1 Introduction and Motivation
The average labor productivity in Bulgaria in the period following the introduction of currency board (1997) is highly pro-cyclical.1 The classical explanation for this stylized fact, which is observed in many developed economies (Jorgenson and Griliches, 1967), is that the major inputs of production, labor, and capital are used more intensively during periods of expansion as compared to periods of recession. In order to quantitatively rationalize this phenomenon, and gain a deeper understanding of the transmission mechanism responsible for economic fluctuations, we introduce an endogenous utilization rate of physical capital stock into a relatively standard real business cycle model in a government sector in detail. We examine the quantitative importance of the variability in capital utilization and its relevance to generate plausible cyclical movements in aggregate variables. More specifically, we investigate whether allowing for cyclical capital utilization helps our augmented real business cycle model match the empirical business cycles in Bulgaria in the period after the introduction of the currency board arrangement. This period was chosen in our investigation due to the fact that the introduction of the hard exchange-rate peg achieved macroeconomic stability in Bulgaria, and thus the time series have good statistical properties.
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.2 In turn, there are two costs to the capital utilization decision that are at play in the current model: (i) a cost in terms of higher energy use and (ii) a cost in terms of a higher depreciation rate of physical capital stock. The first is a direct effect working through the production function, and following from fact that energy becomes a de facto factor of production through the link with utilization rate and capital stock. The second is an indirect channel, which is one of the novelties in this paper. This effect occurs due the presence of a depreciation cost of utilization and the linkage between it and the use of energy, which works indirectly through the accumulated stock of physical capital. We then use this artificial economy with endogenous capital utilization through energy use as a model in order to study the importance of energy price shocks on the main aggregate variables. In order to be able to draw plausible quantitative predictions, we calibrate the theoretical economy to approximate Bulgaria in the period 1999–2016. We find that a positive shock to energy prices is akin to a negative technological shock and propose an explanation for a technological shock.3
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.4 Instead, what takes a central stage in this paper is the fact that energy prices directly affect the productivity and the profitability of all sectors in the economy, and thus aggregate output. Overall, the model with endogenous utilization rate through the use of energy performs better than earlier real business cycle models when data for Bulgaria is considered. In particular, consistent with the observed cyclical fluctuations in Bulgaria, total hours follow output movement. Nevertheless, as with the standard RBC model, the model with endogenous utilization rate of capital falls far short of generating wage variability as in data, and the wage rate in the model is very strongly pro-cyclical, while wages are acyclical in data.
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.
2 Model Description
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.
2.1 Household’s Problem
There is a representative household, which maximizes its expected utility function, as in Finn (2000):
where E0 denotes household’s expectations at period 0, ct denotes household’s private (non-energy) consumption in period t, ht are hours worked in period t, 0 < β < 1 is the discount factor, 0 < ψ < 1 is the relative weight that the household attaches to consumption and σ > 0 is the curvature of the utility function5.
The household starts with an initial stock of physical capital k0 > 0, and has to decide how much to add to it in the form of new investment, as well as the rate at which the stock of physical capital is being utilized. As a result, in every period, physical capital depreciates at an endogenous rate, which depends on the level of utilitization rate ut chosen by the household, so that 0 < δ(ut) < 1. Following Taubman and Wilkinson (1970), Greenwood, Hercowitz, and Huffman (1988), and Finn (1995, 2000), the functional form for the endogenous depreciation rate is as follows:
where ω0 > 0, ω1 > 1. This depreciation function is consistent with Keynes’s (1936) notion of the “user cost of capital,” which argues that higher utilization causes faster depreciation, at an increasing rate, because of faster ”wear and tear” on the aggregate physical capital stock. In addition, as in Finn (2000), we assume that capital utilization requires energy, et.6 More specifically, it will be postulated that energy spending complements the service flow from physical capital as follows:
where ν0 > 0, ν1 > 1. The technical relationship function, a(.), the same as those developed by Finn (1995), postulates that energy is essential to the utilization of capital, with increases in utilization requiring more energy usage per unit of capital, at an increasing rate.7 The law of motion for physical capital is then
and the real interest rate is rt, hence the before-tax effective (utilized) physical capital income of the household
in period t equals rtutkt. In addition to capital income, the household can generate labor income. Hours supplied to the representative firm are rewarded at the hourly wage rate of wt, so pre-tax labor income equals wtht. Lastly, the household owns the firm in the economy and has a legal claim on all the firm’s profit, πt.
Next, the household’s problem can be now simplified to
where τc is the tax on consumption, τy is the proportional income tax rate (0 < τc, τy < 1), levied on both labor and capital income, pt is the relative (to the aggregate consumption price index) energy price, et denotes energy use in period t, and
where λt is the Lagrangean multiplier attached to household’s budget constraint in period t.
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.
2.2 Firm Problem
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, utkt, and labor hours, ht, to maximize static profit
where At denotes the level of technology in period t. Since the firm rents the capital from households, the problem of the firm is a sequence of static profit maximizing problems. In equilibrium, there are no profits, and each input is priced according to its marginal product, i.e.:
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.9
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.
2.4 Dynamic Competitive Equilibrium (DCE)
For a given process followed by technology
3 Data and Model Calibration
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, β= 0.982, is set to match the steady-state capital-to-output ratio in Bulgaria, k/y = 13.964, in the steady-state Euler equation. The labor share parameter, 1 − α = 0.571, is obtained as in Vasilev (2017d), and equals the average value of labor income in aggregate output over the period 1999–2016. This value is slightly higher as compared to other studies on developed economies, due to the overaccumulation of physical capital, which was part of the ideology of the totalitarian regime that was in place until 1989. Next, the average income tax rate was set to τy = 0.1. This is the average effective tax rate on income between 1999 and 2007, when Bulgaria used progressive income taxation, and equal to the proportional income tax rate introduced as of 2008. Similarly, the tax rate on consumption is set to its value over the period, τc = 0.2.
In terms of parameters characterizing the household’s preferences, the curvature of the utility function is set to σ = 2, as in Hansen and Singleton (1983). Note that this parameter does not enter steady-state computation, and only affects cyclical fluctuations. Next, the relative weight attached to the utility out of leisure in the household’s utility function, ψ, is calibrated to match that in steady-state consumers would supply one-third of their time endowment to working. This is in line with the estimates for Bulgaria (Vasilev 2017a) as well over the period studied. In sum, the steady-state depreciation rate of physical capital in Bulgaria, δ = 0.013, was taken from Vasilev (2016). It was estimated as the average quarterly depreciation rate over the period 1999–2014. In addition, the steady-state capital utilization rate is normalized to unity, thus ω0 = 0.013. The curvature parameter, ω1, does not enter the steady state, and only matters for cyclical fluctuations. As in Finn (2000), we set ω1 = 1.25. Next, the scale parameter ν0 set to average value of energy-to-physical capital ratio, e/k. Again, the curvature parameter of the energy-utilization function, ν1, does not enter the steady state, and only matters for cyclical fluctuations. As in Finn (2000), we set ν1 = 1.61.
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.
|α||0.429||Capital share||Data average|
|1 − α||0.571||Labor share||Calibrated|
|ψ||0.873||Relative weight attached to consumption||Calibrated|
|σ||2.000||Curvature parameter, utility function||Set|
|δ||0.013||Depreciation rate on physical capital||Data average|
|ω0||0.013||Scale parameter, depreciation function||Calibrated|
|ω1||1.250||Curvature parameter, depreciation function||Set|
|ν0||0.0143||Scale parameter, energy utilization function||Data average|
|ν1||1.610||Curvature parameter, energy utilization function||Set|
|τy||0.100||Average tax rate on income||Data average|
|τc||0.200||VAT/consumption tax rate||Data average|
|A||0.604||Steady-state value of TFP process||Calibarated|
|p||1.000||Steady-state energy price level||Calibrated|
|ρa||0.701||AR(1) persistence coefficient, TFP process||Estimated|
|ρp||0.980||AR(1) persistence coefficient, energy price process||Estimated|
|σa||0.044||st. error, TFP process||Estimated|
|σp||0.013||st. error, energy process||Estimated|
4 Steady State
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 A differs from 1, which is usually the normalization done in other studies), which greatly simplified the computations. Next, the model matches consumption-to-output and government purchases ratios by construction. The investment ratios are also closely approximated, despite the closed-economy assumption and the absence of foreign trade sector. The shares of income are also identical to those in data, which is an artifact of the assumptions imposed on functional form of the aggregate production function.
Data Averages and Long-run Solution
|c/y||(non-energy) Consumption-to-output ratio||0.624||0.624|
|pe/y||Energy consumption-to-output ratio||0.151||0.151|
|gt/y||Government transfers-to-output ratio||0.220||0.149|
|wh/y||Labor income-to-output ratio||0.571||0.571|
|ruk/y||Capital income-to-output ratio||0.429||0.429|
|h||Share of time spent working||0.333||0.333|
|r¯||After-tax net return on capital||0.014||0.016|
The after-tax return, where r¯ = (1 − τy )r − δ is also relatively well-captured by the model. Lastly, given the absence of debt, and the fact that transfers were chosen residually to balance the government budget constraint, the result along this dimension is understandably not so close to the average ratio in data.
5 Out of Steady-State Model Dynamics
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.
5.1 Impulse Response Analysis
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.10 The impulse response functions (IRFs) are presented in Fig. 1 and 2.
5.1.1 Impulse Responses to Technology Shocks
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.
5.1.2 Impulse Responses to Unanticipated Energy Prices
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.
5.2 Simulation and Moment-Matching
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.11 To minimize the sample error, the simulated moments are averaged out over the computer-generated draws. As in Vasilev (2016, 2017b, 2017c), the model matches quite well the absolute volatility of output and investment. By construction, government consumption in the model varies as much as output. However, the model in this paper underestimates the variability in consumption, due to the presence of energy consumption, which
Business Cycle Moments
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, e.g. Vasilev (2009), does not describe very well the dynamics of labor market variables. In addition, as in Vasilev (2017b, 2017c), the model fails in matching unemployment volatility, which in this model varies as much as the employment rate.12 Next, in terms of contemporaneous correlations, the model systematically over-predicts the pro-cyclicality of the main aggregate variables such as consumption, investment, and government consumption.
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.
5.3 Auto- and Cross-Correlation
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
|Data||corr(ut, ut − k)||1.000||0.765||0.552||0.553|
|Model||corr(ut, ut − k)||1.000||0.955||0.901||0.837|
|Data||corr(nt, nt − k)||1.000||0.484||0.009||0.352|
|Model||corr(nt, nt − k)||1.000||0.955||0.901||0.837|
|Data||corr(yt, yt − k)||1.000||0.810||0.663||0.479|
|Model||corr(yt, yt − k)||1.000||0.955||0.901||0.836|
|Data||corr(at, at − k)||1.000||0.702||0.449||0.277|
|Model||corr(at, at − k)||1.000||0.955||0.900||0.836|
|Data||corr(ct, ct − k)||1.000||0.971||0.952||0.913|
|Model||corr(ct, ct − k)||1.000||0.955||0.903||0.845|
|Data||corr(it, it − k)||1.000||0.810||0.722||0.594|
|Model||corr(it, it − k)||1.000||0.954||0.901||0.841|
|Data||corr(wt, wt − k)||1.000||0.760||0.783||0.554|
|Model||corr(wt, wt − k)||1.000||0.920||0.900||0.836|
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.
Dynamic correlations for Bulgarian data and the model economy
|Data||corr(nt, (y/n)t − k)||-0.342||-0.363||-0.187||-0.144||0.475||0.470||0.346|
|Model||corr(nt, (y/n)t − k)||0.123||0.195||0.292||0.918||0.288||0.221||0.171|
|Data||corr(nt, wt − k)||0.355||0.452||0.447||0.328||-0.040||-0.390||-0.57|
|Model||corr(nt, wt − k)||0.123||0.195||0.292||0.918||0.288||0.221||0.171|
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).
Baxter Marianne and Robert King. 1993. ”Fiscal policy in general equilibrium” American Economic Review 83: 315–334.
Bulgarian National Bank. 2017. Bulgarian National Bank Statistics. Available on-line at www.bnb.bg Accessed on Oct. 21 2017.
Bils Mark and Jang-Ok Cho. 1994. ”Cyclical Factor Utilization” Journal of Monetary Economics 33(2): 319–54.
Canova Fabio. 2007. Methods for Applied Macroeconomic Research Princeton University Press: Princeton NJ.
Cogley Timothy and James Nason. 1995. ”Output dynamics in Real-Business-Cycles.”
American Economic Review 85(3): 492–511.
Finn Mary. 2000. ”Perfect Competition and the Effects of Energy Price Increases on Economic Activity” Journal of Money Credit and Banking 32(3): 400–416.
Finn Mary. 1995. ”Variance Properties of Solow’s Productivity Residual and Their Cyclical Implications” Journal of Economic Dynamics and Control 19: 1249–81.
Greenwood Jeremy Zvi Hercowitz and Gregg Huffman. 1988. ”Investment Capacity Utilization and the Real Business Cycle” American Economic Review 78: 402–17.
Hamilton James 1983. ”Oil and the Macroeconomy since World War II” Journal of Political Economy 91: 228–48.
Hamilton James. 1985. ”Historical Causes of Postwar Oil Shocks and Recessions” The Energy Journal 6: 97–116.
Hamilton James. 1996. ”This Is What Happened to the Oil Price-Macroeconomy Relationship” Journal of Monetary Economics 38: 215–20.
Hansen Lars and Kenneth Singleton. 1983. ”Stochastic Consumption Risk Aversion and the Temporal Behavior of Asset Returns” Journal of Political Economy 91: 249–65.
Hodrick Robert and Edward Prescott. 1980. ”Post-war US business cycles: An empirical investigation.” Unpublished manuscript (Carnegie-Mellon University Pittsburgh PA).
Jorgenson Dale and Zvi Griliches. 1967. ”The Explanation of Productivity Change” Review of Economic Studies 34: 247–83.
Keynes John. 1936. The General Theory of Employment Interest and Money McMillan: London UK.
Kydland Fynn and Edward Prescott. 1988. ”The Workweek of Capital and Its Cyclical Implications” Journal of Monetary Economics 21: 343–60.
National Statistical Institute. 2017. Aggregate Statistical Indicators. Available on-line at www.nsi.bg Accessed on Oct. 21 2017.
Nelson Charles and Charles Plosser. 1982. ”Trends and Random Walks in Macroeconomic Time Series” Journal of Monetary Economics 10(2): 139–62.
Rotemberg Julio and Michael Woodford. 1996a. ”Imperfect Competition and the Effects of Energy Price Increases on Economic Activity” Journal of Money Credit and Banking 28: 549–77.
Rotemberg Julio and Michael Woodford. 1996b. ”Real-Business-Cycle Models and the Forecastable Movements in Output Hours and Consumption” American Economic Review 86: 71–89.
Taubman Paul and Maurice Wilkinson. 1970. ”User Cost Capacity Utilization and Investment Theory” International Economic Review 11: 209–15.
Vasilev Aleksandar. 2017a. ”Business Cycle Accounting: Bulgaria after the introduction of the currency board arrangement (1999–2014) European Journal of Comparative Economics 14(2): 197–219.
Vasilev Aleksandar. 2017b. ”A Real-Business-Cycle model with efficiency wages and a government sector: the case of Bulgaria” Central European Journal of Economics and Econometrics 9(4): 359–377.
Vasilev Aleksandar. 2017c. ”A Real-Business-Cycle model with reciprocity in labor relations and fiscal policy: the case of Bulgaria” Bulgarian Economic Papers BEP 03-2017 Center for Economic Theories and Policies Sofia University St. Kliment Ohridski Faculty of Economics and Business Administration Sofia Bulgaria.
Vasilev Aleksandar. 2017d. ”VAT Evasion in Bulgaria: A General-Equilibrium Approach.” Review of Economics and Institutions 8(2): 2–17.
Vasilev Aleksandar. 2016. ”Search and matching frictions and business cycle fluctuations in Bulgaria” Bulgarian Economic Papers BEP 03-2016 Center for Economic Theories and Policies Sofia University St. Kliment Ohridski Faculty of Economics and Business Administration Sofia Bulgaria.
Vasilev Aleksandar. 2015a. ”Welfare effects of at income tax reform: the case of Bulgaria” Eastern European Economics 53(2): 205–220.
Vasilev Aleksandar. 2015b. ”Welfare gains from the adoption of proportional taxation in a general-equilibrium model with a grey economy: the case of Bulgaria’s 2008 flat tax reform” Economic Change and Restructuring 48(2): 169–185.
Vasilev Aleksandar. 2009. ”Business cycles in Bulgaria and the Baltic countries: an RBC approach” International Journal of Computational Economics and Econometrics 1(2): 148–170.