Do Risk Preferences Really Matter? The Case of Pesticide Use in Agriculture

Abstract

Even if there exists an extensive literature on the modeling of farmers’ behavior under risk, actual measurements of the quantitative impact of risk aversion on input use are rare. In this article, we use simulations to quantify the impact of risk aversion on the optimal quantity of input and farmers’ welfare when production risk depends on how much of the input is used. The assumptions made on the technology and form of farmers’ risk preferences were chosen such that they are fairly representative of crop farming conditions in the USA and Western Europe. In our benchmark scenario featuring a traditional expected utility model, we find that less than 4% of the optimal pesticide expenditure is driven by risk aversion and that risk induces a decrease in welfare that varies from −1.5 to −3.0% for individuals with moderate to normal risk aversion. We find a stronger impact of risk aversion on quantities of input used when farmers’ risk preferences are modeled under the cumulative prospect theory framework. When the reference point is set at the median or maximum profit, and for some levels of the parameters that describe behavior toward losses, the quantity of input used that is driven by risk preferences represents up to 19% of the pesticide expenditure.

Appendices

Appendix 1

1.1 Description of the Crop Data from France Used for Calibration of the Production Function

We use data from crop farms from the Meuse département (a small administrative region in northeastern France). Our sample is a rotating panel which covers years from 1993 to 2010. The major crops produced on these farms are wheat, barley, and rapeseed. In the following, we focus on winter wheat, which was grown on 1,144 farms on our sample over the entire period (with the number of observations per year varying from 255 to 687 farms). We focus on a single-output single-input function, and we choose pesticides as the major input of the production of winter wheat. Information on input use is crop-specific and is measured in real monetary terms (expenditure). Below are two graphs featuring smoothed regressions of wheat yield on pesticides expenditure:

As expected, the relationship is increasing and concave. We try and fit a Just-Pope function on these data. The best fit was obtained with the following function: $y = 16 \times {x^{0.28}} + 30 \times {x^{ - 0.20}} \times \varepsilon$, which we approximate with $y = 15 \times {x^{0.30}} + 30 \times {x^{ - 0.20}} \times \varepsilon$ in our simulation exercise. The price of wheat is set at 110€/t, which corresponds to the median on our sample over the entire period.

Appendix 2

2.1 Review of Technology Parameters: Elasticities of Mean Output and Output Variance to Input Use

We report below estimates from the literature of the elasticity of mean output and output variance to input use, for studies that have estimated production functions and measured output in physical quantities. All these estimates are for arable farms producing crops.

Article	Country and inputs	Elasticity of mean output	Elasticity of output variance
Just and Pope [21]	Country: USA Input: fertilizers	0.3‐0.4	0.1‐0.2
Gardebroek et al. [46]	Country: Netherlands Input: herbicides and pesticides	0.23 (for conventional farms)	0.21 (but not statistically significant)
Koundouri et al. [32]	Country: Finland Inputs: fertilizers and plant protection	0.50 for plant protection 0.77 for fertilizers	−0.05 for plant protection −0.02 for fertilizers
Femenia and Letort [45]	Country: France Inputs: fertilizers and pesticides	0.1 for both fertilizers and pesticides	-

Appendix 3

3.1 Review of Relative Risk Aversion Parameters Derived from Expected Utility Models

Authors	Sample	Coefficient of relative risk aversion
Bontems and Thomas [24]	Corn plots from US Midwest 1990–1992	3.7
Chavas and Holt [31]	US aggregate corn and soybean acreage decisions 1954–1985	[1.4–6.8]
Isik and Khanna [47]	Field-level data (99 fields) from Illinois (USA) 1993–1994	1.5
Lence [48]	US aggregate data 1934–1994	[1.1–2.5]
Love and Buccola [29]	Farm data on corn and soybean production, Iowa (USA)	[2.4–18.8]
Pope et al. [49]	State-level data on net returns and acreages, 8 US states	0.4
Saha et al. [30]	Farm-level data for wheat farms in Kansas (USA), 1979–1982	[3.8–5.4]
Sckokai and Moro [38]	Farm-level data from Italy, 1993–1999	[0–5.5]

Appendix 4

4.1 Sensitivity of the Optimal Input Use to the Mean Function

Fig. 3

Profit functions for different mean production functions (price ratio = 1 to 11, $\varepsilon$ = 0)

Fig. 4

Profit functions for different mean production functions (price ratio = 1 to 11, $\varepsilon$ = 0)

Appendix 5

5.1 Sensitivity of Input Use and Risk Premium to Levels of Relative Risk Aversion and Parameters of the Risk Function

		Risk neutrality	Risk aversion
Risk function		rr = 0	rr = 1	rr = 2	rr = 3	rr = 4
Risk-decreasing input
$g(x) = 20{x^{ - 0.1}}$	Input use (EUR)	264	266	268	270	272
	Self-insurance (%)	0.0	0.8	1.5	2.2	2.9
	RP* (EUR)	0	7	15	22	30
	$\Delta$ CE (%)	-	−0.6	−1.3	−2.0	−2.6
$g(x) = 40{x^{ - 0.1}}$	Input use (EUR)	264	273	283	295	309
	Self-insurance (%)	0.0	3.3	6.7	10.5	14.6
	RP* (EUR)	0	30	63	100	142
	$\Delta$ CE (%)	-	−2.7	−5.7	−9.0	−12.7
$g(x) = 30{x^{ - 0.1}}$	Input use (EUR)	264	268	273	279	285
(Benchmark)	Self-insurance (%)	0.0	1.5	3.3	5.4	7.4
	RP* (EUR)	0	17	34	52	71
	$\Delta$ CE (%)	-	−1.5	−3.0	−4.6	−6.3
$g(x) = 30{x^{ - 0.2}}$	Input use (EUR)	264	267	270	273	276
	Self-insurance (%)	0.0	1.1	2.2	3.3	4.3
	RP* (EUR)	0	5	11	16	21
	$\Delta$ CE (%)	-	−0.5	−1.0	−1.4	−1.9
Risk-increasing input
$g(x) = 10{x^{0.1}}$	Input use (EUR)	264	259	253	247	241
	Self-insurance (%)	0.0	−1.9	−4.3	−6.9	−9.5
	RP* (EUR)	0	17	35	54	73
	$\Delta$ CE (%)	-	−1.5	−3.1	−4.8	−6.6
$g(x) = 15{x^{0.1}}$	Input use (EUR)	264	251	234	213	187
	Self-insurance (%)	0.0	−5.2	−12.8	−23.9	−41.2
	RP* (EUR)	0	41	87	142	207
	$\Delta$ CE (%)	-	−3.6	−7.8	−12.8	−18.7

1. The pesticide-crop price ratio is 1 to 11 and initial wealth ${w_0}$ = 500 EUR

Appendix 6

6.1 Sensitivity to Initial Wealth

		Risk neutrality	Risk aversion
Initial wealth		rr = 0	rr = 1	rr = 2	rr = 3	rr = 4
500 EUR	Input use (EUR)	264	268	273	279	285
(benchmark)	Self-insurance (%)	0.0	1.5	3.3	5.4	7.4
	RP* (EUR)	0	17	34	52	71
	$\Delta$ CE (%)	-	−1.5	−3.0	−4.6	−6.3
1000 EUR	Input use (EUR)	264	267	270	273	277
	Self-insurance (%)	0.0	1.1	2.2	3.3	4.7
	RP* (EUR)	0	11	23	34	46
	$\Delta$ CE (%)	-	−0.7	−1.4	−2.1	−2.8
1500 EUR	Input use (EUR)	264	266	268	271	273
	Self-insurance (%)	0.0	0.8	1.5	2.6	3.3
	RP* (EUR)	0	8	17	26	34
	$\Delta$ CE (%)	-	−0.4	−0.8	−1.2	−1.6
5000 EUR	Input use (EUR)	264	264	265	266	267
	Self-insurance (%)	0.0	0	0.4	0.8	1.1
	RP* (EUR)	0	3	6	10	13
	$\Delta$ CE (%)	−	−0.1	−0.1	−0.2	−0.2

1. The pesticide-crop price ratio is 1 to 11. The production function is the same as the one used in the benchmark scenario

Appendix 7

7.1 Review of Parameters Derived from CPT Models

Authors, sample	Specification	Parameter estimates
Tversky and Kahneman [12] 25 Students	CRRA utility function No preference reversal ${\alpha }^{+}={\alpha }^{-}=\alpha$ Loss aversion $\lambda$ KT probability weighting function ${\gamma }^{+}\ne {\gamma }^{-}$	$\widehat{\alpha }=\widehat{{\alpha }^{+}}=\widehat{{\alpha }^{-}}=0.88$ (Farmers are risk averse) $\widehat{\lambda }\approx 2.25$ (Farmers are loss averse) $\widehat{{\gamma }^{+}}\approx 0.61;\widehat{{\gamma }^{-}}\approx 0.69$ (Less probability distortion in losses than in gains)
Tanaka et al. [50] 181 Villagers, Vietnam	CRRA utility function No preference reversal ${\alpha }^{+}={\alpha }^{-}=\alpha$ Loss aversion $\lambda$ Prelec probability weighting function ${\gamma }^{+}={\gamma }^{-}=\gamma$	$\widehat{\alpha }=0.60$ $\widehat{\lambda }=3.47$ $\widehat{\gamma }=0.74$
Nguyen and Leung [51] 103 Fishermen, Vietnam	CRRA utility function No preference reversal α+ = α− Loss aversion λ Prelec probability weighting function ${\gamma }^{+}={\gamma }^{-}=\gamma$	$\widehat{\alpha }=0.62$ $\widehat{\lambda }=2.63$ $\widehat{\gamma }=0.75$
Harrison et al. [52] 531 Villagers, Ethiopia, India and Uganda	CRRA utility function No loss aversion (gains only) KT probability weighting function γ	$\widehat{\alpha }=0.54$ $\widehat{\gamma }=1.38$
	CRRA utility function No loss aversion (gains only in the experiment) Prelec probability weighting function $\gamma$ (and additional scale parameter $\eta$)	$\widehat{\alpha }=0.50$ $\widehat{\gamma }=0.96$ $\widehat{\eta }=1.20$
Nguyen [53] 181 Fishermen, Vietnam	CRRA utility function No preference reversal α+ = α− = α Loss aversion $\lambda$ Prelec probability weighting function ${\gamma }^{+}={\gamma }^{-}=\gamma$	$\widehat{\alpha }=1.01$ $\widehat{\lambda }=3.26$ $\widehat{\gamma }=0.96$
Liu and Huang [54] 320 Cotton farmers, China	CRRA utility function No preference reversal α+ = α− = α Loss aversion $\lambda$ Prelec probability weighting function ${\gamma }^{+}={\gamma }^{-}=\gamma$	$\widehat{\alpha }=0.52$ $\widehat{\lambda }=3.47$ $\widehat{\gamma }=0.69$
Bocquého et al. [14] Farmers, France	CRRA utility function No preference reversal α+ = α− = α Loss aversion $\lambda$ Prelec probability weighting function ${\gamma }^{+}={\gamma }^{-}=\gamma$	$\widehat{\alpha }=0.28$ $\widehat{\lambda }=2.28$ $\widehat{\gamma }=0.66$
Liebenehm and Waibel [55] 211 Cattle farmers, Mali and Burkina Faso	CRRA utility function No preference reversal α+ = α− = α Loss aversion $\lambda$ Prelec probability weighting function ${\gamma }^{+}={\gamma }^{-}=\gamma$	$\widehat{\alpha }=0.11$ $\widehat{\lambda }=1.35$ $\widehat{\gamma }=0.13$
Bougherara et al. [15] 197 Crop farmers, France	CRRA utility function No preference reversal α+ = α− = α Loss aversion $\lambda$ KT probability weighting function ${\gamma }^{+}\ne {\gamma }^{-}$	$\widehat{\alpha }=0.61$ $\widehat{\lambda }=1.37$ $\widehat{{\gamma }^{+}}=0.79;{\gamma }^{-}=0.84$
	CRRA utility function No preference reversal ${\alpha }^{+}={\alpha }^{-}=\alpha$ Loss aversion $\lambda$ Prelec probability weighting function ${\gamma }^{+}\ne {\gamma }^{-}$	$\widehat{\alpha }=0.63$ $\widehat{\lambda }=1.38$ $\widehat{{\gamma }^{+}}=0.81;\widehat{{\gamma }^{-}}=0.89$
	CRRA utility function Preference reversal ${\alpha }^{+}\ne {\alpha }^{-}$ No loss aversion KT probability weighting function ${\gamma }^{+}\ne {\gamma }^{-}$	$\widehat{{\alpha }^{+}}=0.60;\widehat{{\alpha }^{-}}=0.66$ $\widehat{{\gamma }^{+}}=0.79;\widehat{{\gamma }^{-}}=0.84$

1. KT is for Kahneman and Tversky. See Prelec [56] for Prelec’s probability weighting function