Using Grey Production Functions in the Macroeconomic Modelling: An Empirical Application for Romania

The work is a development of our earlier studies containing empirical application of models with representative agent. The extensions developed in this paper consist of the following: the introduction of the labor market via the use of labor as the second production factor, the use of the GM(1,1) algorithm in order to adjust the capital and labor data series and to compute grey Cobb-Douglas production function, and finally the comparison of the results obtained applying the model to the actual data and the grey data. The grey production function is estimated using GM(1,1) adjusted statistical series of the GDP, capital stock and labor data. For the two vari-ants we computed the predictions of the indicators: real GDP, consumption, government ex-penditures, trade balance, and burden of debt.


Introduction
In the macroeconomic theory, "representative agent" model is a mean to provide the aggregate behavior of the economy based on microeconomic optimization of the agents.Macroeconomic behavioral relationships are derived using inter-temporal optimization of microeconomic agents.This approach belongs to the New Classical Macroeconomic School that has great success around 1970s.Nevertheless this approach can generate some controversial aspects, such that the aggregation and the Pareto consistency problems of the representative agents problems [1].Another aspect to be controversy is the fact that most of the literature employing representative agents framework is deterministic, stochastic intertemporal optimization is difficult and often very formal.Hansen and Sargent [2] formulated a justification of the use of these models that consist of possible consistency between their solutions with the solutions of the aggregated optimal decision models and of the general equilibrium models.The analysis made by Hartley in [3] leads to the conclusion that the representative agent models cannot describe adequately the macroeconomic aggregate behavior.In [4] Kirman holds back against the representative agent, since the choices of the agents are not homogeneous and their reaction to changes are not the same.Other controversial issues on the representative agent can be found in ( [5], [6]).This research fields is still subject to research: [7], [8].Recently Krueger [9] extends the representative agent approach building economic cyclicality models with agents of the same type (households and firms) and uses them in business cycle analysis.A possible alternatives to the approach of representative agent in economics would be agent-based simulation models.In agentbased computational economics, an agent is a physical or virtual entity [10] which acts in a specified environment, communicates with other agents, follows a set of trends representing goals or optimizes a function, etc. Agent-based models are based on autonomous agents which interact with one another according to a set of rules of behavior [11].The applications lie in a large area such as: modeling agent behavior in the stock markets and supply chains, modeling consumer behavior, predictions etc. [12].DOI: 10.12948/issn14531305/18.4.2014.14 In many situations, the assumption of a representative household is not appropriate according to Acemoglu [13] because households do not have an infinite planning horizon and because new households arrive (or are born) over time.Decisions made by older generations will affect the prices faced by younger generations, so, besides the neoclassical growth model, the "overlapping generations' model" is the second great result of research into modern macroeconomics.We use this type of modes due to the facile application on statistical data in order to compute some macroeconomic indicators and to forecast them [14], [15], [16].We organized the paper in six sections beginning with the introduction, continuing with general formulation of the model, solving the model with actual data and grey data, empirical application and the comparison of the results of the two approaches, and final conclusions and further topics.

Open Economy Extended With Labor Market and Investments
We extend the analysis made in our previous paper [14], considering two production factors, labor and capital, so that the technology follows a neoclassic production function: ), , ( with Ls the labor at the time s, s K is the physical capital at the time s, and F(Ks,Ls) has decreasing returns to scale.Assume that the representative firm behaves competitively and it is owned entirely by domestic residents.The consumer's problem consists of the maximization of the sum of the present discounted values of the utilities, on the infinite periods of time: subject to the budget constraint (Honkapoja [17]): where ws are the wages the firm hires the labor at period s; Vs is the price of the claim to the firm's entire profits from the period (s+1) onwards; xs+1 is the share of the domestic firm in the private propriety of the consumer at the end of the date s; ds is the dividend the firm at the end of the date s; Bs are the net foreign assets in period s; Gs denotes the government consumption, period s; Cs is the consumption in period s; r is the world interest rate.The first order conditions are given by: a) Euler equation: r, is the international interest rate, assuming this is constant in time.The firm's market value on date t is the present discounted value of the dividends it will pay to shareholders over the future, starting the date (t+1).For this reason, we will sometimes call Vt the firm's ex dividend market value on date t.
provided that: This condition often is called "no-bubbles" condition.
From (5) we obtain: DOI: 10.12948/issn14531305/18.4.2014.14 At optimum, the second term must be 0, so:  we obtain another form of the condition "without bubbles".Considering =constant, and applying the sum of the infinite geometric series in (10), we obtain: In order to introduce the firm's behavior, consider the firm's dividends as output less salaries and investments: with , the investments function.
The firm maximizes the current and future dividends: where ws is the wage per person in the period s.
The first optimum conditions are: The above conditions have the well-known significance from microeconomics: the equilibrium of the firm is obtained as long as the marginal product equals the marginal cost of production factors.
The current account equals national savings less investments, so, for our model, the equation of current account is: The economy's trade balance is: is the proportion of debt to GDP The current account deficits are sustainable if they will satisfy the relation ( [17]):

The Analytic Solution of the Model Using Logarithmic Utility Function and Cobb-Douglas Production Function
We considered a macroeconomic utility function of the form: and two factor Cobb-Douglas production function, ) with α being the elasticity of output with respect to physical capital stock,  the elasticity of output with respect to labor.
gives the dynamics of technological progress, with the solution: In (23) we denote the value of the individual's financial wealth at the end of period s.The first order conditions for the consumer will be: (24) that is a difference homogenous first order equation with the solution:   From ( 25) and ( 26) we obtain: which is a first order nonhomogeneous difference equation, that could be solved recursively, knowing and the sequence of the dividends: , computed in the firm's problem conforming to .

Firm's problem:
Let us write the dividends as: The firm will maximize the sum of the dividends: The first order conditions for the firm are: 30) and (31), we will determine the values of , where  is the growth rate of the government expenditures, a political decision parameter and 0 G is the statistical value for the period 0. The trade balance and the burden of the debt in total GDP are computed using ( 17) and (18).

Grey Two Factors Cobb-Douglas Production Function
Grey system theory is a method initiated in 1982 by Julong Deng [18] in order to study the models with small databases and partial known information.The systems with lack information are called Grey Systems.In the case of completely unknown information we talk about Black systems, and in the case of completely known information we talk about White systems.Due to the frequent problems in the macroeconomic data that could lead to errors of estimation, such as high volatility, incompleteness of the series, we use GM ( will stand for the GM(1,1) simulated vectors of K Y , .This way grey two factor Cobb-Douglas production function is obtained.Applying the OLS method, the coefficients of the Cobb-Douglas grey function were estimated, on the GM(1,1) approximated data.

GM(1,1) Grey Model
The GM(1,1) model construction process is described below , according to [19], [20], [21].Let us consider the sequence ) 0 ( X is obtained by applying the accumulated generating operation to ) 0 ( X : )) ( ),... where: It is obvious that Z is defined as: where: The following equation for is written as: X at time k has the following form:

The Classic Approach
Because the employment in Romania decreased due to lower birth rates and increased labor migration, we approximated labor input using the method proposed by Altar et al. [22]: N stands for the HP filtered of the active civil population, q is the HP filtered participation The estimators obtained are: . 01357 , We consider the estimators acceptable, because they verify the significance level test, and R square=0.933621461reveal that the model replicate well the economic phenomena.The degree of homogeneity is given by the sum of the two elasticity: 1 6492 , 0      so, multiplying with 0   the two production factors, the production will increase with 6492 , 0  , 01357 , 0  g the total productivity growth rate, is small, which reveals that in terms of technical progress implemented in production, there are great resources in Romania.All the parameters are economically acceptable.We considered r= 0.034, the average world interest rate (HomeFinance.nl,2013), and  =0.968 as the consumer's discount factor, computed as the inverse of the expected inflation index that follows an adaptive expectations mechanism.Unfortunately, the delayed effects of the global economic crisis that were appointed "hysteresis", is present also in the economic processes in Romania, which is manifested by a very slow recovery process.The predicted values of the macroeconomic indicators have positive, but reduced values of the growth rates.The reduction of the capital accumulation and the reduction of potential GDP took place.The last year of the forecasting reveals the increase of the growth rate that will have an average of 4.12% in 2018, higher than the capital growth rate marking a predominantly intensive character of the Romanian economy.We can see that the debt/GDP ratio is decreasing that will result in the reduction of the current account deficits assuring further government solvability.Due to ample variations of the gross investments data, we applied the HP filter to the gross investments data separating the trend that was used as input data to compute the production function parameters.We identified also the cyclical component of that data series, the result could be shown in the figure 5.The prediction of gross investments using representative agent model show an increase in gross investments between 2014Q3-2018Q4 with 3,67%, that is lower than predicted GDP growth rate, that is the capital /output ratio, growth rate is decreasing, meaning the growing productivity of capital.

The Grey Approach
Using quarterly data for real GDP, for the real gross capital formation, and for the working hours of the employed population, between 2000Q1 and 2014Q2, we determined the grey data series, following GM(1,1) grey algorithm.Constructing the indexes series, and applying the same logarithm as in the preceding case, we estimated the parameters of the CD production function, obtaining the grey production function: , and a productivity growth rate g=0,11379 higher than in the case of the previous estimation.
In order to apply the model and to compute the macroeconomic indicators, we used the same average world interest rate r and discount factor , as in the case of the model based on actual data.The computed the real GDP data series using estimated parameters, are compared with the actual data series (Figure 7), and then is computed a 5-year prediction DOI: 10.12948/issn14531305/18.4.2014.14The government expenditures were simulated using the same growth rate and the initial value, as in the case of the model applied to the actual data.
The intensive character of the economy is even proved by the growth rate of the grey gross investments of 3.12% that is lower than the grey GDP growth rate of 3.987%.We mention that the predicted grey real gross capital formation has a reduced growth rate than in the case of the prediction based on the actual data, this having the effect of a lower growth rate than the ratio capital / GDP and so a productivity growth rate, even higher.

Conclusions
In the empirical work we used quarterly data for GDP, gross capital formation and active civil population expressed as hours worked, with the algorithm presented above.
Due to important fluctuations in the original data representing the gross capital formation, we applied the HP filter to the original data, DOI: 10.12948/issn14531305/18.4.2014.14 and remarked a high volatility of the data, sized between -4000 and 4000 monetary units, the limits values being reached especially during 2008-2010 during the economic crisis in our country.Due to the increase of the gross investments growth rate, 2014Q3-2018Q4 (3.67%), the productivity of the capital grew in that period.
The absence of the adequate data for the stock of the capital for Romania, determined us to use GM(1,1) adjustment of the data and grey production function approach.
Regarding the two approaches, we observe some common features: -the increasing growth path of GDP, consumption, investments, with different growth rates and a decreasing rate of the burden of debt; -predicted annual GDP growth rate is higher than gross investments one, that proves a rise of the productivity of the capital and accordingly to a decreasing capital output rate; -the results indicate a slow increase in the growth rate of output after the third quarter of 2010, that followed the decline in 2008Q4 and 2010Q2.For the period 2004Q2-2007Q3, the main driving forces in the GDP growth were the technical progress, and the contribution of the physical capital.