Expected Utility

Participation in an insurance program has been typically modelled by assuming a producer maximizes his/her expected utility of profits (π_t) which is a random variable dependent on the unknown states of nature (θ) as represented by the probability density function g(θ). The expected utility of withdrawal or non-participation (EU_W) is expressed as:

(1)$${\rm E}{\rm U}_W\lpar {\rm \pi} _t\rpar = \mathop \int \nolimits_{{\rm \theta} _{{\rm min}}}^{{\rm \theta} _{{\rm max}}} U \lsqb {\rm \pi} _t\lpar {\rm \theta} \rpar \rsqb \; g\lpar {\rm \theta} \rpar d{\rm \theta} $$

whereas the expected utility of participation (EU_P) is given by

(2)$${\rm E}{\rm U}_P({\rm \pi }_t) = \mathop \int \nolimits_{{\rm \theta }_{{\rm min}}}^{{\rm \theta }} U[{\rm \pi }_t({\rm \theta }) + P({\rm \theta }) - \; F ] {g({\rm \theta })d{\rm \theta } + \mathop \int \nolimits_{{\rm \theta }}^{{\rm \theta }_{{\rm max}}} v[{\rm \pi }_t({\rm \theta }) - \; F} \right]g({\rm \theta })d{\rm \theta }$$

where P(θ) is the indemnity or program payment received, F is the premium or program fee paid, and θ* is the state of nature implicitly defined by the change in program year margin required to trigger a payment The payment function P(θ) for an individual producer based on the structure of the AgriStability program is, for example:

(3)$$P\lpar {\rm \theta} \rpar = {\rm \delta} \times 0.7 \times \lsqb 0.7{\rm R}{\rm M}_t - {\rm PYM\lpar \theta} {\rm \rpar }_t\rsqb $$

where${\rm \; \delta } = \left\{ {\matrix{ {1\; \,{\rm if}\; {\rm PY}{\rm M}_t < 0.7{\rm R}{\rm M}_t} \cr {0\; {\rm if}\; {\rm PY}{\rm M}_t \ge 0.7{\rm R}{\rm M}_t} \cr } } \right.$ and PYM_t is the program margin in year t, and δ indicates if a program payment is made. The program margin is defined as all allowable revenues minus all allowable expenses, and a farm will observe this margin when the farm income tax is filed in the following year. A payment will be received (δ = 1) if the program year margin fell below 70 percent of the five-year reference margin RM_t (five-year moving average of the program margin), and the payment is equal to 70 percent of this gap. By evaluating the expected utility of withdrawing from AgriStability compared to the expected utility of remaining enrolled, the decision to remain enrolled results if EU_Pi —EU_Wi >0 and the producer will withdraw if EU_Pi —EU_Wi <0.

Prospect Theory

Rather than absolute levels of an outcome as is assumed in expected utility theory (EU), an alternative is to assume that an individual's preferences are determined by changes in outcomes relative to a reference point, since individuals tend to be more sensitive to losses in wealth than to gains (Rabin Reference Rabin1998). Prospect theory (PT) was developed by Kahneman and Tversky (Reference Kahneman and Tversky1979) as an alternative to the restrictive assumption embedded within expected utility theory. The core principle of prospect theory is reference dependence in which individuals evaluate a decision based on losses or gains from a reference point or initial endowment, rather than the prospect's effect on final wealth (Eckles and Wise Reference Eckles and Wise2011). In addition, the assumptions of loss aversion, implying losses have a larger effect on welfare than corresponding gains, and diminishing sensitivity, where an individual reacts more strongly to smaller gains and losses than to larger ones, are consistent with empirical findings (Freund and Ozden Reference Freund and Özden2008).

DellaVigna (Reference DellaVigna2007) propose that individuals who exhibit prospect theory preferences evaluate the lottery (x, p;y, 1 − p) as w (p) v (x − r) + w (1 − p) v (y − r), where v is value function which is defined over differences from the reference point r, and w (p) is a probability weighting function which overweights small probabilities and underweights large probabilities. The two key principles of prospect theory relevant to this analysis are (i) that individuals evaluate changes to income relative to a reference point rather than final wealth, and (ii) that individuals value losses more heavily than gains. Given this simplification and the parameters outlined above, individuals will maximize:

(4)$$\mathop \sum \limits_{i = 1}^n p_iv\lpar x_i \vert r\rpar $$

where the value function is defined as

(5)$${\rm \; } v\lpar x_i{\rm \vert} r\rpar = \left\{{\matrix{ {x - r\; \; \; \; \; \; \; \; \; {\rm if}\; x \ge r} \cr {\; \lambda \lpar x - r\rpar \; \; \; {\rm if}\; x \lt r\; } \cr}} \right.$$

and λ is a measure of loss aversion that is greater than 1. Tversky and Kahneman (Reference Tversky and Kahneman1992) estimate this loss aversion measure to be approximately 2.25, or that losses are weighted roughly twice as heavily as gains.

The principles of reference dependence and loss aversion are incorporated into a model of choice under uncertainty to determine how prospect theory preferences affect the decision to withdraw from AgriStability. Expected utility now includes the previous years' income π_t−1 as the reference point, i.e. f(π_t, π_t−1). The value function that a producer maximizes for the case of withdrawal from AgriStability is defined as:

(6)$${\rm E}{\rm V}_W\lpar {{\rm \pi}_t{\rm \vert} {\rm \pi}_{t - 1}} \rpar = \, \mathop \int \nolimits_{{\rm \theta} _{{\rm min}}}^{{\rm \theta} _{{\rm max}}} u\lsqb {\rm \pi} _t\lpar {\rm \theta} \rpar + f\lpar {\rm \pi} _t\lpar {\rm \theta} \rpar - {\rm \pi} _{t - 1}\rpar \rsqb \; g\lpar {\rm \theta} \rpar d{\rm \theta} $$

where

(7)$$\; f\lpar {\rm \pi} _t\lpar {\rm \theta} \rpar - {\rm \pi} _{t - 1}\rpar = \left\{{\matrix{ {\; a\lpar {\rm \pi}_t - {\rm \pi}_{t - 1}\rpar \; \; \; \; \; {\rm if}\; {\rm \pi}_t \ge {\rm \pi}_{t - 1}} \cr {\; b\lpar {\rm \pi}_t - {\rm \pi}_{t - 1}\rpar \; \; \; \; \; \; {\rm if}\; {\rm \pi}_t \lt {\rm \pi}_{t - 1}\; } \cr}} \right.$$

and b is a parameter capturing loss aversion with b >a >0. The measure of loss aversion scales utility so losses are valued more than gains. The expected utility equation for the case of continued enrollment is expressed as:

(8)$$\eqalign{EV_P({\rm \pi }_t|{\rm \pi }_{t - 1}) = & \mathop \int \nolimits_{{\rm \theta }_{{\rm min}}}^{{\rm \theta }} v[{\rm \pi }_t({\rm \theta }) + f\,({\rm \pi }_t({\rm \theta }) - {\rm \pi }_{t - 1}) + P({\rm \theta }) - \; F]g({\rm \theta })d{\rm \theta } \cr & + \mathop \int \nolimits_{{\rm \theta }}^{{\rm \theta }_{{\rm max}}} v[{\rm \pi }_t({\rm \theta }) + f\,({\rm \pi }_t({\rm \theta }) - {\rm \pi }_{t - 1}) - \; bF]g({\rm \theta })d{\rm \theta }}$$

where f(π_t(θ) − π_t−1) is given by (7), and like (3), P(θ) is the payment function, F is the program fee, g(θ) is the probability density, and θ* is the state of nature.

The producer will remain enrolled in AgriStability as long as EV_P —EV_W >0. Variables that decrease EV_P or increaseEV_W, have the effect of increasing the likelihood of withdrawal from AgriStability, and the opposite is true for variables that increase EV_P or decrease EV_W.

Program Margin and the Changes Over Time

To be more detailed, increasing historical net income increases the program margin and subsequently the reference margin (five-year Olympic average excluding the highest and lowest program margins) on which AgriStability payments are based. The higher expected value of a program payment increases the value of remaining in AgriStability and decreases the likelihood of withdrawal. Relative to equations (2) and (8), an increase in π_t−1 will cause EU_W to decrease since the f(π_t(θ) − π_t−1) term will be smaller. This would indicate the producer is more likely to stay enrolled based on the expression EU_P —EU_W >0.

The key principle of sensitivity to losses compared to gains associated with prospect theory is tested by calculating the change in income from the previous year. If (π_t(θ) − π_t−1) is positive (negative), then EV_W will increase (decrease); a producer is more likely to exit the program if they experience a gain in net income and to remain enrolled if they experience a loss. The changes in net income have similar effects on EV_P but result in the opposite participation effect (i.e., gain in net income will cause EV_P to increase, which increases the likelihood of participation; loss in net income will cause EV_P to decrease, which increases likelihood of withdrawal). However, a loss in net income also increases the likelihood of a program payment which will cause EV_P to increase and hence results in an increase in the likelihood of continued participation. The expected sign of the gain and loss variables are therefore ambiguous, with the sign on the loss variable depending on whether the loss is large enough to trigger a payment.

If a farm treats AgriStability as an investment strategy to generate additional income (i.e., payments), it is expected to stay enrolled when experiencing an increase in the program margin. A decrease in the program margin makes the downside risk more salient to the farm. A farm that considers the government business income insurance program as a risk management tool is also expected to stay enrolled if it experiences margin decreases in previous years.

Insurance Payment and Payment History

In addition to the program margin and margin historical changes, previous program payments would be a second source for farm entities' gain-loss evaluation. Based on equation (2), the larger the AgriStability program payment P(θ), the higher the expected utility from enrollment, which means the producer is less likely to withdraw from the program, giving an expected negative sign on the variable.

Brown and Finkelstein (Reference Brown and Finkelstein2008) find that individuals view insurance as an investment and evaluate the expected gains or losses associated with purchasing a policy, or in our case choosing to enroll in AgriStability. Hwang (Reference Hwang2015) echoes this theory, explaining how insurance can be viewed as a risky gamble rather than as a wealth protection measure. In the context of AgriStability, producers who are participating will lose the program fee if payment is not triggered (i.e., if the program year margin does not fall below a certain percentage of the historical reference margin), perhaps viewing the insurance program itself as a risk. Further, Hwang (Reference Hwang2015) asserts that prospect theory individuals ignore diversification effects, meaning they evaluate the insurance decision in isolation for existing risks, instead focusing on the insurance policy's own value. More recently, Du, Feng, and Hennessy (Reference Du, Feng and Hennessy2016) and Babcock (Reference Babcock2015) also investigated farm insurance demand as an investment strategy (for subsidy) versus a risk management tool.

Therefore, no payment in the previous year is expected to increase the likelihood of withdrawal, because the second term of equation (8) $\mathop \int \nolimits_{{\rm \theta }}^{{\rm \theta }_{{\rm max}}} v[{\rm \pi }_t({\rm \theta }) + f({\rm \pi }_t({\rm \theta }) - {\rm \pi }_{t - 1}) - \; bF]\,g({\rm \theta })d({\rm \theta })$ which is used to calculate EV_P will be smaller than if the producer were not enrolled. This means EV_P —EV_W <0, and the producer is more likely to withdraw. Moreover, program payment history would also influence participation decisions via the gain-loss “mental” calculation. The exact payment amount, as well as whether or not there is no payment, matters.

Data Sources

The Ontario Farm Income Database (OFID) is used for the empirical analysis. OFID is an administrative database managed by the Ontario Ministry of Agriculture, Food and Rural Affairs (OMAFRA), which documents the annual farm income tax returns for all agricultural related farm business entities in the province of Ontario, Canada. The database includes detailed yearly farm financial and operational information as well as participation and program payment information for government-sponsored Business Risk Management (BRM) programs including AgriStability. This study focuses on the years between 2003 and 2013, which includes two five-year themes of the BRM program reforms, i.e., Growing Forward (GF 2003–2007) and Growing Forward 2 (GF2 2008–2012). The payment structure of the BRM program and the longitudinal feature of the OFID data jointly allow us to model the gain-loss framework and empirically test against the conventional expected utility framework. For the purpose of this study, the final sample includes a farm in a certain year if this farm participated in the BRM program in the previous year. This sample allows us to model a farm's program withdrawal behaviors conditional on previous participation status.

The beef sector is selected for investigation. Beef producers do not have access to production insurance like crop farmers, meaning AgriStability is the main risk management program available to them from the government. A farm entity is considered a beef farm in OFID if more than 50 percent of its annual operational revenue comes from the sector. It is possible that the same farm is categorized into different sectors across years. In order to obtain a farm entity's dynamic operational information, the study sample includes all existing years of a farm entity in the database so long as the farm entity was considered a beef farm for at least one year over the study period.

Study Sample

The number and percentage of beef farms in OFID by revenue class between 2003 and 2013 are listed in Table 1. The total number of beef farms fell by approximately 20 percent from nearly 8,000 in 2003 to 6,285 in 2013. The decline occurred primarily in the middle three revenue categories, as the number of farms selling less than $10,000 stayed relatively constant over the decade and the number of beef farmers selling more than $500,000 increased from 433 in 2003 to 557 in 2013.

Table 1.

Number (Percentage) of Ontario Beef Farms, 2003–2013

Source: Ontario Farm Income Data base internal calculations, OMAFRA.

The trends in withdrawal from AgriStability for Ontario beef farmers between 2003 and 2013 is given in Table 2. AgriStability is the primary mechanism that individual beef farmers can utilize to protect themselves against large income losses that may arise from low commodity prices, rising input prices, or production losses. The absolute number of farms withdrawing from AgriStability in a given year does not change drastically over the ten-year period. However, the percentage of farms withdrawing is increasing over time across all revenue sales classes. The top-three classes all have a withdrawal rate increase of around 9 percent over the period; for example, the over-$500,000 category starts off with 1.9 percent of farms withdrawing in 2003 (i.e., five farms withdrawing from 262 participating ones) to 11.7 percent of farms withdrawing in 2013. The two lowest income categories experience larger rates of withdrawal, with the largest in the 0 to $10,000 category from 10.3 percent in 2003 to 28.6 percent in 2013. A detailed list of program participation rates specific to year and farm revenue category can be found in the Appendix Table A1.

Table 2.

Number (Percentage) of Producers Withdrawing from AgriStability by Year and Revenue Size, 2003–2013

Note: Withdrawal defined as participating in AgriStability in time period t−1 and not participating in AgriStability in time t.

Source: Ontario Farm Income Data Base author's calculations, OMAFRA.

Summary statistics for all observations of a farm participating in AgriStability in the previous year of observation are given in Table 3. A total of 28,390 farm-year observations, representing 6,138 farm operations, are in the sample over the period between 2003 and 2013. Comparing this conditional sample of farms that participated in AgriStability in the previous year to the overall unconditional distribution of beef farms by revenue category (Table 1), it indicates higher percentages in Table 3 (the conditional sample) for the larger farms, i.e., participation rates in AgriStability are higher for farms with more sales. Conditional on program enrollment in the previous year, the average withdrawal rate was about 13 percent, with larger revenue categories having lower rates than the smaller sales farms.

Table 3.

Summary Statistics

Model Specification and Hypothesis Testing

The conditional probability of withdrawal from AgriStability is estimated with the following regression:

(9)$$\eqalign{{\rm Prob}\left( {{\rm Withdrawa}{\rm l}_t} \right) & = {\rm \beta }_1{\rm Program}\,{\rm Margi}{\rm n}_{t - 1} + {\rm \beta }_2{\rm Margin}\,{\rm Increas}{\rm e}_{t - 1} \cr & \quad + {\rm \beta }_3{\rm Margin}\,{\rm Decreas}{\rm e}_{t - 1} + {\rm \beta }_4{\rm Paymen}{\rm t}_{t - 2} \cr & \quad + {\rm \beta }_5I\left( {{\rm No}\,{\rm Payment}} \right)_{t - 2} + {\rm \theta }\,{\rm Control} + \epsilon } $$

or alternatively,

(10)$$\eqalign{{\rm Prob}\left( {{\rm Withdrawa}{\rm l}_t} \right) & = {\rm \delta }_1{\rm Program}\,{\rm Margi}{\rm n}_{t - 1} + {\rm \delta }_2{\rm Margin}\,{\rm Differenc}{\rm e}_{t - 1} \cr & + {\rm \delta }_3I\left( {{\rm Margin}\,{\rm Increase}} \right)_{t - 1} + {\rm \delta }_4{\rm Paymen}{\rm t}_{t - 2} \cr & + {\rm \delta }_5I\left( {{\rm No}\,{\rm Payment}} \right)_{t - 2} + {\rm \theta Control} + \epsilon } $$

The decision to withdraw from the AgriStability program in a certain year given that the farm was previously enrolled (Withdrawal_t = 1| Participation_t−1 = 1) depends on the value of the potential payment to be received within both the expected utility (2) and prospect theory (8) approach. The value of participation (or withdrawal) hence depends on the farm's program margin in the previous year (Program Margin_t−1), which is calculated as allowable incomes minus allowable expenses. Hypothesis 1 predicts positive value of β₁ and θ₁.

The gain-loss framework from Prospect Theory is tested by the variables measuring the program margin changes. Program margin changes experienced in the last year t−1 are derived by subtracting the program margin_t−2 from the program margin_t−1. Note that in reality at the beginning of each calendar year t, a farm business owner will make a decision on whether or not to participate in the AgriStability program for the current (calendar/program) year when s/he is handling the previous year's farm income tax return. At that time, s/he only knows the revenue (or program) margin and changes in the previous year t−1 and the exact values of program payment for the years before last year, i.e., t−2, t−3 and so on.

In equation (9), Margin Increase_t−1 for a certain farm is defined as the value of margin change if it is positive, and 0 otherwise. Similarly, Margin Decrease_t−1 is defined as the value of margin change if it is negative, and 0 otherwise. Hypothesis 2a and 2b predict that β₂ and β₃ are negative. Hypothesis 2c predicts that β₂ <β₃.

Equation (10) is different from (9), such that Margin Difference_t−1 is defined as the absolute value of margin change, and the dummy indicator I(Margin Increase)_t−1 equals to 1 if margin changes are positive. Margin Difference measures the magnitude of margin fluctuation regardless of increase or decrease, and the margin increase dummy indicator differentiates the farms with positive margin changes from those with negative ones. According to Hypothesis 2a, b, and c, δ₂ is negative (i.e., risk management needs) and δ₃ is positive (i.e., an investment attempt that will offset the risk management needs).

Payment_t−2 is the value of payment in the previous year. I(No Payment)_t−2 indicates if a farm did not get any payment in year t−2. Hypothesis 3a predicts negative β₄ and δ₄. Hypothesis 3b predicts positive β₅ and δ₅. In order to capture the historical effects of no payment, an additional indicator of no payment in year t−3 is also included in the model with the expectation that history of no payment yields different effects across years.

The model also controls farm characteristics such as farm sector concentration, business diversification, farm type, and other government payments, etc. A structural change dummy variable is also included to capture the policy effects after 2008, when Growing Forward 2 was implemented.

Variable Description

According to Table 3, the average program margin from the previous year was approximately $7,970 for the sample. Program margin was negative for the lowest two revenue categories, and the remaining three higher categories ranged from $8,740 to $60,780. The average program payment in the sample was about $6,520. The payment size ranged widely from as low as $2,130 for the smallest farm category to as high as $27,030 for the biggest farm category, even though the probability of having a non-zero payment was higher in the smaller farm categories than the bigger ones.

In addition to program margin changes and payment history, a series of farm operation variables are included as control variables. Sector concentration is calculated as the percentage share of current year operation revenue coming from the beef sector. Sector concentration ranged from 0.84 to 0.93 across revenue categories. The Diversification index measures the balance and diversification of farm operation among alternative revenue streams, where 0 means no diversification and 1 means perfectly diversified. The Diversification index ranged from 0.34 to 0.57. Larger farms (proportionally more of which are feedlot farms) are less diversified. Farm type includes three categories: cow-calf and feedlot operations within the beef sector, and other type, when a beef farm switches sectors in some certain years; 13.3 percent of the sample were feedlot farms, 3.6 percent were others (i.e., when a farm was categorized as other than beef sector in certain program years), and the remaining 80 percent were cow-calf farms. The farm type distribution varied significantly across revenue categories, with the majority of cow-calf operations being in the three lowest revenue categories, and the majority of feedlot operations in the two highest revenue categories. Two measures of government ad hoc payments, i.e., targeting to the beef sector and targeting to farm entities in general, are also included. Year fixed effects and an indicator for years 2008 and later (i.e., after program reform) are also controlled. As with AgriStability, larger farms received substantially larger ad hoc payments from other government programs targeting the beef sector. These payments tended to be associated with disease outbreaks that negatively affected market revenue to the whole beef sector.