The average labor productivity in Bulgaria in the period following the introduction of currency board (1997) is highly pro-cyclical. This stylized fact is similar to the finding documented in Bils and Cho (1994) for the United States as well.
The other novelty in this paper is the particular way capital utilization enters the model. We follow Finn (2000) by adopting the empirical regularity that capital utilization requires energy and argue for the importance of energy in the transmission of technological shocks. Earlier studies, The novelty, however, is that the transmission mechanism of energy price shocks stems from a relatively little explored relationship between energy usage and services provided by physical capital, described in this paper. Put differently, since energy enters the production function only because it is essential to the utilization of capital, the endogenous variations in utilization and energy use would be inter-related.
It comes as no surprise that unexpected changes in world energy prices are very important for an energy-intensive production in Bulgaria, a small open economy. Energy price hikes or drops can have important real effects on the Bulgarian economy as Bulgaria imports most of its energy inputs (oil and natural gas in particular) from the Russian Federation. Next, from the perspective of the Bulgarian economy, the price of the aggregate energy input is taken as given. Thus, the industry structure of the energy production is not of central importance and will be ignored in this paper. More specifically, we abstract away from the issue, as it is of limited relevance for the international transmission of how changes in the price of imported energy inputs affect the Bulgarian economy. In another line of research, Rotemberg and Woodford (1996a) present a theory based on imperfect competition in the oil market to explain business cycle fluctuations in the US economy. Hamilton (1983, 1985, 1996) studies the effect of oil price on real output in the US.
The rest of the paper is organized as follows. Section 2 describes the model framework and describes the decentralized competitive equilibrium system. Section 3 discusses the calibration procedure, and Section 4 presents the steady-state model solution. Sections 5 proceeds with the out-of-steady-state dynamics of model variables, and compares the simulated second moments of theoretical variables against their empirical counterparts. Section 6 concludes the paper.
There is a representative household that derives utility out of consumption and leisure. The time available to households can be spent in productive use or as leisure. In addition, the household chooses optimally the rate at which capital stock is being utilized. The government taxes consumption spending and levies a common tax on all income, in order to finance wasteful purchases of government consumption goods, and government transfers. On the production side, there is a representative firm, which hires labor and utilized capital to produce a homogenous final good, which could be used for consumption, investment, government purchases, or energy consumption. Depreciation rate is endogenous and is a function of the endogenous capital utilization rate, and depends on the energy use.
There is a representative household, which maximizes its expected utility function, as in Finn (2000):
where This utility function is equivalent to a specification with a separable term containing government consumption, e.g. Baxter and King (1993).Since in this paper we focus on the exogenous (observed) policies, and the household takes government spending as given, the presence of such a term is irrelevant. For the sake of brevity, we skip this term in the utility representation above.
The household starts with an initial stock of physical capital
where This channel is missing from earlier studies, such as Taubman and Wilkinson (1970), Greenwood, Hercowitz, and Huffman (1988), and is one of the novelties of this paper.
where This modelling choice could be traced back to Jorgen and Grilliches (1967), who find that capital and electricity are complements in production. In addition, after some algebra, one can show that
and the real interest rate is
in period
Next, the household’s problem can be now simplified to
s.t.
where Note that by choosing
where λ
The interpretation of the first-order conditions above is as follows: the first one states that for each household, the marginal utility of consumption equals the marginal utility of wealth, corrected for the consumption tax rate. The second equation states that when choosing labor supply optimally, at the margin, each hour spent by the household working for the firm should balance the benefit from doing so in terms of additional income generated, and the cost measured in terms of lower utility of leisure. The third equation describes the optimal utilization rate, which requires that the change in the depreciation rate, or the marginal cost in terms of an increased depreciation rate resulting from utilizing capital at a higher rate, together with the marginal cost in terms of additional energy used in the capital utilization, equal the after tax return on utilized capital. In other words, the marginal benefit resulting from physical capital services should balance with the user cost of capital at the margin. The fourth equation is the so-called Euler condition, which describes how the household chooses to allocate physical capital over time. The last condition is called the “transversality condition” (TVC): it states that at the end of the horizon, the value of physical capital should be zero.
There is a representative firm in the economy, which produces a homogeneous product. The price of output is normalized to unity. The production technology is Cobb-Douglas and uses both utilized (effective) physical capital,
where
In addition, using the link between energy, capital, and utilization we can express output as follows:
The equation specifies output as a function of labor, capital, and energy, showing the direct effect of energy on output. Note that if the depreciation rate is held constant, then the transmission of energy price shocks is restrained only to the effect of energy input on output through the production function channel. However, when depreciation rate is endogenous and depends on the utilization of physical capital, and then in turn through it on the use of energy, then energy has an additional indirecteffect on output, which operates through the capital stock. As we show later in the paper, the combination of those direct and indirect effects produces important difference in the dynamics of model variables
In the model setup, the government is levying taxes on labor and capital income, as well as consumption, in order to finance spending on wasteful government purchases, and government transfers. The government budget constraint is as follows:
Tax rates and government consumption-to-output ratio would be chosen to match the average share in data, and government transfers would be determined residually in each period so that the government budget is always balanced.
For a given process followed by technology
To characterize business cycle fluctuations with an endogenous depreciation rate in Bulgaria, we will focus on the period following the introduction of the currency board (1999–2016). Quarterly data on output, consumption, and investment was collected from National Statistical Institute (2017), while the real interest rate is taken from Bulgarian National Bank Statistical Database (2017). The calibration strategy described in this section follows a long-established tradition in modern macroeconomics: first, as in Vasilev (2016), the discount factor,
In terms of parameters characterizing the household’s preferences, the curvature of the utility function is set to
Finally, the processes followed by TFP processes and energy prices are estimated from the detrended series by running an AR(1) regression and saving the residuals. Tab. 1 summarizes the values of all model parameters used in the paper.
Model Parameters
Parameter | Value | Description | Method |
---|---|---|---|
0.982 | Discount factor | Calibrated | |
0.429 | Capital share | Data average | |
1 | 0.571 | Labor share | Calibrated |
0.873 | Relative weight attached to consumption | Calibrated | |
2.000 | Curvature parameter, utility function | Set | |
0.013 | Depreciation rate on physical capital | Data average | |
0.013 | Scale parameter, depreciation function | Calibrated | |
1.250 | Curvature parameter, depreciation function | Set | |
0.0143 | Scale parameter, energy utilization function | Data average | |
1.610 | Curvature parameter, energy utilization function | Set | |
0.100 | Average tax rate on income | Data average | |
0.200 | VAT/consumption tax rate | Data average | |
0.604 | Steady-state value of TFP process | Calibarated | |
1.000 | Steady-state energy price level | Calibrated | |
0.701 | AR(1) persistence coefficient, TFP process | Estimated | |
0.980 | AR(1) persistence coefficient, energy price process | Estimated | |
0.044 | st. error, TFP process | Estimated | |
σ | 0.013 | st. error, energy process | Estimated |
Once the values of model parameters were obtained, the steady-state equilibrium system solved, the “big ratios” can be compared to their averages in Bulgarian data. The results are reported in Tab. 2. The steady-state level of output was normalized to unity (hence the level of technology
Data Averages and Long-run Solution
Variable | Description | Data | Model |
---|---|---|---|
Steady-state output | N/A | 1.000 | |
(non-energy) Consumption-to-output ratio | 0.624 | 0.624 | |
Investment-to-output ratio | 0.201 | 0.175 | |
Energy consumption-to-output ratio | 0.151 | 0.151 | |
Government transfers-to-output ratio | 0.220 | 0.149 | |
Labor income-to-output ratio | 0.571 | 0.571 | |
Capital income-to-output ratio | 0.429 | 0.429 | |
Share of time spent working | 0.333 | 0.333 | |
After-tax net return on capital | 0.014 | 0.016 |
The after-tax return, where
Since the model does not have an analytical solution for the equilibrium behavior of variables outside their steady-state values, we need to solve the model numerically. This is done by log-linearizing the original equilibrium (non-linear) system of equations around the steady state. This transformation produces a first-order system of stochastic difference equations. First, we study the dynamic behavior of model variables to an isolated shock to the total factor productivity process, and then we fully simulate the model to compare how the second moments of the model perform when compared against their empirical counterparts.
This subsection documents the impulse responses of model variables to a 1% surprise innovation to technology, as well as an unexpected 1% change in energy prices. This price is to be interpreter as an aggregate category, comprosing electricity, coal, natural gas, and petroleum.
As a result of the one-time unexpected positive shock to total factor productivity, output increases upon impact. This expands the availability of resources in the economy, so the use of output consumption, investment, energy use and government consumption also increase contemporaneously. At the same time, the increase in productivity increases the after-tax return on the two factors of production, labor, and capital. The representative households then respond to the incentives contained in prices and start accumulating capital, and supplies more hours worked. In turn, the increase in capital input feeds back in output through the production function and that further adds to the positive effect of the technology shock. Lastly, the utilization rate increases as well, following the increase in the return on capital, but this also increases the endogenous depreciation rate. In the labor market, the wage rate increases, and the household increases its hours worked. In turn, the increase in total hours further increases output, again indirectly.
Over time, as capital is being accumulated, its after-tax marginal product starts to decrease, which lowers the households’ incentives to save. As a result, physical capital stock eventually returns to its steady-state and exhibits a hump-shaped dynamics over its transition path. The rest of the model variables return to their old steady states in a monotone fashion as the effect of the one-time surprise innovation in technology dies out.
As a result of an unexpected one-time increase in the price of the aggregate energy input, illustrated in Fig. 2, the consumption of energy decreases, while its substitute, the non-energy private consumption, increases. Due to the relative scarcity of energy, illustrated in the increased valuation of energy, capital utilization rate increases. In turn, the time-varying endogenous depreciation rate increases as well, which in turn decreases capital accumulation and investment. As a result of lower capital availability, real interest rate goes up.
Next, from the complementarity of capital and labor in the Cobb-Douglas production function, hours fall as well, and the wage rate in the economy increases. Interestingly, aggregate output falls as well upon impact of the energy price shock, so an increase in energy prices is akin to a negative productivity shock, as energy could be expressed as a direct input in the production function. Next, as government consumption and transfers follow private out-put, both fall as well. Over time, all variables return to their steady state, but the negative effects from one-time unexpected increases in energy prices has a long-term negative effect on the economy.
As in Vasilev (2017b), we will now simulate the model 10,000 times for the length of the data horizon. Both empirical and model simulated data are detrended using the Hodrick-Prescott (1980) filter. Tab. 3 summarizes the second moments of data (relative volatilities to output, and contemporaneous correlations with output) versus the same moments computed from the model-simulated data at quarterly frequency. The model-predicted 95% confidence intervals are available request.
Business Cycle Moments
σ | Data | Model |
---|---|---|
σ | 0.55 | 0.14 |
σ | 1.77 | 1.97 |
σ | 1.21 | 1.00 |
σ | 0.63 | 0.63 |
σ | 0.83 | 0.39 |
σ | 0.86 | 0.39 |
σ | 3.22 | 0.63 |
σ | 1.32 | 1.61 |
0.85 | 0.47 | |
0.61 | 0.75 | |
0.31 | 1.00 | |
0.49 | 0.96 | |
-0.01 | 0.97 | |
-0.47 | -0.96 | |
-0.14 | 0.92 |
acts as a substitute for non-energy consumption. Still, the model is qualitatively consistent with the stylized fact that consumption generally varies less than output, while investment is more volatile than output.
With respect to the labor market variables, the variability of employment predicted by the model is almost identical to that in data, but the variability of wages in the model is much lower than that in data. This is yet another confirmation that the perfectly competitive assumption, The reason behind this mismatch could be driven by several possible explanatory factors: the fact that the model misses the “out-of the-labor-force” segment, as well as the significant emigration to the older EU member states.
This, however, is a common limitation of this class of models. However, along the labor market dimension, the contemporaneous correlation of employment with output, and unemployment with output, is relatively well-matched. With respect to wages, the model predicts strong cyclicality, while wages in data are acyclical. This shortcoming is well-known in the literature and an artifact of the wage being equal to the labor productivity in the model.
In the next subsection, as in Vasilev (2016), we investigate the dynamic correlation between labor market variables at different leads and lags, thus evaluating how well the model matches the phase dynamics among variables. In addition, the autocorrelation functions (ACFs) of empirical data, obtained from an unrestricted VAR(1) are put under scrutiny and compared and contrasted to the simulated counterparts generated from the model.
This subsection discusses the auto-(ACFs) and cross-correlation functions (CCFs) of the major model variables. The coefficients such as empirical ACFs and CCFs at different leads and lags are presented in Tab. 4 against the averaged simulated AFCs and CCFs. Following Canova (2007), this is used as a goodness-of-fit measure.
Autocorrelations for Bulgarian data and the model economy
k | |||||
---|---|---|---|---|---|
Method | Statistic | 0 | 1 | 2 | 3 |
1.000 | 0.765 | 0.552 | 0.553 | ||
1.000 | 0.955 | 0.901 | 0.837 | ||
(s.e.) | (0.000) | (0.027) | (0.051) | (0.073) | |
1.000 | 0.484 | 0.009 | 0.352 | ||
1.000 | 0.955 | 0.901 | 0.837 | ||
(s.e.) | (0.000) | (0.027) | (0.051) | (0.074) | |
1.000 | 0.810 | 0.663 | 0.479 | ||
1.000 | 0.955 | 0.901 | 0.836 | ||
(s.e.) | (0.000) | (0.027) | (0.050) | (0.073) | |
1.000 | 0.702 | 0.449 | 0.277 | ||
1.000 | 0.955 | 0.900 | 0.836 | ||
(s.e.) | (0.000) | (0.026) | (0.050) | (0.072) | |
1.000 | 0.971 | 0.952 | 0.913 | ||
1.000 | 0.955 | 0.903 | 0.845 | ||
(s.e.) | (0.000) | (0.026) | (0.050) | (0.073) | |
1.000 | 0.810 | 0.722 | 0.594 | ||
1.000 | 0.954 | 0.901 | 0.841 | ||
(s.e.) | (0.000) | (0.026) | (0.050) | (0.073) | |
1.000 | 0.760 | 0.783 | 0.554 | ||
1.000 | 0.920 | 0.900 | 0.836 | ||
(s.e.) | (0.000) | (0.026) | (0.050) | (0.073) |
As seen from Tab. 4, the model compares relatively well vis-a-vis data. Empirical ACFs for output and investment are slightly outside the confidence band predicted by the model, while the ACFs for total factor productivity and household consumption are well-approximated by the model. The persistence of labor market variables are also relatively well-described by the model dynamics. Overall, the model with energy-utilization channel generates too much persistence in output and both employment and unemployment and is subject to the criticism in Nelson and Plosser (1992), Cogley and Nason (1995) and Rotemberg and Woodford (1996b), who argue that the RBC class of models do not have a strong internal propagation mechanism besides the strong persistence in the TFP process. In those models, e.g. Vasilev (2009), and in the current one, labor market is modelled in the Walrasian market-clearing spirit, and output and unemployment persistence is low.
Next, as seen from Tab. 5, over the business cycle, increase in data labor productivity leads to employment. The model, however, cannot account for this fact. As in the standard RBC model a technology shock can be regarded as a factor shifting the labor demand curve, while holding the labor supply curve constant. Therefore, the effect between employment and labor productivity is only a contemporaneous one.
Dynamic correlations for Bulgarian data and the model economy
k | ||||||||
---|---|---|---|---|---|---|---|---|
Method | Statistic | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
-0.342 | -0.363 | -0.187 | -0.144 | 0.475 | 0.470 | 0.346 | ||
0.123 | 0.195 | 0.292 | 0.918 | 0.288 | 0.221 | 0.171 | ||
(s.e.) | (0.320) | (0.286) | (0.250) | (0.146) | (0.243) | (0.281) | (0.317) | |
0.355 | 0.452 | 0.447 | 0.328 | -0.040 | -0.390 | -0.57 | ||
0.123 | 0.195 | 0.292 | 0.918 | 0.288 | 0.221 | 0.171 | ||
(s.e.) | (0.320) | (0.286) | (0.250) | (0.146) | (0.243) | (0.281) | (0.317) |
We introduce a pro-cyclical endogenous utilization rate of physical capital stock into a real business cycle model augmented with a detailed government sector. We calibrate the model to Bulgarian data for the period following the introduction of the currency board arrangement (1999–2016). We investigate the quantitative importance of the endogenous depreciation rate, and the capital utilization mechanism working through the use of energy for cyclical fluctuations in Bulgaria. In particular, a positive shock to energy prices in the model works like a negative technological shock. Allowing for variations in factor utilization and the presence of energy as a factor of production improves the model performance against data, and in addition this extended setup dominates the standard RBC model framework with constant depreciation and a fixed utilization rate of physical capital (e.g., Vasilev, 2009).
Still, the failure of the model along the labor market dimension – the high pro-cyclicality of wages and the low variability of the price of labor relative to that observed in data – suggest that the setup should depart from the perfectly competitive paradigm. As a suggestion for future research, the model should focus on the important frictions in the labor market, which forms almost two-thirds of total income (and much quantitatively much more important than the share of capital and energy), and extend the model along the lines of Vasilev (2016, 2017b, 2017c).