Political Economy of Crop Insurance Risk Subsidies under Imperfect Information. June 7, Harun Bulut and Keith J. Collins *

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1 Poitica Economy of Crop Insurance Risk Subsidies under Imperfect Information June 7, 213 Harun Buut and Keith J. Coins Seected Paper prepared for presentation at the Agricutura & Appied Economics Association s 213 AAEA & CAES Joint Annua Meeting, Washington, DC, August 4-6, 213. Harun Buut is Senior Economist, Nationa Crop Insurance Services (NCIS), Overand Park, KS ( Keith J. Coins retired as Chief Economist, U.S. Department of Agricuture and serves as Economic and Poicy Advisor, NCIS. Correspondence can be sent to: E-mai: harunb@ag-risk.org. Copyright 213 by Harun Buut and Keith J. Coins. A rights reserved. Readers may make verbatim copies of this document for non-commercia purposes by any means, provided that this copyright notice appears on a such copies. 1

2 Poitica Economy of Crop Insurance Risk Subsidies under Imperfect Information Harun Buut and Keith J. Coins June 7, 213 We consider a poitica economy where government cares about risk-averse farmers oss of income and yet incurs poitica cost if it provides monetary support to farmers. Government evauates three options: 1) ex-post disaster aid; 2) ex-ante insurance option with perfect information; 3) ex-ante insurance with imperfect information (farmers are over-confident about their risk). It is assumed that margina poitica cost is high enough so that the possibiity of monetary support to farmers in the absence of economic oss is rued out. In comparing 1) and 2), we find that government prefers farmers manage their risks through fairy priced insurance In comparing 1) and 3), if the information probems prevent risk-averse farmers to take up fu insurance under actuariay fair rates, government prefers to subsidize farmers insurance ex-ante rather than providing disaster aid ex-post (subject to poitica cost) for a wide range of parameter vaues. Key words: Agricutura risk, crop insurance, disaster assistance JEL codes: D81, G22, Q12, Q18 During the 212 Farm Bi debate, the crop insurance program has undergone an intense scrutiny, and the justification for crop insurance subsidies is being questioned in ight of budget and poicy issues. For exampe, critiques from an invited paper session on crop insurance subsidies at the 212 annua meeting of the Agricutura and Appied Economics Association (AAEA) point out that high risk areas have received a higher portion of subsidies which in turn encourages over-production (Goodwin and Smith, 212). In addition to efficiency concerns, budget costs are another key issue. Gauber (212) aso finds high risk areas get higher net indemnities. This paper theoreticay examines the issue of crop insurance coverage and risk subsidies from the perspective of the government s preference for crop insurance support or Harun Buut is Senior Economist, Nationa Crop Insurance Services (NCIS), Overand Park, KS ( Keith J. Coins retired as Chief Economist, U.S. Department of Agricuture and serves as Economic and Poicy Advisor, NCIS. Correspondence can be sent to: harunb@ag-risk.org. 2

3 disaster assistance, subject to a poitica cost. Government s invovement in crop insurance markets is traditionay expained by the information asymmetries (mora hazard and adverse seection probems) or the catastrophic nature of crop insurance risks. Zuauf (211) argues that the optima subsidy shoud be equa to systemic component of risk. Nevertheess, Duncan and Myers (2) find subsidized reinsurance as a soution to catastrophic (systemic) risk probems in crop insurance. Meanwhie, adverse seection probems (if present) may ead to market faiure in the form of underinsurance. The premium cost refecting the average risk in the market woud drive ower risk farmers out of the program disproportionatey, eaving a higher risk poo of insureds which over time coud ead to higher oss ratios and premium rates. Subsidizing the equiibrium price of insurance is one way for government to intervene to ensure high and diverse participation. The centra presumption of adverse seection probems is that farmers are better informed about their risks compared with the insurers. However, Cobe and Barnett (212) point out that the preceding presumption may not necessariy hod, on the contrary, farmers coud be overconfident. The atter point is aso made in Just (22) and confirmed in empirica studies (Sherrick, 22; Umarov and Sherrick, 25; and Gao et a., 211). The information probems in the form of over-confidence coud be another reason for government support for crop insurance. That issue is the centra focus of this paper and wi be studied within a poitica economy framework deveoped in Innes (23). Innes (23) points out that the ex-post urge to provide disaster reief to farmers has impications for the design of ex-ante government farm poicy. Furthermore, the effects of the reationship between a farmer s perception of risk and the farmer s risk aversion (taste for insurance) for crop insurance purchases wi aso be expored. Menapace, Coson and Raffaei (212) provide some evidence that farmers are risk averse; and 3

4 farmers who are more (ess) risk-averse perceive greater (smaer) farm osses. We present a theoretica mode that inks the modeing of the poitica economy in Innes (23) with the farmer s insurance coverage choice modeing in Buut, Coins and Zacharias (212). In ine with Innes (23), we assume government cares about a farmer s income osses and may consider providing financia assistance at times of financia distress, yet in doing so, the government incurs some poitica cost. Government s interest in the farmer s income osses are reveaed through persistent disaster assistance over time. Every year from fisca years 1989 through 212 (except for the ast two years) farmers with wide-spread production or price osses received ad hoc disaster assistance (Chite, 212). Unike Innes (23), we consider risk averse farmers and do not expicity mode production. 1 The farmer has a inear mean-variance utiity function, pays a premium, and chooses coverage eve provided that an insurance option is avaiabe. The theoretica modeing is suppemented with a numerica anaysis (using MATLAB software) when no anayticay tractabe soution coud be obtained. Farmer s Probem Denote the current income (weath) of a representative farmer with M. The farmer faces the prospect of a oss (denote the oss amount with which may refer to a production or a revenue oss) with probabiity p and no oss with probabiity (1 ). The preceding random variabe is p denoted with. The expected farmer s oss is p and the variance of the farmer s oss is p (1 ) 2 p. As in Duncan and Myers (2), a representative farmer is assumed to have a inear 1 Ex-ante insurance shoud hep with the production because indemnity is certainy paid in a timey manner once a shortfa happens (provided that farmer foow good farming practices). Whie ex-post disaster aid woud impy a great dea of uncertainty to the production (Goodwin and Vado, 27). 4

5 mean-variance preference function specified as U M p.5 p (1 p ), (1) 2 where U denotes the utiity of farmer under the assumption that farmer has perfect information regarding farmer s own risk, M is the initia income and is the risk aversion parameter for the farmer. 2 Denote the reaized vaue of the random variabe with h, then h in case of oss and h in case of no oss. Then, we define the ratio of amount of oss (reaized) to farmer s initia income with r h, that is M h h. (2) r In case of oss, we wi refer to r h as r that is, r h r, otherwise rh. M Government s Preferences (Poitica Economy) In ine with Innes (23), government is benevoent and cares about farmer s osses of income and imposes taxes/transfers (denoted with : where indicates transfer whereas indicates taxing). In doing so, government incurs a poitica cost. For the sake of simpicity, we assume that it is equay poiticay costy to subsidize or tax farmers. Government s preferences can be further specified as G( ) Bv( ) c( ), (3) where B represents the government s utiity in status quo from non-food sectors in the economy 2 The expression in equation (1) corresponds to farmer s certainty equivaent under the assumption that farmer s income is normay distributed and farmer s utiity can be represented with negative exponentia utiity function (Moss, 21, p. 128). Otherwise, it is approximatey equa to the certainty equivaent based on Arrow-Pratt approximation to risk-premium (Goier, 21, p. 22). 5

6 (which wi be normaized to zero), indexes the change in farmer s financia we-being, v represents the vaue government receives from the changes in farmer s financia we-being, and c( ) is the poitica cost of extending federa funds at the amount of to an individua farmer. We wi now focus on v and c( ) functions in turn. We specify the vaue function v (.) as v. (4) In the preceding equation, indexes the change in farmer s we-being as before, whie represents the government s sensitivity to the vaues of. A simpe way to think about the parameter as some monetary vaue per farm e.g., the per-farm net-vaue added created in the economy. 3 The parameter can be specified as 4 w r e. (5) w(, r ) (, ) ( h h ) In the preceding equation, w(, r h ) is the percent change in farmer s financia we-being depending on the amount of government s transfers ( ) and the ratio of farmer s oss to farmer s initia income ( r h ). In case of oss, farmer s ex-post income with government transfer woud be M. 3 Net vaue-added in 212 is forecasted at $164.8 biion (USDA ERS, 213) whie the number of farms in 212 is 2.2 miion (USDA NASS, 213). Per farm vaue added then woud be $74,99 in This function is a specia case of expo-power function given in Saha (1993, p. 96): 2 1w uw ( ) e where 1, 1, and 2. Here, we assume, w is the percent change in farmer s we-being (income), 1, to maintain simpicity and tractabiity. In addition, here the concavity of government objective function does not arise from risk aversion but instead from diminishing margina poitica pressure as the farmer s financia webeing improves. 6

7 Then, the percent change in we-being is (ex-post cacuation) ( M ) M w(,) r 1r 1r. (6) M M M In case of no oss, and so r but with government transfer, the farmer s ex-post income woud be M M. Then, h ( M ) M w(,) 1 1. (7) M M M In the case oss, and so r but without government transfer, the farmer s ex-post income woud be M. Then, ( M ) M w(, r) 1r1 r (8) M Finay, in case of no oss and no government transfers (that is, and so r, and ), which coud be viewed as status quo, there woud be no change in farmer s we-being M M w(,). (9) M In the preceding situation, the parameter in equation (5) woud reduce to w e (,) ( ) ( 1). (1) The remaining eement in equation (5) is the parameter, which may represent the poiticay targeted vaue for the farmer s financia we-being. In ine with Saha (1993), 1. One can further specify 2. Pugging 2 in equation (1) resuts in h w(,) 1. Pugging that further in equation (4) woud yied v( w(,) ( w(,) ( 1) (21). That is, when there is no change in farmer s income (status quo), the parameter (farmer s net vaue added) shows up in the government s objective function (or some vaue to the government may not be the vaue added). 7

8 w Note that function w ( e ) is increasing and concave in w with the foowing critica vaues. ( e ) 1, im w and im w w w the preceding function for 2. We specify the cost function c( ) as foows:. Figure 1 dispays Fd k if c( ) if, (11) where F d represents the fixed poitica cost of providing funds and k is the variabe poitica cost to be incurred in extending support eve. Note that the margina poitica cost is k as in Innes (213). 5 The idea of a fixed poitica cost of program enactment is aso obtained from Innes (213, p. 329). The fixed poitica costs refer to costs such as time spent in deiberations, crafting egisation anguage, foor time needed or costs added during the process of passing the egisation (some egisative maneuvering is needed to gather enough support). In addition, we wi assume that uness a disaster is decared, per farm fixed cost wi be prohibitivey high. This wi ensure that government does not provide ex-post disaster aid to a singe farm when the neighboring farms are faring we. If, on the other hand, a disaster is decared, per farm fixed cost wi be ow enough and workabe. A disaster wi be decared if the 5 Innes (23) defines the margina poitica cost k as the poitica vaue of the government doar directed to another constituency. Presumaby, the margina poitica cost shoud be reated to margina opportunity cost of government spending. Aston and Hurd (199) find that the margina opportunity cost of a doar of U.S. federa spending is not one doar per doar of government spending but rather is ikey to be in the range of $1.2 to $1.5. Innes (23) (see footnote 18) points out that the k (per doar poitica cost of funds) may refer to aternative use of funds as we as deadweight costs of raising the government revenue. Innes (23) aso mention the possibiity of random poitica cost (see footnote 13 in p. 329). Here, both fixed and margina poitica costs parameters coud be random variabes. 8

9 amount of area oss is sufficienty high and wide-spread, that is, the critica mass of farm distress wi be needed (Innes, 23). Thus, a consideration of positive vaue of necessitates disaster decaration, and hence the subscript d in the fixed cost F d. Even though a disaster can be decared due to high enough area oss, the government does not provide a transfer uness the government s vaue of the change in the farmer s we being ( ) exceeds the poitica cost of the transfer. To ensure this, government s utiity with and without any transfers shoud show M (2 e ) F k for a. (12) Re-express equation (12) as d M (1 e ) k F F d hods for a. (13) In the preceding equation, F is the maximum eve of fixed cost that can be accommodated given the other parameter vaues. When, both terms on the eft-hand side of equation (13) is zero. Then, the inequaity hods for any Fd. Equating the eft hand side in equation (13) to M zero yieds k (1 e ). Then, taking the imit of k as goes to zero yieds a benchmark eve of margina cost as k. (14) M M M Note that when k k, the foowing hods (1 e ) 1 ( e ) F M M F 1 d for any F. Thus, for any k k, margina poitica cost is high enough to deter ex-post disaster aid d to the farmer when farmer does not show any oss (despite the event of disaster decaration due 9

10 to wide-spread osses of other farmers in the same area). Because disaster decaration requires the event of area oss, we now specify the uncertainty in this regard. Denote the event of area oss with S a which is either 1 (area has a oss) or (area does not have a oss). Use the short-hand notation A to denote the event of Sa 1. We denote the probabiity of such an event A wi happen with PA ( ) pl. 6 Assume that the probabiity that a disaster is decared without an area oss is zero. Denote whether a disaster is decared or not with the event S d. Now, S d is either 1 (a disaster is decared) or (a disaster is not decared). Use the short-hand notation D to denote the event of Sd 1. Disaster wi be decared when the size of oss in the area is sufficienty arge. Once the area oss happens, the conditiona probabiity that a disaster wi be decared can be denoted with PD ( A. ) And the joint probabiity that both an area has a oss and a disaster is decared is 7 PD ( A) PD ( A) pl. (15) Actuay, the unconditiona probabiity that a disaster is decared equa the joint probabiity that 6 Because of aggregation invoved, the risk shoud be ower in the area compared to a typica farmer in the area. In rea word, some farmers gains woud offset other farmers osses in the area. Here and in Duncan and Myers (2), the gains in farmers prospects are normaized into the event of no oss to maintain simpicity and tractabiity. Together with the assumption of identica farmers who are identica in terms of risk profie (except perhaps risk preferences and income), the sufficient size of the area oss can be defined with the sufficient number of farmers with a oss in the area. In ine with the systemic risk modeing in Duncan and Myers (2), the probabiities for area oss can be defined as the right tai probabiities of correated Binomia distribution. Moody s (24) provides an approach to obtain the probabiities of the preceding distribution. 7 Goodwin and Vado (27) state: in ight of the consistency of agricutura disaster payments in U.S. agricuture, it is ikey that farmers condition their production decisions based on an estimate of the probabiity that payments wi be forthcoming in the event of poor production or market conditions (see p. 41). Buut and Coins (212) show that the expected suppementa disaster payments and/or avaiabiity of under-priced area-insurance woud dampen the crop insurance demand. 1

11 area has a oss and a disaster is decared (that is, PD ( ) PD ( A) ) and it is ess than the unconditiona probabiity of area has a oss ( p L ) from equation (15), that is, PD ( ) pl. Use the foowing short-hand notations: p dl for PD ( A; ) pd L for PD ( A) ; and p d for PD ( ). Regarding individua farmer s situation vis-a-vis the area, we denote the random outcome of whether a oss occurred for the farmer with S i. The random outcome of whether a oss occurred for the area is denoted with S a as defined earier. Then, the joint events can be denoted with ( S, S ). The joint distribution of the individua and the area osses is as foows: both i a individua and area see a oss, (1,1) with probabiity L p ; individua sees a oss but area does not, (1, ) with probabiity p N ; individua does not see a oss but area does, (,1) with probabiity p nl and neither individua nor area sees a oss, (,) with probabiity p nn. Furthermore, the probabiities for joint events can be written as pl pp L rr L ; pn p (1 pl) rr L ; p (1 p ) p rr ; and p (1 p )(1 p ) rr, where is the correation coefficient nl L L nn L L from above, s is the standard deviation of event S i and s L is the standard deviation of event S a. The standard deviations are defined as s p (1 p ) and s p (1 p ). In addition, the i a i a L L L L covariance term between the events ( S, S ) is Cov( S, S ) s s. Note that the vaue of the correation coefficient parameter must be consistent with the fact that probabiities are a nonnegative. We further assume that pn and pnl. From these reationships, one can reobtain the margina probabiity of osses as p pl pn and pl pl pnl. Against the prospect of farmer s oss, government is evauating two options: First option is to rey on ad-hoc disaster aid, which may happen after the farmer s oss, thus it is an ex-post instrument. Second option is having farmer covered ex-ante through insurance. The ex-ante 11

12 insurance option (a much more compex form of which is currenty the main risk-management program in the U.S.) wi be evauated against the ex-post disaster aid option under perfect and imperfect information environments, which are defined beow in turn. The superscripts and 1 henceforth indicate perfect and imperfect information environments. Finay, subscripts EA and EP henceforth indicate the ex-ante and ex-post situations, respectivey. Perfect Information Perfect information refers to a farmer being abe to accuratey estimate the risk the farmer is facing. Specificay, the farmer accuratey estimates the probabiities p, p L, p L, p N, p nl, pnn, p dl and the correation coefficient. Ex-Post Disaster Aid under Perfect Information In the absence of ex-ante insurance coverage, when oss happens, the farmer s ex-post income with government transfer woud become M. The government s objective function is written as (, ) () ( w r G B e ) Fd k (16) where w(, r) is as defined in equation (6) and c( ) is as defined in equation (11). The government s probem is to maximize its objective function in equation (16) by choosing a nonnegative eve of transfer. Soving the F.O.C. (which is necessary and sufficient) yieds. M r n km. In addition to the ower bound defined in equation (14), we now define an upper bound for the margina poitica cost as r e k. (17) M 12

13 Note that so ong as k k and it is monotonicay decreasing in k as k increases within [ kk, ]. Furthermore, increasing in, increasing in r and does not depend on p. Note that even though becomes negative when k k, it wi be set to zero because government is not interested in taxing the farmer. Reca the assumption that it is equay poiticay costy to subsidize or tax farmers and the variabe poitica cost can be defined as k where is the absoute vaue operator. Based on F.O.C., one then obtains km M rn k k, k k. (18) In addition to the variabe cost, the government shoud take into account the fixed cost. In the w(, r) fina anaysis, the government s utiity with a farm oss and no transfer, G() y ( e ) shoud be ess than shoud be ess than its utiity with a farm oss and the optima transfer, G y e F k w(, r) ( ) ( ) ( d ) in order for government to extend to the farmer. The preceding condition can be expressed in the foowing: r M Fd e (1 e ) k, (19) F ( k) EP where F ( k ) denotes the impied maximum eve of fixed cost that can be accommodated. The EP right hand side of the preceding equation shows the additiona vaue gained by extending ˆ to the farmer whie the eft hand side shows the fixed cost of doing so. For a high enough poitica cost k k,, that is, margina poitica cost is so high, just based on margina anaysis aone, the government does not extend any ex-post disaster aid. For k k, the righthand side of equation (19) is zero. Given that Fd, that woud ead to contradiction. At k k, 13

14 where k is defined earier in equation (14), F EP( k ) becomes F ( ) r EP k e (1 r), which is positive for a r and the magnitude of it increases with higher vaues of r, suggesting that the fixed cost of ex-post disaster aid is justified for bigger osses. One can further verify that F ( k ) is a convex function of k and monotonicay decreases to zero as k increases towards k EP. Based on the foregoing, there shoud exist foowing hods: k EP in equation (19) such that k k k, and the EP F ( k ) F. (2) EP d Provided that kep k, the government can extend of fixed cost F d. We now summarize the foregoing. for a k k k EP [, ), despite the presence Lemma 1: Suppose that the amount of fixed poitica cost F d is not too high so that there exists k in equation (2) such that k EP EP k. For a margina poitica cost vaues that are beyond k EP and ess than k, that is, k k k [ EP, ) government to extend monetary support to farmers., the mere presence of fixed poitica cost prevents the Based on the preceding emma, it foows then. k [ k, kep), k kep (21) Now, the farmer s ex-ante cacuation wi take the ex-post optima into account. Let DA denote the random variabe of additive change in the farmer s initia income under disaster assistance. The farmer s expected oss under disaster assistance is then E p p p p p. (22) ( DA) L( d L ) N ( ) nl nn 14

15 The preceding expression can be re-expressed as E ( ) ( p p ) p p p p p. (23) DA L N L d L L d L As discussed earier when the farmer does not have a oss but the area has a oss (that event denoted with the subscript nl ) because of high enough margina poitica cost. In addition, disaster aid does not protect against the basis risk (that event denoted with the subscript nl ) as disaster wi not be decared in the event of no area oss. Finay, in the event of, neither the farmer nor the area has a oss, the farmer wi not receive any disaster aid. A component of the farmer s variance of the oss is E ( ) p ( p ) p ( ) p p. (24) DA L d L N nl nn E ( ) E ( ), pugging the 2 2 Because the variance of the farmer s oss is 2 corresponding equations yied DA DA DA p (1 p )( p ) p (1 p )( ) 2 p p ( p ). (25) L L d L N N L N d L DA Let DA U denote the utiity of farmer under the disaster aid option and perfect information. Then, DA U can be expressed as DA U U (, r) M E( ).5. (26) 2 DA DA In the preceding equation, the farmer s ex-ante utiity cacuation takes into account the possibiity of ex-post disaster aid in the event of oss. The government s ex-ante cacuation is DA w (, r) G ( ) B ( e ) c EA ( ). (27) In the preceding equation is the optima soution from equation (18) (subject to area-wide oss) and 15

16 w (, ) (, ) EA r U r M U r U (, ) 1 1. (28) M M M DA where DA U is as obtained in equation (26). Finay, c ( ) EA is the expected cost of extending ex- post disaster assistance eve and equa to EA L d L d c ( ) p p F k. (29) We now deveop the government s objective function under the ex-ante insurance option. Ex-ante Protection through Insurance under Perfect Information We use the insurance choice modeing deveoped in Buut, Coins and Zacharias (212). Denote the premium per unit of insurance coverage eve with. If a farmer hods x units of coverage with individua insurance, farmer s objective function in terms of decision variabe x is Ins U U( x ;, r) M x p(1 x).5 1 ( x 2 x) p (1 p ). (3) In the preceding equation, Ux (; ) 2 2 denotes the utiity with insurance coverage given premium rate, xp denotes the expected indemnity from insurance and ( x 2 x) p (1 p ) denotes the 2 2 risk reduction that can be obtained by hoding coverage x and the other parameters are as defined above. The farmer s probem is to maximize the utiity function in equation (3) by choosing a non-negative eve of coverage x. Soving the necessary and sufficient first-order condition 16

17 (F.O.C.) yieds demand for insurance (denote with x ) 8 as x 1 1 ( p ). (31) 2 If the farmer purchases fu insurance, then the farmer s utiity is stabiized across the states of oss or no oss as Ins U U x M p M p r f ( 1; ) (1 ), (32) which is ess than the farmer s initia income. 9 Note that even though r is aowed to exceed 1, there is sti upper bound to it: 1 r. The preceding condition is equivaent to: p M, p that is, the expected oss amount does not exceed farmer s initia income so that farmer remains sovent when such a oss occurs. Define the percentage change in farmer s financia we-being as Ins U( x ;, r) M U wx ( ; r) 1. (33) M M 8 The demand for coverage decreases with the increases in premium and increases with increases in the expected oss p. If the insurance is actuariay fair, that is, premium rate equas f expected oss, p (so that the premium amount woud equa expected indemnity), then the stricty risk averse individua wi insure competey ( x 1). If the premium rate is p, that is, rates are unfair (overrated), the demand for coverage wi be ess than one and increasing with the risk aversion parameter and the variance of oss. If insurance is underrated, that is, p, then the demand for coverage wi exceed one, that is, the farmer woud ike to over-insure. In this case, the farmer woud be wiing to toerate increased exposure to risk with increased mean income. Nevertheess, the wiingness of the farmer to over insure decreases as the risk aversion parameter and/or the variance of oss increases. 9 Moschini and Hennessy (21) state that stating the obvious might yet be usefu: risk management activities in genera do not seek to increase profits per se but rather invove shifting profits from more favorabe states of nature to ess favorabe ones, thus increasing the expected we-being [utiity] of a risk averse individua. 17

18 Pugging the preceding into the government s objective function in equation (3) resuts f Ins f w( x ;, r ) G G( x 1; ) y( e ). (34) Again, in case of oss the farmer obtains the utiity given in equation (32) whie the government woud derive the utiity given in equation (34) ex ante and ex-post from insurance for a risk aversion eves. In order to compare the farmer s ex-ante utiity from ex-post disaster assistance equation (26) with the farmer s utiity from ex-ante insurance Ins U given in equation (32), re- DA express U in equation (26) after substituting E ( ) from equation (23) as DA DA U in DA 2 U U (, r) M p plpd L.5. (35) DA Ins U H ( ) In the preceding equation, the term in parenthesis H ( ) represents change (gain/oss over) with respect to the insurance option under actuariay fair rates. Now, define a critica vaue of risk aversion as ( k) EA EA 2 p p L d L 2 DA. (36) For each k and F d, is determined, which in turn determines EA EA 2. Reca that aso depends on in equation (25). From the definition of, it foows that for a EA, the DA term H in equation (35). Note that if k k EP, then. Then, EP reduces to zero. In that case, any risk-averse individua is better off with the insurance option because ex-post disaster assistance does not pay due to high poitica cost. We summarize the foregoing. 18

19 Lemma 2. Assume the fixed cost under a disaster decaration ( Fd ) and the margina poitica cost are such that above ( ) EA k k k k EP hods so that. Then, a farmers with risk aversion prefer the fairy priced insurance option over ex-post disaster aid (reca that the atter is free to the farmer). As the margina poitica cost decreases from (meanwhie increases), ( ) EA k preferring fairy priced insurance tends to decrease. k EP towards k increases, therefore the fraction of risk-averse farmers For those farmers who prefer the insurance option over disaster assistance aid, the government s choice is straightforward for insurance because ex-post disaster is poiticay costy whie insurance under perfect information is not. For those farmers (with very ow risk aversion) who prefer ex-post disaster assistance over insurance, the government has to weigh the additiona vaue for such farmers with the cost associated with the disaster assistance option. Intuitivey, government s ex-post income transfer may not efficienty reduce risk of a risk-averse farmer, favoring the ex-ante insurance option. Even though the ex-post disaster aid can increase mean income, the poitica cost of extending ex-post aid may not be economicay justified. So far, the anaysis assumed away information probems which can prevent a farmer from taking up fu insurance under actuariay fair rates. By providing premium subsidies, the government can get the farmer into insurance or encourage purchasing higher eves of coverage. The insurance option under information probems wi be anayzed in the next section. Imperfect Information: Introducing a Farmer s Over-Confidence The previous empirica studies has indicated that farmers can be over-confident (Scherrick, 19

20 22; Umarov and Sherrick, 25; Gao et a. 211). 1 We mode a farmer s over-confidence as q max{ p,}, (37) where q denotes the farmer s assessment of risk and represents the discrepancy with respect to true risk ( p as defined earier). 11 Furthermore, Menapace, Coson and Raffaei (212) provide some evidence that farmers are risk averse and farmers who are more (ess) riskaverse perceive greater (smaer) farm osses. 12 The effects of the reationship between a farmer s perception of risk and the farmer s risk aversion (taste for insurance) for crop insurance purchases is modeed pa( ). (38) In the preceding equation, (,1) is a parameter sets the upper imit on farmer s overconfidence the degree of over-confidence and the function A( ) indexes the over-confidence in terms of the degree of risk-aversion. We write A( ) as 1 Based on a survey of mid-western farmers, Scherrick et a. (24) note (see p. 113) Our survey experience indicates that farmers can readiy provide subjective probabiities (and ikey use them intuitivey in decision making), but how we their expectations correspond to actua yied risk is especiay important to consider in the deveopment of effective insurance markets. Providing information to hep caibrate farmers expectations about insurabe risks wi have high vaue in a continued high-risk environment. 11 This can be viewed as a form of decision maker s errors in probabiity weighting in the anguage of Prospect Theory (Kahneman and Tversky, 1979). Appication of Prospect Theory into farm risk-management choices can be interesting avenue of research. A review of the appications of prospect theory in other areas (risk or non-risk based) can be found in Barberis (213). 12 Finkestein and McGarry (26) find that risk-preferences and risk-types can be negativey correated with each other in ong-term care insurance markets. They mention that this may not hod in a insurance markets as there is evidence of positive correation between the two in auto insurance markets. 2

21 max A( ), (39) max min where the parameters max and min are the maximum and minimum risk aversion eves that can be considered (see Tabe 1 for more information). Thus, over-confidence is assumed to be decreasing in the farmer s risk aversion parameter. Now, substituting the expression from equation (38) in equation (37) resuts in q p 1 A( ). (4) One can verify that im p and so min im q p (1 ). (41) min im and so max im q p. (42) max That is, as risk aversion gets higher, the farmer s perception of risk approximates the true risk. Denote the random variabe j as the farmer s oss at the amount of with the probabiity q. Then, the farmer s expected oss is E( j) q and the variance of farmer s oss is q(1 q). Based on E( j ) and j, farmer s preferences in equation (1) can be rewritten j U M q.5 q (1 q ). (43) 1 2 The same definition of area oss from the perfect information environment appies here as we. Denote the farmer s assessment of area oss under imperfect information with q L. We wi assume that the farmer hods a simiar attitude towards the area risk as the farmer views one s own risk, that is, q max{ p,}. (44) L L L In the preceding equation, the parameter L represents the farmer s over-confidence towards 21

22 area risk and it is defined as simiar to in equation (38) pa( ). (45) L L Simiary, one can obtain that as the risk aversion increases towards max, L goes to zero and the perceived area risk q L becomes equa to true area risk p L. In addition, as the risk aversion goes to min, L increases towards pl, whie q L decreases to pl(1 ). Furthermore, we wi assume that the farmer wi accuratey estimate the correation between area and farm yieds (so remains as before). The joint distribution of the individua and the area osses is re-expressed in terms of farmer s perceptions as: q L repaces repaces p N L L L, q nl repaces p nl and q nn.repaces p nn with the foowing formuations: q qq z z ; q q (1 q ) z z ; q (1 q ) q z z ; and N L L nl L L p L, q N qnn (1 q )(1 ql) zzl, where is the correation coefficient, z is the standard deviation of event S i and z L is the standard deviation of event S a under imperfect information. The standard deviations z and z L are defined as z q(1 q) and zl ql(1 ql) and they repace s and s L, respectivey. In addition, the covariance term between the events ( Si, S a) becomes Cov( S, S ) z z. As before, the vaue of the correation coefficient parameter must i a L be consistent with the fact that probabiities are a non-negative. The assumption that qn and qnl remains. From these reationships, one can again re-obtain the margina probabiity of osses as q ql qn and ql ql qnl. Comparing the joint probabiities under perfect and imperfect information environments, one can deduce that ql pl because q p from equation (4) and the covariance term is perceived to be ower zz L ss L. The atter foows from i) q p from equation (4) and 22

23 the assumption that p.5 impy that z s and ii) ql pl from equation (44) and the assumption that pl.5 impy that zl sl. Meanwhie, the direction of bias regarding the basis risk can not be determined in the formuations of q N and p N because (1 ql) (1 pl) and that counters the perception through covariance term. Finay, we wi assume that the conditiona probabiity that a disaster wi be decared wi remain the same under the two information environments, that is, q p. (46) da / da / Ex-Post Disaster Aid under Imperfect Information The farmer s utiity under the disaster aid option shoud be revised in ine with the farmer s perceptions of probabiity of farmer s own oss and area oss given in equations (4) and (44), respectivey. Combined with the estimated vaue for q dl above, farmer s expected oss under disaster assistance option under imperfect information is then E j q q q q q. (47) ( DA) L( d L ) N ( ) nl nn Reca that in equation (21) does not depend on a farmer s perceived risk, whether that be or q. A component of the farmer s variance of the oss is p E( j ) q ( q ) q ( ) q q. (48) DA L d L N nl nn E( j ) E( j ), pugging the j 2 2 Because the variance of the farmer s oss is 2 corresponding equations yied DA DA DA q (1 q )( q ) q (1 q )( ) 2 q q ( q ). (49) j DA L L d L N N L N d L 23

24 2 Based on E( j DA) and j DA, one can then write the farmer s preference under the disaster aid option as 1 DA U M E( j ).5. (5). DA 2 jda The preceding equation represents farmer s perception of farmer s financia we-being. However, the government wi continue to use farmer s utiity DA U given in equation (26) as it has the knowedge of farmer s true risk and wants to accuratey evauate the farmer s we-being. In addition, the expected cost of disaster assistance under imperfect information (denote it with 1 DA EA ( ) c ) is equa to DA EA ( ) c under perfect information given in equation (29) as the government (unike the farmer) accuratey estimates p L. Denote further that the government s utiity from disaster assistance under imperfect information with G 1 DA EA. Then, G 1 DA DA EA GEA ( ) ( ) (51) where DA EA ( ) G is the government utiity from disaster assistance under perfect information given in equation (27). Ex-ante Protection through Insurance under Imperfect Information In ine with probabiity of oss q from equation (37), the farmer s objective function with insurance coverage (given in equation (3)) can be written as U ( x; ) M xq(1 x).5 1 ( x 2 x) q (1 q ) From F.O.C.s, the demand for coverage woud be x 1 q 1. (52) 2 q (1 q ) 24

25 Suppose further that the government sets the actuariay fair rates consistent with the probabiity of oss given in equation (37), that is 13, f pq. (53) Pugging the preceding rate in equation (31) yieds x (54) 2 q (1 q ) The equation indicates that the farmer s over-confidence resuts in under-insurance at the true actuariay fair rate. Note that the farmer not ony under-estimates the expected oss but aso the risk (variance of the oss) provided that q.5 ). Moreover, in equation (54) 1 x 1 x when q (1 q ) and (55) when q (1 q ). Note that [ min, max ] and min can be a very sma number. Thus, it is quite possibe that the right hand side of equation (55) can indeed be ower than the eft hand side. That is, if the farmer s over-confidence is sufficienty high, it can reduce the coverage demand a the way to zero. Suppose that crop insurance premium subsidies are avaiabe. Denote the tota amount of subsidy with T. The amount of subsidy is modeed as a inear function of a constant subsidy rate, that is, T t x where t is the subsidy rate. 14 Pugging that in the farmer s objective function yieds 13 Gao et a. 211 (in p. 27 of their paper, which we obtained through persona communication) state that Since it is perceived risk that governs demand and actua risk that governs suppy it is no wonder that market imperfections arise and that government invovement is required to encourage participation. 14 This is consistent with modeing the amount of premium as a inear function of premium rate. 25

26 U ( x;, t) M (1 t) xq(1 x).5 ( x 2x1) q (1 q ). (56) Maximization of the preceding utiity function with respect to x yieds x 1 q (1 t) 1, (57) 2 q (1 q ) where the constant subsidy rate creates a proportiona outward shift in coverage demand. Now, suppose that government sets the premium rate based on the true probabiity of oss given in equation (53). The demand becomes x 1 q(1 t)( q) tp q(1 q) q(1 q) q(1 q). (58) Note that in the preceding equation, if t, then x 1 ; if p t, then x 1 p ; and if t, p then x 1. Thus, if the subsidy rate is sufficienty ow, that is, insurances. 15 t p, then the farmer under By using equation (38), one can re-express the subsidy eve that induces farmer to take up fu insurance in terms of parameter as A( ) where A( ) is from equation (39). As p risk aversion increases, goes to zero, whereas as risk aversion decreases, then goes to, p p the maximum eve of over-confidence We do not consider coverage restrictions, aso known as deductibe. For exampe, the maximum coverage eve avaiabe in Revenue Protection (RP) insurance pan is 85%. Furthermore, the subsidy rates in federa crop insurance are decreasing at the very high coverage eves instead of constant rate throughout coverage eves used in here. 16 Note that this subsidy rate depends on individua farmer s risk preferences because does so in equation (38). In actuaity, crop insurance subsidy rates are fixed for a farmers, regardess of 26

27 Ex-ante Protection through Insurance in the Presence of Ex-Post Disaster Assistance under Imperfect Information Having shown that farmer with over-confidence can demand ess than fu coverage at the actuariay fair premium rates in the previous section, we now extend the insurance modeing to recognize that a sufficienty ow insurance coverage can trigger disaster assistance given the poitica economy from equation (3). We begin with the ex-post situation where the insurance coverage is 1 x from equation (58), which is a function of a subsidy rate t. In case of oss, farmer s ex-post income with insurance coverage 1 x woud be the initia income minus the farmer paid premium minus part of the farmer s oss not covered by insurance, that is, M t x x 1 1 (1 ) (1 ). (The preceding cacuation is in ine with the Suppementa Revenue Assistance (SURE) revenue cacuation reported in Zuauf, Schnitkey and Langemeier, 21, p. 52, coumn 2.). 17 Then, the ratio of oss with insurance coverage defined as 1 x to the initia income can be (1 t) x (1 x ) (1 t) x 1 rx (, t) r(1 x). (59) M M If government considers a transfer amount in the post insurance situation, the percent change in farmer s financia we-being with is ( M (1 t) x (1 x ) ) M 1 w(, x,,) t r r( x,) t, (6) M M their risk preferences. At the same time, farmer s risk preference can be farmer s private information. Within our modeing, if government knew ony up to the distribution of risk preferences in the popuation, it coud offer a constant subsidy rate based on the expected risk preference. 17 Even though farmer (ex-ante) enjoys some peace of mind with the risk reduction obtained from purchased coverage eve, this is ignored in the ex-post situation. 27

28 where rx t 1 (, ) is as defined earier. Pugging w 1 (, x,,) t r in the government s utiity in equation (16), from F.O.C.s, one can obtain the ex-post disaster assistance in the post insurance situation as 1 1 ( x, t) M r( x, t) n km. (61) In addition to the variabe cost, the government shoud take into account the fixed cost in 1 deciding whether to extend ( x, t) to the farmer. Simiar to the approach used earier in equation (19), the amount of fixed cost (so ong it is not too high) picks an upper bound for the margina poitica cost (denote it with k EP, which can be different than k EP in equation (2)). Then, 1 ( x, t) 1 ( x, t) k [ k, kep) k kep.. (62) In case of no oss, the random variabe takes the vaue of zero and so rh, but with the insurance coverage x in hand, the post-insurance oss woud be the farmer paid premium rx t 1 (, ) r h (1 t) x M 1. (63) Going back to the perfect information environment for a moment (setting in equation (38) and L in equation (45) to zero at a risk aversion eves wi do) and aso assuming the f actuariay fair premium rate p and t, the farmer woud choose fu coverage eve and the demands in equations (57) and (31) woud coincide ( 1 x x 1). One then obtains p M rx ( 1, t ) pr f rh,., (64) Based on preceding equation, if government considers extending transfer amount, the 28

29 percent change in farmer s financia we-being with woud be w x t r pr (, 1,, h ) M. In order to prevent any government transfer in the preceding situation (where ony oss incurred is the farmer paid premium), it is sufficient to revise the minimum eve of margina poitica cost in equation (14) upward to k pr e. (65) M Note that for an initia income of M $5,, $74,99 (see footnote 3), probabiity of farmer s oss is p.2 and r.25 (which impies that.25 e ), the boundary vaues for k ( k in equation (65) and k in equation (17) ) woud reduce to k , which is in ine with the poitica cost numbers reported in Aston and Hurd (199). Having identified the benchmark eve of margina poitica cost k in equation (65), we revert back to our anaysis under imperfect information. From equation (62), we identify the optima ex-post disaster aid as function of ex-ante insurance subsidy and so ex-ante insurance coverage. We know deveop government s ex-ante objective function under insurance by recognizing the poitica reaity of potentia ex-post disaster assistance. Note that even though government takes into account the farmer s perception in order to arrive at the farmer s coverage demand ( x 1 from equation (57)), it wi continue to use farmer s true probabiity of oss in wefare cacuations. That is because the government is interested in farmer s true financia webeing. From the government s point of view, the farmer s true expected oss under insurance pus disaster assistance option (when farmer has imperfect information) is E ( ) p ( x p ( x, t )) p ( x ) p p, (66) 1 1 Ins DA L d L N nl nn 29

30 where 1 x 1 is from equation (57) and ( x, t) is from equation (62). Moreover, a component of the farmer s variance of the oss can be obtained as E ( ) p ( x p ( x, t )) p ( x ) p p. (67) Ins DA L d L N nl nn Government then obtains the variance of the farmer s oss as E ( 1 1 ) E ( 1 1). (68) Ins 1 DA 1 Ins DA Ins DA where the expressions for E ( 1 1) from equation (66) and E ( 1 12 ) from equation (67) are Ins DA substituted. Finay, combining E ( 1 1) and Ins DA 2 Ins 1 DA 1 Ins DA from above, government arrives at the farmer s true financia we-being under insurance pus disaster aid option as U M (1 t) x E( ).5. (69) Ins DA Ins DA 2 Ins 1 DA 1 Again, it is the government that cacuates the preceding utiity using the information regarding true risk whie aso taking into account the farmer s perceived coverage demand given in (57).. Using U from equation (69), the government evauates ex ante the percentage change in 1 1 Ins DA the farmer s financia we-being as wx tr x t U M U (,,, Ins DA Ins DA (, )) 1 M M. Finay, the government s ex-ante objective function when farmer has an imperfect information is w( x ; t, r, ( x, t)) G 1 1 G( x ; t, r, ( x, t)) B( e ) Ins DA 1 ktpx p p F k x t I x t 1 1 L d / a( d (, )) ( (, )). (7) f where insurance premium is set at the actuariay fair rate p (priced at the true risk), 1 ( x, t) is from equation (62) and I 1 ( ( x, t)) is the indicator variabe which takes vaue of one 1 when ( x, t) 1 or zero when ( x, t). That is, the government optimay chooses a 3

31 subsidy eve by taking into account that the induced coverage demand may eiminate the need for some disaster assistance down the road. Note that we are not considering a fixed cost in providing subsidies with crop insurance. 18 Denote the optima soution to the maximization of the objective function given in equation (7) with t. Now, whenever t, x 1 p woud hod (the atter condition can be imposed per actuaria standards). Note that p is monotonicay increasing in and if the margina poitica cost is sufficienty high, then government may not fuy subsidize insurance coverage. The government s optima subsidy under information probems (per requirement that the coverage eve wi not exceed 1%) is t = minimum t, p 1 1 and so x minimum x,1. The government s objective function in equation (7) once evauated at with G and is 1 1 Ins DA ( t, x ) wi be denoted w( x, t, r, ( x, t )) G 1 1 G( x, t, r, ( x, t )) B( e ) Ins DA kt p x 1 p p F k x t I x t 1 1 L d / a( d (, )) ( (, )), (71) f where again insurance premium is set at the actuariay fair rate p, 1 I( ( x, t )) is the 1 indicator variabe which takes vaue of one when ( x, t ) 1 or zero when ( x, t ). Simuation Anaysis Numerica anaysis is used to gain insights into the government choices among mutipe 18 Crop insurance s authorization is under permanent aw, the Federa Crop Insurance Act of 198. Its egisative fixed costs are sunk. The infrastructure and abor force for the deivery of crop insurance have deveoped and evoved over time. 31

32 instruments to provide financia support to farmers at times of distress. Government s utiity in equation (51) and that in equation (71) are simuated under perfect and imperfect information environments. For the perfect information environment, the farmer s perceived probabiities are repaced with the actua probabiities of oss (by setting in equation (38) and L in equation (44) to zero for a risk aversion eves) and the government s optimization probem equation (7) is re-soved, based on which then equation (71) is re-cacuated. We know from equation (51) that government objective function under soe disaster assistance option is the same under both information environments. The preceding government s utiity functions are simuated based on the parameter vaues given in Tabe 1 (the necessary programs are written and run using MATLAB software). Particuary, the upper imit on the over-confidence is restricted to 75%, that is,.75 in equations (38) and (44). That means as the farmer s risk aversion decreases towards min, the over-confidence woud approach to 75% of the farmer s probabiity of oss (for both individua and the area). Otherwise, the over-confidence eve monotonicay decreases in risk aversion with zero, which can be verified in equations (41) and (42). The resuts for perfect information case are dispayed in in Figures 2a, 3a and 4a under the ow oss situation ( r.25 ); medium oss situation ( r.5 ); and high oss situation ( r 1. ), respectivey. The y axis in a figures is the farmer s risk aversion eve ( ) and the x- axis in a figures is the margina poitica cost ( k ) (see Tabe 1). One can verify that the government prefers the insurance option for virtuay a parameter vaues considered. This preference hods even though some farmers may prefer ex-post disaster aid over insurance for a sma range of parameters (ow risk aversion eves and ow margina poitica cost). No insurance subsidy is needed. Farmer take up fu insurance at the actuariay fair rate and ex-post disaster assistance competey deterred. Simiary, the resuts for the government net utiity under 32

33 imperfect information environment are dispayed in Figures 5a, 6a and 7a (for the ow, medium and high oss situations, respectivey). The preceding Figures indicate that the government s net utiity with subsidized insurance exceeds its utiity with ex-post disaster payments over most parameter vaues. In addition, Tabes 2 and 3 ist extreme vaues of the government s maximum net utiity from insurance and disaster assistance options under perfect and imperfect information environments, respectivey. One thing that is apparent from these tabes is that the insurance option provides more stabe (in terms of the difference between extreme vaues) maximum net utiity to government than that under the disaster aid option ony. Furthermore, Figures 5b, 6b, and 7b dispay optima subsidy rate for ow, medium and high oss situations under imperfect information. One can verify that the mode generates substantia subsidy rates from ow to moderate risk aversion eves, which induces farmers to participate or buy up higher crop insurance coverage. Finay, Figures 5c, 6c, and 7c report expost disaster assistance amounts with insurance and without insurance, as we as ex-ante subsidy amount for ow, medium and high oss situations under imperfect information. By inducing higher coverage eves, ex-ante insurance subsidies accompish: First, they reduce (in most cases they totay deter) ex-post disaster assistance. Second, they provide the ony safety net when the government can not extend ex-post disaster assistance due to the fixed cost invoved and high margina poitica cost. Figure 5c and 6c ceary iustrate these effects. In Figure 7c, fixed cost amount is reativey sma given the arge amount of oss considered. As a resut, in the absence of insurance, government woud provide ex-post disaster assistance (dispayed in green) on a arge portion of margina poitica cost vaues considered. In the presence of subsidized insurance; ex-post disaster assistance is mosty deterred with the exception of farmers who are at the owest end of the risk aversion spectrum considered. 33

34 Summary and Concusion The justification for crop insurance subsidies is being questioned in ight of budget and poicy issues. Crop insurance has received government support for a variety of reasons. One reason for expanding government support has been to discourage ad hoc, ex-post disaster payments, which grew sharpy in the 198s and 199s, were costy and discouraged the purchase of crop insurance. Congress restructured and further increased subsidies through egisation in the Federa Crop Insurance Reform Act of 1994 and the Agricutura Risk Protection Act of 2 (ARPA), in a sense, buying up the farmer s participation (Cobe and Barnett, 212). Better data arising from high and diverse participation heped improve underwriting and ratemaking. Perhaps, upfront payment of subsidies crediby communicated the Federa Government s commitment to the insurance program and its preference over ex-post disaster aid. 19 In fact, after many years of Congress passing ad hoc disaster egisation to dea with weather misfortunes in agricuture, there were no cas for crop disaster egisation in 211 and 212. We have focused on information probems in the form of a farmer s over-confidence as a reason that for government support for crop insurance by assuming government has an interest in farmers wefare and considers to make transfer payments to farmers to hep them dea with income osses. However, the government faces both fixed and margina poitica costs in making transfer payments to farmers. In this framework, the government s objective function (which depends on the farmer s income osses) is written to evauate three options: 1) ex-post disaster aid; 2) ex-ante insurance option with perfect information; and 3) ex-ante insurance option with 19 House Majority Leader John Boehner tod Agri-Puse (213) during a taped interview Over the ast 15 years, crop insurance is where we have been trying to hep move farmers in terms of taking advantage of risk management toos for their crops. Mr. Boehner aso noted It is sti the centra focus of where we think farmers ought to be abe to have easy access to insure their crops and insure some type of revenue out of it. It makes the most sense to me and aways has. 34

35 imperfect information (farmers are over-confident about their risk). For option 1), an optima eve of ex-post transfer payment is derived for a given margina poitica cost, fixed cost and farmer s risk. For option 2), government s utiity is obtained when the farmer s risk is protected ex-ante through unsubsidized, actuariay fair crop insurance. In comparing 1) and 2), numerica anaysis shows the government prefers unsubsidized crop insurance at fair premium rates. Regarding option 3), the mode is augmented by considering a farmer who underestimates the actua probabiity of oss for yied or revenue. The degree of over-confidence is assumed to be decreasing in the farmer s risk aversion parameter. Assuming that the government sets premium rates on the true probabiity of oss for the farmer, the anaysis indicates the farmer wi reduce insurance coverage beow the eve that the farmer woud chose under fair rates and fu information on the probabiity of oss. Government prefers to subsidize the premium rates to induce the over-confident farmer to take up more insurance but must baance that against a margina poitica cost of providing insurance subsidies and a given eve of fixed cost. When providing subsidies, government aso takes into account the disaster assistance impications of induced coverage eves. The optima subsidy eve is numericay soved. Numerica anaysis indicates that the government s net utiity with subsidized insurance exceeds its utiity with ex-post disaster payments over most parameter vaues (Figures 5a, 6a, and 7a). The ex-ante poitica cost arising from the insurance subsidy appears to be much smaer compared with the ex-post poitica cost arising from the optima disaster aid whenever they are both positive (Figures 5c, 6c, and 7c). As the 213 Farm Bi process is underway, the debate on the degree and form of government support of agricuture wi continue. The modeing framework here is fairy fexibe to study (given government s preferences) the effectiveness of suppementa revenue programs 35

36 as found in egisation deveoped during the summer of 212. These suppementa programs, (which are typicay area based pans and being offered either free or highy subsidized, under - priced in short) tend to repace crop insurance at high coverage eves (Buut, Coins, and Zacharias, 212; Buut and Coins, 212; and Buut and Coins, 213). 36

37 Tabe 1. Parameter Vaues Used in the Simuation Parameters Vaues M : Farmer s initia income $5, B : Normaized vaue of government s utiity in status quo (perhaps from non-food reated sectors) : Net vaue-added to the U.S. economy, per farmer : Target (reference) eve of for the farmer s financia we-being r : The ratio of amount of farmer s oss to the M farmer s initia income Mr: Amount of farmer s oss : Maximum over-confidence.75 p : Probabiity of farmer s oss.2 $74,99 a 2.25 in the sma oss scenario;.5 in the medium oss scenario; and 1. in the arge oss scenario. $12,5 for the sma oss scenario; $25, for the medium oss scenario; $5, for the arge oss scenario. pl p( A) : Probabiity of area oss (perfect information) pd/ a PD ( A) : The conditiona probabiity of disaster decaration under perfect information qda / Q( D A) : The conditiona probabiity of disaster decaration under imperfect information p PD ( ) PD ( A) PD ( APA ) ( ): The d unconditiona probabiity of disaster decaration

38 Tabe 1. Parameter Vaues Used in the Simuation, continued for the sma oss scenario, pr k e : Lower bound for the margina M for the medium oss scenario, poitica cost for the arge oss scenario. r k e : Upper bound for the margina M poitica cost k : margina poitica cost F d : The amount of fixed cost (per farmer with a oss in the area in case of disaster decaration). : Correation between farmer s and area osses min : Lower bound for the risk aversion for the sma oss scenario, 2.47 for the medium oss scenario, for the arge oss scenario. Varies between k and k with equa increments (3 data points are used). $1,.5 It corresponds to the eve of risk aversion where the ratio of risk-premium (the variance component of the farmer s utiity function.5 (1 ) 2 min p p ) to the size of the oss ( ) is.1. Note that min decreases as the size of max : Upper bound for the risk aversion oss increases. It corresponds to the eve of risk aversion where the ratio of risk-premium (the variance component of the farmer s utiity function.5 (1 ) 2 min p p ) to the size of the oss ( ) is 1. Note that max decreases as the size of oss increases. : The risk aversion eve It varies min between max with equa increments (3 data points are used). 38

39 Tabe 2. Extreme Vaues of Government s Maximized Net Utiity (in $) under Perfect Information Insurance Option (No Subsidy is Needed) Lowest Highest Sma oss ( r.25 ) $71,68 $71,68 Medium oss ( r.5 ) $67,31 $67,31 Large oss ( r 1) $58,324 $58,324 Disaster Assistance Option Lowest Highest Sma oss ( r.25 ) $48,71 $7,88 Medium oss ( r.5 ) $13,325 $66,685 Large oss ( r 1) $-98,889 $57,595 Note. Government s utiity in status-quo (when there is no change in the farmer s financia webeing is the per farmer net vaue added ( $74,99 ). Lowest and highest vaues are of those maximized vaues of the government net utiity function for a given option on a grid of risk aversion and margina poitica cost eves (3 data points for each; see Tabe 1). That is, for a given option, 9 maximized vaues of the government net utiity function are obtained, and the tabe reports the owest and highest of those vaues for that option. 39

40 Tabe 3. Extreme Vaues of Government s Maximized Net Utiity (in $) under Imperfect Information Insurance Option with Optima Subsidy Lowest Highest Sma oss ( r.25 ) $7,542 $71,68 Medium oss ( r.5 ) $65,52 $67,31 Large oss ( r 1) $5,97 $58,324 Disaster Assistance Option Lowest Highest Sma oss ( r.25 ) $48,71 $7,88 Medium oss ( r.5 ) $13,325 $66,685 Large oss ( r 1) $-98,889 $57,595 Note. Government s utiity in status-quo (when there is no change in the farmer s financia webeing is the per farmer net vaue added ( $74,99 ). Lowest and highest vaues are of those maximized vaues of the government net utiity function in question on a grid of risk aversion and margina poitica cost eves (3 data points for each; see Tabe 1). That is, for a given option, 9 maximized vaues of the government net utiity function are obtained, and the tabe reports the owest and highest of those vaues for that option. 4

41 References: Agri-Puse, 213. Avaiabe at Forum asp, March 4. Aston, J. M. and B.H. Hurd Some Negected Socia Costs of Government Spending in Farm Programs. American Journa of Agricutura Economics 72(1): Babcock, B.A., E.K. Choi and E. Feinerman Risk and Probabiity Premiums for CARA Utiity Functions. Journa of Agricutura and Resource Economics. 18(1): Babcock, B. A. 29. Shoud Government Subsidize Farmer s Risk Management? Iowa Ag Review, Vo. 15 No 2. Spring. Barberis, N.C Thirty Years of Prospect Theory in Economics: A Review and Assessment. Journa of Economic Perspectives. 27(1): Buut, H., K.J. Coins, and T.P. Zacharias Optima Coverage Demand with Individua and Area Pans of Insurance. American Journa of Agricutura Economics. 94(4): Buut, H. and K.J. Coins The Impact of a Suppementa Disaster Assistance Program on the Reative Demands for Individua and Area Pans of Insurance. Seected Poster presented at the Annua Meeting of the Agricutura and Appied Economics Association (AAEA), August 11-14, Seatte, WA. Buut, H. and K.J. Coins Substitution and Interaction Effects of Farm Suppementa Revenue Coverage Options. Seected Paper presented at the Annua Meeting of the SCC- 76 Group, March 14-16, Pensacoa, FL. Chambers, R.G On the Design of Agricutura Poicy Mechanisms. American Journa of Agricutura Economics. 74(3): Chambers, R.G. and J. Quiggin. 21.Decomposing Input Adjustments Under Price and Production Uncertainty. American Journa of Agricutura Economics. 83(1): Chambers, R.G. and J. Quiggin. 22. Optima Producer Behavior in the Presence of Area- Yied Crop Insurance. American Journa of Agricutura Economics. 84(2): Coins, K.J. and H. Buut Crop Insurance and the Future Farm Safety Net. Choices. 26(4), 4 th Quarter. Cummins, J. D. and N.A. Doherty. 25. The Economics of Insurance Intermediaries. Working Paper. Wharton Schoo. University of Pennsyvania. Avaiabe at 41

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43 Kahneman, D. and A. Tversky Prospect Theory: An Anaysis of Decision under Risk. Econometrica 47(2): Kostad, C.D., T.S. Uen, and G.V. Johnson Ex Post Liabiity for Harm vs. Ex Ante Safety Reguation: Substitutes or Compements? American Economic Review. 8(4): Keindorfer, P.R. and R.B. Kein. 23. Reguation and Markets for Catastrophe Insurance. In Advances in Economic Design. Eds. M.R. Serte and S. Koray. Springer. Mas-Coe, A., M.D. Winston and J.R. Green Microeconomic Theory. Oxford University Press, New York. Menapace, L., G. Coson, and R. Raffaei Risk Aversion, Subjective Beiefs and Farmer Risk Management Strategies. Invited Paper presented at the Annua Meeting of the Agricutura and Appied Economics Association (AAEA), August 11-14, Seatte, WA. Moody s. 24. Moody s Correated Binomia Defaut Distribution, August 1. Avaiabe onine at Last accessed on Apri 25, 213. Moschini, G. and D.H. Hennessy. 21. Uncertainty, Risk Aversion and Risk Management of for Agricutura Producers. In Eds. B.L.Gardner and G.C. Rausser. Handbook of Agricutura Economics Vo 1A Agricutura Production, North-Hoand. Moss, C.B. 21. Risk, Uncertainty and the Agricutura Firm. Word Scientific Papers, Singapore. Pease J., Wade E., Skees J., and Shrestha M Comparisons between Subjective and Statistica Forecasts of Crop Yied. Review of Agricutura Economics, v 15, n2. Gao, X., Turvey, C.G., R. Nie, L. Wang, and R. Kong Subjective versus Objective Risk Divergence and the Crop Insurance Probem. Seected Paper presented at the Annua Meeting of SCC-76 Group, March 17-19, Atanta, GA. Saha, A Expo-Power Utiity: A Fexibe Form for Absoute and Reative Risk Aversion. American Journa of Agricutura Economics. 75: Schnitkey, G Impacts of Limits on Crop Insurance Risk Subsidies. University of Iinois Department of Agricutura and Consumer Economics Farmdoc Pubication, May 1. Sherrick, B. 22. The Accuracy of Producers Probabiity Beiefs: Evidence and Impications for Insurance Vauation. Journa of Agricutura and Resource Economics, v 27 (1). Shieds, D.A. and R.M Chite. 21. Agricutura Disaster Assistance. Congressiona Research Service Report for Congress. RS

44 Umarov, A. and B.J. Sherrick. 25. Farmers Subjective Yied Distributions: Caibration and Impication for Crop Insurance Vauation. Seected Paper presented at the Annua Meeting of the Agricutura and Appied Economics Association (AAEA), Rhode Isand, Juy USDA, Economic Research Service (ERS) Amber Waves, 11(1), February. Avaiabe onine at USDA. Nationa Agricutura Statistics Service (NASS) Farms, Land in Farms, and Livestock Operations 212 Summary, February. Avaiabe onine at Zuauf, C., G. Schnitkey and M. Langemeier. 21. Average Crop Revenue Eection, Crop Insurance, and Suppementa Revenue Assistance: Interactions and Overap for Iinois and Kansas Farm Program Crops. Journa of Agricutura and Appied Economics, 42, 3 : Zuauf, C A Coordinated Framework for a 21 st Century Farm Safety Net. Presented at the Agricutura and Appied Economics Association Meeting in Pittsburgh, PA, Juy. 44

45 e w % 18% 16% 139% 119% 99% 79% 59% 38% 18% 2% 22% 42% 63% 83% 13% 123% 143% 164% 184% w Figure 1. A pot of the function that indexes the changes in farmer s financia we-being. The x-axis dispays the percent change in w farmer s financia we-being ( w ), whie y-axis dispays the vaue of the indexing function ( e where 2). It can be verified in the figure that the function takes the vaue of one when there is no change in farmer s financia we-being. 45

46 Figure 2a. Government s net utiity under insurance option (in red) and that under disaster assistance option (in green) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.25 and the information is perfect. Exante insurance option takes into account the ex-post disaster assistance impications. No subsidy under insurance option is needed. 46

47 Figure 3a. Government s net utiity under insurance option (in red) and that under disaster assistance option (in green) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.5 and the information is perfect. Exante insurance option takes into account the ex-post disaster assistance impications. No subsidy under insurance option is needed. 47

48 Figure 4a. Government s net utiity under insurance option (in red) and that under disaster assistance option (in green) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r 1. and the information is perfect. Exante insurance option takes into account the ex-post disaster assistance impications. No subsidy under insurance option is needed. 48

49 Figure 5a. Government s net utiity under insurance option (in red) and that under disaster assistance option (in green) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.25 and the information is imperfect. Exante insurance option takes into account the ex-post disaster assistance impications. Some subsidy under insurance option is used.. 49

50 Figure 5b. Optimum subsidy rate (in bue) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.25 and the information is imperfect. Ex-ante insurance option takes into account the ex-post disaster assistance impications. 5

51 Figure 5c. Ex-post disaster assistance in case of a oss with and without insurance option (in red and in green, respectivey), and the ex-ante subsidy amount under insurance option (in bue) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.25 and the information is imperfect. Ex-ante insurance option takes into account the ex-post disaster assistance impications. 51

52 Figure 6a. Government s net utiity under insurance option (in red) and that under disaster assistance option (in green) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.5 and the information is imperfect. Exante insurance option takes into account the ex-post disaster assistance impications. Some subsidy under insurance option is needed. 52

53 Figure 6b. Optimum subsidy rate (in bue) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.5 and the information is imperfect. Ex-ante insurance option takes into account the ex-post disaster assistance impications. 53

54 Figure 6c. Ex-post disaster assistance in case of a oss with and without insurance option (in red and in green, respectivey), and the ex-ante subsidy amount under insurance option (in bue) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r.5 and the information is imperfect. Ex-ante insurance option takes into account the ex-post disaster assistance impications. 54

55 Figure 7a. Government s net utiity under insurance option (in red) and that under disaster assistance option (in green) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r 1. and the information is imperfect. Exante insurance option takes into account the ex-post disaster assistance impications. Some subsidy under insurance option is needed. 55

56 Figure 7b. Optimum subsidy rate (in bue) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r 1. and the information is imperfect. Ex-ante insurance option takes into account the ex-post disaster assistance impications. 56

57 Figure 7c. Ex-post disaster assistance in case of a oss with and without insurance option (in red and in green, respectivey), and the ex-ante subsidy amount under insurance option (in bue) in z-axis, margina poitica cost ( k ) is on the x-axis and risk-aversion ( ) is on the y-axis. Note: r 1. and the information is imperfect. Ex-ante insurance option takes into account the ex-post disaster assistance impications. 57