Silviase/jgov_v1.4_full_ann huggingface dataset:
where $ X_{t} $ is aggregate investment, $ w_{t} $ is the wage rate, $ r_{t} $ is the rental rate of capital, and $ K_{t} $ is aggregate capital. The capital stock depreciates geometrically so that

$$ K_{t+1}=(1-\delta)K_{t}+X_{t}. $$

The household chooses the infinite sequences of $ \{c_{t}, k_{t+1}, h_{t}\} $ to maximize life-time utility (1) subject to the budget constraint (2) and the capital law of motion (3) given the initial level of capital stock.

The representative firm decides how much capital stock it rents, how much labor it employs, and how much energy it imports from abroad in each period to maximize profit $ \pi_{t} $ ,

$$ \pi_{t}\equiv Y_{t}-w_{t}H_{t}-r_{t}K_{t}-p_{t}E_{t}, $$

where $ Y_{t} $ is gross output, $ p_{t} $ is the relative price of energy, and $ E_{t} $ is aggregate energy imported from abroad. To make the analysis as simple as possible, it is assumed that the variation in $ p_{t} $ is exogenous to the firm so that it can import as much energy as it desires at the given price $ p_{t} $ . In this sense, the model is a small open economy.

Gross output is defined by a nested constant elasticity of substitution production function with constant returns to scale:

$$ Y_{t}=(\Gamma_{t}H_{t})^{1-\theta}\bigg[(1-\mu)K_{t}^{\frac{\varepsilon-1}{\varepsilon}}+\mu(z_{t}E_{t})^{\frac{\varepsilon-1}{\varepsilon}}\bigg]^{\frac{\varepsilon}{\varepsilon-1}\theta}. $$

$ \Gamma_{t} $ is an index of labor-augmenting technological progress, $ z_{t} $ is the level of energy-saving technology, and $ \varepsilon $ is the elasticity of substitution between capital and energy use. The exogenous time-varying energy-saving technology is incorporated since it is shown in Niizeki (2012) that rapid growth of energy-saving technology is observed following the first oil crisis in Japan and that it is essential to reproduce the downward trend in energy use. The next subsection illustrates how the series of $ z_{t} $ are estimated. To close the model, the economy-wide resource constraint is

$$ C_{t}+X_{t}=Y_{t}-p_{t}E_{t}\equiv V_{t}, $$

where $ V_{t} $ is aggregate value added.

On the balanced growth path, all aggregate variables (except labor input) grow at rate $ \gamma_{t} + n_{t} $ ,
where $ \gamma_{t} \equiv \Gamma_{t+1}/\Gamma_{t} $ and $ n_{t} \equiv N_{t+1}/N_{t} $ , whereas aggregate hours worked grow at rate $ n_{t} $ . Compared to standard neoclassical growth models (see, e.g., Prescott 1986), this model has the following additional optimality condition regarding energy use:

$$ p_{t}=\theta\mu\left(\frac{Y_{t}}{B_{t}}\right)\left(\frac{B_{t}}{z_{t}E_{t}}\right)^{\frac{1}{\varepsilon}}z_{t}, $$

where $ B_{t} \equiv \left[(1 - \mu)K_{t}^{\frac{\varepsilon-1}{\varepsilon}} + \mu(z_{t}E_{t})^{\frac{\varepsilon-1}{\varepsilon}}\right]^{\frac{\varepsilon}{\varepsilon-1}} $ .

The right hand side of Equation (7) is the marginal product of energy use. Thus, the firm decides how much energy it imports from abroad by equating the marginal cost and the marginal benefit of energy use, taking the relative price of energy as given.

### 2.1 Energy-saving technological change

In his seminal work, Hicks (1932) put forward the so-called “induced innovation hypothesis,” which states that a change in relative factor prices will lead to innovation to enhance the efficient use of relatively expensive factors. In the context of energy use, the induced innovation hypothesis predicts improvements in energy-saving technology following an upsurge in the relative price of energy. There are several empirical studies that have sought to capture the causality from changes in the relative price of energy to energy-saving technological change. For instance, Popp (2002), using U.S. patent data from 1970 to 1994, looks at the impact of increases in energy prices on energy-efficiency innovations. He finds that rises in energy prices have a statistically significant positive impact on energy-efficiency innovations. On the other hand, Newell et al. (1999) investigate whether energy prices affect the energy efficiency of new models of energy-using consumer durables, such as room air conditioners and gas water heaters, and conclude that for some products the direction of innovation is influenced by changes in energy prices. For Japan, Fukunaga and Osada (2009) measure energy-saving technological change by estimating time-varying biases of technical change. They report that the bias of technical change for energy input in the 1980s was energy-saving. Niizeki (2012) also estimates energy-saving technological change and shows that it is required to replicate the declining energy use observed following the first oil crisis in Japan.

In this paper, the level of energy-saving technology is estimated using the production function and the first order condition for energy use shown below:
Figure 2: The level of energy-saving technology normalized to 100 in 1973.

$$ Y_{t}=(\Gamma_{t}H_{t})^{1-\theta}\Big[(1-\mu)K_{t}^{\frac{\varepsilon-1}{\varepsilon}}+\mu(z_{t}E_{t})^{\frac{\varepsilon-1}{\varepsilon}}\Big]^{\frac{\varepsilon}{\varepsilon-1}\theta} $$

$$ p_{t}=\theta\mu\left(\frac{Y_{t}}{B_{t}}\right)\left(\frac{B_{t}}{z_{t}E_{t}}\right)^{\frac{1}{\varepsilon}}z_{t}. $$

Given the parameter values obtained in Section 3 and the actual time series for $ Y_{t}, H_{t}, K_{t}, E_{t} $ , and $ p_{t} $ , there are two unknown variables, $ \Gamma_{t} $ and $ z_{t} $ , and two equations, so that the series of $ \{\Gamma_{t}, z_{t}\}_{1973}^{1978} $ can be obtained by solving the system of these equations for $ \Gamma_{t} $ and $ z_{t} $ each year. $ ^{7} $

Figure 2 displays the estimated level of energy-saving technology normalized to 100 in 1973. As can be seen, rapid growth of energy-saving technology is observed following the first oil shock in 1973. In fact, the estimated average annual growth rate of energy-saving technology over the period 1973–1978 is 8.76%. These improvements in energy-saving technology are included in the original TFP growth rate estimated using a production function without energy-saving technological change. That is, the growth rate of TFP when assuming that $ z_{t} $ is unity throughout in Equation (5) is 1.67%. On the other hand, the modified TFP growth rate, which strips out
improvements in energy-saving technology, that is, when $ z_{t} $ takes the estimated time-varying values, is 0.72%. Throughout this paper, the estimated time-varying levels of energy-saving technology are always fed into the model, so that the effect of energy-saving technological change is always excluded from the TFP series.

## 3 Data and calibration

### 3.1 Energy

In this paper, the energy-related variables are real energy use, $ E_{t} $ , and the relative price of energy, $ p_{t} $ . These variables are constructed based on the methodology developed by Atkeson and Kehoe (1999). Real energy use $ E_{t} $ at time t is calculated as follows:

$$ E_{t}\equiv\sum_{i}P_{i,0}Q_{i,t}, $$

where i denotes the type of energy. In the analysis here, there are three types of energy: petroleum, coal, and liquid natural gas. $ P_{i,0} $ is the price of type i energy in the base year, which is 1990. Note that $ P_{i,0} $ is the CIF (cost, insurance, and freight) price converted into Japanese yen, so that exchange rate changes are already taken into account. $ Q_{i,t} $ is the amount of imported type i energy in year t. Note that $ Q_{i,t} $ is the amount of imports, not the amount of consumption of type i energy. However, since in Japan most of the energy imported in any given year is consumed within the year, $ Q_{i,t} $ is treated as the amount of consumption of type i energy in year t.

To construct the relative price of energy, the energy price deflator at time t, denoted as $ DEF_{t}^{P} $ , is derived as follows:

$$ \boldsymbol{D E F}_{t}^{P}\equiv\frac{\sum_{i}P_{i,t}Q_{i,t}}{\sum_{i}P_{i,0}Q_{i,t}} $$

The relative price of energy, $ p_{t} $ , is then constructed by dividing the energy price deflator by the GNP deflator, whose base year is also 1990:

$$ p_{t}\equiv\frac{D E F_{t}^{P}}{D E F_{t}^{V}}, $$
where $ DEF_{t}^{V} $ is the GNP deflator at time t.

### 3.2 Other variables

Other variables are defined as follows. The working age population is defined as the number of people aged 15–64, which is obtained from the Population Census published by the Ministry of Internal Affairs and Communications. Average weekly hours worked per employed person $ (\ell_{t}) $ are taken from the Monthly Labour Survey published by the Ministry of Health, Labour and Welfare, while the number of employed persons $ (M_{t}) $ is taken from the 1968 SNA (System of National Accounts) produced by the Economic and Social Research Institute, Cabinet Office. Using these data, the empirical counterpart of $ h_{t} $ is then calculated as follows:

$$ h_{t}=\frac{\ell_{t}*M_{t}}{N_{t}*16*7}. $$

Following Otsu (2009), it is assumed that the maximum number of hours worked per day is 16.

The remaining variables are obtained from the 1968 SNA. The 1968 SNA is used for analysis since data based on the 1993 SNA are not available for the period before 1980. As the models described in this paper contain no government sector, it is necessary to adjust the data from the 1968 SNA to match up the series in the models. $ ^{8} $ In the 1968 SNA, real value added (real GNP in this paper) is decomposed into the following parts:

$$ V_{t}=C_{t}+X_{t}+G_{t}+NX_{t}+NFP_{t} $$

where $ V_{t} $ is real value added, $ C_{t} $ is real “Private final consumption expenditure,” $ X_{t} $ is the sum of real “Gross fixed capital formation” and real “Change in inventories,” $ G_{t} $ is real “Final consumption expenditure of government,” $ NX_{t} $ is real “Net exports (excluding real energy imports),” and $ NFP_{t} $ is real “Net factor payments.” The nominal values are converted into real values by dividing them by the constant 1990 yen deflator.

Following Hayashi and Prescott (2002), real “Final consumption expenditure of government” is included in $ C_{t} $ and both real “Net exports (excluding real energy imports)” and real “Net factor payments” are incorporated in $ X_{t} $ . That is,
$$ V_{t}=C_{t}^{^{\prime}}+X_{t}^{^{\prime}} $$

where $ C_{t}^{'} $ is the sum of $ C_{t} $ and $ G_{t} $ , and $ X_{t}^{'} $ is the sum of $ X_{t} $ , $ NX_{t} $ , and $ NFP_{t} $ . $ V_{t} $ , $ C_{t}^{'} $ , and $ X_{t}^{'} $ are the empirical counterparts of value added, aggregate consumption, and aggregate investment in this paper, respectively.

Finally, the capital stock series are constructed using the perpetual inventory method. The initial capital stock, $ K_{1973} $ is set so that $ K_{1973}/V_{1973} = 1.57,^{9} $ which is taken from the value in the dataset constructed by Hayashi and Prescott (2002). The series of subsequent values of capital stock is obtained by the law of motion for capital stock, Equation (3).

### 3.3 Calibration

In standard RBC models such as those by Prescott (1986) and King and Rebelo (2000), parameter values are set so that the steady state in the models is consistent with growth observations. This strategy is reasonable, since it is assumed in standard RBC models that per-capita variables are basically on a balanced growth path. They deviate from the balanced growth path only if the economy is hit by shocks. On the other hand, in the case of the Japanese economy in the 1970s, it seems more reasonable to assume that it was in a transition to a balanced growth path rather than on a balanced growth path. $ ^{10} $ Therefore, using growth observations to calibrate parameters would probably not be appropriate here. As an alternative strategy, therefore, some parameter values are calibrated using only data for the period 1973–1978, while the values of other parameters, which are usually regarded as constant over time, are either set at conventional values or taken from previous studies.

Combining the production function and the first order condition for energy use gives
$$ \frac{1-\mu}{\mu}=\left(\frac{\theta Y_{t}-p_{t}E_{t}}{p_{t}E_{t}}\right)\left(\frac{E_{t}}{K_{t}}\right)^{\frac{\varepsilon-1}{\varepsilon}},\quad for\quad t=1973,1974,...,1978. $$

Equation (16) $ ^{11} $ is solved for $ \mu $ each year and $ \mu $ is then averaged over 1973–1978, yielding $ \mu = 0.005 $ . $ ^{12} $ The first order condition for $ h_{t} $ condition gives

$$ \left(\frac{\alpha}{1-\alpha}\right)\left(\frac{h_{t}}{1-h_{t}}\right)=(1-\theta)\frac{y_{t}}{c_{t}},\quad for\quad t=1973,1974,...,1978. $$

Equation (17) is solved for $ \alpha $ and $ \alpha $ is then averaged over 1973–1978, resulting in $ \alpha = 0.709 $ . $ \beta $ and $ \theta $ are set at 0.976 and 0.362, respectively, which are the values used in Hayashi and Prescott (2002). $ \delta $ is set at the conventional value for annual data, 0.100.

The most important parameter to quantify the impact of the upsurge in the relative price of energy in this model is the elasticity of substitution between capital stock and energy use. It is well known that a high elasticity of substitution such as unity provides unrealistically volatile movements in energy use. For instance, Backus and Crucini (2000) claim that the elasticity of substitution between capital stock and oil should be around 0.09 to produce the realistic movements of oil prices and oil quantities in their model. On the empirical side, Hassler et al. (2012), using data for the United States and employing maximum likelihood estimation, arrive at an elasticity of substitution between energy and the capital/labor composite of 0.0044. Meanwhile, Miyazawa (2009), also using data for the United States and conducting a generalized method of moments estimation, arrives at values of $ \varepsilon = 0.100 $ and $ \varepsilon = 0.086 $ . Given these results, $ \varepsilon $ in this paper is set to 0.1. Table 1 summarizes the calibration results.

## 4 Simulation results with benchmark model

As in the studies by Hayashi and Prescott (2002, 2008) and Chen et al. (2006), it is assumed that agents have perfect foresight about the sequence of four exogenous variables: the relative price of energy $ (p_{t}) $ , the growth rate of TFP $ (\gamma_{t}) $ , the level of energy-saving technology $ (z_{t}) $ , and the growth rate of the working-age population $ (n_{t}) $ . The model is then solved numerically by
Parameter	Description	Value
$ \varepsilon $	Elasticity of subst. btw. capital and energy	0.100
$ \theta $	Capital/Energy composite share	0.362
$ \delta $	Depreciation rate of capital	0.100
$ \beta $	Discount factor	0.976
$ \alpha $	Leisure weight in preferences	0.709
$ \mu $	Share of energy in capital-energy composite	0.005

Table 1: Parameter values

Figure 3: Detrended value added per working-age population (benchmark model vs. data) normalized to 100 in 1973.
applying a shooting algorithm given the initial capital stock level and the path of the exogenous variables. The levels of the exogenous variables after the period 1973–1978 are assumed to take the average value for the period 1973–1978 except in the case of energy-saving technology. The level of energy-saving technology is assumed to continue to take the 1978 value after the period 1973–1978.

Figure 3 shows three simulation results for value added. The first is the simulated path with the actual time series for the relative price of energy only, labeled “Price only.” The second is the simulated path with the actual time series for the TFP growth rate only, labeled “TFP only.” Finally, the third is the simulated path with the actual time series for the relative price of energy and the TFP growth rate, labeled “Price and TFP.” To generate the path labeled “Price only,” the TFP growth rate is set to its geometric mean over the period 1973–1978, while to obtain the path labeled “TFP only” the relative price of energy is set to its mean over the same period. $ ^{13} $ A notable feature in Figure 3 is that the upsurge in the relative price of energy alone fails to account for the drop in value added in 1974. In the “Price only” simulation, value added in 1974 declines by only 0.7%, while in the actual data it falls by about 4%. Finally, in the “Price and TFP” simulation, the sluggish growth rate of TFP adds to the negative effect on value added, but this is insufficient to account for the severe recession following the first oil shock in 1973. Next, Figure 4 displays the simulated paths for other aggregate variables. As can be seen, the simulated path for energy use reproduces the downward trend in energy use well because of the improvements in energy-saving technology. $ ^{14} $

In sum, the simulation analysis using the benchmark model shows that the sharp rise in the relative price of energy plays a limited role in accounting for the decline in value added. In addition, the analysis suggests that sluggish TFP growth also does not appear to have played an important role in the recession following the first oil crisis, which contrasts with the findings by Hayashi and Prescott (2002) and Chen et al. (2006) that trends in TFP growth provide a good explanation of developments in macroeconomic variables in Japan. A likely reason why the relative price of energy in the benchmark model only plays a limited role in explaining the drop in value added is
Figure 4: Other aggregate variables per working-age population (benchmark model vs. data) normalized to 100 in 1973.

the small ratio of real energy use to real value added $ (p_{t}E_{t}/V_{t}) $ . In Japan, the average value of this ratio is only 3.7% during the period 1973–1978. That is, the relative price of energy does not play an important role in accounting for recessions simply because only a small amount of energy is needed for production.

In the next section, therefore, following Aguiar–Conraria and Wen (2007) and Finn (2000), endogenous capacity utilization is incorporated into the benchmark model as an amplification mechanism in order to examine whether this can help to explain the actually observed impact of the jump in the relative price of energy.

## 5 Endogenous capacity utilization

In the model with endogenous capacity utilization, the representative household faces the same problem as in the benchmark model, except that the depreciation cost of capital is now borne by the representative firm instead of the household. Thus, the budget constraint of the household at time t is
$$ C_{t}+K_{t+1}-K_{t}=w_{t}H_{t}+r_{t}K_{t}. $$

The firm now is able to change its capacity utilization rate endogenously, so that the time t gross output is produced according to

$$ Y_{t}=(\Gamma_{t}H_{t})^{1-\theta}\bigg[(1-\mu)(u_{t}K_{t})^{\frac{\varepsilon-1}{\varepsilon}}+\mu(z_{t}E_{t})^{\frac{\varepsilon-1}{\varepsilon}}\bigg]^{\frac{\varepsilon}{\varepsilon-1}\theta}, $$

where $ u_{t} $ is the capacity utilization rate. As in Greenwood at el. (1988), it is assumed that a more intensive use of the capital stock depreciates it more quickly: $ ^{15} $

$$ \delta_{t}=\frac{1}{\phi}u_{t}^{\phi}, $$

where $ \phi > 1 $ . Capital stock is accumulated according to the following equation:

$$ K_{t+1}=(1-\delta_{t})K_{t}+X_{t}. $$

In sum, the representative firm maximizes the following profit function:

$$ \pi_{t}\equiv Y_{t}-w_{t}H_{t}-(r_{t}+\delta_{t})K_{t}-p_{t}E_{t} $$

subject to Equations (19), (20), and (21) given the initial capital stock and the actual time series of all exogenous variables. An additional optimality condition in this model is the first order condition for the capacity utilization rate. That is,

$$ u_{t}^{\phi-1}K_{t}=\theta(1-\mu)\bigg(\frac{Y_{t}}{\tilde{B}_{t}}\bigg)\bigg(\frac{\tilde{B}_{t}}{u_{t}K_{t}}\bigg)^{\frac{1}{\varepsilon}}K_{t}, $$

where $ \tilde{B}_{t} \equiv \left[(1 - \mu)(u_{t}K_{t})^{\frac{\varepsilon-1}{\varepsilon}} + \mu(z_{t}E_{t})^{\frac{\varepsilon-1}{\varepsilon}}\right]^{\frac{\varepsilon}{\varepsilon-1}} $ . The left hand side of Equation (23) shows the additional depreciation of capital stock if the firm increases the capacity utilization rate by one unit. The right hand side represents the additional output the firm gains by raising the capacity utilization rate by one unit. Equation (23) requires that the marginal cost and benefit must be equal at optimum.
In order to obtain the TFP series taking the effect of capacity utilization into account, the actual time series for the capacity utilization rate is required. According to previous studies, there are at least three ways to obtain these series. The first is to use official statistics. The Ministry of Economy, Trade, and Industry (METI) provides the "Operating Ratio," which is calculated by dividing the actual production level by production capacity. Since the "Operating Ratio" published by METI is based on a survey of firms, the data are likely to be quite reliable. The shortcoming of these data, however, is that they only cover certain industries of the manufacturing sector, so that they do not provide an appropriate indicator of capacity utilization in the economy as a whole. The second way would be to use a proxy variable for the capacity utilization rate. Burnside et al. (1995), for example, use electricity consumption as a proxy. This seems like a reasonable assumption, since firms use more electricity when operating more machines. On the other hand, this indicator is likely to be downwardly biased when there are improvements in energy-saving technology. That is, electricity consumption can decline due not only to a drop in the capacity utilization rate, but also to improvements in energy-saving technology.

The third way to obtain the actual series for capacity utilization is to use the first order condition for the capacity utilization rate. In this paper, following Burnside and Eichenbaum (1996), this third methodology is employed. Specifically, the empirical counterpart of the capacity utilization rate is obtained by exploiting Equation (23). This means that it will also be necessary to recalculate the capital stock series, since the depreciation rate is no longer constant over time when applying the perpetual inventory method. In the analysis here, the time series of the capacity utilization rate and capital stock are obtained simultaneously as follows. First, the first order condition

$$ u_{1973}^{\phi-1}=\theta(1-\mu)\Bigg(\frac{Y_{1973}}{\tilde{B}_{1973}}\Bigg)\Bigg(\frac{\tilde{B}_{1973}}{u_{1973}K_{1973}}\Bigg)^{\frac{1}{\varepsilon}} $$

is solved for the capacity utilization rate in 1973. Next, the depreciation rate for 1973 is calculated as follows:

$$ \delta_{1973}=\frac{1}{\phi}u_{1973}^{\phi}. $$

Finally, plugging $ \delta_{1973} $ into the law of motion for capital stock,

$$ K_{1974}=(1-\delta_{1973})K_{1973}+X_{1973}, $$
Figure 5: Imputed capacity utilization rate and METI’s “Operating Ratio.” The values for both series in 1973 are normalized to 100.

yields the capital stock in 1974. This procedure is repeated for each year up to 1978. $ \phi $ is calibrated so that the series of $ \delta_{t} $ generated by the procedure above is equal to 0.100, which is the depreciation rate for capital stock in the benchmark model. This procedure leads to $ \phi = 2.001 $ .

Figure 5 displays the imputed capacity utilization rate and METI's “Operating Ratio” for comparison. As can be seen, although both series show a decreasing trend after the first oil shock, the trend in the imputed capacity utilization rate is much smoother than that in the “Operating Ratio.” The substantial drop in the capacity utilization rate implied by the “Operating Ratio” may look more realistic, but it needs to be remembered that the “Operating Ratio” is constructed using data only for certain industries in the manufacturing sector. The Appendix provides further justification for using the imputed capacity utilization rate instead of the “Operating Ratio” by constructing a crude measure of the capacity utilization rate in the non-manufacturing sector and showing that this series is much less volatile than the “Operating Ratio.” $ ^{16} $

The TFP series calculated by solving Equations (5) and (19) for $ \Gamma_{t} $ respectively are displayed in Figure 6. A notable feature in Figure 6 is that TFP in the benchmark model continues to be