Catherine Benjamin

Alexandre Gohin

Hervé Guyomard

The Future of EU Dairy Policy

Even if the

Introduction

Supply management policies are often advocated as a possible response to a situation of overproduction that protects producers' incomes without boosting budgetary costs. This was effectively the rationale for the introduction of milk quotas in the European Union (EU) in 1984. Even if the quota regime has been extended until 2007 /08 as part of the Agenda 2000 reform of the Common Agricultural Policy (CAP) adopted in March 1999, the EU dairy sector is still nowadays experiencing large uncertainty over future policy.

In this paper, main economic characteristics of the EU dairy sector are surveyed along with policy arrangements such as Uruguay Round Agreement on Agriculture (URAA) commitments and Agenda 2000 measures. After current dairy policy issues have been analyzed, some thoughts on the central question of an eventual dismantling of quotas are offered.

The EU dairy sector: time for radical policy change?

The first principle underlying the current EU dairy policy is the management of the markets for dairy products in order to secure product prices that permit domestic milk producers to obtain the target price for milk. This mainly involves the annual fixing of both a target price for farm milk and intervention prices for butter and skimmed milk powder (SMP); the public purchase of butter and SMP which is triggered when the market price falls below 92 percent of the intervention price for two consecutive weeks; the payment of aids for the private storage of butter and cream, SMP and certain cheeses; the payment of aids for domestic disposa! of some dairy products (mainly SMP for animal feed, butter for pastry and ice cream manufacturers, butter for non-profit making organizations, and rnilk and milk products supplied to school children); the payments of variable export refunds for a large set of dairy More than 90 percent of the 120 million tons of EU mille output is delivered to dairies while the remainder goes to direct sales (farm products) or on-farm use (animal feed and human consumption). The EU produces roughly 12 million tons more dairy products (expressed in whole mille equivalent) than the domestic market can absorb and this surplus has to be exported thanks to subsidies which represent about 45 percent of the dairy policy budgetary cost (1605 million Ecus over a total of 3635 million Ecus in 1996). In addition about 10 percent of domestic consumption can only be sustained by butter and SMP internai subsidies which represent more than 40 percent of the dairy policy budgetary cost (1508 million Ecus in 1996). After a long period of decline since the mid 1970's, per capita consumption of butter in the EU-15 has stabilized recently at around 4.7 kilograms per head with however very large disparities across member states (less than 0.5 kilograms per head in Spain and about 8.3 kilograms per head in France). Butter manufacture still absorbs one third of the total rnilk produced in the EU and 30 percent of domestic consumption of butter is subsidized. The shase of SM:P used in animal feed is continuously decrnasing although about 40-45 percent of domestic consumption is subsidized. In contrast domestic production and consurnption of WMP is increasing. Less than 45 percent of production is used within the EU, rnainly for human consumption in the food processing industry. EU production and consurnption of cheese is on a long-terrn increasing terrn since the mid 1970's. Per capita consumption levels are however still very different among mernber states (less than 7 kilograms per capita in Portugal and more than 23 kilograrns per capita in France). Fresh dairy products are the third dairy products for which dornestic consurnption is going up.

While output of drinking mille has been more or less stable over the last decade, other fresh dairy products (mainly crearn, acidified mille and milk-based drinks) are on an upward trend.

The EU is still nowadays the major exporter of dairy products with more than 40-45 percent of world trade (Table 1 ). Its market share is steadily decreasing since the rnid 1980's rnainly package since a quota elimination option is politically feasible only if dairy producers are, at least partially, compensated for the reduction in price support. If the whole of the difference between current and reduced prices is compensated, the budgetary cost of any compensation package is likely to be prohibitive. On the other hand, mille production would significantly decline, at the limit disappear, in the poorer dairy regions of the EU if compensatory payn1ents are not modulated in their favor. In other words, the extent, nature and permanence of compensation and the subsequent distribution and justification of payments would play a large role in determining the EU dairy sector after quota elimination (O' Leary and Fingleton, 1998).

Agenda 2000 reforms and consequences for the EU dairy sector

In a general way the Agenda 2000 CAP reform will deepen ( cereals and beef) and extend (dairy products) the 1992 MacSharry reform through further shifts from price support to direct payments. The dairy reform may be summarized as follows (European Commission, March 1999a and[START_REF]Agriculture Council: Policy Agreement on CAP Reform. Directorate General VI Newsletter[END_REF]. Intervention prices for butter and SMP will be reduced by 15 percent in three equal steps starting on 1 July 2005. Support price cuts will be compensated by the introduction of yearly direct payments on a fiat rate basis per ton of quota and by an increase in milk quotas. The compensation scheme will include a national envelope which member states are free to allocate according to objective criteria. Dairy producers will also receive some benefit from the beef compensation payments. Quotas will be increased in each member state by 1.5 percent in three steps starting on 1 April 2005. In addition five countries (Greece, Italy, Spain, Ireland and Northem Ireland) will receive specific quota increases in two steps, i.e., 2000/01 and 2001/02. In the whole this corresponds to an increase in milk quotas by 2.4 percent. The quota regime is thus extended, officially up to the marketing year 2005/06, in practice up to the year 2007/08. i.e., non-subsidized, exports of cheese. Table 3 shows that the ultimate impact of the Agenda 2000 reform on fann milk price will crucially depend on the management of export subsidies and on substitution possibilities between intervention dairy products (butter and Sl\tfP) and other dairy products. The second column of this table shows that a 15 percent reduction in butter and Sl\tfP intervention prices leads to a 9 .3 percent reduction in milk price at the farm gate in France, other dairy policy instruments being maintained unchanged at base period levels. The additional effect of a 15 percent decrease in unit export subsidies on cheese, WPM and fresh dairy products is to reduce the fann milk price by 2.2 points (-11.5 percent instead of -9.3 percent) and the supplementary effect of a 2 percent quota increase is to reduce the farm milk price by 3 points (-14.5 percent instead of -11.5 percent). The last column of this table shows that when substitution possibilities between intervention dairy products and other dairy products are increased, the effects of a reduction in butter and SMP prices on farrn milk price are also increased. This last scenario illustrates the recent move from bulk cornmodities towards higher-value dairy products more closely focused on consumer needs.

The previous analysis is confirmed by the FAPRI-UMC study about the implications of the Agenda 2000 reform for EU agriculture [START_REF] Fapri-Umc | Implications of the Berlin accord for EU agriculture[END_REF]. In an attempt to represent not only producer and consumer behaviors but also EC behavior, F APRI-UMC modelers clearly illustrate the trade-off faced by the EC which may want to "reduce the Euro value of export subsidies, in part because of smaller gaps between EU and world prices" and simultaneously "keep domestic market prices high enough to avoid the accumulation of stocks, so that markets prices are likely to fall less than support prices." One can add that the maintenance of sufficiently high milk market prices is politically essential to sustain the incarne of smaller, less efficient dairy farmers.

to fall to world levels and third country imports to significantly enter the EU market.

Simulation results also depend on two key behavioral pararneters, i.e., marginal cost and milk supply elasticity estimates. In addition, these studies do not include any incarne enhancement from a compensation prograrn. More generally income effects of a quota removal are not explicitly considered in these studies. Finally they ignore the related highly sensitive issue of the number of dairy farmers who could abandon rnilk production under a quota elimination scenario. It remains to check if taking these elements into account will not paint a black picture of the situation.

In order to illustrate the sensitivity of simulation results of a quota elimination scenario to the different parameters identi:fied above, we use a computable general equilibrium (CGE) mode! of the French economy benchmarked to data for 1994 (Gohin, 1998;Gohin and Guyomard, 1999;Gohin et al., 1999). The mode! is a static, single-country, multi-sector CGE modeL in the tradition of Shoven of Whalley (1984). It highlights agricultural and food sectors, with special attention given to CAP instrument modeling. Two export and import zones are distinguished, i.e., the rest of the European Union (RoEU) and the rest of the world (Ro W). This allows us to consider policy scenarios which apply at the EU level. Experiments are detailed in Table 4 and results are surnmarized in Table 5.

Reference experiments

In experiment 1, dairy quotas, intervention support prograrns and domestic subsidy measures are eliminated. Import tariffs on all dairy products and unitary export subsidies on cheese, WMP and fresh dairy products (hereafter called higher-value dairy products) are maintained at base period levels. There are no compensation payments. Relative to the base year, this first experiment triggers a milk price decrease by 26.1 percent and a rnilk production increase by 4.3 percent. With the elimination of intervention support programs, butter and SMP value dairy products are maintained, milk output increases relative to the base year ( experiment 1 ). When these programs are simultaneously eliminated, the EU milk price decrease is sufficient to induce a milk production decline relative to the base year ( experiment 2). In both experiments, the decrease in dairy producer incarnes is too important ta imagine a quota elimination policy without a compensation scheme to offset at least part of the decline in incarnes.

Robustness to the dairy quota rent value

Experiments 3 and 4 allow us ta analyze the robustness of simulation results to the dairy quota rent value, initially estimated to 27.5 percent of the EU average milk price. The rent equals 17.5 percent of the milk price in experiment 3 and 37.5 percent in experiment 4. When the value of the rent decreases (respectively increases), the effects of a quota elimination policy on domestic prices are reduced (augmented) for farm mi&, lower-value dairy products and higher-value dairy products. Although the mil.le price decline is lower in experiment 3 (-18.9 percent) than in experiment 2 (-28.2 percent), milk supply drops by a larger percentage (-4.8 percent instead of -2.9 percent) due to the upward shift of the milk supply function induced by the lower value of the quota rent. The negative incarne effect of a lower milk production is more than offset by the positive impact of higher milk prices sa that dairy farmer incarnes increase in experiment 3 relative to experiment 2. Of course, they fall with respect to the base year and a quota elimination option is still unlikely if it is not accompanied by a compensation package.

As noted by [START_REF] Westhoff | Elimination or expansion of milk quotas: impacts on European mille supply, demand and prices[END_REF], "in approaching the analysis of (a) quota elimination scenario, the analysts face a major challenge (because) the quota program has been in place so long that historical data provides little guidance on likely supply response across the (EU) member states without a quota." Our second set of experiments shows that 1?

A speedy elimination of dairy quotas and intervention support measures seems politically feasible only in the context of acceptable compensation, at least over a certain transition period. In experiment 7, compensatory payments offered to dairy producers are tied to production levels. In experiment 8, they are based on acreage declarations. To make easier result comparison, the total amount of compensatory payments is the same in both experiments, i.e., 14.74 billion 1994 French francs which corresponds to the incarne loss of dairy producers in experiment 2.

The coupled compensation package of experiment 7 does relatively little to support dairy producer incarnes because it translates into a milk price decrease which nearly offsets the positive income effect of payments. Dairy producer incomes fall by 20.4 relative to the base year. In this experiment, production decisions of dairy farmers are based on an equivalent price which is the sum of the market price and the average compensatory payment. As a result, despite the decrease by 53.2 percent in the milk market price, French milk output increases by 6.1 percent with respect to the base ycar. This increase mainly benefits the dairy industry. For the latter, output price declines (-37.6 percent for lower-value dairy products and -22.2 percent for higher-value dairy products) are more than offset by the positive incarne eff ects linked to both the milk price decrease and the milk output increase. As a result, dairy industry incomes raise by 7.7 percent with respect to the base year. By contrast, experiment 8 allows French dairy producers to maintain their incomes at base period levels.

Relative to experiment 7, dairy producers now capture most of the benefits of the compensation package to the detriment of the dairy industry and final consumers. However the cost of this full compensation scheme is likely to be prohibitive in the context of a very tight common agricultural budget. 3

Any dairy quota elimination scenario should include compensation for a least part of the decline in dairy farmer incarnes. What our last set of experiments shows is that any :rn

2.

3.

4.

5.

6.

7.

8. Experiment 2 assumptions, and area based compensatory payments for a total amount equal to the incarne loss of dairy producers in experiment 2 (14,74 billion 1994 French francs)

  dairy quotas, elimination of intervention support and domestic subsidy programs, no compensatory payments, quota rent equal to 27.5 percent of the milk price Experiment 1 assumptions, and variable export subsidies on higher-value dairy products set to zero (see end note 1) Experiment 2 assurnptions, except that the dairy quota rent is equal ta 17.5 percent of the mille price Experiment 2 assumptions, except that the dairy quota rent is equal ta 3 7 .5 percent of the mille price Experiment 2 assurnptions, and import tariffs on ail dairy products set to zero with an infinite import supply elasticity Experiment 2 assurnptions, and import tariffs on all dairy products set to zero with a strictly positive import supply elasticity Experiment 2 assumptions, and production linked compensatory payments for a total amount equal to the income loss of dairy producers in expei:iment 2 (14,74 billion 1994 French francs)

Table 2 .

 2 Impacts of the Agenda 2000 reform on the EU dairy sector in 2007 according to CARD-FAPRI-ISU: Changes ii: percent with respect to the baseline scenario

		EU production	World price	EU price	EU consumption	EU experts
	Mill<	+l.68	na	-9.48	na	na
	Butter	-0.08	+0.40	-10.95	+2.77	-8.98
	SMP	-6.07	+3.30	-11.54	+3.86	-17.18
	Cheese	+3.06	-1.36	-7.76	+2.61	+5.73
	WMP	+2.77	na	-9.32	na	+5.71
	Source: CARD-FAPRI-ISU, 1999 (appendix tables). Na: Not available.

Table 4 .

 4 Experiment designNo.

	1.

End notes 1 The mixed-complementarity approach is used to model the quota policy and the intervention support mechanism prevailing for butter and SMP. For these lower-value dairy products, the price regime is endogenously determined as part of the mode! solution. The unit export subsidy is thus an endogenous variable which ensures that the market clears when the market price is equal to the intervention price. By contrast, unit export subsidies for highervalue dairy products are exogenous since corresponding markets are not regulated by an intervention support mechanism. For more details, see Gohin and Guyomard, 1999. 2 The standard Annington approach is modified in order to allow undifferentiated dairy imports to enter French and EU markets in a regime where tariffs are set to zero. For more details, see Gohin et al., 1999. 3 The same remark applies to experiment 7.

?4


N R A -Rennes

Olivier Mahul

ECONOMIE RURALE B18LIOTHEQUE SELF-INSURANCE IN THE PRESENCE OF BACKGROUND RISK

This paper addresses the question of whether independent and undesirable background risk will lead farmers to use more input against another risk that is selfinsurable. Background risk generally has an ambiguous etfect on the optimal input decision. However, if preferences exhibit standard risk ,wersion, the qualitative effect of the added background risk is unambiguous and depends whether the input is risk increasing or risk decreasing. Natural disaster can be such an exogenous risk. The limitation of individual liability for financial Joss by prograrns of public disaster relief or the reduction of the Joss probability influence the farmer's decision to self-insure against another independent risk.

Cet article montre comment la présence d'un risque sous-jacent inassurable peut inciter un agriculteur à augmenter sa consommation d'intrant destiné à lutter contre un autre risque indépendant. Un risque sous-jacent inassurable a un effet ambigu sur le niveau optimal de l'intrant. Cependant, si les préférences de l'agriculteur vérifient la propriété de "standard risk aversion", l'effet de l'introduction d'un risque sous-jacent inassurable n'est plus ambigu et varie selon que la productivité marginale de l'intrant est croissante ou décroissante avec les états de la nature. Un risque catastrophique peut s'apparenter à un tel risque exogène. L'introduction d'une responsabilité limitée en cas de sinistre à travers la mise en place de programmes d'indemnisation des victimes ou la réduction de la probabilité d'occurrence d'un sinistre influencent les décisions d'autoassurance d'un agriculteur envers un autre risque indépendant.

Introduction

In the standard mode! of self-insurance, the stochastic production function is defined by one input and one single source of uncertainty which characterizes the states of nature [START_REF] Just | Production function estimation and related risk considerations[END_REF]. Market insurance is assumed unavailable. In the context of a simple expected utility mode!, the individual decision 10 self-insure is influenced by his attitude toward risk and the nature of the technology. This technology is described by the relation between the marginal return to input on self-insurance and the occurrences of the random variable. Optimal application of risk decreasing (increasing) input is found to be ail the more excessive (inadequate) as the preferences are risk averse.

The mode! described above assumes that the decision marker's profit contains a single source of uncertainty. Such a framework is obviously at odds with the more realistic environment in which farmers must cope with multiple risks. In this paper, we consider the analytically simpler problem of choosing optimal level of input use in the presence of an undesirable and independent background risk which affects a second output. This output is considered as exogenous because it is not under the control of the agent when he chooses the level of input use about endogenous risk of his first output. Any correlation between the background risk and the risk which is to be self-insured will obviously be significant. It is straightforward to see that a positive correlation increases the effectiveness of self-insurance, whereas a negative correlation provides a partial insurance of the risk against each other [START_REF] Mayers | The interdependence of individual portfolio decisions and thedemand for insu rance[END_REF][START_REF] Eeckhoudt | Background risk, prudence and the demand for insurance[END_REF].

The main question considered in this paper is : How does an undesirable and independent background risk affect the level of input use selected by a risk averse farmer ? Although there is no correlation effect under independence, risk averse behavior is often altered [START_REF] Nachman | Preservation of "more risk averse" under expectations[END_REF]. A sufficient condition on underlying preferences, which allows to define unambiguously the optimal application of input, is standard risk aversion [START_REF] Kimball | Precautionary saving in the small and in the large[END_REF]. We show how a natural disaster can be treated as background risk in the farmer's maximization problern.

The modification of probability of Joss or the introduction of public financial compensation to victims of the natural disasters based on deductible or coinsurance are particular cases of a change in the distribution of the background risk. These programs may affect farmer's optimal application of input. Thus the authorities can incite the farmer to modify his consumption of input by changing the riskiness of exogenous wealth.

Our paper is organized as follows. The second section presents the standard model of self-insurance and we define the relation between the types of input, the farmer's attitudes toward risk, and his optimal level of input use. In the third section we introduce an undesirable and independent background risk and we analyze the effects on the optimal application of input when the preferences exhibit standard risk aversion. The fourth section is devoted to analyzing changes in background risk on the farmer's input decision.

Self-insurance and risk aversion

Consider an economy in which each fariner is endowed with an exogenous value of monetary wealth, ~' > 0 , and a stochastic technology êj = f (x , iv ), where q is output,

x is input and m is a random variable. Because of iv, output is random. lndividual beliefs about the distribution of the random variable iv are represented by the arbitrary cumulative density function <t>( ro) defined over the support [ w 1 ,(V J where co 1 < w 2 •

We assume that the marginal productivity of input is positive and decreasing, so we have + = 0 f > 0 and f. .. = 0 2 •( < 0 . Greater realizations of ro are assumed to

J x iJx • •'" cJx- of correspond to greater level of output, so .f,, = -• -> 0 .
êrv

The marginal return of input, f 0 , will vary with severity of iv . We can consider two types of inputs (Leather and Quiggin, 199 1 ). Input is called "risk increasing" when the marginal productivity of input is higher in more fa vorable states of nature (fxro = ~2;w > 0 ). It is called "risk decreasing" when the marginal productivity of input is higher in more adverse states of nature ( f "" < 0 ). This input corresponds to self-insu rance.

In an agricultural context, fertilizers are often risk increasing in their impact on output risk [START_REF] Just | Production function estimation and related risk considerations[END_REF] Leathers and Quiggin 199 1) white pesticides, herbicides, irrigation and vaccination against epidemics are risk decreasing [START_REF] Lewis | Self-insu rance against na tu rai disasters[END_REF].

A farmer will select the application of input x which maximizes his expected utility of profit MaxEU(W(x,w)) [START_REF] Nachman | Preservation of "more risk averse" under expectations[END_REF] with U a von-Neumann-Morgenstern utility function defined on profit, W(x,w) = ~ + f(x,ro)-pxwhere output price is nonnalized at unity and input price is p. The first order condition of the maximization problem above is

E[U'(W(x,ro))(fx(x,w)-p)] = O (2)
This condition is necessary and sufficient if we assume that the fariner prefers more to less ( U' > 0) and he is risk averse ( U" < 0 ). Suppose that a former with utility function U 2 is more risk averse than another farmer with utility function U 1 (Pratt 1964). U 2 is a concave transformation of U 1 .

Then it exists an increasing and concave function g such as U 2 (.) = (g O U 1 )(.) .

We define x 1 ( x 2 ) as the optimal level of input for farmer with utility U 1 ( U 2 ). Then X;

for i= 1,2 is solution of

(2')
We assume that both farmers have the same beliefs about the distribution of the random variable w. We are now ready to state and prove our first result.

Proposition 1

Let define X;= arg max EU;(W(x,ro)) for i= L2. Given two utility fonctions U 1 and U 2 X such that U 2 is more risk averse than U 1 , (i) If for ail levels of input x and prices p, it exists rv 0 E[rv 1 ;rv 2 ]such as (J,:(x,w) -pXw -wn):?: 0 V ctJ E[rv 1 ;rv 2 ] , then x 1 > x 2 ;

(ii) If for ail levels of input x and prices p, it exists rv,, E(rv 1 ;rv 2 ]such as

(J . . . . (x,w)-p)(rv-rv 0 )~0 "i (tJ E[(v 1 ;(ùJ,then x 1 <X 2 .

Proof

We prove result (i). We can write U 2 (.) = (g 0 U 1 )(.) with i' > 0 and g" < 0. To prove (i), we must show that For cv::::; cv 0 We have

f/xi,w)-p S. 0 We have also W(x 1 , w) < W(x 1 ,w 0 ) because f,u > 0 then g'(U 1 (W(x 1 ,w)))>g'(UiCW(x 1 ,w 1 ' ) ) ) because l/ 1 is increasingand g isconcave. Thus V w $ r.v 0 g'(lli (W(x 1 ,w ))X u;(W(x 1 ,w))Jf.(x 1 ,ro )-p] <g'(l'i (W(x 1 ,w ")))[ l 1;(W(x 1 ,ro ))J.f . . (x 1 ,w )-p]
For OJ 2 rv 0

We have

fr ( x, w) -p 2 0 and g'(U 1 (W(x 1 ,w )) ) < g'(U 1 (W(xi, ro 0 ))) thus V w 2 w 0 g'(lli (W(Xi,W )) )[ u;(w(xl ,0)) )Jf,(Xi,ù) )-P] < g'(U, (W(x1, W ")))[ { 1;(W(x1 ,W ))If, (xi ,W )-P] Consequently E{u;(W(xl ,w))[fx(X1,ro)-p]} < g'(f 11 (W(xl ,ro 11 )))E{U/(W(x, ,w)][.f,(x1 ,w)-PJ}
But the right hand side of this inequality equals zero because of the first order condition.

The result (ii) follows in a similar manner. Q.E.D.

We can note that the marginal productivity of input must verify the single crossing condition.

If fr is monotonie in r.v, then the single crossing condition is verified because of equation ( 2). From proposition 1, we can deduce the following corollary.

Corollary 1 (i) If the input is risk increasing then the more a farmer risk averse is, the smaller his optimal level of input use is ;

(ii) If the input is risk decreasing then the more a farmer risk averse is, the greater his optimal level of input use is.

A particular case of this corollary is that the optimal application of a risk increasing ( decreasing) input under risk averse preferences is always less (more) than the risk neutral level of input use (Quiggin, ! 992; Ramaswami. 1993 ).

Self-insurance with an independent background risk

As mentioned previously, the literature on self-insurance and self-protection mostly has considered individuals facing a single source of uncer1ainty. The optimal application of input is determined by the first-order condition (2). However, agricultural activities are framed by a multiplicity of risks. To capture this reality, we introduce an undesirable background risk, è, verifying

EU(W + e) $ U(W) (3)
This background risk is undesirable for an individual with utility U and wealth W. If background risk is "universally undesirable" (Kim ball. I 993 ), i. e. undesirable for ail risk averters, its expectation is nonpositive. Individual beliefs about the distribution of the random variable 7: are represented by the arbitrary cumulative density function 'Y defined over the support [a,h]where a <h. Farmer has no preventive means for thîs risk. To keep matters simple, we assume that 7: is independent of production risk w.

The farmer' s expected utility of profit becomes

(U, b E[u(~ + f(x, w )-px + c)] = f J u(w;, + f(XJù) -px+ t'),:lct>(rv )d'I-'(ë) (4)
Following Nachman (I 982), the farmer's objective can be rewritten by defining the derived utility fun ction as

h Û(W) = E[U(W + B)] = f U(W +ë)lfl'(e) (5) (1 The farmer's maximization problem becomes ~ax E[û(W 0 + f(x, <ù) -px)] (6)
This is the standard formulation of the single-risk mode! with ( I replacing U .

Let x • denote the optimal application of input without background risk (the solution to (2)) and x .. the optimal application of input in the presence of background risk (the solution to (6)). Moreover let xN denote the risk neutral level of input use. This optimal level is the same with or without background risk. The optimal application x •• can generally be higher or lower than the optimal application x • . However, this ambiguity is resolved if the underlying utility function, U, exhibits standard risk aversion. This notion is defined below. [START_REF] Kimball | Precautionary saving in the small and in the large[END_REF] defines prudence as the propensity to prepare and forearm oneself in the face of uncertainty. For instance, in a mode! of saving under uncertainty, prudence represents the intensity of the precautionary saving motive. An individual is prudent if the third derivative of his utility function is positive ( U"' > 0 ). From Jensen's inequality, precautionary premium, vr, is defined as the quantity satisfying

U'(W -1/1(.Y, W)) = EU'(W + y) with Ej = O (7)
As we define the degree of absolute risk aversion as A (W) --U"(W) we note ,, -U'(W) ' r !"'(W) P 0 (W) = -the degree of absolu te prudence.

U"(W)

~ [START_REF] Kimball | Standard risk aversion[END_REF] develops the concept of standard risk aversion as « a specific formalization of the notion that bearing one risk makes an agent less willing to bear another risk, even when the two risks are independent». Thus standard risk aversion holds if « any risk that makes a small reduction in wealth more painful also makes any undesirable, statistically-independent risk more painful», where "pain" is measured as utility Ioss. He shows that this condition holds if and only if preferences exhibit decreasing absolute risk aversion (DARA) and decreasing absolute prudence (DAP). This concept is a natural generalization of Pratt and Zeckhauser's proper risk aversion (198 7) : two independent risks that are individually undesirable are also jointly undesirable. The property of standard risk aversion is satisfied by preferences exhibiting constant relative risk aversion (CRRA), or constant absolute ri sk aversion (CARA) in the weaker version of standardness which requires only nonincreasing absolute risk aversion and nonincreasing absolute prudence. Kihlstrom et al. ( l 981) have shown that the derived utility lÎ inherits properties of the original utility fünction : it is monotone and concave ; it exhibits DARA if U is DARA ; it exhibits DAP if U is DAP. Eeckhoudt and Kimball ( 1992) have shown that if preferences exhibit standard risk aversion, then the derived utility function l.Î is more risk averse than the utility function U. Proposition 2 follows from applying coro!lary 1.

Proposition 2

Given an undesirable, nondegenerate, independent background risk, and preferences satisfying standard risk aversion, (i)

x ... < x• < xN if the input is risk increasing; (ii) xN < x• < x .. if the input is risk decreasing.
Note that, with an undesirable background risk, standardness is sufficient but not necessary on utility function . If background risk is universally undesirable, risk vulnerability (Gollier and Pratt, 1995) is a necessary and sufficient condition. This concept includes proper risk aversion but it is more demanding than DARA. It is verified by all hyperbolic absolute risk aversion (HARA) fonctions.

Let define over-consumption as the difference between the level of input use of a risk averse farmer and the level of input use of a risk neutral farmer, and let define under-consumption as the opposite of over-consumption. The above proposition says that background risk increases over-consumption (under-consumption) if the input is risk decreasing (increasing). In other words, the presence of an undesirable and independent background risk increases (in an absolute sense) the difference between the level of input use of a risk averse fariner and the level of input use of a risk neutral farmer, if preferences exhibit standard risk aversion.

Choices about endogenous risk sometimes must be made while simultaneously facing one immuable exogenous risk that is not under the contrai of the agent, and that is independent of endogenous risk. At first, we have assumed that each former was endowed a certain monetary wealth ~1 • In the real world , beside his produ ction activity, farmer has other sources of income (off-farm earnings, tinancial assets ... ) which are random. We can also consider that human capital is random. So we can define random initial wealth as W = ~ + & .

In the standard mode! of self-insurance, one production function is generally introduced. Thus farmer is assumed to manage only one risky output. ln the real world, his business are diversified, so he has to manage two risky yields or more. When farmer choices his optimal level of input use about endogenous risk of the first output, the second output may face exogenous risk which is not under the farmer's contrai. In our static mode!, a risk can be considered as exogenous if choices about this risk and those about endogenous risk are not made simultaneously. This second source of risk is also exogenous if farmer has not means of defense against it such a risk can be called natural disaster (hurricanes, epidemics .. . ). It is also the case when preventive measures exist but their uses are forbidden by the authorities. For example, prophylactic vaccination against foot-and-mouth disease is available but it was stopped in 1991 in European Union. We suppose that both risks are independent. Thus the random wealth W characterizes this risky output.

Consequently, not taking this background risk into account can lead the theoretical rnodel to misestimate optimal risk-taking. This would lead an overestimation or an underestimation of the farmer's optimal level of input use.

Changes in background risk and self-insurance

It is well known that the government is able to influence farmer's optimal consumption of input through output and/or input prices ( Leathers and Quiggin,I 991 ), by proposing an actuarially fair crop insurance (Ramaswami, 1993) or a public financial compensation [START_REF] Lewis | Self-insu rance against na tu rai disasters[END_REF].

Proposition 2 proves that the presence of an independent and undesirable background risk can modify farmer's optimal level of input use. But how does a change in the riskiness of exogenous random variable affect his optimal consumption ? Let us consider any couple of randorn variables (i\ , t: 2 ) with supports in [a, b], with cumulative distribution functions \J' 1 ( 6 ) and \J' 2 ( t'). Denoti ng the monotone likelihood ratio (l'v1LR) dominance criterion by >-, we have : h(e)suchthat 'f\(e)=0 for Ve<c and <-14' 1 (e)= h(e)d4\ (é')for Vt•?.c.

The farmer, facing background risk e with distribution \J' , . i = 1,2, will select the level of input use which maximizes his expected utility given in (4)

Proposition 3

If preferences exhibit decreasing absolute risk aversion (DARA) and the independent and undesirable background risk becomes less risky in the sen se of MLR criterion, then (i) the optimal application of risk increasing input increases and (ii) the optimal application of risk decreasing input decreases.

Pro(?/ /,

We define the derived utility function as {Î; = EU(W + i,) = f l/(W + e)d'f';(ê)and the .,

{l"(W) ~ (Î."(W)
degree of risk aversion as A . , (W) = ----and

A ( W) = -~' for ; = l,2 . ' U'(W) ' U/(W)
First, we must prove that if e, >-e, then (Î, is more risk averse than U 2 at W, i.e.

-MIR

 1 (W) ?.  2 (W) . Observe that b Â,(W)-Â2(W)::::: I a U'(W + 6)d'P 1 (6) U'(W + 1-:)d't\(6) A(l (W + 6) b --, , ------" -- f U'(W + x)d'P 1 (x) f (! '(W + x )d'l\ (x) (l " == J A,,(W+s) bU'(W+s)~(c) + j f U'(W+x)~(x) a h(E) Aa(W+&)U'(W+& _b _ _ _ _ _ -:-b _ _ _ _ _ f U(W+x)clf:(x) JU(W+x)df~(x) a a .,
Since h > 0 and h is nonincreasing, it exists 6* 2". c such that the bracketed expression above is negative for ail 6 2". & *. Consequently, since we assume DARA,

(l'(W + 1;)d4':(ë) A =0 I u 1 ( w + X )d4\ (X)
We deduce proposition 3 by applying corollary 1.

Q.E.D.

If the random variable of background risk has two point support ( -~, < 0 with probability p and O with probability 1-p ), the decreasing of the probability of the bad state is an :MLR irnprovement. 4' 1 puts more probability mass on the lower outcome than does \{/ 2 . Bence we deduce the following corolla,y

Corollary 2 Given a farmer whose utility function is DARA facing an independent background risk with probability of Joss p. Ifthis probability decreases, then (i) the optimal application of risk decreasing input decreases and (ii) the optimal application of risk increasing input increases. [START_REF] Doherty | Optimal insu rance in incomplete markets[END_REF] have highlighted the consistent and apparent overestimation of subjective low probability. The former may overestimate this probability because of a lack of information about his probability distribution. If the government has a cost advantage in collecting and processing information about these phenomena, the dissemination of its information may decrease farmer's subjective probability of Joss and then modify his optimal level of input use. If this exogenous risk is a natural disaster affecting another output, a decrease of the probability of loss can affect the optimal level of input use. Such a policy might serve as an alternative to the direct regulation of input use.

Suppose that "i represents the random lasses on initial wealth W., due to a natural disaster. Its support is [-Wr,,O]. In most of the cou nt ries, individual liability for financial Joss due to natural disaster is limited by compensation from programs of public disaster relief In France, such a program is called "Fonds national de garantie des calamités agricoles" and exists since 1964. Lewis and Nickerson ( l 989) have shown that the introduction of this program for the endogenous risk modifies fanner's optimal application of input. Suppose that this public insurance program affects the exogenous random production W =Ut;)+ Îi. It ensures that the value of exogenous property after the occurrence of a disaster does not fall below some minimal value W = ~ + "'i with "'i < 0. The absolute valueiej can be considered as a deductible. This institutional arrangement on the exogenous risk can be interpreted as a special monotone probability ratio (MPR) change in the distribution of background risk (Eeckhoudt and Gallier, 1995). Notice that the ranking between the two random variables "i 1 and e 2 cannot be achieved by the MLR criterion since d'-P 1 / <.fIJ 2 at e = ë is zero, so that this ratio may not be decreasing. Consequently, we cannot apply proposition 3. But we deduce the following proposition.

Proposition 4

Given an individual whose utility function is DARA.

If the government introduces a limited liability for loss on the independent background wealth by implementing a compensation program based on deductible, then (i) the optimal application of risk increasing input increases and (ii) the optimal application of risk decreasing input decreases.

Proof

The degree of risk aversion of the derived utility functions are defined by Then we apply corollary 1.

Q.E.D.

Of course, this result is also valid if the government imposes a minimum price for the exogenous production by implementing a trading program when the price goes below a minimum price. Suppose that the former has got pig farm and corn field. Proposition 4 says that he will be incited to decrease his consumption of pesticides if a minimum price of pig meat is guaranteed by the authorities or if the lasses due to a disease (against which there is no prophylactic vaccination) are limited by a compensation program. Suppose now that the government replaces this insurance prograrn with deductible by a coinsurance program. Fanner bears a percentage of total lasses. It is called "rate of retentie n".

Proposition 5

Given a farmer whose preferences verify P;,(W+ë)~A 0 (W+ë 1) Ve,ë'E[-w;,;o] (9) If the government introduces a system of coinsurance for loss on the independent background wealth, then (i) the optimal application of risk increasing input increases and (ii) the optimal application of risk decreasing input decreases.

Proof

Note a E ( 0,1) the rate of retention. The background risk becomes e 2 = ae, .

Thus we have \f'

1 (B) = Prob[e 1 se]= Prob[a'i, s aë] = Prob(i 2 s ae] = '4\(ae) for ail ê E[-Wa,0] .
Consequently, \f' 2 (e)s\J',(e) for alleE [-w;.,o] since cumulative functions are nondecreasing and ë < 0. The introduction of the coinsurance is a shift in the distribution of the background risk Ë from 'I', to \JI,, where lf-1 1 is dominated by \f/ 2 in the sense of the first stochastic dominance (FSD). Eeckhoudt et al. ( 199 5) show that [ \f/ 2 ; ; \f/ 1 implies Â, (W) ~ Â 2 (W)] <=> utility U verifies prope11y (9). We obtain propo~ sition 5 by applying corollary 1.

Q.E.D.

If preferences exhibit constant absolute risk aversion, the derived utility function is also CARA and the introduction of a coinsurance system does not modify the optimal level of input use. The inequality of property (9} is strict if ( I is the "one switch" utility function, U(W)=k,w-e-1:,w,k, > 0. k 1 > 0 or if U is CRRA with relative risk aversion parameter y and W ~ (1 + y )Wc). Note that this proposition deals with only risk aversion of the derived utility function at a particular value of W. To obtain definitive comparative static results we would need to apply proposition 5 at each relevant level of wealth. Propositions 3, 4 and 5 show that a change in the riskiness of the exogenous risk can incite farmer to modify his level of input use about an endogenous risk, even if bath risks are independent.

Conclusion

The literature on self-insurance has considered an individual facing a single source of uncertainty. We have focused on the influence of two important factors on the level of input use selected by an expected utility maximizing former. This first factor is the farmer's degree of risk aversion. The second factor is the nature of technology characterized by the single crossing condition of production function. They lead to the well-known result : the optimal application of a risk decreasing (increasing) input is ail the higher (Iower) as the farmer is risk averse.

But the single risk assumption imposes severe limitations upon the relevance of the mode!. Thus this paper has examined a simple mode! of the individual decision to selfinsure when we introduce a background risk that is not under the control of the agent, and that is independent of endogenous risk. Two impo11ant implications arise from our model. First, the presence of a background risk increases (decreases) the optimal level of risk decreasing (increasing) input use if preferences exhibit standard risk aversion. Thus the level of input use selected by a risk averse farmer can be explained in part by the presence of background risk. Second, a shift in the distribution of background risk, in the sense of MLR criterion, MPR criterion or FSD criterion, affects the optimal application of input. Hence the introduction of a public relief program characterized by a deductible or a coinsurance, or a change in the fanner's subjective probability of loss modifies the optimal input decision.

While the simplicity of our mode! requires our results to be viewed as exploratory, our analysis suggests that a change in the riskiness that affects an output which is exogenous when farmers make choices about endogenous risk can incite them to modify their optimal level of input use.

  E{u;(W(xl ,ro) )[./, (xi ,ro) -pl}::; 0 with E{Ui'(W(xl ,CD) X.tJx1 'm) -pl}= 0 We can write E{u;(w(xl ,ro))[./,(xl ,ro)-p]} = E{g'(U1 (W(xl ,m)))[Ui'(W(x, ,m))][./,.(x, ,w)-Pl} .

  -e, if there exists a scalar c E[a,h] and a nonnegative nonincreasing function MU?

hJ

  U"(W + e)<.fIJ 1 (e) ÂI (W) = -: -----J U'(W + e)d'-P 1 (e) a h U"(W + s) 4\ (Ïi) + f [ I "(W + e)<.fIJI (ë) of proposition 3, it is straightforward to show that A 1 (W):?: A 1 (W).