329
Primary submission: 26.11.2013 | Final acceptance: 16.03.2014
How Taxes and Spending on Education
Influence Economic Growth in Poland
Michał Konopczyński1
This paper investigates the relationship between economic growth in Poland and four types of
ABSTRACT
taxes and human capital investment. We primarily rely on an exogenous growth model that merg-
es 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.
KEY WORDS: fiscal policy; income taxes; labor taxes; capital taxes; VAT; economic growth; human capital
JEL Classification: E62; H21; H52
1
Poznań University of Economics, Poland
Introduction ( jump ) variables (e.g., savings and consumption) in
The standard approach in modern growth theory is response to policy changes. In our view, it would be
to describe the savings and consumption decisions of overly optimistic (unjustified) to assume that CEE
households as an intertemporal optimization problem. economies are already in this type of equilibrium.
However, in our view, in the case of Central and East- These countries remain in transition from centrally
ern European (CEE) countries, the calibration (or es- planned, Eastern-oriented economies to market-based
timation) of such models would be difficult for several economies integrated with the West (the EU). More-
reasons. First, to the best of our knowledge, there are over, over the last 20 years, the CEE economies have
no reliable empirical estimates of the parameters of the experienced intense structural changes coupled with
intertemporal utility function for most CEE countries. significant changes in economic policies. Furthermore,
Second, optimal control models assume that economic external conditions have also rapidly evolved, with the
agents are consistently optimizing, adjusting control expansion of the EU in 2004 arguably representing the
greatest (revolutionary) change.
For the above reasons, our analysis is deliberately
Correspondence concerning this article should be addressed to: based on a simple exogenous growth model, with numer-
Michał Konopczyński, Poznań University of Economics, ous elements borrowed from the Mankiw-Romer-Weil
al. Niepodległości 10, 61-875 Poznań, Poland. tel. [phone] (1992) growth model. For example, we incorporate the
fax: [phone] Email: [e-mail] power production function with constant economies of
scale and exogenous rates of investment and savings. We
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
330
also conceptualize human capital as a stock that requires assume positive externalities related to learning-by-
investment and depreciates over time. (A thorough re- doing and spillover-effects; see, e.g., Romer (1986) and
view of human capital research is presented by Cichy Barro and Sala-i-Martin (2004). These externalities are
(2008) and Acemoglu (2008).) Furthermore, mathemati- reflected in the labor-augmenting technology index E,
cal rules describing the public sector are taken from the which is proportional to the capital per worker ratio,
literature on optimal fiscal policy; see (e.g., Agenor, 2007; i.e., E = x K L , where x = const. > 0 . Thus, the pro-
Dhont & Heylen, 2009; Lee & Gordon, 2005). Four types duction function can be written as
of taxes are considered: taxes on capital, labor, human
capital and consumption. The tax revenues are expended Y = A KÄ… +² H1-Ä… -² , (4)
on public consumption and human capital investment,
and the remainder is transferred back to households. The where A = ax² = const > 0. Therefore, aggregate out-
government budget is permanently balanced, which is put in the economy is described by a Cobb-Douglas
a standard assumption in most research on optimal fis- function with constant returns to scale for both types
cal policy, initiated by Barro (1990), and developed in of capital (physical and human). The assumption of
numerous subsequent studies, including the abovemen- constant returns to scale is supported by strong empir-
tioned Lee & Gordon (2005), Agenor (2007), Dhont and ical evidence. See, e.g., (Balisteri, McDaniel, & Wong,
Heylen (2009). The assumption of a balanced budget is 2003; Cichy, 2008; Mankiw, Romer, & Weil, 1992;
fully justified for closed economy models (as in our pa- Manuelli & Seshadri, 2005; Próchniak, 2013; Will-
per) by the well-known effect of Ricardian equivalence. man, 2002). Nevertheless, we note that by considering
However, in light of recent empirical data, such an as- increasing or decreasing returns to scale, our analysis
sumption may appear unrealistic. Therefore, in the near could lead to different conclusions.
future, we intend to generalize the model presented here We assume that the labor supply in the country is
by allowing the government to borrow both internally growing exponentially:
and from abroad. A simple example of such a model with
perfect capital mobility was presented by Konopczyński L = L0ent, (5)
(2013). The paper is organized as follows. Section 1 pres-
ents the benchmark private economy model. In section where L0 > 0 denotes the initial stock of labor (at t = 0),
2, this model is augmented with a government that col- whereas t e" 0 is a continuous time index. Demand for
lects four types of taxes and invests in education. Section all three factors of production results from the rational
3 contains a qualitative sensitivity analysis. In section 4, decisions of firms maximizing profits in perfectly com-
we calibrate the model on the basis of statistical data on petitive markets. Let wK and wH denote the real rental
the Polish economy in the period 2000  2011. In section price of physical capital and human capital, respectively,
5, we present the baseline scenario, corresponding to the and let w denote the real wage rate. In the profit maxi-
results of the calibration exercise. Sections 6  9 present mizing equilibrium, all three factors are paid their mar-
scenarios analyzing tax cuts and increased educational ginal products, i.e.,
expenditures by both the government and the private sec-
tor. The concluding section synthesizes the main results. MPK = "Y "K = Ä…Y K = wK = r + ´ , (6)
K
Mathematical proofs are included in the appendix.
MPH = "Y "H = (1- Ä… - ² )Y H = wH , (7)
1. The private economy
The aggregate output of the country is described by the MPL = "Y "L = ²Y L = w , (8)
following production function:
Obviously, in equilibrium, the real rental rate of physi-
1-Ä… -²
Y = aKÄ… H (E )² , (2) cal capital is equal to the sum of the real interest rate
L
r and the rate of depreciation ´K . We assume that
where K denotes the stock of physical capital, H repre- a constant, exogenous fraction of national income is
sents the stock of human capital, and L is raw labor. We saved: S = Å‚ Å"Y , where 0 < Å‚ < 1. Savings are invested
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
331
in physical and human capital, with a fixed share coef- minus a depreciation allowance. The total sum of all
ficient 0 <È <1: income taxes is expressed as
IK = (1-È ) Å" S , (9) T1 = Ä w L+Ä wH H +Ä r K, (13)
L H K
IH =È S , (10) In addition, the government collects consumption
taxes equal to
The accumulation equations are:
T2 = ÄCC , (14)

K = IK - ´K K , 0 < ´K <1, (11)
where C is aggregate consumption. Total government

H = IH -´H H , 0 < ´H < 1, (12) revenue is T = T1 + T2 . For simplicity, the government
is assumed to maintain a balanced budget in each
where ´K and ´H denote depreciation rates. Through- period, i.e., G = T . This assumption is justified by
out the text, a dot over the symbol for a variable de- Ricardian equivalence  see, for example, Elmendorf

notes the time derivative, e.g., K = "K(t) "t. and Mankiw (1998), and it is commonly applied in
the literature; see for example Lee & Gordon (2005),
Proposition 1. (proof in the Appendix) Dhont & Heylen (2009), and Turnovsky (2009). Public
In the long run, the private economy converges towards expenditures include three components:
the balanced growth path, with K, H and Y growing
at the same, constant rate (the balanced growth rate, G = GT + GE + GC , (15)
BGR). This balanced growth equilibrium is unique and
globally asymptotically stable. The BGR cannot be de- where GT denotes cash transfers to the private sec-
termined analytically. It can only be identified numeri- tor (primarily social transfers, i.e., pensions, various
cally by solving a particular non-linear equation. De- benefits, social assistance, etc.), GE represents public
spite this difficulty, it is possible to prove that the BGR spending on education, and GC is public consumption
is an increasing function of the rate of savings Å‚ and (primarily health care, national defense, and public
a decreasing function of both depreciation rates. Most safety). By assumption, transfers and expenditures on
important, the relationship between the BGR and the education are proportional to GDP:
share coefficient È is ambiguous.
GT = Å‚T Å"Y , where 0 < Å‚T < 1. (16)
2. The economy with the government
investing in human capital GE = Å‚ Å"Y , where 0 < Å‚ < 1. (17)
E E
Now, we augment the above model by introducing the
public sector (hereafter referred to as the government), In a closed economy, the total compensation of all
which levies income and consumption taxes and in- production factors is equal to output. Therefore,
vests in human capital. households disposable income Yd is equal to GDP
The optimality conditions (6)  (8) remain valid, net of taxes, plus transfers. A fraction of that income
but the variables w, wH and wK = r + ´K hereafter is saved, and the remainder is consumed; hence the
represent gross rates, i.e., the unit costs of labor, hu- budget constraint of the private sector is expressed
man capital and physical capital from the perspective as follows:
of the representative firm. Let Ä , Ä , and Ä denote
L H K
the average tax rates. Taxes on labor and human capi- Yd = Y -T1 -T2 + GT = C + S . (18)
tal are levied on gross wage rates, i.e., the government
collects Ä w and Ä wH . The income tax on capital is We assume a constant, exogenous rate of savings:
L H
calculated as follows: Ä (wK - ´K ) = Ä r , i.e., the tax is
K K
levied on net capital income, defined as gross income S = Å‚Yd = Å‚ (Y - T1 - T2 + GT ) . (19)
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
332
Savings are invested in physical and human capital, Substituting (25), equation (28) can be transformed
with a fixed coefficient È , according to equations (9) into the following form:
and (10). From (18), it follows that private consump-
Y
Ć
tion is equal to: K = (1 -È )A2 A3 + A4 , (30)
K
C = Yd - S = Y - T1 - T2 + GT - S . (20) where
Notice that equations (19) and (20) are interconnected A3 =1-Ä…Ä - ²Ä - (1-Ä… - ² )Ä + Å‚ , (31)
K L H T
because of (14). According to (19), savings depend on
consumption, and simultaneously, according to (20) A4 = [(1-È )A2Ä -1]´K , (32)
K
consumption depends on savings. For convenience,
we solve this system of equations. Simple algebraic Similarly, using (17) and (24) in equation (29) yields:
manipulation yields:
Y K
$
= A5 + A6 -´H , (33)
H H
1- Å‚
C = A1 Å"(Y -T1 + GT ) , where A1 = (21)
1 +ÄC (1 - Å‚ )
where
Å‚
S = A2 Å"(Y -T1 + GT ) , where A2 = (22)
1 + ÄC (1- Å‚ )
A5 = Å‚ +ÈA2 A3 , (34)
E
Henceforth, for simplicity, certain expressions (func-
tions of parameters) will be denoted by A , A , etc. A6 =ÈA2Ä ´ , (35)
1 2 K K
Substituting (13) and (16), and using (6)  (8), equa-
tion (22) can be written as: Finally, using (4), the growth rates (30) and (33) can
be written as:
S =A2Å"[(1-Ä…Ä -²Ä -(1-Ä… -² )Ä +Å‚T)Y+ÄK´KK]. (23)
K L H
Ä… +² -1
K
Ć
K = (1-È )A2 A3 AëÅ‚ öÅ‚ + A4 , (36)
ìÅ‚ ÷Å‚
H
íÅ‚ Å‚Å‚
From equations (19), (9), (10) and (23), it follows that:
Ä… +²
K K
IH =ÈS =ÈA2Å"[(1-Ä… ÄK -²Ä - (1-Ä… -² )Ä +Å‚ )Y +Ä ´K K] $
= A5 AëÅ‚ öÅ‚ + A6 -´H . (37)
ìÅ‚ ÷Å‚
L H T K
H H
íÅ‚ Å‚Å‚
ÈS =ÈA2Å"[(1-Ä… ÄK -²Ä - (1-Ä… -² )Ä +Å‚T)Y +Ä ´K K]. (24)
L H K
Ä ´K K]
IK =(1 -È ) Å"S = (1 -È )A2 Å"[(1 -Ä…Ä -²Ä - (1 -Ä… -² )Ä + Å‚T)Y +Proposition 2 (proof in the Appendix)
K L H K
Å"S = (1 -È )A2 Å"[(1 -Ä…Ä -²Ä - (1 -Ä… -² )Ä + Å‚ )Y + Ä ´K K]. (25) In the long run, the economy converges towards the
K L H T K
balanced growth path, with K, H and Y growing at the
The dynamic equations for physical and human capital same, constant rate (the balanced growth rate, BGR).
are of the form: This balanced growth equilibrium is unique and glob-
ally asymptotically stable.
Ć
K = IK - ´K K , 0 < ´K <1, (26) In equilibrium, it holds that v
= $
= K . Thus,
the BGR can be obtained by equating the right-

H = GE + IH -´H H , 0 < ´H < 1. (27) hand sides of equations (36) and (37). The resulting
equation (except for certain special cases) cannot be
Dividing both sides of these equations by K and H (re- solved analytically  it can only be solved numerical-
spectively) yields the following growth rates: ly, after substituting certain values for all parameters.
Although it is not possible to derive an explicit for-

K IK
Ć
K = = - ´K , (28) mula for the BGR, it is perfectly possible (and worth-
K K
while) to perform a qualitative sensitivity analysis to

H GE + IH
$
= = -´H , (29) determine the relationship between the parameters of
H H
the model and the BGR.
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
333
Table 1. Qualitative sensitivity analysis
ÄC Ä™!
Ä Ä™! Ä Ä™! Ä Ä™! Å‚ Ä™! Å‚T Ä™! Å‚ Ä™!
È Ä™!
K H L E
A2
“! Ä™!
= = = = = =
A3
“! “! “! Ä™!
= = = =
A4
Ä™! “! Ä™! “!
= = = =
A5 “! “! “! “! Ä™! Ä™! Ä™! Ä™!
A6 “! “! Ä™! Ä™!
= = = =
graphof
“! “! “! Ä™! Ä™! “!
? =
Ć
K(K / H )
graph of
“! “! “! Ä™! Ä™! Ä™! Ä™!
?
$
(K / H )
“! “! “! Ä™! Ä™! Ä™!
BGR ? ?
3. Qualitative sensitivity analysis the BGR and the rate of financial transfers to the private
In this section, we wish to determine how changes sector Å‚T requires explanation. Due to the assumption
in parameter values influence the BGR. Specifically, of a permanently balanced government budget, higher
we account for all (four) tax rates, the rate of private transfers to the private sector (with no change in taxes)
savings Å‚ , the rate of public transfers Å‚T , the rate of are automatically offset by reduced public consumption,
spending on education , and the share coefficient with no change in public spending on education. These
Å‚
E
È . The analysis is performed in 3 steps. First, we structural changes result in higher disposable income in
investigate whether an increase in the value of a pa- the private sector. Therefore, private investment in educa-
rameter increases or reduces the values of expressions tion and physical capital increases, whereas public spend-
A2 , & , A6 . Second, using formulas (36) and (37), we ing on education remains unchanged. The total effect is
Ć
investigate whether the graphs of functions K(K / H ) unambiguous  the BGR increases.
and $
(K / H ) shift up or down. Third, based on these The effect of increasing the share parameter È is
observations, we conclude whether the intersection quite interesting. Recall that È represents the share
of these curves, which corresponds to the BGR (see of private savings invested in education. Therefore,
Appendix, fig. A2), moves up or down. The results are increasing È raises the rate of human capital ac-
summarized in table 1. cumulation and simultaneously reduces the rate of
Notice that increasing any tax rate reduces the BGR, physical capital growth. Technically, the graph of
Ć
with one noticeable exception. The effect of raising the $
(K / H ) shifts up, whereas the graph of K(K / H )
tax rate on capital is ambiguous, as without additional shifts down (see Appendix, fig. A2). The intersection
assumptions, we cannot determine whether the graphs of these curves unambiguously moves to the left, but it
Ć
of $
(K / H ) and K(K / H ) shift up or down. is uncertain whether it moves up or down. Therefore,
It is unsurprising that the higher the rate of private sav- a higher value of È reduces the balanced growth ratio
ings Å‚ , the higher the BGR. Similarly, there is a positive of K / H  there is more human capital per each unit
relationship between the rate of public spending on edu- of physical capital. However, the relationship between
cation Å‚ and the BGR. The positive relationship between È and the BGR is ambiguous.
E
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
334
Based on table 1, we can formulate the following. various research papers, physical capital depreciation
varies from approximately 3.5% to 7%. As the focus of
Proposition 3 our analysis is on the long-run equilibrium, we set the
First, the balanced growth rate (BGR) is an increas- depreciation rate at a relatively low level of ´K = 4% ,
ing function of the rate of private savings Å‚ , the rate following Nehru & Dhareshwar (1993). The rate of hu-
of public transfers Å‚T , and the rate of public spend- man capital depreciation has been estimated by Manu-
ing on education Å‚ . Second, the BGR is a decreasing elli & Seshadri (2005), Arrazola & de Hevia (2004) and
E
function of the tax rates on labor, human capital and others. Following these authors, we set ´ = 1.5% .
H
consumption. Third, the relationship between the BGR Next, we must estimate the real rate of return on
and the tax rate on capital income, as well as the share capital (r). From (6), it follows that r = Ä… Å"Y K - ´K (if
coefficient (the percentage of private savings invested firms maximize profits, which we assume). The ratio of
in education), is ambiguous. Y K is very difficult to estimate for Poland, as (to the
These qualitative results, though interesting per se, best of our knowledge) there are no data on the stock
only enhance our desire for quantitative results. More- of productive capital in Poland. The Polish Main Sta-
over, as the BGR cannot be determined analytically, tistical Office only registers  gross value of fixed assets ,
it is not possible to determine how strongly changes which is a far narrower category than  productive capi-
in the values of parameters influence the BGR. Sim- tal . This situation becomes obvious when consider the
ply, we know the direction of the effect, but we know useful data collected by the Kiel Institute for the World
nothing of the size of the effect. Answering these ques- Economy in the  Database on Capital Stocks in OECD
tions is not possible without establishing (estimating Countries . It contains capital stock estimates for 22
or calibrating) certain (benchmark) parameter values OECD countries for the period 1960-2001. Poland is
and performing numerical analyses. Achieving these not included. For the 22 countries that are included,
outcomes is possible for any country or group of coun- the average ratio of capital to GDP was very close to 3
tries. In our view, Poland represents an interesting case throughout the period 1960-2001  it varies between
 it experienced tremendous growth in education over 2.9 and 3.3. In certain countries it was slightly lower,
the past 20 years, coupled with a substantial increase e.g., for the U.S., Canada, and the United Kingdom, it
in physical capital. Both of these factors contributed to was close to 2.5. In most of continental Europe, how-
rapid (and relatively stable) economic growth. In what ever, it is close to 3 or slightly higher, e.g., for Germany,
follows, we calibrate the model for Poland and numeri- Switzerland and Greece, it is approximately 3.5. Gen-
cally analyze optimal fiscal policy, as well as optimal erally, these ratios are very close to generally accepted
private sector parameters. The calibration is based on stylized facts.
macroeconomic data for Poland for the period 2000  However, there is an issue regarding the case of
2011, published by the Eurostat, OECD, and the Kiel Poland. If we employ the data provided by the Polish
Institute for the World Economy. Main Statistical Office and calculate the ratio of  gross
value of fixed assets to GDP, it is approximately 1.7-
4. Model calibration for Poland 1.8. Clearly, the data available for Poland only reflect
a share of all productive capital. To the best of our
Technological parameters knowledge, there are no data available for Poland that
The elasticities of the production function (2) have would better satisfy our requirements. Therefore, as
been estimated in numerous empirical papers; see, a reasonable calibration, we will use the average ra-
(e.g., Cichy, 2008; Mankiw, et al., 1992; Manuelli & tio from the Kiel database, i.e., we set Y K = 1 3. (In
Seshadri, 2005; Próchniak, 2013). They are typically the appendix, we present the sensitivity analysis, as-
close to 1/3; hence we set: Ä… = ² =1-Ä… - ² =1 3. The suming higher and lower Y K values.) Substituting
rate of physical capital depreciation is difficult to esti- this number into (6) yields the real rate of return on
1
mate for Poland, due to its economic transformation capital equal to r = 1 3Å"1 3 - 0.04 = 7.1 %. This result
and massive stock of useless machinery, infrastructure, is very similar to most empirical estimates for OECD
etc. inherited from the centrally  planned economy. In countries. For example, Campbell, Diamond & Shoven
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
335
(2001) report that the average real rate of return on on formula (18), we set Å‚ = 5.84%. In the same pe-
E
stocks in the U.S. over the period 1900  1995 is 7%. riod, consumption taxes were equal to 12.1% of GDP.
Similar indicators for the Polish stock market exist; Thus, the ratio of income taxes to GDP is equal to
however, Poland s stock market has only existed for ap- T1 Y = T Y -T2 Y = 32.7% -1 2.1% = 20.6%.
proximately 23 years, and most of that period should
be regarded as one of intense transformation and Average tax rates
privatization of the economy. Thus, in our view, Pol- Eurostat reports  implicit tax rates on capital, labor
ish stock market indicators do not reflect the long-run and consumption. In Poland during the period 2000-
equilibrium and cannot be used to calibrate our model. 2010 (the latest data), these rates were on average
equal to: Ä = 2 1.2% Ä = 3 2.8% Ä = 1 9.4%
, , and ,
K L C
Social transfers and the rates of savings and respectively. Note that the implicit tax rate on labor is
investment defined as the  Ratio of taxes and social security contri-
According to Eurostat, cash transfers to the private butions on employed labor income to total compensa-
sector (primarily social transfers, i.e., pensions, vari- tion of employees . To the best of our knowledge, there
ous benefits, social assistance, etc.) were on average are no data on the average tax rates on human capital.
equal to 15.5% of GDP over the period 2000-2011. However, certain research papers provide valuable in-
Thus, we set Å‚ = 15.5%. dications, (e.g., Gordon & Tchilinguirian, 1998; Heck-
T
The average rate of savings can be calibrated on the man & Jacobs, 2010). These authors note the strong
basis of equation (19), which can be transformed into correlation between the level of education (human
the following formula: capital) and individual income. Therefore, in countries
with highly progressive taxes on personal income, tax
S IK + IH IK Y + IH Y
. (38) rates on human capital must be higher than tax rates
Å‚ = = =
Yd Y -T + GT 1-T Y + GT Y
on (raw) labor. Apart from these types of general (and
According to Eurostat, gross fixed capital formation in obviously correct) indications, the literature provides
Poland in the period 2000-2011 was on average 20,1% virtually no methods for measuring average tax rates
of GDP. Moreover, private spending on education in on human capital. Fortunately, we can obtain valu-
the period 2000-2009 (the latest data available form able information from the OECD Tax Database, which
Eurostat) was on average 0.62% of GDP. The ratio of contains average tax rates on wages (precisely:  the
 total receipts from taxes and social contributions to average personal income tax and social security con-
GDP in 2000-2011 was equal to 32.7% (and very sta- tribution rates on gross labor income ) for several lev-
ble). Substituting these numbers into (38) yields els of country-wide average wages: 67%, 100%, 133%,
and 167%. In certain countries, tax rates on wages are
I Y + I Y 2 0,1% + 0,62%
K H
Å‚ = = = 25.02%. (39) highly progressive, e.g., in Finland in 2012, the aver-
1- T Y + GT Y 1- 32,7% +15,5%
age tax wedge for 67% of average income is equal to
The share parameter È can be directly calculated from 36%, whereas for 167%, it increases to 48%. In Poland,
equation (10): the analogous tax wedges are 33.3% (for 67% of the av-
erage income) and 35% (for 167%). These figures are
I I Y 0.62%
H H
È = = = = 2.9 %. (40) very similar throughout the period 2000-2011. There-
9
S I Y + I Y 20.1% + 0.62%
K H
fore, in Poland, the size of tax wedge on labor is nearly
Clearly, in Poland, a mere 3% of private savings is in- independent of the level of income, i.e., effective tax
vested in education. (It is possible that private spend- rates on wages are nearly linear. Thus, it is reasonable
ing on education is underestimated in official sta- to assume that average tax rates on human capital and
tistics  a substantial share of it is likely classified as raw labor in Poland are identical, i.e., Ä = Ä .
H L
consumption, e.g., the cost of accommodation, travel, Recall that according to Eurostat, Ä = 3 2.8%. Un-
L
books, etc.). However, public outlays on education in fortunately, if we set Ä = Ä = 3 2.8%, and perform
H L
Poland during the period 2002-2010 were on aver- the entire calibration as follows, the model significantly
age equal to 5.84% of GDP (Eurostat); hence based overestimates the total revenue from income taxes (by
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
336
approximately 5% of GDP). Presumably, this problem the number of students, PhDs, etc. It is worth noting
arises because our (model s) concepts of human capi- that a 5% growth rate of human capital implies that its
tal and raw labor are not identical to the definitions stock doubles in only 15 years.
employed by Eurostat. In particular, Eurostat classi- To perform the calculations (simulations), it is neces-
fies  taxes on income and social contributions of the sary to have an estimate of the value of the parameter A.
self-employed as part of the capital income tax  a de- First, from equation (33), we calculate the proportion
tailed explanation can be found in the methodologi-
K $
+ ´
H
cal publication by Eurostat (2010), Annex B. However, = = 3.0371. (43)
H A5 Y K + A6
self-employed entrepreneurs definitely correspond to
our concept of human capital (as well as part of raw Transforming formula (4) and substituting the above
labor). Self-employment is very popular in Poland  ratio yields
not only are there millions of small, family businesses,
1-Ä… -²
Y Y K
ëÅ‚ öÅ‚
but very often individuals operate single-person firms A = = Å" = 0.4827 . (44)
ìÅ‚ ÷Å‚
Ä… +² 1-Ä… -²
K H K H
íÅ‚ Å‚Å‚
and provide services for larger enterprises. Moreover,
the tax rate on capital income published by Eurostat is To perform the simulations, we should also assume
much lower (21.2%) than the tax rate on labor (32.8%). certain initial values of the variables K, H and L. Two
Therefore, in our model, the tax rate on human capi- of them (K and L) can be determined completely freely,
tal and labor should be somewhere between these two provided that we confine our interest to the rates of
numbers. As there are no additional statistics, we cali- growth and relationships (the proportions) among
brate both rates at this level, for which the model pro- variables. Therefore, we set L(0) = 1 and K(0) = 300 .
duces a total share of taxes in GDP that is consistent This particular choice is convenient, as the initial level
with statistics (32.7%, see above). In so doing, we ob- of GDP is then equal to 100, and the initial values of
tain Ä = Ä = 2 4.2%, i.e., rates that are approximately all the other variables are identical to their percentage
H L
ź lower than those reported by Eurostat. shares of GDP. From (43), it follows that H (0) = 98.78.
The next step in the calibration is computing the In summary, we have the following base set of val-
values of expressions Ai . We do not report these val- ues for the parameters and initial values of the factors
ues here, as they do not have any economic interpreta- of production:
tion. Knowing these values, and using formula (30), we
compute the average capital growth during the period A = 0.4827, Ä… =1 3, ² = 1 3, ´K = 4%, ´ = 1.5%,
H
2000-2011: Å‚ = 2 5.02%, È = 2.99 % , Å‚ = 5.84%, Å‚ = 15.5%,
E T
Ä = 2 1.2%, Ä = 1 9.4%, Ä = Ä = 2 4.2%, L(0) =1,
K C H L
Y
Ć
K = (1-È )A2 A3 + A4 = 2.70%. (41) K(0) = 300 , H (0) = 98.78. (45)
K
The average GDP growth rate in Poland during the pe- 5. Baseline scenario
riod 2000-2011 was 3.48% (geometric mean). Knowing The baseline set of parameters (45) generates results
this, we can estimate the human capital growth rate, on that precisely correspond to actual statistics on the
the basis of equation (A4), from which it follows that Polish economy during the period 2000  2011. Spe-
2
cifically, the baseline scenario reproduces factual (av-
3.48% - Å" 2.7 0%
Ć
v
- (Ä… + ² )K
3
$
= = = 5.04%. (42) erage) ratios of the following variables to GDP: C , IK ,
(1- Ä… - ² ) 1 3
IH , T1 , T2 , GT , GE , as well as the (average) rate of
These results suggest that in the period 2000-2011, GDP growth observed between 2000 and 2011. There
economic growth in Poland was primarily driven by is nothing surprising in this  the result is precisely ob-
rapid growth in the stock of human capital and only tained due to the method of calibration. The rates of
secondarily by the accumulation of productive capital. growth for t = 0 generated by the model in the base-
This impressive increase in human capital in Poland line scenario are equal to
is a well-known  stylized fact confirmed by numerous
Ć
indicators concerning education  a sharp increase in v
= 3.48%, K = 2.70%, $
= 5.04%.
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
337
Fig. 1. Convergence to the balanced growth path.
5,0%
rate of growth of H
4,5%
rate of growth of Y
rate of growth of K
4,0%
3,5%
3,0%
2,5%
0 10 20 30 40 50 60 70 80 90 100
Figure 1. Convergence to the balanced growth path.
These rates are not equal, and hence the Polish econ- GDP after 30 years (in table 2, numbers in bold and
omy is not yet on the balanced growth path. By (nu- italics). These indicators are calculated as follows:
merically) solving the equation formed by equating
Y(t = 30) i selected scenario
n
right-hand sides of equations (36) and (37), we obtain gain after 30 years = - 1. (46)
Y (t=30)i the baseline scenario
n
the BGR in the baseline scenario. It is equal to 3.58%,
slightly higher than the average growth rate recorded In each scenario, the tax rates are reduced at t = 0 .
during the period 2000-2011. To depict the process Unsurprisingly, the most favorable results are asso-
of convergence towards the balanced growth path, we ciated with the largest tax cuts, i.e., the scenario of re-
present a graph illustrating the trajectories of the above ducing all tax rates by 5 pp. After 30 years, GDP would
growth rates in the baseline scenario. be 11.9% higher than under the baseline scenario. Let
us analyze this specific scenario in greater depth. Table
6. Selected tax-cut scenarios in 3 summarizes selected structural macroeconomic in-
Poland dicators under that scenario, relative to those in the
Let us determine the effects of reducing various types baseline scenario.
of taxes in the model calibrated for Poland. We con- After lowering all tax rates by 5 pp, the overall tax
sider 2 types of scenarios: burden would decline from current 33% to 26.1% of
a) reducing a given tax rate by 1 or 5 percentage GDP, which would be similar to those currently ob-
points (pp), served in the United States (approx. 25%), South Korea
b) reducing all tax rates by 1 or 5 pp. (26%) and Japan (27%). The immediate effect of the re-
Table 2 contains the BGRs calculated under all of these duction in taxes would be an increase in private sector
scenarios. In all cases, the economy grows more rap- savings relative to GDP (from 20.7% to 22.4%), an in-
idly (on the balanced growth path) than in the baseline crease in investment (from 20% to 21.7% of GDP), and
scenario. To better visualize the long-term (welfare) finally, a rise in private expenditures on education. The
effect of reducing taxes, we also include numbers in- accelerated accumulation of both physical and human
dicating by how many percent GDP exceeds baseline capital would shift the economy to a higher balanced
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
338
Table 2. Simulation results for Poland - different scenarios of tax cuts
1 pp reduction 5 pp reduction
3.59% 3.66%
Ä
L
0.5% 2.5%
3.59% 3.63%
Ä
K
0.3% 1.6%
3.59% 3.66%
Ä
H
0.5% 2.5%
3.61% 3.73%
ÄC
0.9% 4.8%
3.65% 3.95%
reduction of all tax rates simultaneously
2.2% 11.9%
Table 3. The scenario of simultaneously reducing all tax rates by 5 pp.
the BGR and structural indicators (%) baseline scenario reduction of all tax rates by 5 pp
the BGR 3.58 3.95 (the effect after 30 years= +11.9%)
C / Y
61.9 67.0
T / Y
33.0 26.1
S / Y
20.7 22.4
IK / Y
20.0 21.7
GE / Y
5.8 5.8
IH / Y
0.6 0.7
growth path. As a result, the BGR would increase by Under the scenario presented in table 3, the share of
approximately 0.38 percentage points. private consumption in GDP increases from 61.9% to
It is worth noting that this scenario is associated 67.0%. Again, this would bring the Polish economy
with significant structural changes in the economy. Re- structurally closer to the United States, where private
duced tax receipts, while maintaining a 15.5% share of consumption is equal to approximately 70% of GDP.
cash social transfers in GDP (primarily pensions) and
a 5.8% share of public expenditures on education in 7. Changing the structure of tax
GDP, would negatively affect public consumption ex- revenue
penditures, i.e., national defense, public safety, health The scenario of significant tax cuts presented in
care, public administration, environmental protec- the previous paragraph would be quite difficult
tion, etc. This gap would have to be (partially) offset to achieve in practice due to the abovementioned
by increased consumption spending in the private sec- structural changes induced by the reduction in pub-
tor. Thanks to the tax cuts, this would occur naturally. lic spending. It is tempting, therefore, to consider
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
339
Table 3. The scenario of simultaneously reducing all tax rates by 5 pp.
C
A B
The BGR and Baseline scenario Increase in private
Increase in public Increase in private
Å‚ = 5,84%
structural spending on education
E
spending on education savings by 1 pp of GDP
Å‚ = 2 5,02%
indicators by 1 pp of GDP
Å‚ = 2 6,17%
by 1 pp of GDP
È = 2,99 % Å‚ = 2 6,2 1%
(%)
È = 2,99 %
Å‚ = 6,84%
E
È = 7,47%
the BGR 3,58 3,89 3,80 3,90
GDP effect after 30 years GDP effect after 30 years GDP effect after 30 years
+9,4% +6,8% +9,5%
C / Y
61,9 61,8 61,1 61,0
T / Y
33,0 33,0 32,8 32,9
S / Y
20,65 20,65 21,65 21,65
IK / Y
20,04 20,04 21,01 20,04
GE / Y
5,84 6,84 5,84 5,84
IH / Y
0,62 0,62 0,65 1,62
alternative scenarios with unchanged levels of taxa- (i.e., an unchanged value of È ). As a result, private
tion (and public spending) but a modified tax struc- investment in physical and human capital would
ture. Under this scenario, all three income tax rates increase by a total of 1 pp of GDP.
are reduced by 5 percentage points and the con- C) private sector savings increase by 1 pp of GDP (at
sumption tax rate is increased, and hence the share the expense of individual consumption), but addi-
of taxes in GDP is identical to that in the baseline tional savings are spent solely on education. (For
scenario, i.e., 26.62% instead of 19.4%. The results this purpose, the value of È has been appropriately
of the calculations reveal that such a change in the amended). In other words, private spending on ed-
tax structure would be neutral for the economy. Nei- ucation increases by 1 pp of GDP at the expense of
ther the BGR nor any of structural indicators (listed private consumption.
in table 3) would change. Simply, in our model, the Table 4 presents the results.
structure of taxes is neutral  the important factor is With respect to the BGR, all three scenarios signifi-
the level of taxation. cantly outperform the baseline scenario. However, the
effect of additional spending on education (scenarios
8. Selected scenarios of increasing A and C) is stronger than the effect of a similar in-
public and private spending on crease in private savings, with additional resources be-
education ing primarily spent on investments in physical capital
In this section, 3 scenarios are presented: (97%). These simulations suggest that it is much more
A) the government increases public spending on edu- preferable to spend additional money on education
cation by 1 pp of GDP at the expense of public con- rather than on physical capital. Moreover, from the
sumption. comparison of scenarios A and C, it follows that it is
B) private sector savings increase by 1 pp of GDP (at relatively unimportant whether the additional funds
the expense of individual consumption), with an for education come from a reduction in public or pri-
unchanged structure of investment expenditures vate consumption.
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
340
Fig. 2. The BGR as a function of the share parameter È
4,0%
BGR
3,0%
2,0%
1,0%
0,0%
0 0,2 0,4 0,6 0,8 1
Figure 2.The BGR as a function of the share parameter È
9. The optimal structure of private is globally asymptotically stable. Despite the simplic-
investment ity of the model, the balanced growth rate (BGR) can
Clearly, investing in human capital (education) is of only be calculated numerically, as it requires solving
crucial importance for economic growth. However, a complex, non-linear equation. On the one hand, the
in section 3, we were unable to analytically establish BGR is an increasing function of the rate of private
the relationship between the BGR and the share pa- savings, the rate of public transfers, and the rate of
rameter È (precisely, the share of private savings spent public spending on education. On the other hand, the
on education). Now, using the baseline scenario as BGR is a decreasing function of tax rates on labor, hu-
a benchmark, we can calculate the BGR corresponding man capital and consumption. Finally, the relationship
to any value of È from 0% to 100%. Figure 2 pres- between the BGR and the tax rate on capital income,
ents the results. The BGR reaches a maximum (equal as well as the share coefficient (the percentage of pri-
to 3.695%) at È = 14 %. According to Eurostat, at pres- vate savings invested in education), is ambiguous. It is
ent only 3% of private savings in Poland is spent on important to recall that this ambiguity is a property of
education. Therefore, the current structure of private the theoretical model and implies that these relation-
investment in Poland is far from optimal. Households ships are dependent on specific parameter values. In
should spend 14% of their savings on education, rather other words, the relationship between the tax rate on
than only 3%. However, in our view, it appears nearly capital and the BGR can be positive or negative - it
certain that private spending on education is underes- depends on the parameter values. Therefore, this ques-
timated in official statistics  a substantial share of it is tion can only be addressed after establishing the values
classified as consumption (e.g., the cost of accommo- of all parameters  as we do for Poland in the second
dation, travel, books, etc.). part of the paper. The central empirical conclusions re-
garding Poland can be summarized as follows. In the
Summary period 2000-2011, economic growth in Poland was
In the long run, the economy is trending toward a dy- primarily driven by a very rapid increase in the stock
namic equilibrium, characterized by so-called bal- of human capital (at a rate of 5% per annum) and only
anced growth. We demonstrated that the equilibrium secondarily by the accumulation of productive capital
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
341
(2.7% annually). Income taxes and consumption taxes young persons from Poland to other EU countries was
restrict economic growth to an equally burdensome observed. The growth effects of these two phenomena re-
extent. Therefore, if the government must collect a cer- main under investigation, but it is reasonable to contend
tain amount of tax revenue, it is irrelevant what type of that they offset one another out to some extent.
tax will be used for that purpose.
Reducing income and consumption tax rates by 5 References
percentage points in Poland should increase annual Acemoglu, D. (2008). Introduction to Modern Eco-
GDP growth by approximately 0.4 percentage points, nomic Growth. Princeton, NJ: Princeton Univer-
which after 30 years would entail a cumulative effect of sity Press.
a 12% increase in GDP relative to a scenario assuming Agenor, P. R. (2007). Fiscal policy and endogenous
current tax rates. Structurally, this scenario would bring growth with public infrastructure. Oxford Eco-
the Polish economy closer to such countries as the Unit- nomic Papers, 60 (1), 57-87.
ed States or South Korea, where the tax burden is sig- Arrazola, M., de Hevia, J. (2004). More on the estima-
nificantly lower (approximately 25% of GDP, whereas in tion of the human capital depreciation rate. Ap-
Poland it is 33%) and the share of individual consump- plied Economic Letters, 11 (3), 145-148.
tion is significantly higher (approximately 70% of GDP, Balistreri, E. J., McDaniel, C. A., Wong, E. V. (2003).
whereas the figure for Poland is approximately 62%). An estimation of US industry-level capital-labor
Investing in human capital (education) is crucial to substitution elasticities: support for Cobb-Doug-
economic growth. An increase in education expendi- las. The North American Journal of Economics and
tures by 1 percentage point of GDP would have a similar Finance, 14 (3), 343-356.
long-run effect as simultaneously reducing all tax rates Barro, R. J. (1990). Government spending in a simple
by 4 percentage points. The GDP growth rate would in- model of economic growth. Journal of Political
crease by approximately 0.3 percentage points. Whether Economy, 98 (5), 103-125.
the additional funding for education comes from a re- Barro, R. J., Sala-i-Martin, X. (2004). Economic Growth
duction in public or private consumption is irrelevant. (2nd ed.). Cambrigde, MA: MIT Press
At present, households in Poland save approxi- Campbell, J., Diamond, P., & Shoven, J. (2001). Estimat-
mately 21% of GDP, but only 3% of private savings ing the Real Rate of Return on Stocks Over the Long
is spent on education, and 97% is invested in capital. Term. Social Security Advisory Board, Washington.
However, the optimal composition of savings, holding Retrieved from: http://www.ssab.gov/publications/
all other parameters constant, is approximately 14% / financing/estimated%20rate%20of%20return.pdf
86%. Therefore, the current structure of private invest- Cichy, K. (2008). Kapitał ludzki i postęp techniczny jako
ment in Poland is far from the optimum. It is possible, determinanty wzrostu gospodarczego [Human capital
however, that private spending on education is under- and technological progress as determinants of econom-
estimated in official statistics  a substantial share of ic growth]. Warsaw: Instytut Wiedzy i Innowacji.
it is likely classified as consumption (e.g., the cost of Dhont, T., Heylen, F. (2009). Employment and growth in
accommodation, travel, books, etc.). Europe and the US  the role of fiscal policy com-
Despite certain obvious simplifications, our analysis position. Oxford Economic Papers, 61 (3), 538-565.
provides qualitative and quantitative insights into the Elmendorf, D. W., Mankiw, N. G. (1998). Government
negative effects of taxes and positive influence of educa- Debt (Working Papers No. 6470). National Bureau
tion on economic growth in Poland. It appears to capture of Economic Research.
certain important  stylized facts
, especially the rapid ac- Eurostat (2010). Taxation trends in the European Union
cumulation of human capital over the past 20 years. Nev- - Data for the EU Member States, Iceland and Nor-
ertheless, one should recall that our model neglects cer- way. Luxembourg: Publications Office of the Eu-
tain important, though transitory, factors of growth. For ropean Union. Retrieved from: http://ec.europa.
example, over the past decade, Poland experienced large eu/taxation_customs/resources/documents/
inflows of foreign capital, in the form of both FDI and taxation/gen_info/economic_analysis/tax_struc-
portfolio investment. However, a significant migration of tures/2010/2010_full_text_en.pdf
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
342
Gordon, K., Tchilinguirian, H. (1998). Marginal Ef-
fective Tax Rates on Physical, Human and R&D
Capital (Working Papers No. ECO/WKP(98)12).
OECD Economic Department.
Heckman, J. J., Jacobs, B. (2010). Policies to Create
and Destroy Human Capital in Europe (Working
Papers No. 15742). National Bureau of Economic
Research.
Konopczyński, M. (2013). Fiscal policy within a com-
mon currency area  growth implications in the
light of neoclassical theory. Contemporary Eco-
nomics, 7 (3), 5-16.
Lee, Y., Gordon, R. (2005). Tax structure and econom-
ic growth. Journal of Public Economics, 89 (5-6),
1027-1043.
Mankiw, G. N., Romer, D., & Weil, D. N. (1992).
A Contribution to the Empirics of Economic
Growth. Quarterly Journal of Economics, 107 (2),
407-437.
Manuelli, R., Seshadri, A. (2005). Human Capital and
the Wealth of Nations (Meeting Papers No. 56). So-
ciety for Economic Dynamics.
Nehru, V., Dhareshwar, A. (1993). A new database on
physical capital stock: sources, methodology and
results. Revista de análisis económico, 8 (1), 37-59.
Próchniak, M. (2013). To What Extent Is the Institu-
tional Environment Responsible for Worldwide
Differences in Economic Development. Contem-
porary Economics, 7(3), 17-38.
Romer, P. M. (1986). Increasing returns and long-run
growth. Journal of Political Economy, 94 (5), 1002-
1037.
Turnovsky, S. J. (2009). Capital Accumulation and Eco-
nomic Growth in a Small Open Economy. Cam-
bridge, UK: Cambridge University Press.
Willman A. (2002). Euro Area Production Function
and Potential Output: A Supply Side System Ap-
proach (Working Papers No. 153). European Cen-
tral Bank.
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
343
Appendix
Ä… +² -1
K
ëÅ‚ öÅ‚
Ć
K = (1 -È )Å‚A - ´ , (A5)
ìÅ‚ ÷Å‚
K
H
íÅ‚ Å‚Å‚
Proof of Proposition 1.
Ä… +²
K
Dividing both sides of equation (11) by K and substi- $
=ÈÅ‚AëÅ‚ öÅ‚ - ´ , (A6)
ìÅ‚ ÷Å‚
H
H
íÅ‚ Å‚Å‚
tuting (9), we obtain a formula for the growth rate of
productive capital: We can treat the ratio K H as a single variable (the un-
known). Then, from (A5) and (A6), it follows that the

K Y
Ć Ć
K = = (1-È )Å‚ -´ , (A1) equation K = $
can only be solved numerically, after
K
K K
substituting certain values for all parameters. (Only in
Similarly, by dividing both sides of equation (12) by H special cases can this equation be solved analytically.
and substituting (10), we obtain a formula for the rate For example, if we set Ä… =² =1 3, then this equation
of human capital growth: can be transformed into a polynomial equation of the
fourth degree and solved analytically.) Nevertheless,

H Y
$
= = È Å‚ - ´ , (A2) it is possible to  solve it graphically, by graphing the
H
H H
right-hand sides of equations (A5) and (A6). Strictly
Ć Ć
By assumption, È " (0;1) and Å‚ " (0;1) . Thus, if K < v
, speaking, we graph the rates of growth K and $
as
then capital grows more slowly than output, and con- functions of K H . It is easy to show that the function
Ć
sequently, the ratio Y K increases over time, which ac- K(K / H ) is decreasing and strictly convex. Moreover,
Ć Ć Ć
çÅ‚ çÅ‚
cording to (A1) implies that K also increases over time. K çÅ‚çÅ‚çÅ‚çÅ‚ +", and K çÅ‚çÅ‚çÅ‚çÅ‚ -´ . However,
K
K / H 0+ K / H +"
Ć
Conversely, if K > v
, then capital grows more rapidly the function $
(K / H ) is increasing, strictly concave,
Ć
çÅ‚
than output, and hence the ratio Y K , as well as K , will $
(K / H = 0) = -´ , and $
çÅ‚çÅ‚çÅ‚çÅ‚ +". The graphs of
H
K / H +"
decrease over time. Therefore, from equation (A1), it these functions are illustrated in figure A1. Due to the
follows that over time, the economy is converging to- properties of these functions, there is exactly one point
Ć
ward a balanced state, in which K = v
. A similar con- of intersection, i.e., there exists exactly one ratio K H
Ć
clusion follows from equation (A2): with the passage of for which K = $
. The values of both functions at this
time, the economy is converging toward the balanced point determine the balanced growth rate (the BGR).
state, in which $
= v
. Therefore, in the long term, the Figure A1 also indicates that the balanced state is
economy converges toward the balanced growth path, globally asymptotically stable. Notice that to the left
Ć
on which the following equality holds: of the point of intersection of the graphs, K > $
, and
hence over time, K H increases, which implies that
Ć
K = $
= v
, (A3) with the passage of time, the economy moves to the
right. However, to the right of the point of intersec-
Ć
From equations (A1) and (A2), if follows directly that tion of the graphs, K < $
, and hence over time, K H
there exists exactly one such equilibrium (it is unique), decreases, which implies that with the passage of time,
and it is globally asymptotically stable. To determine the economy moves to the left. (The direction of mo-
the equilibrium, one must solve the system of equa- tion is illustrated by the arrows in fig. A1.)
tions (A3). First, note that from the equation (4), it Notice that an increase in the value of the param-
Å‚
follows that: eter and/or a decrease in the value of ´ shifts the
K
Ć
graph of the function K(K / H ) upwards. Similarly, an
Ć
v
= (Ä… + ² )K + (1 - Ä… - ² )$
, (A4) increase in the value of the parameter Å‚ and/or a de-
crease in the value of ´ shifts the graph of the func-
H
Ć Ć
Thus, if K = $
, then K = $
= v
, and hence the system tion $
(K / H ) upwards. Thus, the BGR is an increasing
Å‚
of equations (A3) can be reduced to a single equation: function of and, simultaneously, a decreasing func-
Ć
K = $
. Unfortunately, except for certain special cases, tion of both rates of depreciation.
this equation cannot be solved analytically. To see why, However, when the share parameter È increas-
Ć
let us use equation (4) to write growth rates (A1) and es, the graph of K(K / H ) shifts up, but the graph of
(A2) in the following equivalent form: $
(K / H ) simultaneously shifts down. Therefore, the
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
344
Ć
Fig. A1. Graphs of the functions K(K / H ) and $
(K / H ) in the private ec
Ć
K, $

Ć $

K
BGR
0
K H
K H
- ´H
- ´K
Ć
Figure A1. Graphs of the functions K(K / H ) and $
(K / H ) in the private economy
Ć
relationship between the BGR and È is ambiguous. It ratio K H for which K = $
. The values of both func-
can only be established numerically, after substituting tions at this point determine the balanced growth rate
values for all parameters. (the BGR). The balanced state is globally asymptoti-
cally stable, which is illustrated in figure A2. In equi-
Ć
Proof of Proposition 2. librium K = $
, which together with (4), implies that
Ć
First, let us determine the signs of all expressions that v
= K = $
.
are marked with symbols Ai . Under the assumptions
adopted regarding the signs and the values of tax rates, Sensitivity of the results to the K/Y ratio
rates of savings, and other parameters, it can easily be Due to lack of suitable statistics, the ratio of K / Y for
shown that: Poland was calibrated based on the average value for
22 OECD countries, which is equal to 3.0. However,
, and , , (A7) in certain OECD countries, the K / Y ratio is higher,
A2 , A3, A5 , A6 > 0 A4 < 0 A2 < 1 A6 <1
while it is lower in others. This section presents the
Similarly as above, let us graph the rates of growth most important results of the paper; we set the ratio
Ć
K and $
given by (36) and (37) as the functions of of K / Y for Poland at the level of 3.3 or 2.7, instead of
K H . Using (A7) it is easy to prove that the function 3.0 (as we do in the main text). Tables 2A  4A are the
Ć
K(K / H ) is decreasing and strictly convex. Moreover, counterparts of tables 2  4 if we set K / Y = 3.3 . Simi-
Ć Ć
K çÅ‚çÅ‚çÅ‚çÅ‚ +", and K çÅ‚çÅ‚çÅ‚çÅ‚ A4. However, the larly, tables 2B  4B are the counterparts of tables 2  4
çÅ‚ çÅ‚
K / H 0+ K / H +"
function $
(K / H ) is increasing, strictly concave, if we set K / Y = 2.7 . The general conclusion is that the
$
(K / H = 0) = -´ , and $
çÅ‚çÅ‚çÅ‚çÅ‚ +". The graphs results are very insensitive to the initial value of K / Y .
çÅ‚
H
K / H +"
of these functions are illustrated in figure A2. Due All welfare gains  as measured by the GDP effect after
to the properties of these functions, there is exactly 30 years  are very similar to the results obtained in
one point of intersection, i.e., there exists exactly one the main text.
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
345
Ć
Fig. A2. Graphs of the functions K(K / H ) and $
(K / H ) in the economy with the gove
Ć
K, $

Ć
$

K
BGR
0
K H
K H
- ´H
A4
Ć
Figure A2. Graphs of the functions K(K / H ) and $
(K / H ) in the economy with the government
Table 2A. Simulation results for Poland - different tax-cut scenarios, .
K / Y = 3.3
1 pp reduction 5 pp reduction
3.54% 3.60%
Ä
L
0.5% 2.4%
3.53% 3.58%
Ä
K
0.3% 1.5%
3.54% 3.60%
Ä
H
0.5% 2.4%
3.55% 3.68%
ÄC
0.9% 4.6%
3.60% 3.90%
reduction of all tax rates simultaneously
2.2% 11.6%
Table 2A. The scenario of simultaneously reducing all tax rates by 5 pp, K / Y = 3.3 .
the BGR and structural indicators (%) baseline scenario reduction of all tax rates by 5 pp
the BGR 3.52 3.90 (the effect after 30 years= +11.6%)
C / Y
61.7 66.9
T / Y
33.2 26.3
S / Y
20.6 22.3
IK / Y
20.0 21.6
GE / Y
5.8 5.8
IH / Y
0.6 0.7
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
346
Table 4A. Scenarios of increasing public and private spending on education, K / Y = 3.3
C
A B
Baseline scenario Increase in private
The BGR and structural Increase in public Increase in private
Å‚ = 5.84%
E
spending on education
indicators spending on education savings by 1 pp of GDP
Å‚ = 2 5.0 2%
by 1 pp of GDP
Å‚ = 2 6.17 %
(%) by 1 pp of GDP
È = 2.99 %
Å‚ = 2 6.21%
È = 2.99 %
Å‚ = 6.8 4%
E
È = 7.47%
3.84 3.74 3.84
GDP effect GDP effect GDP effect
the BGR 3.52
after 30 years after 30 years after 30 years
+9.8% +6.6% +9.9%
C / Y
61.7 61.7 60.9 60.8
T / Y
33.2 33.3 33.0 33.1
S / Y
20.6 20.6 21.6 21.6
IK / Y
20.0 20.0 21.0 20.0
GE / Y
5.84 6.84 5.84 5.84
IH / Y
0.62 0.62 0.65 1.62
Table 2B. Simulation results for Poland  different tax-cut scenarios, K / Y = 2.7 .
1 pp reduction 5 pp reduction
3.50% 3.57%
Ä
L
0.5% 2.5%
3.50% 3.54%
Ä
K
0.3% 1.7%
3.50% 3.57%
Ä
H
0.5% 2.5%
3.52% 3.64%
ÄC
0.9% 4.8%
3.56% 3.86%
reduction of all tax rates simultaneously
2.3% 12.1%
Table 3B. The scenario of simultaneously reducing all tax rates by 5 pp, K / Y = 2.7 .
the BGR and structural indicators (%) baseline scenario reduction of all tax rates by 5 pp
the BGR 3.49 3.86 (the effect after 30 years= +12.1%)
C / Y
62.1 67.2
T / Y
32.7 25.8
S / Y
20.7 22.4
IK / Y
20.1 21.8
GE / Y
5.84 5.84
IH / Y
0.62 0.67
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149
How Taxes and Spending on Education Influence Economic Growth in Poland
347
K / Y = 2.7
Table 4B. Scenarios of increasing public and private spending on education, .
C
A B
Baseline scenario Increase in private
The BGR and structural Increase in public Increase in private
Å‚ = 5.84%
E spending on education
indicators spending on education savings by 1 pp of GDP
Å‚ = 2 5.0 2%
by 1 pp of GDP
(%) by 1 pp of GDP Å‚ = 2 6.17%
Å‚ = 2 6.21%
È = 2.99 %
È = 2.99 %
Å‚ = 6.84%
E È = 7.47%
the BGR 3.49 3.80 3.70 3.81
GDP effect after 30 GDP effect after 30 GDP effect after 30
years +8.6% years +6.9% years +8.8%
C / Y
62.1 62.0 61.3 61.2
T / Y
32.7 32.8 32.5 32.6
S / Y
20.7 20.70 21.7 21.7
IK / Y
20.1 20.1 21.1 20.1
GE / Y
5.84 6.84 5.84 5.84
IH /Y
0.62 0.62 0.65 1.62
www.ce.vizja.pl Vizja Press&IT
Vol.8 Issue 3 2014 329-348 Michał Konopczyński
348
CONTEMPORARY ECONOMICS DOI: 10.5709/ce.1897-9254.149