Fuzzy mathematical risk preferences based on stochastic production function among medium-scale hog producers

Abstract

Farmers’ risk preferences and degree of risk aversion affect their production and management decisions. According to Just-Pope stochastic production function model, we get the expression of the single element risk-aversion coefficients that include input element and hog slaughter absolute price, compared with the expression of relative price mean risk-aversion coefficients, it can directly observe the influence of the element and output price on single element risk-aversion coefficients. Based on the regression procedures and the calculation method of the average value of the element risk-aversion coefficients, mean risk-aversion coefficients of per household medium-scale hog producers are calculated in 76 households, 11 counties, Heilongjiang province. The results show that medium-scale hog producers are risk-averse, accounting for 96%; newborn animal weight and feed consumption affect hog producers’ degree of risk aversion. The former is the risk-reducing input element, while the latter is the risk-increasing input element.

Keywords

Medium-scale hog producers just-pope stochastic production function newborn animal weight mean risk-aversion coefficients

1 Introduction

In practical production, if a producer’s preference was found, this preference will guide us to discover the producer’s choice behaviors, and its choice will tend to produce specific decisions [1]. Many empirical researches on producers of livestock, fishery and poultry indicate that their risk preferences affect their business decisions and input choices [2 –6].

In recent years, the price of hog fluctuates greatly in the market, and the price level remains high. The government has introduced a series of measures to stabilize hog production and strengthen insurance support for the breeding industry in China. Analyzing the degree of risk aversion of medium-hog producers and their supply decisions can provide references for the government to formulate policies to stabilize the supply of medium-hog from the perspective of hog producers’ risk aversion.

2 Related work

First, the farmers’ risk preferences and degree of risk aversion could affect their production and management decisions which have been proved by some researches

Gunjal and Legault applied power utility function that relative risk-aversion coefficients are constant to compare the risk preferences of dairy producers with hog producers in Canada Quebec Province, and pointed out that the production, finance and marketing decisions that agricultural producers faced were related to variable risk degree, and hog producers were more risk-averse than dairy producers [7]. Therefore, government needs to fully understand the farmers’ risk behavior to reduce instability in agricultural production. Bwala and Bila proposed the risk-averse farmers should be educated and the negative impact of risk aversion on farmers’ decision-making ability should be considered in the study of farmers’ risk aversion in Borno [8]. The forestry farmers’ renegotiation and investment decisions were influenced by their degree of risk aversion [9]. Brennan deems that the different farmers’ degree of risk aversion have different options for the best savings strategy when he studied savings and technology selection of risk-averse farmers in Vietnam rice shrimp farm [10]. El-Nazer and McCarl pointed out that the farmers’ risk attitudes affect the choice of optimal crop rotation when their predetermined rotation and free rotation were analyzed in the northeastern Oregon irrigation farming [11]. Pennings and Garcia obtained certainty equivalence through designing problems, under the condition that without considering the variable input by using negative exponential utility function and power utility function analyzed the risk preferences of Holland hog producers [3].

Second, hog producers are the takers of hog slaughter price. If there are no obvious differences in the lean meat rate, breeding methods or breeds, the hog producers are price takers. The hog slaughter price is determined by the market supply and demand. In the following computational process, hog slaughter price based on market purchase price that was provided by producers. The surveyed producers are medium-scale hog (100–499 heads) individual operators. The slaughter amounts of 100–499 heads is the highest in each scales, so it is determined as the object of study. Otherwise, producers take advantage of own resources to carry out production in the research, do not need to hire other labor forces. Only 6 households investigated have more than 400 heads annual slaughter amounts, but there are no labor forces to be hired [12].

The newborn animal weight and feed consumption affect degree of risk aversion of medium-scale hog producers. The former is risk-reducing input element, while the latter is risk-increasing input element. The medium-scale hog producers are risk-averse.

3 Stochastic production function

The researchers more used Just-Pope (1978) stochastic production function model (y = f (X) + ɛh (X), E (ɛ) =0) to study the mean risk of farmers. The model is consist of mean production function f (X) and production risk function h (X) ɛ, where x is input, y is output and ɛ is the error term and obey normal distribution N(0, 1) [13]. Based on the studies of Picazo-Tadeo and Wall [14] and Harvey [15], the stochastic production function is specified as $y = f (x_{ki}; α) + g (x_{ki}; β) \cdot ɛ$ (1)

The inputs x_ki are input quantity of ith producers kth products, which have influence on both mean output (E (y) = f (x_ki ; α)) and variance of output ( $(\hat{u})^{2} = (y - f (x_{ki}; α))^{2} = (g (x_{ki}; β) \cdot ɛ)^{2}$ ). However, marginal risk output is ∂Var (y)/∂x_ki = 2 · g (x_ki ; β) · ∂g (x_ki ; β)/∂x_ki, whose positive and negative depends on sign of ∂g (x_ki ; β)/∂x_ki. If ∂g (x_ki ; β)/∂x_ki is greater than zero (less than zero), the x_ki is risk-increasing (risk-reducing) input element [4, 14]. Saha et al. indicated that the combined estimation of the production function parameter with the utility function parameter is more efficient than separate estimation of each [15].

3.1 Model selection of mean production function and production risk function

The linear-quadratic function $f {(x}_{ki}; α) = α_{0} + \sum_{k = 1}^{K} α_{k} x_{ki} + \frac{1}{2} \sum_{k = 1}^{K} \sum_{h = 1}^{K} α_{kh} x_{ki} x_{hi} + u_{i}$ is used as mean production function [14 , 16–18]; while the logarithm Cobb-Douglas production function $ln ({\hat{u}}_{i}^{2}) = β_{0} + \sum_{k = 1}^{K} β_{k} ln x_{ki} + v_{i}$ is used as risk function. The function form selection of f (x_ki ; α) and g (x_ki ; β) is based on the following:

First, the Cobb-Douglas production function was used in mean production function and risk function when some scholars used Just-pope (1978) stochastic production function model. The Just-pope (1978) stochastic production function model y = f (X) + ɛh (X) was used to estimated positive or negative marginal risk based on panel data. The form of $f (X) = γ \prod_{k = 1}^{k} X_{k}^{β h}$ was used as mean production function f (X), and the form of $h (X) = \prod_{k = 1}^{k} X_{k}^{β h}$ was used as risk function h (X) [19]. Kumbhakar used stochastic production function model y = f (x, z) + g (x, z) ɛ - q (x, z) u when he studied farmers’ production risk, risk preferences and technical efficiency in Norway salmon farm, among them, the Cobb-Douglas production function was used in f (x, z), g (x, z) and q (x, z). Initially, the writer used the linear-quadratic function to regress the three (production) risk functions, but the square and cross terms didn’t pass the test. Therefore, he chose the Cobb-Douglas production function instead of the linear-quadratic function [4].

Second, the linear-quadratic function form was used in mean production function and the Cobb-Douglas production function or its logarithm function form was used for risk function when some scholars used Just-pope (1978) stochastic production function model. The linear-quadratic function f (x, α) = α₀+ $\sum_{k = 1}^{K} α_{k} x_{ki} + \frac{1}{2} \sum_{k = 1}^{K} \sum_{h = 1}^{K} α_{kh} x_{ki} x_{hi} + u_{i}$ , u ≡ g (x ; β) · ɛ could avoid the trouble of non-linear calculation method. The stochastic production function model $y^{I} = f (x^{I}, z^{I}, α^{I}) + \sqrt{g (x^{I}, z^{I}, β^{I})} ɛ$ was used to compare the risk attitudes of the traditional farm farmers with organic farm farmers based on the case of Spanish cultivated farmers. The function form $f (x, z, α) = (α_{0} + α_{0}^{o} o) \sum_{i = 1}^{2} (α_{i} + α_{i}^{o} o) x_{i} + (α_{3} + α_{3}^{o} o) z_{II} + \sum_{i = 1}^{2}$ $\sum_{j = 1}^{2} (α_{ij} + α_{ij}^{o} o) x_{ij} + \sum_{j = 1}^{2} (α_{i 3} + α_{i}^{o} o) x_{i} z_{I}$ and g (x, z, $β) = (β_{0} + β_{0}^{o} o) x_{1}^{(β_{1} + β_{1}^{o} o)} x_{2}^{(β_{2} + β_{1}^{o} o)} z_{1}^{(β_{3} + β_{3}^{o} o)}$ were adopted as mean production function and risk function respectively, which was composed of linear-quadratic function and Cobb-Douglas production function [20]. The effect of agricultural policy changes on the farmer’s risk attitudes was studied by the Just-Pope (1978) stochastic production function model y = f (x, A ; z) + g (x, A) e. Among them, the linear-quadratic function was used as f (x, A ; z); the Cobb-Douglas production function was used as g (x, A). Gardebroek et al. used Just-pope (1978) stochastic production function model $y_{it} (X_{it}, ɛ_{it}) = f (X_{it}) + ɛ_{it} \sqrt{h (X_{it})}$ , $E (ɛ_{it}) = 0 var (ɛ_{it}) = σ_{ɛ}^{2} > 0$ , analyzing the production technology and production risk between organic farm owners and traditional farm owners. Among them, $y_{it} = α_{i} + \sum_{k = 1}^{7} α_{i} X_{kit} + \frac{1}{2} \sum_{k = 1}^{7} \sum_{j = i}^{7} α_{kj} X_{kit} X_{jit} + u_{it}$ was adopted as f (X_it), $h (X_{it}) = β_{i} \prod_{k = 1}^{7} X_{kit}^{β_{k}} \prod_{k = 1}^{7} \prod_{j = 1}^{7} e^{1 / 2 β_{kj} {ln}_{xj}}$ was adopted as h (X_it) [16]. Picazo-Tadeo and Wall (2011) used Just-pope (1978) stochastic production function model y = f (x ; α) + g (x ; β) · ɛ, taking rice output as the dependent variable, taking land, labor, capital, seed, fertilizer and fitosanitary as the independent variables to estimate the stochastic production function of rice farmers in the eastern Spain Albufera Natural Park. Among them, $f (x, α) = α_{0} + \sum_{k = 1}^{K} α_{k} x_{ki} + \frac{1}{2} \sum_{k = 1}^{K} \sum_{h = 1}^{K} α_{kh} x_{ki} x_{hi} + u_{i}$ was used for f (x ; α), the logarithm Cobb-Douglas production function $ln ({\hat{u}}_{i}^{2}) = β_{0} + \sum_{k = 1}^{K} β_{k} ln x_{ki} + v_{i}$ was used for g (x ; β) [14]. The effects of linear-quadratic function and Cobb-Douglas production function on mean production function were compared, and the former was chose to estimate the mean production function finally. In the estimation of risk function, the writer originally used the linear-quadratic function, but all regression results were not significant, so they chose the logarithm Cobb-Douglas production function.

3.2 Calculation of producers’ mean risk-aversion coefficients

Based on equation (1), the anticipated profits is calculated: $\begin{matrix} π = p \cdot y - ω x = p_{i} \cdot f (x_{ki}; α) + p_{i} \cdot g (x_{ki}; β) ɛ \\ - ω_{ki} x_{ki} \equiv μ_{π} + p_{i} \cdot g (x_{ki}; β) ɛ ɛ \sim (0, 1) \\ μ_{π} = E (π^{e}) = p \cdot f (x_{ij}; α) - ω_{ki} \cdot x_{ki} \end{matrix}$ where ω_ki is input element price, p_i is output price.

The expected utility maximization of producers’ anticipated profits is assumed as Max (U (π)), where U(·) is a continuous and differentiable function of anticipated profits. The first-order condition expected utility maximization [21] (Max (U (π))) is $\begin{matrix} E (U^{'} (π)) \cdot (p_{i} \cdot \frac{\partial f (x_{ki}; α)}{\partial x_{ki}} + p_{i} \cdot \frac{\partial g (x_{ki}; β)}{\partial x_{ki}} \cdot ɛ - ω_{ki}) = 0 \\ E (U^{'} (π)) \cdot p_{i} \cdot \frac{\partial f (x_{ki}; α)}{\partial x_{ki}} + E (U^{'} (π)) \cdot p_{i} \cdot \frac{\partial g (x_{ki}; β)}{\partial x_{ki}} \\ \cdot ɛ - E (U^{'} (π)) \cdot ω_{ki} = 0 \end{matrix}$

Divide by E (U′ (π)) · p_i and obtain: $\frac{\partial f (x_{ki}; α)}{\partial x_{ki}} + \frac{E (U^{'} (π) \cdot ɛ)}{E (U^{'} (π))} \cdot \frac{\partial g (x_{ki}; β)}{\partial x_{ki}} - \frac{ω_{ki}}{p_{i}} = 0$ $\frac{\partial f (x_{ki}; α)}{\partial x_{ki}} - \frac{ω_{ki}}{p_{i}} + \frac{\partial g (x_{ki}; β)}{\partial x_{ki}} θ (\cdot) = 0$ (2)

According to Taylor expansion [14 , 21–22], θ (·) could be written as $\begin{matrix} θ (\cdot) & = \frac{E (U^{'} (π) \cdot ɛ)}{E (U^{'} (π))} \\ = \frac{E [(U^{'} (μ_{π}) + U^{″} (μ_{π}) \cdot p_{i} \cdot g (x_{ki}; β) + \dots) ɛ]}{E [(U^{'} (μ_{π}) + U^{″} (μ_{π}) \cdot p_{i} \cdot g (x_{ki}; β) + \dots)]} \\ = \frac{U^{″} (μ_{π}) \cdot p_{i} \cdot g (x_{ki}; β)}{U^{'} (μ_{π})} \end{matrix}$ $r_{A_{ki}} (μ_{π}) = - \frac{U^{″} (μ_{π})}{U^{'} (μ_{π})}$ is Arrow-Pratt absolute risk-aversion coefficients [23, 24], when r_{A
_ki} (μ_π) <0(>0, = 0) producers expressed as risk-loving (risk-averse, risk-neutral) for the input elements. That is θ (·) = - r_{A
_ki} (μ_π) · p_i · g (x_ki ; β). The positive or negative of the risk preferences function θ (·) value can reflect producers’ risk attitudes. The θ (·) value greater than 0 indicates that producers are risk-loving, θ (·) value less than 0 indicates that producers are risk- averse, θ (·) value equal to 0 indicates that producers are risk- neutral [4, 25].

Substituting θ (·) into the utility maximization equation (2), the utility maximization input equation become as $\frac{\partial f (x_{ki}; α)}{\partial x_{ki}} = \frac{ω_{ki}}{p_{i}} + r_{A_{ki}} (μ_{π}) \cdot g (x_{ki}; β) \cdot p_{i} \frac{\partial g (x_{ki}; β)}{\partial x_{ki}}$ (3)

Because based on the estimation function of f (x_ki ; α) and g (x_ki ; β) to calculate r_{A
_ki} (μ_π), r_{A
_ki} (μ_π) will be rewritten the form as $r_{A_{ki}} (μ_{π}) = \frac{\frac{\partial f (x_{ki}; \hat{α})}{\partial x_{ki}} - \frac{ω_{ki}}{p_{i}}}{p_{i} \cdot g (x_{ki}; \hat{β}) \cdot \frac{\partial g (x_{ki}; \hat{β})}{\partial x_{ki}}}$ (4)

Based on mean values of input elements risk-aversion coefficients obtain the mean risk-aversion coefficients [14]: $r_{A_{i}} = \frac{1}{k} \sum_{k = 1}^{k} r_{A_{ki}}$

The risk preferences of surveyed objects are judged based on the sign of r_{A
_i}.

Picazo-Tadeo and Wall directly used the rice output and input elements costs to express anticipated profits in the estimation of risk function, and calculated the mean risk-aversion coefficients of rice farmers [14]. Serra et al. [26], Koundouri [27] and Kumbhakar [21] introduced established price to calculate the anticipated profits in the estimation of farmers and barley, wheat and rice farmers’ risk function. This paper introduces actual slaughter price and input elements price of per households hog producers to obtain profit expression, whose advantage is based on the changes of their absolute price to analysis the changes of hog producers’ degree of risk aversion. The risk function is estimated by using hog actual slaughter weight, through analyzing changes of elements input in risk function to reflect the risk attitudes of producers for elements and the mean risk attitudes of hog producers.

4 Empirical analysis

The date were collected from a survey carried out from May 19, 2015 to January 6, 2016, the final sample included 76 households medium-scale hog producers in 45 counties (Jixian, Longjiang, Keshan and so on), Heilongjiang Province.

4.1 Regression procedure

4.1.1 Descriptive statistics

In this paper, recently hog actual slaughter total weight as output variable Y, the newborn animal weight (x₁), feed consumption (x₂), other direct and indirect costs (x₃) and labor costs (x₄) as input variables. The descriptive statistics values of 76 households medium-scale hog producers are shown in Table 1.

Table 1
Descriptive statistics of production variables

Variable X Units Mean Standard deviation Min. Max.

Household Head Household Head Household Head Household Head

y Kg 9585.43 112.62 5544.71 9.83 5512 100 27000 150

x ₁ Kg 1223.91 14.33 726.417 3.04 495 9 3800 25

x ₂ Kg 26571.44 311.86 15548.22 29.78 15000 250 75900 406

x ₃ Yuan 6281.30 72.49 4730.24 29.68 1195 23.29 31574 179.87

x ₄ Yuan 14149.22 191.09 2766.93 47.57 11200 102.40 23552 240

Variable X	Units	Mean	Standard deviation	Min.	Max.
y	Kg	9585.43	112.62	5544.71	9.83	5512	100	27000	150
x ₁	Kg	1223.91	14.33	726.417	3.04	495	9	3800	25
x ₂	Kg	26571.44	311.86	15548.22	29.78	15000	250	75900	406
x ₃	Yuan	6281.30	72.49	4730.24	29.68	1195	23.29	31574	179.87
x ₄	Yuan	14149.22	191.09	2766.93	47.57	11200	102.40	23552	240

The standard deviation of per head hog slaughter weight (y), newborn weight (x₁), feed consumption (x₂), other direct and indirect costs (x₃) and labor costs (x₄) are shown in Table 1; the 5 indicators of each household slaughter hog have the large differences (especially feed consumption) in the 5th column, which indicates that hog slaughter amount of each household have the large differences.

4.1.2 Data test

The dw correlation test for the two variables combination, three variables combination and four variables combination of ln(y), ln(x₁), ln(x₂), ln(x₃) and ln(x₄) is carried out by using Eviews 6.0. The results are shown in Table 2. In accordance with dw test Table could find the critical value of dl and du, which could know all test results for the residual sequence are no correlation, so the four groups data can carry out the regression analysis.

Table 2
DW test results

Variable combination k s dl du dw 4-du 4-dl

x₁, x₂ 2 76 1.57 1.68 2.1632 2.32 2.43

x₁, x₂, x₃ 3 76 1.54 1.71 2.0420 2.29 2.46

x₁, x₂, x₄ 3 76 1.54 1.71 1.9395 2.29 2.46

x₁, x₂, x₃, x₄ 4 76 1.51 1.74 2.1843 2.26 2.49

Variable combination	k	s	dl	du	dw	4-du	4-dl
x₁, x₂	2	76	1.57	1.68	2.1632	2.32	2.43
x₁, x₂, x₃	3	76	1.54	1.71	2.0420	2.29	2.46
x₁, x₂, x₄	3	76	1.54	1.71	1.9395	2.29	2.46
x₁, x₂, x₃, x₄	4	76	1.51	1.74	2.1843	2.26	2.49

Explain: Descriptive statistic values are the indicators of each household hog and per head hog in the Table.

Explain: If 0 ≤ dw ≤ dl, the residual sequence is positive correlation; if du < dw < 4 - du, the residual sequence is no correlation; if 4 - dl < dw ≤ 4, the residual sequence is negative correlation. Among them, k is variables quantity, s is sample size.

4.2 Regression results and analysis

4.2.1 Regression results and analysis of mean production function

Based on FLGS estimation principle, the mean production function of four groups input variables is estimated, the results are shown in Table 3.

Table 3
Mean production function regression results of four groups variables

Variable combination x₁, x₂ x₁, x₂, x₃ x₁, x₂, x₄ x₁, x₂, x₃, x₄

Coefficient Standard error p-value Coefficient Standard error p-value Coefficient Standard error p-value Coefficient Standard error p-value

Constant 175.5299 225.8188 0.4396 –98.5113 220.5422 0.6566 –2913.39 3037.5170 0.3410 –3171.49 2691.474 0.2432

x ₁ 0.0215 0.3139 0.9457 0.4253 0.3053 0.1683 –2.1803 1.1618 0.0650 –2.0263 1.1344 0.0790

x ₂ 0.34735 0.0195 1.13E–27 0.3491 0.0186 3.26E-28 0.3564 0.1083 0.0016 0.3483 0.1036 0.0013

x ₃ – – – –0.0048 0.0349 0.8904 – – – –0.2106 0.1812 0.2497

x ₄ – – – – – – 0.6089 0.6004 0.3142 0.7508 0.5306 0.1622

$x_{1}^{2}$ 0.00065 0.0002 0.0094 0.0006 0.0002 0.0041 0.0004 0.0003 0.1146 0.0003 0.0002 0.2089

$x_{2}^{2}$ 6.03E-07 4.35E-07 0.1695 2.06E-07 7.26E-07 0.7778 1.79E-06 1.28E-06 0.1667 1.62E-07 1.55E-06 0.9169

$x_{3}^{2}$ – – – 8.53E-06 5.93E-06 0.1552 – – – 1.4E-05 6.39E-06 0.0321

$x_{4}^{2}$ – – – – – – –2.8E-05 3.12E-05 0.3649 –5E-05 2.76E-05 0.0760

x₁ · x₂ –4E-05 1.93E-05 0.0419 –3.2E-05 2.36E-05 0.1758 –6.2E-05 2.15E-05 0.0052 –5.8E-05 2.91E-05 0.0498

x₁ · x₃ – – – –0.0001 7.8E-05 0.1594 – – – –0.0001 9.89E-05 0.1863

x₁ · x₄ – – – – – – 0.0002 0.0001 0.0472 0.0003 0.0001 0.0161

x₂ · x₃ – – – 1.84E-06 3.53E-06 0.6035 – – – –3E-06 3.92E-06 0.4542

x₂ · x₄ – – – – – – –3.3E-06 1.18E-05 0.7798 5.57E-06 1.09E-05 0.6129

x₃ · x₄ – – – – – – – – – 2.09E-05 1.76E-05 0.2401

R ² 0.9965 0.9973 0.9968 0.9979

Adjusted R² 0.9962 0.9970 0.9964 0.9974

Variable combination	x₁, x₂	x₁, x₂, x₃	x₁, x₂, x₄	x₁, x₂, x₃, x₄
Constant	175.5299	225.8188	0.4396	–98.5113	220.5422	0.6566	–2913.39	3037.5170	0.3410	–3171.49	2691.474	0.2432
x ₁	0.0215	0.3139	0.9457	0.4253	0.3053	0.1683	–2.1803	1.1618	0.0650	–2.0263	1.1344	0.0790
x ₂	0.34735	0.0195	1.13E–27	0.3491	0.0186	3.26E-28	0.3564	0.1083	0.0016	0.3483	0.1036	0.0013
x ₃	–	–	–	–0.0048	0.0349	0.8904	–	–	–	–0.2106	0.1812	0.2497
x ₄	–	–	–	–	–	–	0.6089	0.6004	0.3142	0.7508	0.5306	0.1622
$x_{1}^{2}$	0.00065	0.0002	0.0094	0.0006	0.0002	0.0041	0.0004	0.0003	0.1146	0.0003	0.0002	0.2089
$x_{2}^{2}$	6.03E-07	4.35E-07	0.1695	2.06E-07	7.26E-07	0.7778	1.79E-06	1.28E-06	0.1667	1.62E-07	1.55E-06	0.9169
$x_{3}^{2}$	–	–	–	8.53E-06	5.93E-06	0.1552	–	–	–	1.4E-05	6.39E-06	0.0321
$x_{4}^{2}$	–	–	–	–	–	–	–2.8E-05	3.12E-05	0.3649	–5E-05	2.76E-05	0.0760
x₁ · x₂	–4E-05	1.93E-05	0.0419	–3.2E-05	2.36E-05	0.1758	–6.2E-05	2.15E-05	0.0052	–5.8E-05	2.91E-05	0.0498
x₁ · x₃	–	–	–	–0.0001	7.8E-05	0.1594	–	–	–	–0.0001	9.89E-05	0.1863
x₁ · x₄	–	–	–	–	–	–	0.0002	0.0001	0.0472	0.0003	0.0001	0.0161
x₂ · x₃	–	–	–	1.84E-06	3.53E-06	0.6035	–	–	–	–3E-06	3.92E-06	0.4542
x₂ · x₄	–	–	–	–	–	–	–3.3E-06	1.18E-05	0.7798	5.57E-06	1.09E-05	0.6129
x₃ · x₄	–	–	–	–	–	–	–	–	–	2.09E-05	1.76E-05	0.2401
R ²	0.9965	0.9973	0.9968	0.9979
Adjusted R²	0.9962	0.9970	0.9964	0.9974

According to the regression results of Table 3, the mean production function analysis as follows:

The mean production function has p-values doesn’t pass test under four groups input variables. This result is consistent with the research result of Picazo-Tadeo and Wall, the estimation of mean production function has p-values of a number of variables (labor, labor square, seed square, land and capital cross terms, etc) don’t pass the test in the study of production risk, risk aversion and risk attitudes of Spanish rice farmers. The mean production function doesn’t pass test, which means that y - f (x_ki ; α) is not residual, the random fluctuation doesn’t equal to 0. Therefore, according to the research procedure of Picazo-Tadeo and Wall (2011), the y - f (x_ki ; α) can be used as the dependent variable to regress risk function [14].

4.2.2 Regression results and analysis of risk function

According to the four groups of independent variables in Table 3, dependent variable [g (x_ki ; β)] ² adopt Cobb-Douglas production function to regress, and there is $\ln [g (x; β)] = \frac{1}{2} lng {[(x; β)]}^{2}$ , the results are shown in Table 4.

Table 4
Risk function regression results of four groups variables

Variable groups Constant ln(x₁) ln(x₂) ln(x₃) ln(x₄) R ² Adjusted R²

x₁, x₂ Coefficient –7.9321 –1.3666 2.2169 – – 0.1396 0.1161

Standard error 3.8251 0.6041 0.6828 – –

t-Stat –2.0737 –2.2623 3.2470 – –

p-value 0.0416 0.0267 0.0018 – –

x₁, x₂, x₃ Coefficient –6.7748 –0.3666 1.3534 0.0603 – 0.1740 0.1396

Standard error 3.1999 0.4977 0.6301 0.2919 –

t-Stat –2.1171 –0.7366 2.1481 0.2066 –

p-value 0.0377 0.4638 0.0351 0.8369 –

x₁, x₂, x₄ Coefficient 10.9311 –0.6321 1.9198 – –2.2064 0.0941 0.0564

Standard error 12.1501 0.5316 0.7593 – 1.8652

t-Stat 0.8997 –1.1890 2.5285 – –1.1829

p-value 0.3713 0.2383 0.0136 – 0.2407

x₁, x₂, x₃, x₄ Coefficient –1.8765 0.8497 –0.7016 –0.0887 0.8802 0.0383 –0.0159

Standard error 16.2420 0.7112 1.0888 0.3988 2.4915

t-Stat –0.1155 1.1948 –0.6443 –0.2225 0.3533

p-value 0.9083 0.2362 0.5214 0.8246 0.7249

Variable groups		Constant	ln(x₁)	ln(x₂)	ln(x₃)	ln(x₄)	R ²	Adjusted R²
x₁, x₂	Coefficient	–7.9321	–1.3666	2.2169	–	–	0.1396	0.1161
	Standard error	3.8251	0.6041	0.6828	–	–
	t-Stat	–2.0737	–2.2623	3.2470	–	–
	p-value	0.0416	0.0267	0.0018	–	–
x₁, x₂, x₃	Coefficient	–6.7748	–0.3666	1.3534	0.0603	–	0.1740	0.1396
	Standard error	3.1999	0.4977	0.6301	0.2919	–
	t-Stat	–2.1171	–0.7366	2.1481	0.2066	–
	p-value	0.0377	0.4638	0.0351	0.8369	–
x₁, x₂, x₄	Coefficient	10.9311	–0.6321	1.9198	–	–2.2064	0.0941	0.0564
	Standard error	12.1501	0.5316	0.7593	–	1.8652
	t-Stat	0.8997	–1.1890	2.5285	–	–1.1829
	p-value	0.3713	0.2383	0.0136	–	0.2407
x₁, x₂, x₃, x₄	Coefficient	–1.8765	0.8497	–0.7016	–0.0887	0.8802	0.0383	–0.0159
	Standard error	16.2420	0.7112	1.0888	0.3988	2.4915
	t-Stat	–0.1155	1.1948	–0.6443	–0.2225	0.3533
	p-value	0.9083	0.2362	0.5214	0.8246	0.7249

According to the regression results of Table 4, the risk function analysis as follows:

First, the newborn animal weight and feed consumption pass test.

In the estimation of the risk function, when the input variables are newborn animal weight and feed consumption, the newborn animal weight at 5% level and feed consumption at 1% level both pass the significance test (see x₁,x₂ combination). Only the feed consumption variable passes test at 5% significant level in the two groups of the three input variables combination (see x₁, x₂, x₃ and x₁, x₂, x₄ combinations), others don’t pass the test. Under the four input variables combination (see x₁, x₂, x₃, x₄ combination), all the variables don’t pass the test. Therefore, the newborn animal weight and feed consumption have influence on the production input decisions of medium-scale hog producers.

Second, newborn animal weight is risk-reducing input element, while feed consumption is risk-increasing input element.

According to the influence of inputs on the variance of output is increased, reduced or no effect, inputs can be divided into risk-increasing, risk-reducing or risk-neutral. The regression result of risk function g (x_ki ; β) shows that the output elasticity value of newborn animal weight is –1.3666, which shows that increase of newborn animal weight will reduce the total weight of hog slaughter, the reduction of newborn animal weight will increase the risk output, which belongs to risk-reducing input element; while output elasticity value of feed consumption is 2.2169, which suggests that the increase of feed consumption will increase the hog slaughter total weight, the increase of feed consumption can increase the risk output, which belongs to risk-increasing input element. The fertilizer belongs to risk-increasing input element in organic farm, while it belongs to risk-reducing input element in traditional farm; for the farms of two types, other variable inputs and labor are risk-increasing input elements, while capital and land are risk-reducing input elements concluded that the increase of land area, fitosanitar and labor will reduce rice output, which belong to risk-reducing input elements, that is, the reduction of land, fitosanitar and labor can increase the risk output; while the increase of capital, seed and fertilizer will increase rice output, which belong to risk-increasing input elements, that is, the increase of these elements can increase the risk output.

5 Analysis of risk preferences of hog producers

Based on the mean production function f (x_ki ; α) (Table 3), production risk function g (x_ki ; β) (Table 4) and ∂g (x_ki ; β)/∂x_ki, according to equation (4), risk-aversion coefficients of newborn animal weight and feed consumption as well as mean risk-aversion coefficients are calculated, and the relations of the three coefficients are shown in Fig. 1. The mean risk-aversion coefficients of producers are calculated by formula $r_{A_{i}} = \frac{1}{2} (r_{{Ax}_{1} i} + r_{{Ax}_{2 i}})$ .

Fig. 1

The relations of the three risk-aversion coefficients.

The mean, standard deviation, minimum value and maximum value of the risk-aversion coefficients of newborn animal weight and feed consumption as well as mean risk-aversion coefficients of producers are shown in Table 5. The mean risk-aversion coefficients r_{A
_i} of the producers is the mean value of the risk-aversion coefficients of input elements, so the mean of r_{A
_i} is between r_{Ax
_1i} and r_{Ax
_2i}.

First, the risk attitudes of hog producers to element input are analyzed. There are 8 households hog producers for newborn animal weight are risk-loving (r_{Ax
_1i} < 0), while 68 households hog producers for newborn animal weight are risk-averse (r_{Ax
_1i} > 0), that is, there are 8 households hog producers tend to buy lighter newborn animal to breed, while 68 households hog producers tend to buy heavier newborn animal to breed. However, there are 4 households hog producers for feed consumption are risk-loving (r_{Ax
_2i} < 0), while 72 households hog producers for feed consumption are risk-averse (r_{Ax
_2i} > 0), namely there are 4 households hog producers tend to buy heavier feed to breed, while 72 households hog producers tend to buy lighter feed to breed.

Second, the volatility of risk-aversion coefficients is analyzed. As shown in Table 5, the minimum and maximum value of r_{Ax
_1i}, r_{Ax
_2i} and r_{A
_i} overall volatility is small, which indicates that the differences of degree of risk aversion among producers are relatively small. Among them, the fluctuation range of mean risk-aversion coefficients is from –0.0022 to 0.0199, the mean value is 0.0044. This result is consistent with study result of Picazo-Tadeo and Wall [14]. The fluctuation range of mean risk-aversion coefficients is from –0.00043 to 0.2767, the mean value is 0.0027. The difference between the maximum and minimum values of newborn animal weigh is 0.0547 greater than the difference of feed consumption is 0.0201, which indicates that the risk aversion fluctuation of newborn animal weight is greater than feed consumption. The mean value of r_{Ax
_1i} and r_{Ax
_2i} shows that the producers’ degree of risk aversion for feed consumption is higher than the newborn animal weight. Standard deviation of r_{Ax
_2i} is less than r_{Ax
_1i}, which shows the difference of producers’ degree of risk aversion for feed consumption is less than the newborn animal weight.

Table 5

Descriptive statistics of risk-aversion coefficients

Risk-aversion coefficients	Mean	Standard deviation	Min.	Max.
r _{Ax _1i}	0.0036	0.0060	–0.0206	0.0341
r _{Ax _2i}	0.0052	0.0043	–0.0016	0.0185
r _{A _i}	0.0044	0.0036	–0.0022	0.0199

Third, the risk preferences of medium-scale hog producers are analyzed. The mean risk-aversion coefficients r_{A
_i} of 73 households hog producers are positive in 76 households, accounting for 96% of surveyed producers, while the mean risk-aversion coefficients of 3 households (Serial number 35, 38 and 49, respectively) are negative (there are three negative bubbles in Fig. 1), accounting for 4% of surveyed producers. The producers that mean risk-aversion coefficients are positive account for a higher proportion, so there is no need test to illustrate the surveyed medium-scale hog producers are risk-aversion.

6 Discussion

6.1 Discussion on input variables

First, the variables x₁ (newborn animal weight) and x₂ (feed consumption) pass test in risk function g (x_ki ; β). The result is consistent with the result that by telephone communicate with producers. Producers thought that they only considered newborn animal cost and feed cost in the calculating hogs breeding cost. Other expenses such as water fees, fuel and power fees are not taken into account and labor costs are understood as labor remuneration. So, the total profit values understood by hog producers are total revenue minus the expenses of newborn animal and feed. Therefore, the factors that could influence hog producers’ risk attitudes are newborn animal fees and feed fees. At the same time, producers also pointed out that corn price that account for the largest portion of the feed costs continued to fall, leading to feed costs reduce, so the risk-averse producers’ enthusiasm on hog breeding was improved, but the rise of newborn animal price, producers’ total investment for hog will depend on the comprehensive consideration of the two. Otherwise, this result is consistent with the existing research results. Yin and Wang are based on 2001–2012 Statistical Yearbook data through the co-integration test to analyze the factors that influence Jilin Province scale hog breeding efficiency [28]. The feed was important costs of hog breeding in the study of hog supply fluctuation, which will influence the production decisions of producers [29].

Second, the input elements that can’t pass the test are discussed. The other direct and indirect costs that doesn’t pass the test is analyzed. In this paper, other direct and indirect costs include the 3 kinds of fixed costs (fixed assets depreciation, epidemic prevention costs, technical service fees) don’t pass the test in the regression of the production risk function [14]. The labor that doesn’t pass the test is analyzed. The land, capital, labor and so on 6 kinds of elements passed the risk function test in the study of Spanish rice farmers’ the degree of risk aversion; Serra et al. used the seeds, fertilizer and other 4 kinds of variable input elements and a fixed input element labor and all passed the test when they studied farmers’ risk attitudes in traditional farm and organic farm [20]. The capital, labor and feed were selected, and they passed the test in the study of salmon farmers’ production risk, risk preferences and technical efficiency; Villano and Fleming (2006) selected the area, pesticides, labor, and so on and all passed the test in the study of rice planting technology efficiency and production risk. It is obvious that labor is an important variable in the risk function of the existing researches [30].

6.2 Discussion on the p-value

When Stuart et al. analyzed returns to scale of cutting wood in the eastern United States, and the p-value of insurance variable is 0.054 and p-value of administrative variable is 0.065, while the p-value of the other 5 variables are less than 0.01. It can be considered that the 7 variables passed the test at 6.5% significant level [31]. When Picazo-Tadeo and Wall (2011) study Spanish rice farmers’ production risk, risk aversion and risk attitudes, the p-value of seed variable is 0.149, the p-value of fitosanitary variable is 0.068, the 6 variables in the model can be considered have passed the test at a significant level of 14.9% [14]. However, risk function g (x_ki ; β) pass the test at significant level of 5% (see x₁, x₂ combination in Table 4) when the variables are the newborn animal weight (p = 0.0267) and feed consumption (p = 0.0018). The other three groups have individual p-value can not reach the 15% significant level, such as: ln(x₃) (p = 0.8369) of the second combination, ln(x₄) (p = 0.2407) of the third combination, ln(x₂) (p = 0.5214) of the forth combination in the Table 4.

6.3 Discussion on risk-averse producers

The calculation results show that medium-scale hog producers of 96% are risk-averse. Other scholars also suggest that farmers are more inclined to risk aversion. The linear regression model and constant absolute risk-aversion coefficients were used to test the risk preferences of dairy producers in New York, there are 26% are risk-loving, 39% are risk-neutral and 34% are risk-averse in 72 producers. Zuhair et al. measured risk-aversion coefficients of 30 farmers by using the exponential, quadratic and the cubic utility function in Kandy farm and Matale farm. Research result showed that the absolute risk-aversion coefficients of 72 households were positive in 90 households, which means that farmers of 80% were risk-averse [32]. Pennings and Garcia used negative exponential utility function and power utility function to analyze the risk preferences of Holland hog producers, and pointed out that producers of 73% were risk-averse [3]. The woodland individual operators of 74% were risk-averse when risk preferences of ecological forest and economic forest individual operators were analyzed [33]. The farmers of 80% were risk-averse in the study that maize farmers’ risk preferences had influence on elements input.

7 Conclusion

The main factors that influence degree of risk aversion and risk preferences of the medium-scale hog producers are newborn animal weight and feed consumption. In the producer’s stochastic production function regression, newborn animal weight and feed consumption pass the significance test of production risk function, while other direct and indirect costs and labor costs don’t pass the test. Therefore, newborn animal weight and feed consumption are main factors that affect medium-scale hog producers’ degree of risk aversion and risk preferences.

The newborn animal weight is risk-reducing input element; the feed consumption is risk-increasing input element. The elasticity value of newborn animal weight is negative in the regression result of risk function, which shows that the increase of newborn animal weight will reduce the hog slaughter total weight, that is, reduction of newborn animal weight can increase the risk output; while the elasticity value of feed consumption is positive, which means that the increase of feed consumption will increase the hog slaughter total weight, that is, increase of feed consumption can increase the risk output.

The medium-scale hog producers are risk-averse. On the basis of mean risk-aversion coefficients’ calculation results show that the absolute risk-aversion coefficients of 73 households producers are positive in 76 households. According to the judgment of Arrow-Pratt absolute risk-aversion coefficients on risk preferences, we know that the medium-scale hog producers are risk-averse. The newborn animal weight, feed consumption pass test of medium-scale hog producers’ stochastic production function, and the results show that could propose some relevant suggestions for newborn animal and feed.

Footnotes

Acknowledgment

This work was supported by National Planing Office of Philosphy and Social Science (Project name: Research on Effective Supply of Hogs Based on the Perspective of Risk Aversion; Grant No.13BJY113) and Heilongjiang Bayi Agricultural University Support Program for San Heng San Zong (Grant No. RRCPY201801).

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