP, private consumption, private investment, public investment, and price level, sourced from Chang et al. (Reference Chang, Chen, Waggoner and Zha2015, Reference Chang, Chen, Waggoner and Zha2023). Their dataset aligns with U.S. national accounts definitions while retaining the specific features of Chinese data, making it well-suited for our Bayesian estimation. Additionally, the quarterly average interest rate is derived from monthly data (discount rate for China) obtained from the Fred Economic Data (2023).

Public investment is measured as the sum of gross fixed capital formation by the government and by SOEs, ensuring that the measure captures both direct government investment and investment undertaken through SOEs. Private investment is defined as total gross fixed capital formation excluding inventories, net of public investment. All data are represented in log-deviations from their Hodrick-Prescott (Reference Hodrick and Prescott1997) trends (using a smoothing parameter of 1600). The estimation process employs data spanning from 1995Q1 to 2017Q4, since investment series such as gross fixed capital formation by the government and SOEs are available at a quarterly frequency only for this period. The stochastic dynamics of the model are determined by seven exogenous disturbances: three kinds of public spending shocks (consumption, investment, and transfers), a monetary policy shock, a private IS, a time preference shock, and a cost-push shock.

Watson and Tervala (Reference Watson and Tervala2021) estimated a LBD equation using Australian data. They found the persistence of TFP to be 0.91, and the elasticity of TFP relative to employment averaged 0.25. The prior for the persistence of TFP ( $\rho _{x}$ ) is set at 0.91 with a standard deviation of 0.05. The persistence of all AR(1) processes follows a beta distribution. The prior for the elasticity of TFP relative to employment ( $\mu _{l}$ ) is assumed to be 0.25, characterized by a Normal distribution with a standard deviation of 0.1.

The prior for the elasticity of taxes in relation to public debt ( $\Phi _{d}$ ) is set at 0.075, reflecting the average of the values used by Lieberknecht and Wieland (Reference Lieberknecht and Wieland2019). Its standard deviation is 0.1. Furthermore, the parameter is assigned a lower bound of 0.0001 to guarantee long-term stability in the public debt level. Numerous studies have estimated the output elasticity of public capital ( $\phi _{kg}$ ) in advanced economies, as outlined in a survey by Bom and Ligthart (Reference Bom and Ligthart2013). Calderón et al. (Reference Calderón, Moral-Benito and Servén2015) estimated the output elasticity of public capital across a sample of 88 countries spanning various income levels. Their findings (referenced in Tables 3 and 4) indicate an average productivity of public capital at 0.08, showing no significant variation in relation to a country’s per capita income, infrastructure endowment, or population size. Based on this, we have selected 8% as our parameter value, with an accompanying standard deviation of 0.1. Additionally, a lower bound of 0.0001 is assigned to the parameter.

The priors for the quarterly depreciation rates of private ( $\delta$ ) and public capital are set at 0.025 and 0.0125, respectively, to align the annual depreciation rates with the estimates of 10% and roughly 5% as determined by the IMF (2015). The standard deviation for these rates is set at a low value of 0.05, reflecting the minimal uncertainty associated with these parameters.

Smets and Wouters (Reference Smets and Wouters2007) initially set the priors for the persistence of all types of public spending shocks at 0.5. However, numerous studies have identified significantly higher values, reaching up to 0.94, as found by Sims and Wolff (Reference Sims and Wolff2018). As a compromise, we adopt a value of 0.75, similar to what Iwata (Reference Iwata2013) discovered for public consumption and investment. The standard deviation for these rates is set at 0.1. The priors for the persistence of other shocks, time preference, cost-push, and investment, are set at 0.5, following Smets and Wouters (Reference Smets and Wouters2007), with standard deviations assigned a value of 0.2. We select a gamma distribution for the standard deviation of all shocks, setting the mean at 0.5 and the standard deviation at 0.4.

The Calvo parameter ( $\gamma$ ) is set at 0.75, a standard value in DSGE macro modeling, with its standard deviation fixed at 0.05. The priors for parameters characterizing monetary policy are in line with Smets and Wouters (Reference Smets and Wouters2007). The responses to inflation ( $\mu _{p}$ ) and output ( $\mu _{y}$ ) are described by a Normal distribution with means of 1.5 and 0.125 (a quarter of 0.5), and standard errors of 0.1 and 0.05, respectively. The smoothing parameter ( $\mu _{s}$ ) is set at 0.75, with a standard deviation of 0.05, following a Beta distribution. The prior for the investment adjustment cost parameter ( $\phi$ ) is set at 4, as per Smets and Wouters (Reference Smets and Wouters2007), with a standard deviation of 1.

4.3. Posterior Estimates of the Parameters

Table 1 presents the posterior means and credible intervals of the model parameters, determined using Bayesian Highest Density Intervals (HDI). Figure 3 illustrates the prior (in gray) and posterior (in black) distributions of key variables. The Metropolis-Hastings (MH) algorithm chain’s initial value is set at 0.3, and the scale parameter for the covariance matrix in the Random Walk MH algorithm is 0.4. This configuration results in an acceptance rate of 24%, aligning well with the optimal acceptance ratio of 23% as suggested by Roberts et al. (Reference Roberts, Gelman and Gilks1997). For the MH algorithm, five parallel chains are utilized. In accordance with Smets and Wouters (Reference Smets and Wouters2007), the MH algorithm runs through 250,000 Markov chain Monte Carlo replications. Additionally, we set the fraction of initially generated parameter vectors to be excluded at 0.5, thereby discarding the first 125,000 draws.

Table 1. Prior and posterior distributions of model parameters

Figure 3. Prior (gray) and posterior (black) distributions of main variables.

The model lends support to the LBD equation: greater employment raises TFP. The posterior persistence of TFP is somewhat lower than both the prior estimate and the findings of Watson and Tervala (Reference Watson and Tervala2022) for Australia. By contrast, the elasticity of TFP with respect to employment is higher than both the prior and their results, indicating a significant role of employment in driving TFP. As Rodrik (Reference Rodrik2014) notes, rapid growth in developing economies is often driven by labor reallocation into modern, productivity-enhancing industries where workers accumulate know-how through production. Our model captures this process by linking TFP to employment, reflecting how structural transformation and industrialization boost productivity through experience-based gains.

For many variables, the mean of the posterior distribution is relatively close to the mean of the prior. However, the tax elasticity with respect to public debt is relatively small in China. In addition, the posterior estimate for the productivity of public capital is significantly lower, at 2.9%, compared to the prior of 8%. Additionally, the confidence interval for this estimate is quite narrow. The productivity of public capital may not be constant but has likely declined over time as the capital stock has accumulated. The likely cause of the low efficiency of public investment is the extensive scale of public investment, including real estate, which has resulted in an excess public capital stock, including housing. Lam and Moreno-Badia (Reference Lam and Moreno-Badia2023) find weak performance and returns of SOEs and declining infrastructure returns in China.

It is plausible that the productivity of public capital has declined over time as China has undertaken massive infrastructure development. The full-sample estimate of 3% likely reflects an average that masks a higher elasticity in the early years and a lower one more recently. To test this hypothesis, we re-estimated the model separately for 1995Q1–2008Q4 and 2009Q1–2017Q4. The results show that the output elasticity of public capital was nearly 4% (0.036) in the earlier period but fell to around 2% in the later period. These findings offer quantitative support for the view that the marginal productivity of public capital has declined as the scale of investment has expanded, consistent with the narrative of diminishing returns to China’s public investment boom.

While our model treats public investment as a single aggregate, in reality it consists of heterogeneous components, including infrastructure (transport, energy) and real estate, each with potentially different levels of productivity. As documented by Rogoff and Yang (Reference Rogoff and Yang2024), China’s investment boom has been heavily concentrated in real estate and infrastructure. Much of the new construction has taken place in lower-tier cities, where population growth and economic activity have been weak. This pattern may partly explain why we find a declining output elasticity of public capital over time: the earlier phase of investment likely delivered higher-return infrastructure, while the later phase has increasingly produced real estate projects with limited economic payoff, consistent with diminishing returns at the aggregate level.

The posterior estimate of the depreciation rate of public capital (1.26%) closely aligns with its prior estimate of 1.25%. This corresponds to an annual depreciation rate of 4.9% $(1-(1-0.0126) \,^{\hat {}}\, 4=0.049)$ , which is only slightly higher than the 4.6% annual rate used in the calculations in Section 2. As discussed in Section 2, possible explanations for the large disparity between the stock of public capital and the level implied by the investment rate include a very low efficiency of public investment, a higher depreciation rate of public capital, or measurement errors in the public capital or investment data. Hence, the bulk of the discrepancy is likely explained by the low efficiency of public investment. Using the estimated annual depreciation rate of 4.9%, we calculate that the efficiency of public investment in China is 46%, which represents only a very small change compared with the 43% reported in Section 2.

The annual depreciation rate of private capital is about 7%, which is fairly close to the IMF’s (2015) estimate for middle-income countries. The estimated persistence of public ISs is 0.75. The posterior estimate for the Calvo parameter is 0.45, indicating that the average age of prices is approximately two quarters. This value is quite typical and aligns closely with the result of 0.39 obtained by Liu and Ou (Reference Liu and Ou2020) for China. The posterior estimate for the investment adjustment cost parameter is 6.2, which is relatively close to the estimate of 5.7 provided by Smets and Wouters (Reference Smets and Wouters2007). The posterior estimates for the coefficients of the monetary policy rule imply relatively modest smoothing and a fairly strong response to inflation and output fluctuations.