How Taxes and Spending on Education Influence Economic Growth in Poland

This paper investigates the relationship between economic growth in Poland and four types of taxes and human capital investment. We primarily rely on an exogenous growth model that merges the Mankiw-Romer-Weil model, augmented with learning-by-doing and spillover-effects, with selected elements from the literature on optimal taxation. We demonstrate that in the period 2000-2011, economic growth in Poland was primarily due to a rapid increase in the human capital stock (at a rate of 5% per annum) and only secondarily due to the accumulation of productive capital (2.7% annually). Simulations of tax cuts suggest that income taxes and consumption taxes restrict economic growth equally heavily. Simultaneously reducing all tax rates by 5 percentage points (pp) in Poland should increase annual GDP growth by approximately 0.4 pp. Increasing spending on education by 1 pp of GDP would increase the growth rate by approximately 0.3 pp.


Introduction
The standard approach in modern growth theory is to describe the savings and consumption decisions of households as an intertemporal optimization problem.
However, in our view, in the case of Central and Eastern European (CEE) countries, the calibration (or estimation) of such models would be difficult for several reasons. First, to the best of our knowledge, there are no reliable empirical estimates of the parameters of the intertemporal utility function for most CEE countries.
Second, optimal control models assume that economic agents are consistently optimizing, adjusting control ('jump') variables (e.g., savings and consumption) in response to policy changes. In our view, it would be overly optimistic (unjustified) to assume that CEE economies are already in this type of equilibrium.
These countries remain in transition from centrally planned, Eastern-oriented economies to market-based economies integrated with the West (the EU). Moreover, over the last 20 years, the CEE economies have experienced intense structural changes coupled with significant changes in economic policies. Furthermore, external conditions have also rapidly evolved, with the expansion of the EU in 2004 arguably representing the greatest (revolutionary) change.
For the above reasons, our analysis is deliberately based on a simple exogenous growth model, with numerous elements borrowed from the Mankiw-Romer-Weil (1992) growth model. For example, we incorporate the power production function with constant economies of scale and exogenous rates of investment and savings. We

The private economy
The aggregate output of the country is described by the following production function: where K denotes the stock of physical capital, H represents the stock of human capital, and L is raw labor. We assume positive externalities related to learning-bydoing and spillover-effects; see, e.g., Romer (1986) and Barro and Sala-i-Martin (2004). These externalities are reflected in the labor-augmenting technology index E, which is proportional to the capital per worker ratio, i.e., . Thus, the production function can be written as where . Therefore, aggregate output in the economy is described by a Cobb-Douglas function with constant returns to scale for both types of capital (physical and human). The assumption of constant returns to scale is supported by strong empirical evidence. See, e.g., (Balisteri, McDaniel, & Wong, 2003;Cichy, 2008;Mankiw, Romer, & Weil, 1992;Manuelli & Seshadri, 2005;Próchniak, 2013;Willman, 2002). Nevertheless, we note that by considering increasing or decreasing returns to scale, our analysis could lead to different conclusions.
We assume that the labor supply in the country is growing exponentially: where 0 0 > L denotes the initial stock of labor (at whereas 0 ≥ t is a continuous time index. Demand for all three factors of production results from the rational decisions of firms maximizing profits in perfectly competitive markets. Let K w and H w denote the real rental price of physical capital and human capital, respectively, and let w denote the real wage rate. In the profit maximizing equilibrium, all three factors are paid their marginal products, i.e., The accumulation equations are: where K δ and H δ denote depreciation rates. Throughout the text, a dot over the symbol for a variable denotes the time derivative, e.g.,

Proposition 1. (proof in the Appendix)
In the long run, the private economy converges towards the balanced growth path, with K, H and Y growing at the same, constant rate (the balanced growth rate, BGR). This balanced growth equilibrium is unique and globally asymptotically stable. The BGR cannot be determined analytically. It can only be identified numerically by solving a particular non-linear equation. Despite this difficulty, it is possible to prove that the BGR is an increasing function of the rate of savings γ and a decreasing function of both depreciation rates. Most important, the relationship between the BGR and the share coefficient ψ is ambiguous.

The economy with the government investing in human capital
Now, we augment the above model by introducing the public sector (hereafter referred to as the government), which levies income and consumption taxes and invests in human capital.
The optimality conditions (6) In addition, the government collects consumption taxes equal to where C is aggregate consumption. Total government revenue is For simplicity, the government is assumed to maintain a balanced budget in each period, i.e., T G = . This assumption is justified by Ricardian equivalence -see, for example, Elmendorf and Mankiw (1998), and it is commonly applied in the literature; see for example Lee & Gordon (2005), Dhont &Heylen (2009), andTurnovsky (2009). Public expenditures include three components: where T G denotes cash transfers to the private sector (primarily social transfers, i.e., pensions, various benefits, social assistance, etc.), E G represents public spending on education, and C G is public consumption (primarily health care, national defense, and public safety). By assumption, transfers and expenditures on education are proportional to GDP: In a closed economy, the total compensation of all production factors is equal to output. Therefore, households' disposable income d Y is equal to GDP net of taxes, plus transfers. A fraction of that income is saved, and the remainder is consumed; hence the budget constraint of the private sector is expressed as follows: We assume a constant, exogenous rate of savings: Savings are invested in physical and human capital, with a fixed coefficient ψ , according to equations (9) and (10). From (18), it follows that private consumption is equal to: Notice that equations (19) and (20) are interconnected because of (14). According to (19), savings depend on consumption, and simultaneously, according to (20) consumption depends on savings. For convenience, we solve this system of equations. Simple algebraic manipulation yields: Henceforth, for simplicity, certain expressions (functions of parameters) will be denoted by 1 A , 2 A , etc. Substituting (13) and (16), and using (6) -(8), equation (22) can be written as: From equations (19), (9), (10) and (23), it follows that: The dynamic equations for physical and human capital are of the form: Dividing both sides of these equations by K and H (respectively) yields the following growth rates: Substituting (25), equation (28) can be transformed into the following form: where Similarly, using (17) and (24) in equation (29) yields: where Finally, using (4), the growth rates (30) and (33) can be written as: (37)

Proposition 2 (proof in the Appendix)
In the long run, the economy converges towards the balanced growth path, with K, H and Y growing at the same, constant rate (the balanced growth rate, BGR).
This balanced growth equilibrium is unique and globally asymptotically stable. Although it is not possible to derive an explicit formula for the BGR, it is perfectly possible (and worthwhile) to perform a qualitative sensitivity analysis to determine the relationship between the parameters of the model and the BGR.

Qualitative sensitivity analysis
In this section, we wish to determine how changes in parameter values influence the BGR. Specifically, we account for all (four) tax rates, the rate of private savings γ , the rate of public transfers T γ , the rate of spending on education E γ , and the share coefficient ψ . The analysis is performed in 3 steps. First, we investigate whether an increase in the value of a parameter increases or reduces the values of expressions 2 A , …, 6 A . Second, using formulas (36) and (37) are automatically offset by reduced public consumption, with no change in public spending on education. These structural changes result in higher disposable income in the private sector. Therefore, private investment in education and physical capital increases, whereas public spending on education remains unchanged. The total effect is unambiguous -the BGR increases.
The effect of increasing the share parameter ψ is quite interesting. Recall that ψ represents the share of private savings invested in education. Therefore, increasing ψ raises the rate of human capital accumulation and simultaneously reduces the rate of physical capital growth. Technically, the graph of ) / (  H  K  H shifts up, whereas the graph of fig. A2). The intersection of these curves unambiguously moves to the left, but it is uncertain whether it moves up or down. Therefore, a higher value of ψ reduces the balanced growth ratio of H K / -there is more human capital per each unit of physical capital. However, the relationship between ψ and the BGR is ambiguous.

Social transfers and the rates of savings and investment
The share parameter ψ can be directly calculated from equation (10)

Average tax rates
Eurostat reports 'implicit tax rates' on capital, labor and consumption. In Poland during the period 2000-2010 (the latest data), these rates were on average equal to: approximately 5% of GDP). Presumably, this problem arises because our (model's) concepts of human capital and raw labor are not identical to the definitions employed by Eurostat. In particular, Eurostat classifies "taxes on income and social contributions of the self-employed" as part of the capital income tax -a detailed explanation can be found in the methodological publication by Eurostat (2010), Annex B. However, self-employed entrepreneurs definitely correspond to our concept of human capital (as well as part of raw labor). Self-employment is very popular in Polandnot only are there millions of small, family businesses, but very often individuals operate single-person firms and provide services for larger enterprises. Moreover, the tax rate on capital income published by Eurostat is much lower (21.2%) than the tax rate on labor (32.8%).
Therefore, in our model, the tax rate on human capital and labor should be somewhere between these two numbers. As there are no additional statistics, we calibrate both rates at this level, for which the model produces a total share of taxes in GDP that is consistent with statistics (32.7%, see above). In so doing, we ob- , i.e., rates that are approximately ¼ lower than those reported by Eurostat.
The next step in the calibration is computing the values of expressions i A . We do not report these values here, as they do not have any economic interpretation. Knowing these values, and using formula (30), we compute the average capital growth during the period 2000-2011: Transforming formula (4) and substituting the above ratio yields In summary, we have the following base set of values for the parameters and initial values of the factors of production:

Baseline scenario
The baseline set of parameters (45) (36) and (37), we obtain the BGR in the baseline scenario. It is equal to 3.58%, slightly higher than the average growth rate recorded during the period 2000-2011. To depict the process of convergence towards the balanced growth path, we present a graph illustrating the trajectories of the above growth rates in the baseline scenario.

Selected tax-cut scenarios in Poland
Let us determine the effects of reducing various types of taxes in the model calibrated for Poland. We consider 2 types of scenarios: a) reducing a given tax rate by 1 or 5 percentage points (pp), b) reducing all tax rates by 1 or 5 pp.  In each scenario, the tax rates are reduced at 0 = t .
Unsurprisingly, the most favorable results are associated with the largest tax cuts, i.e., the scenario of reducing all tax rates by 5 pp. After 30 years, GDP would be 11.9% higher than under the baseline scenario. Let us analyze this specific scenario in greater depth. Table   3 summarizes selected structural macroeconomic indicators under that scenario, relative to those in the baseline scenario.
After lowering all tax rates by 5 pp, the overall tax burden would decline from current 33% to 26.1% of GDP, which would be similar to those currently ob-

Changing the structure of tax revenue
The scenario of significant tax cuts presented in the previous paragraph would be quite difficult to achieve in practice due to the abovementioned structural changes induced by the reduction in public spending. It is tempting, therefore, to consider

Selected scenarios of increasing public and private spending on education
In this section, 3 scenarios are presented:  With respect to the BGR, all three scenarios significantly outperform the baseline scenario. However, the effect of additional spending on education (scenarios A and C) is stronger than the effect of a similar increase in private savings, with additional resources being primarily spent on investments in physical capital (97%). These simulations suggest that it is much more preferable to spend additional money on education rather than on physical capital. Moreover, from the comparison of scenarios A and C, it follows that it is relatively unimportant whether the additional funds for education come from a reduction in public or private consumption.

The optimal structure of private investment
Clearly, investing in human capital (education) is of crucial importance for economic growth. However, in section 3, we were unable to analytically establish the relationship between the BGR and the share parameter ψ (precisely, the share of private savings spent on education). Now, using the baseline scenario as a benchmark, we can calculate the BGR corresponding to any value of ψ from 0% to 100%. Figure 2