Changes in Farm Production in the Context of Overinvestment: A Case Study from Poland

Abstract

Investments are the development core of every economic operator and the driving force for national economic development. While finding the optimal investment point is extremely difficult, every operator may seek to attain a level of fixed assets which allows them to maintain an increase in production and keep their income flowing. The difficulties in finding this optimum may lead to overinvestment. Hence, this paper estimates the parameters of the production function at various investment levels. Its purpose is to identify the relationship between the investment levels, the use of productive inputs, and growth or decline in production (output) levels. This study used microdata for 3273 Polish farms, as retrieved from the FADN (Farm Accountancy Data Network). The differences found at various farm investment levels allow for highlighting some problematic areas, e.g., in the context of excessive capital use relative to yearly increments in production values. The proposed method of analyzing overinvestment based on the production function can be extended to other countries. According to this paper, farms that invested too little or too much in relation to their production potential showed the worst economic performances, including the greatest declines in production.

1. Introduction

Investments play a crucial role in agriculture, as they have the capacity to increase farm productivity and thus can positively affect agricultural production volumes and incomes. However, when financial resources used in supporting the investment process can be accessed too easily, farms might face what is referred to as overinvestment. Therefore, the purpose of this study is to identify the economic consequences of excessive and insufficient investments in fixed assets in the Polish farming sector. The first step in achieving this goal will be to present the background by showing how farms change their production processes, and by highlighting their essential role in ensuring food security. This section will also describe farm investments and define the problem of overinvestment. Next, this study will indicate the changes in farm production volumes by estimating the initial degree of farm overinvestment and the parameters of the production function at different investment levels. This study addresses farm overinvestment on a comprehensive basis and analyzes the way it affects changes in production.

2. Background

2.1. The Significance of Investment in Agriculture

In accordance with the neoclassical growth model, the aggregate investment level is believed to be influenced by the increase in incomes, the relative price of capital, taxes, and other market processes [1]. Investments in agricultural fixed assets provide a framework for reducing poverty in the long run, increase land productivity, and reinforce environmental protection measures [2,3]. Numerous studies point to the need for agricultural investments because improvements in agricultural performance have always been the basis for increasing total food production [4]. Efficiency plays an important role in ensuring food security [5,6,7] and in raising farm revenues. Although both current and strategic (including investment-related) decisions made by individual agricultural producers have a microeconomic dimension and pursue individual goals, when looked at from a social perspective, they also affect the overarching objective of agriculture, which is to ensure food security. This is one of the reasons for using public funds to support farm investments [8].

2.2. Agriculture in the EU and in Poland

Agricultural production intrinsically depends on both macro- and microeconomic conditions [9], and it is considered a major factor in ensuring food self-sufficiency, preserving natural resources (mainly land), driving rural development, and delivering social and cultural benefits [10]. It also forms part of the landscape, as it is the main form of land use in the European Union (EU), accounting for more than 47% of its total territory [11].

Compared to the EU, Poland has a fragmented agriculture. In 2021, there were 1,302,300 farms in Poland, accounting for 14.4% of all EU farms. The only EU member state with an even greater number of agricultural holdings is Romania. As regards the structure of employment, as much as 8.4% of the Polish population is employed in agriculture. Once again, higher ratios are recorded only in Romania (18.6%) and Greece (11.4%) [12]. In Poland, the contribution of agriculture to the total GDP is ca. 2.3% but has been on a decline for several years now (and so has the share of people employed in it). The sector plays an important role in the Polish economy and follows a trend characteristic of developing countries, namely, a decreasing share of agriculture in both employment and GDP.

2.3. Agriculture, Agricultural Investment, and Global Food Security

Ensuring food security itself is a matter of socioeconomic importance because the lack of it has multiple consequences that affect every aspect of life [13,14,15]. Therefore, food security is the key challenge facing agriculture. Unfortunately, climate change—a major environmental problem [16,17]—further accentuates the imbalance in food security [18,19,20]. Moreover, agriculture is a strategic sector which ensures food security, alleviates the consequences of poverty, and enables preserving core natural resources [21]. Today, most people on the planet have enough food [22], but the ongoing changes (in the climate, for instance) make it reasonable to focus research on how to assure the production of sufficient volumes of food. Considering the limited area of arable land, the methods for making agriculture more efficient while improving food security include the genetic modification of seeds; crop diversification, which leads to the sustainable development of agricultural production [23]; a more efficient use of productive inputs such as fertilizers; and agricultural mechanization [24]. All of this can be impacted by another dimension of improving food security, i.e., aid policies [25,26]. Other aspects include agricultural education or agricultural extension offered to improve production technology and, as a consequence, contribute to food security [27], and the development of smarter, more crisis-proof food supply chains [28]. Also, having control over food waste could be yet another way of reducing food insecurity [29]. However, investments in fixed assets and in upgrading production technology are the key measures that may be taken to make agriculture more productive because they can significantly affect agricultural productivity.

This paper is about Poland, where food security is viewed at a general, three-dimensional level. Hence, food security is believed to be assured if food is physically and economically available to everyone and is also safe [30]. The economic dimension is thus strictly related to growing production costs, and these, as a consequence, may cause an increase in food prices across the sector and deteriorate food security, especially for vulnerable social groups. This context gives rise to an important problem: how to determine an economically and socially sound investment amount. Insufficient investments may contribute to reducing the production volume and thus to making food less physically available (or lead to a significant increase in the labor intensity of farming, which, however, seems unlikely under the current social conditions because the labor-intensive path of agriculture is being abandoned).

In Poland, the essential obstacle to investments is agrarian fragmentation. Also, just like in other Central and Eastern European countries, many farms struggle with insufficient amounts of capital and face structural problems [31].

A significant role is played by investments in innovative high technologies, including in the agricultural sector (e.g., improvements to supply chains) [32,33]. Rather than replacing labor, they make it more efficient [34]. A number of technological innovations are intended to increase agricultural productivity; in some parts of the world (where climate change significantly affects agricultural production), they help people adapt to climate change [35]. This is all the more important since, according to estimations, agriculture is supposed to be accountable for as much as one-quarter of all greenhouse gas emissions [36]. It was also discovered that the COVID-19 pandemic and the war in Ukraine caused major disruptions in supply chains and thus undermined food security.

2.4. The Specifics of Overinvestment in the Economy and in Agriculture

It is important to be aware of the risks associated with the investment process, which takes on a special dimension in agriculture. A frequent problem is that the importance of investments is underestimated. This is due to the particularities of agriculture, which requires an intense though relatively short use of fixed assets because of production seasonality. In this context, accurately assessing the materiality of investments is essential to achieving production maximization. Investments can be economically unviable, which means a decline in economic performance in the long run for the business concerned. When attempting to estimate the investment viability, it can be noted that excessive investment (which primarily means a mismatch between the investment level and production scale) ultimately results in dysfunctional economic agents, which means that the labor productivity either falls or grows less than proportionally to the increase in the fixed assets employed. In this paper, the above phenomenon is referred to as overinvestment.

Different kinds of enterprises overinvest by making irrational use of their assets. Essentially, this is based on an overly optimistic evaluation of the market conditions. The good situation in the markets can cause an uncontrolled increase in investment, and sometimes debt. As a consequence, the investments do not meet the expected economic effects [37,38,39]; often, the expected return on investment projects is below the interest rate offered in capital markets [40,41,42].

Irrational, reckless investments cause tremendous negative effects both in the sector concerned and in the economy as a whole. Excessive investment can result in an excess production capacity, production inefficiency, the distortion of profits (too much debt service), and unemployment [43]. In agriculture, the consequences of over- or underinvestment include fluctuations in agricultural production levels (due to cash flow disruption); variability in production and prices [44,45]; increased production costs; and a decline in competitiveness [45]. Indeed, production and price volatility (in the form of production surpluses and price reductions, as mentioned above) are ultimately reflected not only in revenues but also in farms becoming less competitive. In a broader context, economists indicate that when inflation arises, the prices paid by farmers grow faster than those they obtain for their products, which has a deteriorating effect on the price-to-cost ratio [46]. As farmers are price-takers, they do not have the direct capacity to transfer the higher costs of inputs to consumers [47]. In addition to deteriorating the quality of the goods produced, this can undermine the international competitiveness of the whole sector. The increase in production costs is related, for instance, to the maintenance costs of excessive assets. All this is indirectly related to overinvestment, which can generate additional, unnecessary production costs and thereby affect production efficiency.

The rationality of farm investments has been researched and addressed in the literature. According to Musshoff and Hirschauer [48], farmers are limited in their cognitive abilities when making financial decisions. In turn, the study by Boahene [49] was focused on economic efficiency issues at the farm level. His socioeconomic model explained the decision-making process of smaller farmers in developing countries. As pointed out by Wang et al. [50], there is a gap between farmers’ willingness to invest and what they actually do. This may lead one to conclude that overinvestment is often derived from farmers’ irrationality. Overinvestment at the farm level leads to increased depreciation and, as a consequence, a drop in agricultural incomes [51]. At the economy level, the simplest description of overinvestment is the case where economic operators invest more than they should [52]. This can have an indirect impact on and disrupt food security; this is the case when farms generate smaller incomes than before and sometimes have no other option but to curb production because of excessive investments because farmers are not able to increase their production costs. Hence, in extreme cases, overinvestment leads to a paradox where investing too much money results in reducing the production volume.

Currently, scientists approach overinvestment within a broad range of contexts—from overinvestment in capital markets to overinvestment in fixed assets. For example, Xia et al. [53] argue that overinvestment, caused by heavy subsidization, is one of the most serious barriers to wind energy development. Chen et al. [54] conclude that overinvestment is mainly the fault of managers who feel more tempted to gain higher profits than they are afraid of the consequences of overinvestment. In contrast, in the environmental context, Bilyay-Erdogan et al. [55] noted that ESG (environment, social responsibility, corporate governance) reduces the problem associated with overinvestment. Also, corporations which rely on digitization are less prone to underinvestment and overinvestment [56]. Such a relationship indicates the importance of progress in optimizing investment decisions. However, case studies on overinvestment in farm fixed assets are less common in the literature. Yet, such a risk exists, especially with the greater availability of subsidies under the Common Agricultural Policy. These (non-refundable) funds create the temptation to make investments that are not always needed on farms. This paper bridges this knowledge gap by juxtaposing farm overinvestment and the effects it has on production.

Attention should therefore be paid to the effects of investments, especially those that are not justified. One of the methods for conducting this kind of research is the analysis of the production function. The Cobb–Douglas production function combined with data analysis is used in describing the relationship between productive inputs and production itself [57]. In the context of overinvestment, this allows for a comparison of the output achieved at different levels of investment. The production function and the estimates of the productivity of the inputs are of major importance not only to agriculture but to the economy as a whole [58]. Every economic operator who intends to increase production seeks an optimum investment level [59]. Agricultural economists dealing with the optimization of production processes often use the Cobb–Douglas production function when examining the efficiency of agricultural businesses. Researchers who have employed the Cobb–Douglas function in addressing this issue include Constantin et al. [60], Bezat [61], Salehi et al. [62], Ghoshal and Goswami [63], Umar et al. [64], and Omar and Fatah [65]. The assumption behind the production function is that all companies are technically efficient, and the function can be used in gauging the level of inputs needed to attain a given production volume (output). Therefore, the aim of this study is to identify the economic consequences of excessive investment and underinvestment in fixed assets on farms in Poland.

3. Materials and Methods

3.1. The FADN and the Research Sample

This research used farm microdata from the European Union’s Farm Accountancy Data Network (FADN), a European system for accounting data collection from all member countries of the EU. Data were collected from commercial farms in accordance with a unified methodology [66]. The farms presented in this study keep accounting records in accordance with the unified FADN methodology, which applies consistently to all entities surveyed and is invariable over time. This makes it possible both to compare farms between each other and to study the dynamics of phenomena. The FADN covers commercial farms that collectively account for 90% of a country’s standard output (SO) (the standard output is the 5-year average of the production value of a specific production activity (crop or animal) obtained in 1 year from 1 hectare or from 1 animal (with the exceptions of edible mushrooms—100 m²; poultry—100 heads; bees—1 bee trunk, i.e., 1 bee colony), under average production conditions for the region). This provided a basis for selecting the research sample, which, in the case of Poland, comprises ca. 12,000 holdings. Two basic sampling criteria were used: the economic size—determined by the absolute value of the standard output (SO)—and the type of farming, which depends on how much each farming activity contributes to the total SO. The farms selected for this study form part of the Polish FADN research sample. Due to the fundamental importance of changes in the assets-to-labor ratio and labor productivity, the continuity of record keeping throughout the period was essential in the sampling procedure. This study needed to determine how the phenomena evolved over time, and therefore data were retrieved from farms which kept continuous records in 2010–2019. In total, there were 3273 of them in the database, which accounts for ca. 27% of farms covered by the system each year (12,167 in 2019).

The following variables retrieved from the database were used in order to determine how the production function changes at various investment levels: gross value added, depreciation, investment subsidy installments, operating subsidies, total labor inputs, fixed assets, land, permanent crops and production quotas, total output, total labor inputs, total costs, and agricultural land area.

3.2. Farm Investment Levels

Agricultural overinvestment can be defined as a condition wherein investments are excessively high compared to the production potential [67]. Two essential parameters need to be developed in order to determine the levels of overinvestment: the assets-to-labor ratio and labor productivity. This paper assumes that increasing the value of farm assets through investments is reasonable if it results in a proportional growth in labor productivity. Higher labor productivity determines the development of farms, including by increasing their efficiency. Therefore, overinvestment is defined as a situation wherein the following apply:

• The increase in the value of assets results in an absolute decline in labor productivity, which may be due to the high maintenance costs of particular assets (e.g., depreciation, insurance, repairs). The above is defined as absolute overinvestment;

−

Labor productivity grows at a slower rate than the value of assets. This is referred to as relative overinvestment.

The calculations were based on microdata and used the FADN’s system variables. The panel data were created by calculating two-year arithmetic means for each farm, resulting in a dataset spanning 5 periods. Averaging was used to eliminate possible distortions in agricultural markets, such as those caused by productive input prices. Growth and decline rates were then calculated as the next step:

$${LP}_t={\displaystyle \frac{{\sum }_t^{t+1}\left(\frac{SE410 - SE360 - SE406 - SE605}{SE010}\right)}{2}}$$

(1)

$$∆LP=\left({\displaystyle \frac{{LP}_{t4}-{LP}_{t0}}{{LP}_{t0}}}\right)\ast 100\%$$

(2)

where LP is the labor productivity; LP_t4 is the labor productivity in period t4; LP_t0 is the labor productivity in the base period; ∆LP is the change in labor productivity; SE410 is the gross value added (includes total production less intermediate consumption, plus or minus the balance of surcharges and taxes on operating activities) (PLN); SE360 is the depreciation (determined based on the replacement value) (PLN); SE406 is the investment subsidy installments (portions of investment grants to be settled within 12 months) (PLN); SE605 is the operating subsidies (other than investment subsidies) (PLN); SE010 is the total labor inputs (AWU).

The gross value added is a key metric of the agricultural efficiency. After the depreciation is deducted, it becomes the net value added, the basic category of agricultural income [68]. Investment subsidy installments and operating subsidies were deducted as the next step. The reason for using the net added value (rather than the family farm income) was the need to eliminate the cost of external factors (hired labor, rents, and interest on loans) from the calculations in order to have a standardized metric of the economic performances of farms which rely on both their own and external productive inputs in their operations. In turn, subsidies were removed from the calculations because public support should not be considered a metric of labor productivity in economic terms.

Changes in the assets-to-labor ratio were calculated as the next step, with the value of the fixed assets less the value of land per FTEs (full-time employees) used as the metric. The FTE ratio was used so as not to disturb the estimation of the relevant parameters. The rationale behind the above approach is that overinvestment is a problem which ultimately boils down to a mismatch between the output and the extent of investments in machinery and buildings. Just like in the case of labor productivity, the study calculated the average values for the 5 selected periods (1 period is the averages of two years), and defined the growth/decline rate:

$${ALR}_t={\displaystyle \frac{{\sum }_t^{t+1}\left(\frac{SE441 - SE446}{SE010}\right)}{2}}$$

(3)

$$∆ALR=\left({\displaystyle \frac{{ALR}_{t4}-{ALR}_{t0}}{{ALR}_{t0}}}\right)\ast 100\%$$

(4)

where ALR is the assets-to-labor ratio; ALR_t4 is the assets-to-labor ratio in period t4; ALR_t0 is the assets-to-labor ratio in the base period; ∆ALR is the change in the assets-to-labor ratio; SE441 is the fixed assets (including agricultural land, farm buildings, forest plantings, machinery and equipment, and livestock) (PLN); SE446 is the land, permanent crops, and production quotas (PLN); SE010 is the total labor inputs (AWU).

After calculating the two specifications necessary to determine the investment levels, farm data were distributed between the groups in accordance with the authors’ own methodology:

• Absolute overinvestment: this is the case for farms where the labor productivity drops while the assets-to-labor ratio grows:

  ΔLP < 0 ∧ ΔALR > 0
• Relative overinvestment: this is the case for farms where both the labor productivity and the assets-to-labor ratio are on an increase but the increase in the assets-to-labor ratio is greater than the increase in the labor productivity:

  ΔLP > 0 ∧ ΔALR > 0 ∧ ΔLP < ΔALR
• Underinvestment: this is the case for farms where both the labor productivity and the assets-to-labor ratio are on a decline:

  ΔLP < 0 ∧ ΔALR < 0
• Optimum investments: this is the case for farms where both the labor productivity and the assets-to-labor ratio are on an increase, and the labor productivity grows faster than the assets-to-labor ratio:

  ΔLP > 0 ∧ ΔALR > 0 ∧ ΔLP > ΔALR
• Optimum investments with no increase in the assets-to-labor ratio: this is the case for farms where the labor productivity grows while the assets-to-labor neither grows nor declines:

  ΔLP > 0 ∧ ΔALR < 0 ∧ ΔLP > ΔALR

The five investment levels were drilled down using all possible combinations of labor productivities and assets-to-labor ratios. The assets-to-labor ratio is an important part of the analysis of overinvestment because the production volume has a fundamental impact on labor productivity and depends on the mix of capital and labor inputs [69]. It is assumed that low levels of the assets-to-labor ratio have an adverse effect on how efficiently labor is used [70]. As regards the capital-to-labor ratio, higher values are also desirable, as they suggest greater amounts of investment. According to Lewis’ dual economic model [71,72], an increase in agricultural labor productivity causes the release of surplus labor to other sectors of the economy and is therefore a prerequisite for economic development. Thus, it is important to take these two variables into account in the context of analyzing the degree of overinvestment.

Table 1. Labor productivity (PLN/AWU) on farms by level of overinvestment in 2010–2019.

Farm Type by Overinvestment Level	Number of Farms	Labor Productivity (PLN/AWU) in t0	Labor Productivity (PLN/AWU) in t4
Absolute (I)	653	135,367	54,078
Relative (II)	209	152,050	64,477
Underinvested (III)	997	94,677	56,110
Optimum investment, no increase in ALR (IV)	600	103,957	91,531
Optimum investment, increase in LP and ALR (V)	496	121,826	154,134
Other (VI)	318	67,233	81,500

Source: own compilation based on FADN microdata, n = 3273.

presents the labor productivity for each overinvestment group to show the scale of overinvestment.

Labor productivity decreased on farms at the absolute and relative overinvestment levels, in the underinvested group, and in those demonstrating optimum investment amounts with no increase in the ALR. This proves that only optimum investment provides an increase in labor productivity. Thus, these farms initially also recorded the highest labor productivity. This means that farms which increased their labor productivity achieved faster growth in their total output than in their labor resources. This is one of the conditions for the development of farms. The assets-to-labor ratio is addressed in the next step (

Table 2. Assets-to-labor ratios (PLN/AWU) for farms grouped by level of overinvestment in 2010–2019.

Farm Type by Overinvestment Level	Number of Farms	Assets-to-Labor Ratio (PLN/AWU) in t0	Assets-to-Labor Ratio (PLN/AWU) in t4
Absolute (I)	653	114,350	143,598
Relative (II)	209	124,628	160,911
Underinvested (III)	997	74,959	50,806
Optimum investment, no increase in ALR (IV)	600	83,525	70,100
Optimum investment, increase in LP and ALR (V)	496	103,539	152,622
Other (VI)	318	55,565	63,136

Source: own compilation based on FADN microdata, n = 3273.

).

The results for the assets-to-labor ratio are similar to what was established for labor productivity. A higher value contributes to improvements in labor productivity. The assets-to-labor relationship is an important part of the analysis of overinvestment because the fundamental influence on labor productivity is the volume of production, which depends on the combination of capital and labor inputs [73]. It is assumed that low levels of the assets-to-labor ratio have an adverse effect on how efficiently labor is used [74]. As regards the capital-to-labor ratio, higher values are also desirable, as they suggest greater amounts of investment. This, in turn, is to some extent related to the implementation of technical progress in agriculture, which results in the attainment of higher levels of production efficiency [75].

3.3. Methodology for Using the Cobb–Douglas Production Function in General and in Research on Farm Overinvestment

Douglas’ research was focused on the elasticity of the labor and capital supply, and on the impact of changes in it on income distribution [76]. He investigated different structures of the labor market and their impacts on the wage levels and competition in that market. In turn, his research conducted together with Cobb resulted in the development of an innovative production function which describes the relationship between the output (product) and productive inputs, such as capital and labor. They analyzed the elasticity of the production efficiency in relation to different combinations of productive inputs. This allowed for understanding which changes in inputs may affect the output [77]. The Cobb–Douglas production function has been broadly used in economic and empirical analyses as a way to model and assess the production efficiency in different industries and economies. The classic Cobb–Douglas power function takes the following form [78]:

$$Y={\beta }_0{L}^{{\beta }_1}{K}^{{\beta }_2}$$

(5)

where Y is the output; L is the labor inputs; K is the capital inputs β₀ is the constant determined by technological and organizational progress; ${\beta }_{1}i{\beta }_2$ are the output elasticities of the capital and labor, respectively.

It was only a refinement of the production function which made it more suitable for agriculture but did not change its formula or interpretation. Supplemented as described above, it can be presented as follows [78]:

$$Y={\beta }_0{L}^{{\beta }_1}{K}^{{\beta }_2}{Z}^{{\beta }_2}$$

(6)

where Y is the output; L is the labor inputs (expressed as the number of employees, man-days, or FTEs); K is the capital inputs; Z is the land inputs (expressed as the area of agricultural land in ha); β₀ is the constant determined by technological and organizational progress; β₁, β₂, and β₃ β₁, β₂ i β₃ are the output elasticities of labor and capital, respectively. The elasticity values depend on whether they are determined by the available technology [78].

An important part of the Cobb–Douglas production function is the ability to estimate economies of scale. As reported by Osti [79], some authors have used it for this purpose. The economies of scale of production are expressed by the sum of the elasticities of capital, labor, and land. If these productive inputs grow by one percent, the output should be expected to increase by the total resulting percentage. The economies of scale were studied by Griliches and Ringstad [80], who also focused on the way that the production function is formulated. This allowed them to estimate the significance of each parameter comprising the production function. In this paper, it is important to demonstrate the economies of scale in the context of capital, labor, and land. Note, however, that capital itself plays a particular role in the investment t.

Overinvestment is one of the dimensions of inefficiency [81]. In turn, efficiency can be assessed by analyzing the Cobb–Douglas production function, a mathematical formalization of the relationships between the output and inputs used in the production process [82]. In this paper, the Cobb–Douglas function was used to investigate the impact of inputs on production volumes at different levels of overinvestment. The variables for the model were selected based on the relevant literature and on data availability in the FADN. The expenditure incurred at the farm level means the use of three productive inputs: labor, land, and capital. The most frequently used mathematical relationship, as mentioned above, is the Cobb–Douglas function, which takes the time factor into account:

$$Q={\beta }_0{L}^{{\beta }_1}{K}^{{\beta }_2}{Z}^{{\beta }_3}{\beta }_4^t$$

(7)

Its linearized form is given as follows:

$$\mathrm{l}\mathrm{n} \left(Q\right)={\mathrm{l}\mathrm{n} (\beta }_0{L}^{{\beta }_1}{K}^{{\beta }_2}{Z}^{{\beta }_3}{\beta }_4^t)$$

(8)

From the properties of logarithms, the following equation is obtained:

$$\begin{gathered} \ln \left(Q\right)=\ln \left({\beta }_0\right)+\ln \left(L^{{\beta }_1}\right)+\ln \left(K^{{\beta }_2}\right)+\ln \left(Z^{{\beta }_3}\right)+\ln \left({\beta }_4^t\right)\ln \left(Q\right) \\ =\ln \left({\beta }_0\right)+{\beta }_1\ln \left(L\right)+{\beta }_2\ln \left(K\right)+{\beta }_3\ln \left(Z\right)+tln\left({\beta }_4\right) \end{gathered}$$

(9)

where Q—SE 131 is the total output (PLN); L—SE010 is the total labor inputs (AWU); K—SE270 is the total costs (PLN); Z—SE025 is the agricultural land area (ha); t is the time (years, t = 1, 2, …, 10, t = 1 for 2010); β₀, β₁, β₂, β₃, and β₄ are the output elasticities of labor, capital, and land, respectively.

The parameters β₁, β₂, and β₃, as indicated earlier, represent the production elasticity for each productive input. The parameter associated with the time variable (t) should be interpreted as the average growth rate in the study period (calculated as (β₄ − 1) × 100%).

The above model was estimated using the least-squares method. The Cobb–Douglas function models were estimated for all levels of overinvestment. The dataset consisting of microdata for each farm covered by the analysis (and entered into the database) was used to estimate changes in the Cobb–Douglas function model. The farms were attributed to classes in accordance with the previously identified overinvestment levels in order to standardize the database and the results. The model was initially estimated based on 32,730 observations, but 74 of them were excluded due to the incompleteness of the logarithmized data. Thus, 32,656 observations were used in estimating the form of the function and in further analysis.

Table 3. Basic statistics of variables.

Variable	Q (PLN)	K (PLN)	L (AWU)	Z (ha)
MEAN	308,475.89	256,332.79	2.15	43.18
Q50 (median)	149,801.99	117,791.50	1.85	23.37
MIN	−368,172.39	4299.00	0.11	0.00
MAX	30,233,247.81	32,639,524.00	111.35	3487.70
Q25	74,638.93	62,163.00	1.41	14.16
Q75	307,111.00	238,864.00	2.25	43.10
SD	779,236.75	776,589.87	2.85	106.89
Coefficient of variation (SD/mean)	253%	303%	133%	248%
Skewness	16.68	19.84	19.57	16.85
Kurtosis	435.11	569.85	538.51	410.52
N	32,730	32,730	32,730	32,730

Source: own compilation based on FADN microdata, n = 3273.

shows the basic descriptive statistics for the variables used in Cobb–Douglas functions. As shown by the analysis of the descriptive statistics for the variables Q, K, L, and Z, their mean values are 308,475.89, 256,332.79, 2.15, and 43.18, respectively. Their medians are lower than the means, which suggests an asymmetric distribution with long right tails. There are high levels of coefficients of variation, with 253% for Q, 303% for K, 133% for L, and 248% for Z. The ranges for the variables Q, K, L, and Z are very wide, indicating the presence of extreme outliers (from −368,172.39 to 30,233,247.81 for Q, from 4299.00 to 32,639,524.00 for K, from 0.11 to 111.35 for L, and from 0.00 to 3487.70 for Z). The skewness values for the variables Q, K, L, and Z are 16.68, 19.84, 19.57, and 16.85, respectively, which means that they are strongly skewed to the right. The right distribution tail is longer than the left one, indicating that the values tend to concentrate on the left side of the distribution. The kurtosis values (435.11 for Q, 569.85 for K, 538.51 for L, and 410.52 for Z) suggest that the distribution of these variables has a higher peak and heavier tails than those of a normal distribution. Once again, it also reveals a great number of extreme values (outliers). Due to the high variability and asymmetry of the sample data and the presence of extreme outliers, the decision was made to explain the variable Q using a Cobb–Douglas function, with the explanatory variables K, L, and Z expressed as terms of a power function and the time (t) as a term of an exponential function. The logarithmic transformation of the variables led to a reduction in the variability and provided a more homogeneous dataset while it also eliminated negative values. The model was cleaned of outliers before further calculations were made, making the results more reliable and universal.

4. Results and Discussion

The estimated production functions for the investment levels (labeled as farm types) are presented in

Table 4. Estimated farm production models grouped by overinvestment level (linearized form).

Farm Type by Overinvestment Level	Variable	Coefficient	Standard Error	p-Value
Absolute (I)	const	0.669	0.062	<0.0001
	lnK	0.960	0.007	<0.0001
	lnL	0.043	0.009	<0.0001
	lnZ	0.022	0.007	0.0011
	t	−0.026	0.001	<0.0001
Relative (II)	const	1.208	0.095	<0.0001
	lnK	0.924	0.010	<0.0001
	lnL	0.056	0.014	<0.0001
	lnZ	0.000	0.008	0.9820
	t	−0.005	0.002	0.0221
Underinvested (III)	const	0.011	0.049	0.8194
	lnK	1.026	0.005	<0.0001
	lnL	0.081	0.008	<0.0001
	lnZ	−0.042	0.005	<0.0001
	t	−0.015	0.001	<0.0001
Optimum investment, no increase in ALR (IV)	const	0.677	0.057	<0.0001
	lnK	0.953	0.006	<0.0001
	lnL	0.155	0.010	<0.0001
	lnZ	−0.009	0.005	0.068
	t	0.011	0.001	<0.0001
Optimum investment, increase in LP and ALR (V)	const	0.675	0.064	<0.0001
	lnK	0.964	0.007	<0.0001
	lnL	0.100	0.011	<0.0001
	lnZ	−0.032	0.006	<0.0001
	t	0.013	0.002	<0.0001
Other (VI)	const	0.128	0.085	0.1293
	lnK	1.014	0.009	<0.0001
	lnL	0.083	0.014	<0.0001
	lnZ	−0.067	0.008	<0.0001
	t	0.005	0.002	0.0162

Source: own compilation based on FADN microdata, n = 3273.

(linearized form) and

Table 5. Estimated farm production models grouped by overinvestment level (original form).

Farm Type by Overinvestment Level	Estimated Form of the Function	R2	ve	n
Absolute (I)	$\widehat{Q}=1.952·K^{0.960}·L^{0.043}·Z^{0.022}·{0.974}^t$	94.0%	1.9%	6573
Relative (II)	$\widehat{Q}=3.348·K^{0.924}·L^{0.056}·Z^{-0.0002}·{0.995}^t$	93.5%	2.1%	2141
Underinvested (III)	$\widehat{Q}=1.011·K^{1.026}·L^{0.081}·Z^{-0.042}·{0.985}^t$	91.4%	2.5%	9931
Optimum investment, no increase in ALR (IV)	$\widehat{Q}=1.969·K^{0.953}·L^{0.155}·Z^{-0.009}·{1.011}^t$	90.4%	2.4%	5801
Optimum investment, increase in LP and ALR (V)	$\widehat{Q}=1.965·K^{0.964}·L^{0.100}·Z^{-0.032}·{1.014}^t$	92.0%	2.4%	4897
Other (VI)	$\widehat{Q}=1.137·K^{1.014}·L^{0.083}·Z^{-0.067}·{1.005}^t$	88.2%	2.8%	3313

Source: own compilation based on FADN microdata, n = 3273.

(original form). The quality of each model was assessed using metrics which show how much of the total variation is explained by it (the R2 coefficient of determination) and how well it matches empirical data (the v_e coefficient of variation in the random effect). The coefficients of determination for the estimated functions fell within the interval of 88.2% to 94%; i.e., they exceeded 80% for all the models. This means that all of them explained a large portion of the total variation through the set of variables they used. The greater the coefficient of determination, the better the match between the regression function and the data. In turn, the coefficients of variation in the random effect fluctuated between 1.9% and 2.8%. Since they did not exceed the threshold of 20%, this means that the model fits the empirical data very well (Table 4). Note that the good match between the models and empirical data, as well as the statistical significance of it, reflect the importance of overinvestment analysis. Table 4 presents the estimated parameters of the linearized-form models for each farm type. On this basis, Table 5 presents the estimated models in their original form after the appropriate transformations.

The statistical verification of the models was complemented by analyzing the statistical significance of the estimated parameters of the explanatory variables (based on the p-value).

The p-value is used in statistical testing because of its versatility, generality, and accuracy. During the testing procedure, it is compared against the α level of significance. If the p-value is smaller than the α, the null hypothesis is rejected in favor of the alternative hypothesis, suggesting that the former is not probable. This, in turn, means that the parameter is statistically significant (i.e., the variable related to it has an impact on the dependent variable). Usually, the α is set at 0.05 (5%). The statistical verification revealed which productive inputs are statistically significant to production volumes for each type of farm grouped by investment level. In each of the analyzed levels of overinvestment, the substitution of labor for capital is noted. By the same token, however, it should be emphasized that underinvested farms still require more capital. In contrast, land in agriculture itself is essential for production. Over the years studied, this value was constant for most farms and the model does not account for its changes. A summary of the significance levels for particular variables in the estimated model is presented in

Table 6. Significance of identified variables for farms grouped by investment level.

Farm Type by Investment Level	Variable (Productive Input)
	Capital (K)	Labor (L)	Land (Z)
Absolute (I)	Below 0.0001	Below 0.0001	0.001
Relative (II)	Below 0.0001	Below 0.0001	0.982
Underinvested (III)	Below 0.0001	Below 0.0001	Below 0.0001
Optimum investment, no increase in ALR (IV)	Below 0.0001	Below 0.0001	0.068
Optimum investment, no increase in LP or ALR (V)	Below 0.0001	Below 0.0001	0.068
Other (VI)	Below 0.0001	Below 0.0001	Below 0.0001

Source: own compilation based on FADN microdata, n = 3273.

.

The impacts of capital (K) and labor (L) proved to be statistically significant in all the estimated models. This means that these two inputs have a continuous and significant impact on output. In turn, land (Z) was statistically significant at 0.05 for four farm types and was not for two: farms at optimum investment levels with no increase in the assets-to-labor ratio and in those affected by relative overinvestment. Despite this variable not being statistically significant, the decision was made to include it in the model because of its importance to the agricultural production process. However, the results show that most farms covered by this study have no rational reason to continue investing in land. This might suggest that they should shift to another production model (i.e., embark on the capital-intensive development path). Also, if land (Z) was not covered by the model, it could have resulted in overestimating the contribution of capital (K) and labor (L) to farm production processes [81]. The results also show that it is reasonable to gauge overinvestment with the assets-to-labor ratio and labor productivity. A study by Czyżewski and Kryszak [82] found labor to be less significant, which led them to conclude that it could be indicative of reaching the end of the labor-intensive development path. However, at the same time, a consistent increase in agricultural labor productivity contributes to reducing overinvestment in this sector (as the farms can allocate more and more resources to investments) [83]. This is a guideline for further research on overinvestment, which may suggest that the importance of each input is likely to vary in the future. As regards capital (K), differences in relationships can be seen between the relative and absolute overinvestment classes. The former are on the normal development path, meaning that—at least in the first period following the investment—the increase in the assets-to-labor ratio is greater than the increase in labor productivity. However, in this case, labor productivity is likely to grow faster than the assets-to-labor ratio in the future (after the farms overcome the difficulties resulting from the need to adjust to new technical and technological conditions). Conversely, on farms affected by absolute overinvestment, the labor productivity declines, which suggests that wrong decisions were made regarding the subject or scope of investments. The analysis of the production function also allowed for calculating the elasticities of the productive inputs, and thus to determine the economies of scale.

Despite land (Z) being statistically insignificant in two estimated models (on farms at optimum investment levels with no increase in the assets-to-labor ratio and on farms affected by relative overinvestment), the decision was made to have it covered in further analysis too. Thus, differences were discovered between the investment levels covered by this study, for instance, in the output elasticities of the inputs. The highest value was recorded for farms at optimum investment levels with no increases in the assets-to-labor ratio; this means that a simultaneous 1% growth in all inputs will increase the output by 1.098%. Hence, the farms witnessed growing economies of scale. In turn, declining economies of scale were found on farms affected by relative overinvestment. A simultaneous 1% growth in inputs will cause a 0.979% increase in output. This makes it unreasonable for these farms to make further investments. They should seek optimum amounts of inputs instead of increasing them (which has proven to be unnecessary). As shown by previous research on overinvested farms [84], farms from the “relative overinvestment” group are the ones who have increased their capital in recent years while also increasing their labor productivity (in accordance with the methodology used), which, however, grew slower than the assets-to-labor ratio. The effects of overinvestment are due to capital saturation—a natural situation occurring when businesses shift to a capital-intensive production path. On farms affected by relative overinvestment, the output grows slower than the inputs. Farms at other levels of investment attain similar growth in their output in response to a simultaneous 1% increase in all inputs. In members of the absolute overinvestment group, the economies of scale were also growing but reached the lowest level of all the overinvestment types.

Note that the farm types identified in this study significantly differ in the share of labor (L) and capital (K) in the elasticity coefficient. The highest share of labor (L) was recorded for farms at optimum investment levels with no increases in the assets-to-labor ratios. This can be explained by the fact that labor productivity grew faster than the assets-to-labor ratio in the study period at this level of investment. A high share of labor (L) was also found for farms at optimum investment levels, which recorded increases in both the assets-to-labor ratio and labor productivity. This, in turn, is indicative of sound investment decisions because labor productivity growth determines farm development. The study also found that labor (L) inputs lose their importance (have an increasingly smaller impact on production volumes) on overinvested farms.

The highest share of capital (K) in the elasticity coefficient was observed for underinvested and other farms. The general remark is that on most farms covered by this study, capital (K) is the input which has an impact on the output. Agricultural mechanization plays a major role in this context because improvements in farm productivity largely depend on it. Indeed, farm development is mostly determined by technical and technological progress. The estimated time-related parameter of the model (the variable t) provides an insight into the average annual production growth/decline rate (

Table 7. Average annual production growth/decline rate in 2010–2019 (%).

Overinvestment Level	Average Difference in Production Value
Absolute (I)	−2.58%
Relative (II)	−0.47%
Underinvested (III)	−1.49%
Optimum investment, no increase in ALR (IV)	1.07%
Optimum investment, increase in LP and ALR (V)	1.36%
Other (VI)	0.47%

Source: own compilation based on FADN microdata, n = 3273.

).

The analysis of the average production growth/decline rate for the identified farm types at different overinvestment levels (Table 7) reveals that only three of them witnessed consistent growth in their output on a year-over-year basis. The first two types are farms which make optimum investments, whether or not accompanied by growth in the assets-to-labor ratio. The third is from the “other farms” group. Conversely, on an average annual basis, there was a decline in output on the farms at both the relative and absolute overinvestment levels, and on underinvested farms. It was established that unjustified investments are not accompanied by growth in production values, and that a lack of investments does not boost production either. Another important finding is that the average annual production decline rate was only 0.47% on farms affected by relative overinvestment, but it was as much as 2.58% for members of the “absolute overinvestment” group. In the context of the analysis of their economic performance, it can be expected that members of the “relative overinvestment” group are not currently witnessing any increases in output due to the considerable growth in capital inputs. However, in the long run, they can be reasonably expected to move to the “optimum investment” group.

5. Conclusions

The analysis of the production function allowed for identifying how significant and strong the impacts of the productive inputs are on the resultant output. The differences found between the identified farm types highlight some problem areas, such as the excessive use of capital relative to annual production gains. Attention was also paid to the increasingly smaller use of labor in the production process, which could be indicative of a shift away from the labor-intensive agricultural development path. Some of the key findings of this study are as follows:

• This study demonstrated the existence of a series of farm groups which are more or less rational in their approach to investments. The worst production performance, reflected in an absolute decline in labor productivity, was witnessed in holdings which invested either too little or too much in relation to their production capacity. Underinvested farms can experience a gradual decline in agricultural production caused by being non-competitive. Overinvested farms, in turn, can make major mistakes in planning and implementing their investments, which is a more serious problem;
• It is also worth noting that overinvestment is indicative of having allocated considerable financial resources to investments which ultimately did not contribute to improving the farm’s economic situation. In fact, it was quite the opposite; they brought about a significant deterioration in it;
• Another equally important remark is that overinvested holdings are intended to develop their activity rather than restrict it. As a result, misallocated investment funds in the future may require such farms to correct an unplanned development path or even cease farming operations.

Investigating long-term development prospects for farms with a view to determining appropriate investment goals and using econometric models to simulate different investment scenarios and assess their impacts on the farms’ production and financial performances are the next fields of research the authors intend to address. This kind of research is being undertaken to avoid situations where agricultural operators have no choice but to make haphazard, ill-considered investment decisions, and to develop more effective investment management strategies that will drive sustainable agricultural development.

It should also be borne in mind that in agricultural investment processes (as a result of technological dreck), the growing resources of fixed assets do not always lead to a significant increase in income. Indeed, incomes may remain constant or grow at a significantly slower rate than the increase in the value of the fixed assets (and therefore this study relies on the net value added). Rather than implying overinvestment, this situation is a consequence of the need to compensate for unfavorable conditions in the macroeconomic environment. This means a change in the price relationships between productive inputs, which forces farmers to restructure the mix of productive inputs used in the production process (to substitute labor with capital). What can also be observed is the changing relationship between agricultural production costs and agricultural product prices, resulting in a permanent decline in the profitability of agricultural production. In the absence of investment, which could be seen as excessive, the situation of farmers would undoubtedly be much more difficult.