The Tangled Web of Agricultural Insurance: Evaluating the Impacts of Government Policy

Transcription

1 The Tangled Web o Agricultural Insurance: Evaluating the Impacts o Government Policy Jason Pearcy Vincent Smith July 7, 2015 Abstract This paper examines the eects o changes in major elements o the U.S. ederal crop insurance program on the structure o the agricultural insurance industry. We model the interactions between armers, insurance agents and insurance companies. Two symmetric equilibria are determined: one with competitive insurance companies and one where insurance companies orm a collusive monopsony. We evaluate how marginal changes in government policy (changes in the premium subsidy rate, A&O subsidy rate, and loading actor) aect the insurance premium rate, agent compensation rates, agent eort levels, and market demand or crop insurance. Conditional on no prior government policy, armers preer a marginal increase in the premium subsidy rate. This change has the lowest associated net social cost, but is the policy least preerred by insurance companies. The insurance companies most preerred policy is a marginal increase in the A&O subsidy rate, which has the highest associated net social cost, the highest cost to the government, and does not beneit armers. We also evaluate the consequences o changes in crop prices. I the market or insurance agent services is competitive, then a change in crop prices does not change agent compensation rates, but otherwise the agent compensation rate will change. This result suggests an empirical test regarding insurance company market perormance. JEL Classiication: Q18, H53, L10. Keywords: Agriculture Subsidy, Crop Insurance, Vertical Restraints. For helpul comments, we thank the editor Christopher McIntosh, two anonymous reviewers, Mark Anderson, Anton Bekkerman, Joseph Glauber, Wendy Stock, Carly Urban, and seminar participants at Montana State University. We also thank David Elsea or excellent research assistance. Vincent Smith acknowledges support rom the Montana Agriculture Experiment Station and the U.S. Department o Agriculture under Agreement No. 4W5004. Any opinions, indings, conclusion, or recommendations expressed in this publication are those o the authors and do not necessarily relect the view o the Montana Agriculture Experiment Station or the U.S. Department o Agriculture. Author Posting. c Jason Pearcy and Vincent Smith This is the authors version o the work. It is posted here or personal use, not or redistribution. The deinitive version was published in the Journal o Agricultural and Resource Economics, Vol. 40, No. 1, 2015, pp Montana State University; jason.a.pearcy@gmail.com Montana State University; vsmith@montana.edu 1

2 1 Introduction Subsidized crop insurance programs in general, and the U.S. subsidized crop insurance program in particular, have received considerable attention rom researchers and policymakers since the 1990s. 1 Most o these studies have ocused on moral hazard, impacts on arm income and risk, environmental eects, adverse selection, determinants o the demand or crop insurance, and alternative methods or estimating actuarially air premium rates (Glauber, 2013; Goodwin & Smith, 2013b). Little attention has been given to the supply side o the ederal crop insurance program and the economic implications o the Standard Reinsurance Agreement (SRA) between private insurance companies who deliver ederally subsidized crop insurance products to armers and the U.S. government. 2 The lack o ocus on the supply side o the U.S. subsidized crop insurance program is to some extent surprising. Over the past six years, crop insurance companies received between $2 billion and $3 billion annually rom the ederal government to deliver heavily subsidized crop insurance products to the arm sector (Glauber, 2013), about 15 % o total ederal spending on all arm subsidy programs since To provide an understanding o the supply side o the U.S. ederal crop insurance program and the economic eects o the dierent subsidies embedded in this program, we present a new model o the crop insurance industry. Our model is structured to examine the tradeos between two dierent government objectives: adequate participation in the program and minimizing the taxpayer/net social costs o the program (Gardner & Kramer, 1986; Goodwin & Smith, 1995). The model is used to evaluate the eects o marginal changes in crop insurance policy variables and crop prices on payments to insurance agents and those agent marketing eorts while considering dierent types o competition between insurance compa- 1 See, or example, recent surveys and reviews o crop insurance issues by Coble and Barnett (2013), Goodwin and Smith (2013a), Goodwin and Smith (2013b), Glauber (2013), Miranda and Farrin (2012), Smith and Glauber (2012). 2 Exceptions include Babcock and Hart (2006); Glauber (2004); Ker (2001); Smith, Glauber, and Dismukes (2012); and Smith and Glauber (2012), most o whom have raised concerns about the transer eiciency o providing subsidies to armers through a program delivered by private insurance companies. 2

3 nies. We also evaluate the eects o marginal changes in crop insurance policy variables on overall participation by armers, subsidies to armers, crop insurance companies revenues, and net social cost. The policy variables include subsidy rates on the premiums paid directly by armers, direct subsidies to the crop insurance companies or administration and operating (A&O) costs, and catastrophic risk loading actors on premium rates mandated by legislation. We also examine the eects o increases in crop prices like those that have occurred or corn, wheat, oilseeds, and other heavily insured commodities over the period 2006 to 2012 on agent compensation rates and eort levels. 3 Indirectly, the model also provides insights regarding the consequences o the methods used by the USDA Risk Management Agency (RMA) to establish policy premium rates, which have been criticized as overestimating actuarially air rates (Coble, Knight, Goodwin, Miller, & Rejesus, 2010). 4 The model takes the orm o a sequential game involving crop insurance companies and independent insurance agents. Insurance agents market crop insurance policies to armers and then supply those policies to insurance companies. The model allows or the endogenous entry/exit o insurance agents. Insurance companies demand policies rom insurance agents and either compete with other companies or agent services or orm a monopsony cartel. Symmetric equilibria are determined or each type o insurance company market. The analysis presented here is most closely related to the literature regarding commission sales and vertical restraints. Hart, Tirole, Carlton, and Williamson (1990) and Ordover, Saloner, and Salop (1990) develop vertical restraint models where upstream irms compete in prices and downstream irms compete in quantities. In our model, upstream insurance companies compete in prices, which are the compensation rates set or insurance agents, and downstream insurance agents compete in quantities o eort. Recent studies on commission sales include Armstrong and Zhou (2011) and Inderst and Ottaviani (2012a; 2012b; 2012c). 3 Several economists have argued that these increases in prices or crops such as corn, soybeans, and wheat have had important impacts on agricultural insurance companies proits and the size o the U.S. crop insurance industry (or example, Babcock, 2011; Glauber, 2013; Goodwin & Smith, 2013a). 4 This issue is discussed in Appendix B. 3

4 In these models, irms set prices or consumers and commissions or intermediaries. As in the model presented here, irms (insurance companies ) products are only sold to consumers (armers) through an intermediary (insurance agents) providing advice to consumers. However, this literature ocuses on how consumer heterogeneity leads to dierent equilibrium outcomes and possible policies to address consumer protection issues. Consumers may dier due to dierent levels o knowledge about prices (Armstrong & Zhou, 2011), dierent levels o knowledge about an advisor s incentives (Inderst & Ottaviani, 2012c), or dierentiated products (Inderst & Ottaviani, 2012a). In contrast to the commission sales literature, here insurance companies oer homogeneous, ederally subsidized crop insurance products, and in the model we abstract rom armer heterogeneity. As discussed below, the SRA among the government, insurance companies, and insurance agents codiies vertical restraints and limits the possible behavior o upstream insurance companies and downstream insurance agents. The government sets premium rates so that there is no strategic pricing, and insurance companies are orced to accept any conorming contract so that vertical integration is not relevant. Since insurance products and premium rates are standardized, consumer protection issues relevant or other industries are not applicable. Another dierence in this paper is that we consider competition between intermediaries (agents) or a consumer s (armer s) business. Most o the literature on commission sales abstracts rom the decision by a buyer to choose an intermediary. 5 We ind that a marginal increase in the A&O subsidy rate paid to insurance companies does not change the premium rate or armers, but increases agent compensation rates and agent eort levels. A marginal increase in the premium subsidy rate beneits armers by reducing their out-o-pocket costs or crop insurance. The increase in the premium subsidy rate also has an added eect o reducing the returns to eort on the part o insurance agents, 5 Our analysis is also related to the literature examining revenue sharing contracts (Dana & Spier, 2001). In our model, insurance companies choose agent compensation rates where these rates determine the amount o revenue shared with insurance agents. The type o competition between insurance companies inluences the agent compensation rates that are chosen. In this paper we only consider linear agent compensation rates as non-linear pricing is not allowed under the current SRA. We leave the evaluation o non-linear agent compensation and non-linear subsides or uture consideration. 4

5 leading those agents to expend less eort per armer (or suiciently low compensation rates). On a per policy basis, raising the premium subsidy rate has no eect on insurance company revenues, as the increase in the premium subsidy is exactly oset by the reduction in armer paid premiums. However, government expenditures on subsidies increase both on a per policy basis and because the amount o insurance demanded by armers increases as the price they pay or coverage declines. Inlating actuarially air premium rates through a marginal increase in the catastrophic loading actor has the ollowing eects. On a per policy basis, government premium subsidy payments and A&O subsidy payments increase, as do insurance company revenues. The overall eect on quantity is ambiguous as armers demand less insurance with higher premium rates, and or suiciently low compensation rates insurance agents respond to the rate increase with an increase in eort. The increase in agent eort increases the quantity o insurance purchased by armers, but depending on the parameterization o the model the increase in eort may or may not oset the decrease in quantity due to the increase in the premium rate. Comparing the net social costs o marginal policy changes is more complex, and we evaluate changes in policy parameters using a convenient baseline o no prior government policy. In this context, an increase in the premium subsidy rate is the marginal change most preerred by armers, least preerred by insurance companies, and has the lowest associated net cost. A marginal increase in the A&O subsidy rate yields the highest beneit or insurance companies but has the highest taxpayer cost, provides no beneits to armers, and has the highest net social cost. We also investigate the impacts o changing crop prices on equilibrium outcomes. Holding agent compensation rates constant, a marginal increase in crop prices increases agents eort. I the market or agents services is competitive, then a change in crop prices does not change agent compensation rates. Instead, i insurance companies collude, then agent compensation rates are likely to decrease as crop prices increase. This result suggests an empirical test 5

6 regarding insurance company market perormance. 2 The U.S. Crop Insurance Program The U.S. ederal crop insurance program was established in 1938 and is overseen by the Federal Crop Insurance Corporation (FCIC). Until 1980, Congress or the most part required the FCIC to establish and charge armers actuarially air premium rates that would cover expected indemnities, while the ederal government covered the A&O expenses o the program (Gardner & Kramer, 1986; Kramer, 1983). Prior to 1983, the FCIC relied on independent agents to sell ederal yield insurance products and hired independent crop loss adjusters to assess losses (Goodwin & Smith, 1995; Kramer, 1983). The 1980 Crop Insurance Act mandated a major change in the delivery system, requiring the FCIC to allow private insurance companies to sell and service ederally developed and subsidized crop insurance products. The 1980 Act also explicitly introduced subsidies that lowered armer paid premium rates below actuarially air levels. In addition, insurance companies received a direct subsidy to cover their A&O costs and were able to keep a substantial proportion o any underwriting gains associated with policies or which they retained some risk o loss. Through stop loss provisions (Ker, 2001) and the creation o insurance pools into which companies were able to dispose o policies they perceived to involve atypically high risks o loss, the ederal government also accepted responsibility or a disproportionate share o expected indemnity payments. The disproportionate allocation o risk between the companies and the government is related to two crucial elements o the program. First, companies operating in any given state are required to accept all insurance policies purchased by armers and oered to them by independent insurance agents, regardless o the potential risk o loss. Second, premium rates are established by the government, largely on the basis o the average per acre expected insurable loss ratio in the county in which the arm purchasing the insurance is located, and 6

7 companies are prohibited rom making any adjustments to those premium rates. Since 1983, the relationship between the government and the companies and agents selling and servicing ederally subsidized crop insurance contracts has been codiied in a series o SRAs. The SRA is periodically modiied through negotiations between the private companies and the FCIC, and is altered by Congressional legislation. The speciic terms o the SRA and the levels o premium rate and A&O subsidies have changed over time as a result o the 1994 Crop Insurance Reorm Act, the 2000 Agricultural Risk Protection Act, and various arm bills (or example, the 2008 arm bill reduced A&O subsidy rates). However, the basic structure o the program (A&O subsidies, premium rate subsidies, a mandated catastrophic loading actor, and risk sharing between the companies and the ederal government) has not been altered substantively by legislation since While the FCIC oversees the U.S. ederal crop insurance program, the RMA manages the program. The RMA is responsible or developing and maintaining all ederally subsidized insurance products, including all premium rates. Crop insurance companies and insurance agents are not allowed to adjust those premium rates or any individual armer. I they do so, they ace severe penalties and can be banned rom selling and servicing ederal crop insurance products in the uture. Under the provisions o the 1994 Crop Insurance Reorm Act, the RMA is required to estimate the actuarially air premium rate or each policy (on a county-by-county basis). 7 As discussed above, the RMA increases the premium rate by a legislatively mandated catastrophic risk loading actor o 13.64%. The rationale or the loading actor is that extreme adverse events may not be included in the historical data typically used to estimate the actuarially air premium rates or each county. 8 Farmers are required to pay only a portion o the total premium or their insurance 6 For example, the 2014 arm bill introduced a new area (county) revenue and yield based product called the supplementary coverage option and mandated the development o some new products (or example, a margin insurance program or rice). However, this bill did not change the underlying structure o the major subsidy related components o the ederal crop insurance program. 7 In Appendix B we discuss how some o the methods used by the RMA to estimate premium rates may overestimate the actuarially air rate. 8 For most crops, the USDA National Agricultural Statistical Service only has data on county yields beginning in or ater

8 coverage. The government subsidizes a substantial proportion o that total premium, paying its share directly into the insurance pool rom which any indemnities or losses will be paid. The government s share o the total premium depends on the amount o coverage purchased by the armer. As a result o the 2000 Agricultural Risk Protection Act, which substantially increased such subsidies, the government s share in recent years has averaged 62% (Glauber, 2013). The government also pays a separate direct A&O subsidy to the insurance companies. The A&O subsidy is deined as a proportion o the total premium paid into the insurance pool by both the armer and the government. That subsidy is currently 18% and is substantially lower than beore 2008, when the A&O subsidy rate was reduced under the provisions o the 2008 arm bill. In the 2010 SRA negotiations, the companies claimed that the reduction in the A&O subsidy would create severe inancial diiculties or them and were successul in negotiating a cap on insurance agent compensation rates. The agent compensation rate cap is equal to approximately 15% o total premiums, limiting an insurance company s costs, and is enorced by the RMA, with severe penalties or companies who attempt to exceed it. Typically, independent insurance agents market crop insurance policies to armers in any given state and are then ree to allocate them to any insurance company operating in that state. The companies thereore compete or those policies and, in eect, insurance agent services. Historically, in states in which underwriting gains are large, insurance companies have competed vigorously and oered relatively high payments (compensation rates) to insurance agents or their services and policies (Smith et al., 2012). 3 Model We present a model o agricultural insurance with our types o economic agents: the ederal government, armers, insurance agents, and insurance companies. The government determines the premium rate and reduces the armer paid share o that premium rate through 8

9 additional subsidies. The government also provides direct subsidies to insurance companies, osetting their A&O costs, and may establish the maximum compensation rate that insurance companies can use to pay insurance agents or their services. Farmers buy insurance through insurance agents. Insurance agents exert eort to sell policies to armers and then choose the insurance company through which each policy is issued. Insurance companies set the wage/compensation rate or insurance agents. Companies and agents are assumed to maximize proits. The government s policy actions are assumed to be exogenous to the model, but the costs o these actions are endogenously determined. Two dierent types o symmetric equilibria are examined through a sequential game o complete inormation. In the game, crop insurance companies simultaneously determine their compensation rates or insurance agents. Subsequently, insurance agents choose their eort levels in response to the compensation rates they are oered. The model also accounts or endogenous entry and exit o insurance agents, where agents enter and exit until agent proits are zero. With this ramework it is possible to determine the equilibrium number o insurance agents, but we leave this or uture work. Each symmetric equilibrium in the model is characterized by an equilibrium level o eort that all insurance agents exert in relation to their clients and an equilibrium compensation rate that is oered by all insurance companies. In the competitive equilibrium, each insurance company sets its compensation rate so that all insurance companies make zero economic proit. In the collusive equilibrium, insurance companies coordinate their actions to obtain a monopsony solution in relation to compensation rates or insurance agents. 3.1 Government Farmers pay a premium rate, p, or each unit (dollar) o insurance coverage they purchase. Two distinct government policies aect that premium rate, causing it to dier rom the actuarially air premium rate,, which generates premium payments that would equal expected 9

10 losses/expected indemnity payments. 9 Under the terms o the 1994 Crop Insurance Reorm Act, the USDA RMA is irst required to identiy the actuarially air premium rate or a policy using all available data. The RMA is then required to add a loading actor to account or the possibility that the historical data used to calculate the actuarially air premium rate do not include suicient extreme events that cause losses to be exceptionally high. In practice, RMA accomplishes this goal by dividing every estimated actuarially air rate by a proportion, 1 α, that is less than one, legislatively deined as Thus, i this were the only adjustment to the armer paid premium rate, then armers would pay g, where g = /(1 α) and α (0, 1). However, the legislation (as currently deined by the 2000 Agricultural Risk Protection Act) also requires the ederal government to pay a share, s, o the estimated total premium, g. Thus, the actual armer paid premium rate is thereore p = (1 s)g = (1 s) (1 α). sgq = The cost o this subsidy to the government is s Q, where Q is the total expected crop revenue insured. The subsidy cost o sgq (1 α) is paid directly to the insurance pool established by the insurance companies. In addition, the government also pays an independent direct subsidy to the insurance companies totaling s d Q, where s d is the direct subsidy rate or administration and operations. Throughout the paper, we make the ollowing assumption. Assumption 1. The premium rate set by the government is g where g < 1, and the direct subsidy rate set by the government is s d, where s d <. Assumption 1 ensures that the government does not charge more than one dollar or one dollar o insurance coverage (g > 1). Requiring s d < ensures that the direct subsidy paid to insurance companies (s d Q) does not exceed the basic cost o insurance provision (Q). 10 In act, since 1981 when private companies began to deliver subsidized crop insurance in the United States the A&O subsidy rate has not exceeded 35% o the actuarially air premium rate. 9 Since is the actuarially air premium rate or one dollar o insurance coverage, [0, 1]. 10 Note that Q is the expected loss associated with insurance provision. 10

11 3.2 Farmers Farmers demand agricultural insurance. They are potentially dierentiated by many actors, including risk preerences, the probability o a loss occurring, and transaction costs associated with purchasing agricultural insurance. We abstract rom potential heterogeneity among armers by using general demand unctions that do not identiy the sources o such variation. A armer s demand or insurance is a unction o the premium rate she pays or coverage, p, and the eort level o insurance agents. Each armer observes the price at which she is oered insurance and the eort level o each agent. The armer then chooses an insurance agent and how much insurance to purchase. Each armer is assumed to purchase insurance rom only one agent. Some armers might not purchase agricultural insurance i the premium rate o agricultural insurance is too high compared to their probability o a loss or i their transaction costs are too large. Insurance agents are able to reduce these transaction costs by expending eort. There are K armers, where k = 1,... K indexes each armer. The demand unction or armer k is D k (p, e k ), where p is the premium per unit o insurance and e k is a vector o eort levels expended by insurance agents on armer k. A armer s coverage level is D k : the expected yield a armer insures. The expected price or the insured crop is Θ, so that q k = ΘD k (p, e k ) is the expected revenue rom production that armer k insures and pq k is the amount that armer k pays or insurance. The market demand or insurance is Q = K k=1 q k, where Q is the total amount o insurance purchased by armers. Throughout most o the paper we implicitly assume that the prices o agricultural crops do not change so that it does not matter i armers insure revenue or quantity. In Section 7.2, we consider the case in which prices o agricultural prices change over time causing a shit in the demand or insurance as measured by the dollar amount o coverage (liability). This sort o shit occurs because the armer s insurance coverage is tied to the value o the crop. When crop prices increased substantially in real terms, as between 2002 and 2013, the result is an increase in the real dollar amount o insurance coverage. 11

12 Each armer s demand unction, D k, depends on her characteristics and her preerences. Hence, D k is unspeciied, but we assume that the quantity demanded o insurance is non-increasing in the premium rate. We also assume that the quantity demanded is nondecreasing in agents eort at a decreasing rate, but only or the agent rom who insurance is purchased. The implications o these assumptions are as ollows. Assumption 2. I a armer purchases insurance, she does so rom only one agent. Her demand unction D k (p, e k ), is concave, dierentiable twice in the premium rate and three times in eort levels. I armer k purchases insurance rom agent a D k p 0 D k e ak 0 2 D k e 2 ak 0 2 D k e ak p 0 D k e a k = 0 where e ak is the eort level agent a exerts or armer k and a a. We assume that 2 D k e ak p 0 which implies that agent eort becomes relatively more eective as the premium rate paid by the armer increases; that is, i the premium rate is high (low) relative to expected indemnities, then armers choose a lower (higher) coverage rate and the marginal eect o an increase in agent eort will be relatively large (small). This assumption ensures that a tradeo exists between premium rates and agent eort levels or insurance agents. 11 Note that we also assume D k e a k = 0, which implies that i armer k purchases insurance rom agent a, then other agents eorts do not inluence the armer s purchase decision. 3.3 Insurance Agents There are A insurance agents indexed by a. Each agent chooses her eort level or each armer, e ak, to maximize proits. Agent eort is required to sell insurance policies to armers. Thus e ak 0 and e a is a K element vector indicating the eort level that agent a expends 11 See equations (14) and (15) in the Appendix. By assuming 2 D k eort when a policy change increases the premium rate. e ak p 0, this ensures agents increase their 12

13 on each armer. In addition to choosing eort levels or each armer, agents also choose the insurance company through which each policy is sold. I an agent sells a policy through insurance company i, the agent receives a share, w i, o the total premium associated with the policy. Let qa i be the quantity o insurance issued by agent a through insurance company i, where qa i = k (a,i) q k and is a unction o p and e, the A by K matrix o insurance agents eort levels. 12 An insurance agent s proit maximization problem is max π a = e a 0 I w i gqa i C a (e a ) F a (1) i=1 where C a (e a ) is the joint variable cost o eort and F a is agent a s ixed cost. An agent s total quantity o insurance is q a through all the companies. = I i=1 qi a, the sum o all policies issued by that agent In practice, the variable costs, C a, and ixed costs, F a, incurred by individual agents may dier amongst agents. Since the ocus here is on symmetric equilibria, we leave any potential agent heterogeneity unspeciied and make the ollowing assumptions. Assumption 3. C a (e a ) is a three times dierentiable, increasing, convex, and symmetric unction where Ca(ea) e ak eak =0 = 0. The assumption that Ca(ea) e ak eak =0 = 0 indicates that i agent a exerts no eort or armer k (e ak = 0), then the marginal cost associated with an increase in eort or armer k is zero. The symmetry o C a implies that the elements o e a can be rearranged without changing the value o C a. 12 In equilibrium there is a set o ordered pairs (k, a, i) where each element o the set indicates that armer k bought insurance rom agent a issued through insurance company i. k i is the set o armers whose insurance is issued through company i and k a is the set o armers who bought insurance rom agent a. k (a, i) is the set o armers who bought insurance rom agent a issued through company i. 13

14 3.4 Insurance Companies There are I insurance companies indexed by i. Each insurance company oers a compensation rate, w i, to insurance agents to maximize the company s expected proits. The compensation rate is the share o total premiums rom any policies given to an agent, where w i [0, w] as the government may cap the agent s share at w < 1. Let q i be the quantity insured through insurance company i, where q i = A a=1 qi a as the insurance company s quantity o insurance is the sum o all the policies issued through that company by each agent. Hence q i is a unction o p, the premium rate per unit o insurance, and e, a matrix o insurance agents eort levels. Total market wide purchases o crop insurance equal the sum o all insurance companies quantities, Q = I i=1 qi. Company i collects premium payments o pq i rom armers and receives premium and A&O subsidy payments o (sg + s d )q i rom the government. Insurance company i pays its insurance agents w i gq i, and has an expected loss o k i c kq k. 13 The expected loss rom armer k is the probability o a loss, c k, multiplied by the amount o insurance purchased, and may vary across armers. The proit maximization problem or insurance company i is ( ) (1 max π wi ) i = w i [0, w] (1 α) + s d q i c k q k. (2) k i In equation (2), is the actuarial air premium rate or the market wide portolio o insurance contracts and, by deinition, equals the expected loss among irms where Q K c k q k = 0. (3) k=1 I the actuarially air premium rate is the premium rate to be subsidized, then insurance revenues rom armer paid premiums and the government premium subsidy will equal expected losses. Instead, the ull premium rate is g = (1 α) and the expected loss ratio is the 13 We assume that insurance companies have no ixed costs to ensure the existence o a competitive equilibrium. 14

15 ratio o expected losses to total insurance premiums, K k=1 c kq k gq expected loss ratio is between 1 and 0, and is decreasing in α. = 1 α. Since α (0, 1), the 3.5 Government Policy Throughout the paper, there are three dierent types o government policies we consider: the premium subsidy rate (s), the direct A&O subsidy rate (s d ), and the loading actor (α). In general, the government chooses these policy parameters to balance two dierent policy objectives: adequate participation in the crop insurance program and minimizing the taxpayer/net social costs o the program (Gardner & Kramer, 1986; Goodwin & Smith, 1995). The ollowing corollary considers marginal changes in s, α, and s d on the premium rate armers pay or insurance (p). Corollary 1. 1(a). An increase in the subsidy rate, s, decreases the premium rate armers pay or insurance ( p < 0). 1(b). An increase in the adjustment to the actuarial air premium rate (an increase in α) increases the premium rate armers pay or insurance ( p α > 0). 1(c). An increase in the administration and operations subsidy rate (s d ) does not change the premium rate armers pay or insurance ( p = 0). Note that Corollary 1 just considers changes in the premium rate paid by armers. Any changes in s, α, or s d have other eects on the equilibrium o the model. For instance, a marginal change in α also aects agent eort levels and agent compensation rates. This in turn aects overall demand or insurance and the cost and beneits or each type o agent. Section 4.1 examines the equilibrium eects associated with changes in eort, and Sections 5 and 6 examine the equilibrium eects associated with changes in the agent compensation rate. 15

16 4 Symmetric Equilibrium We ocus on two dierent symmetric equilibria using the model outlined in Section 3. A sequential game o complete inormation is considered in which all insurance companies simultaneously determine their compensation rates or insurance agents and then all insurance agents simultaneously determine their eort levels. Insurance agents compete with one another in the quantity o eort and insurance companies compete with each other in agent compensation rate levels. The irst equilibrium is a competitive equilibrium or insurance companies characterized by each insurance company increasing its compensation rate until it achieves zero economic proit. The second equilibrium allows insurance companies to collude to achieve a monopsony solution. The two equilibria are dierentiated only by the type o competition occurring among insurance companies. Competition among insurance agents remains the same in each case. While both o these scenarios are abstractions, they serve as useul benchmarks regarding how the nature o competition in the agricultural insurance market and government policies aect equilibrium outcomes. Backwards induction o the sequential game starts with the insurance agent s problem. The best response o an insurance agent is a unction o the equilibrium compensation rate set by the insurance companies. Since competition between insurance agents remains the same or the dierent equilibria we consider, the best response o an insurance agent to any given compensation rate is the same or the insurance company competitive and collusive monopsony equilibria. Section 4.1 ocuses on the equilibrium eort level rom an insurance agent s best response unction. Endogenous entry, where agents enter and exit the market until each agent earns zero economic proit, is allowed. To obtain a symmetric equilibrium and avoid issues associated with the assignment o heterogenous armers to heterogeneous insurance agents, we eliminate heterogeneity among armers and insurance agents using the ollowing assumptions. 16

17 Assumption 4. 4(a). The probability o a loss is the same or all armers (c k = c or all k) and all armers have the same demand unction or insurance. 4(b). For any pair o values or p and e k, D k (p, e k ) = D(p, e k ) or all k. 4(c). The ixed cost or all insurance agents is the same (F a = F A or all a) and the joint variable cost unction is the same or all agents. For a particular e a, C a (e a ) = C A (e a ) or all a. In a symmetric equilibrium, all insurance companies set the same compensation rate and all insurance agents exert the same amount o eort or their clients. The symmetric equilibria we ind serve as a useul benchmarks or the analysis. One immediate result o Assumption 4 and equation (3) is that the actuarially air premium rate is equal to the probability o a loss. This standard result in the insurance literature is expressed by the ollowing lemma. Lemma 1. = c and 0 1, since c is a probability. The proo o this lemma and most o the propositions that ollow are included in Section A o the appendix. 4.1 Insurance Agent Eort Backwards induction in the sequential game starts with the insurance agent s proit maximization problem as expressed in equation (1). Incorporating Assumption 4 and Lemma 1 into equation (1), the representative insurance agent s problem is max π a = e a 0 I i=1 w i (1 α) qi a C A (e a ) F A. (4) The insurance agent s problem is used to determine the symmetric equilibrium eort level described in the ollowing proposition. Proposition 1. In a symmetric equilibrium, an insurance agent s eort is e ak = 0 i I k a = 0 17

18 and e ak = e i I k a = 1, where e is implicitly determined rom w i Θ D (1 α) e = C A e. (5) In Proposition 1, I k a is an indicator unction indicating when armer k buys insurance through agent a. In a symmetric equilibrium, i armer k buys insurance through agent a, then agent a expends an eort level o e or armer k. I armer k does not purchase insurance through agent a, then agent a exerts no eort towards armer k. The equilibrium eort level, e, is determined rom the irst order condition o the insurance agent s problem, as deined in equation (4). 14 The equilibrium level o eort expended on the armer who is a client o an agent, e, is a unction o w i,, Θ, α, and s. In equation (5), D is a unction o p = (1 s) (1 α) and e, as implied by Assumption 4. In any symmetric equilibrium however, e k is a vector o all zeros except or one element that takes on the value e. Thus in what ollows, we express D(p, e k ) simply as D(p, e ). Comparative static eects o w i, s, α, and s d in e are summarized by the ollowing proposition. Proposition 2. In a symmetric equilibrium, e w i 0 e 0 e α 0 e = 0. Proposition 2 indicates that insurance agents increase their equilibrium eort or their clients i their share o total premiums increases. The other comparative statics concern the policy variables set by the government: s, α, and s d. Here we only consider the initial or direct eect o these variables on the equilibrium level o eort, which is determined by holding the agent compensation rate, w, constant. In Sections 5 and 6 we determine the total eect which includes the direct eect and any indirect eects through changes in the agent compensation 14 We use superscript to denote any symmetric equilibrium. Superscript c reers to a competitive equilibrium determined in Section 5, and superscript m reers to a collusive monopsony equilibrium determined in Section 6. 18

19 rate. In this context, an increase in the subsidy rate, s, decreases the premium rate armers pay, p, and armers buy more insurance. The increase in insurance sales increases insurance agent proits. Insurance agents respond by reducing their eort level, which lowers their cost o eort and urther increases agent proits (recall that or Assumption 2, 2 D k e ak p 0). An increase in the rate adjustment parameter, α, increases average premium revenue, an eect that beneits insurance agents but also decreases the quantity demanded o insurance, an eect that adversely aects insurance agents. Insurance agents respond by increasing their eort levels. The administration and operations subsidy rate, s d, has no direct eect on agent eort. 5 Competitive Symmetric Equilibrium The competitive symmetric equilibrium or insurance companies is characterized by insurance companies competing with one another by adjusting agent compensation rates until insurance companies receive zero expected economic proit. The expected proit o each insurance company will be zero because otherwise each insurance company will increase its compensation rate to capture all o the market. The representative insurance company s problem is described by equation (2). Simpliying equation (2) using Assumption 4, Lemma 1, and the condition q i = k i ΘD k, the insurance company s problem becomes ( ) (α max π wi ) i = w i [0, w] (1 α) + s d q i (6) where we assume that (α w i) (1 α) + s d non-negative. Note that q i is a unction o p, e, and K. 0 so that insurance company expected proits are A change in w i has two separate eects on an insurance company s proits. An increase in w i decreases the return on every unit o insurance, as a larger share o the company s revenues is paid to insurance agents. However, an increase in w i also increases each agent s 19

20 eort level and the quantity o insurance purchased by each armer. Insurance companies choose compensation rates prior to insurance agents making their choices about the company to which they will allocate policies. Hence, the equilibrium compensation rate will be the same across insurance companies, w i = w c or all i. This result occurs due to the perectly elastic residual supply curve o agent services aced by each insurance company. I any one insurance company oers a compensation rate slightly above the other companies rates, then all agents will supply their services to that company. Given that insurance companies earn zero expected proits, the competitive equilibrium compensation rate is w c = α + (1 α)s d. (7) In a symmetric equilibrium, each insurance company obtains the same market share, 1 I, and issues the same amount o insurance coverage, KΘD I. The competitive equilibrium compensation rate is increasing in α and s d. Increases in α and s d increase the revenue per unit o insurance that each insurance company receives without initially increasing its costs. Insurance companies now earn positive economic proits, which are passed on to insurance agents. Note that wc α = 1 s d w ; thus, c α > 0 rom Assumption 1. An increase in decreases the equilibrium compensation rate as an increase in increases the cost o providing insurance. Note that the subsidy rate, s, does not eect the equilibrium compensation rate. Holding the amount o insurance purchased by armers constant, a change in s does not aect the amount o revenue an insurance company receives, only the source o that revenue. An increase in s implies that more insurance company revenue comes rom the government rather than rom armers. In a competitive market, insurance companies are likely to support a ceiling, w, or agent compensation rates. I insurance agent compensation is limited through a binding maximum compensation rate, then insurance companies would be able to make positive proits. Another consequence o a binding maximum compensation rate would be to lower agent eort 20

21 levels, aecting the market quantity o insurance. 15 Absent a binding compensation rate ceiling, competition among insurance companies drives w up until their proits are zero. The exogenous policy variables in the model are s, α, and s d. Proposition 2 above describes how changes in the policy variables aect the equilibrium level o eort, e c. From equation (7), changes in these variables aect the competitive equilibrium compensation rate. Changes in s, α, and s d have implications or the level o agent eort. The total change in agent eort is a result o the direct eect on eort, as indicated in Proposition 2, and also the indirect eect through the compensation rate. For example, it is straightorward to show that an increase in the subsidy rate has a non-positive eect: that is, dec ds This result ollows because ec = ec + ec w c w c 0. wc 0 rom Proposition 2 and, rom equation (7), = 0. The eects o the exogenous policy variables on the competitive equilibrium compensation rate and eort level are summarized by the ollowing proposition. Proposition 3. 3(a). An increase in the subsidy rate, s, has no eect on the compensation rate but decreases the eort level o agents ( wc dec = 0 and ds 0). 3(b). An increase in the rate adjustment parameter, α, increases both the compensation rate and eort level ( wc α 0 and dec dα 0). 3(c). An increase in the A&O subsidy rate, s d, increases the compensation rate and agent eort level ( wc > 0 and dec ds d 0). 6 Collusive Monopsony Symmetric Equilibrium Here, we consider a collusive equilibrium in which the insurance companies coordinate their actions to behave as a monopsonist. We assume that insurance companies split the monopsony proits equally. This symmetric collusive equilibrium provides a useul contrast to the symmetric competitive equilibrium. The representative insurance company s problem is again described by equation (6). However, in the monopsony equilibrium, the company supplies the entire market with insurance coverage. Thus the individual company s demand 15 Inderst and Ottaviani (2012c) make a similar point in their study, in which they evaluate the eect o capping an intermediary s commission as a policy to promote consumer protection. 21

22 or policies, q i, is replaced by the market demand to establish the monopsonist s proit maximization problem. The monopsonist s problem is ( ) (α w) max π = w [0, w] (1 α) + s d Q (8) where market demand is Q = KΘD(p, e). Again we assume that insurance company proits are non-negative, (α w) (1 α) + s d 0. The symmetric equilibrium agent compensation rate or the monopsony problem is characterized by the ollowing proposition. Proposition 4. I 2 e w 2 0, the monopsony symmetric equilibrium agent compensation rate, w m, exists and is implicitly determined by the ollowing equation. ( ) ( ) (α w m 2 ( ) D 2 (1 α) + s C A d Θ = e m e wm Θ m2 (1 α) 2 D e m2 ) D (9) Note that Proposition 4 depends on e w i requirement that 2 e w 2 as deined in equation (13) in Appendix A. The 0 in Proposition 4 is suicient but not necessary or a solution. We opt or this suiciency condition as it simpliies the analysis and allows or a straightorward interpretation. In Proposition 2, an increase in the equilibrium compensation rate is shown to increase agent eort. The suicient condition that 2 e w 2 compensation rate on eort is diminishing. 0 implies that the eect o the As in the competitive equilibrium, we are interested in how changes in the exogenous policy variables s, α, and s d change the monopsony compensation rate and agents eort levels. The eects o the exogenous policy variables on the monopsony equilibrium compensation rate are summarized in the ollowing proposition. Proposition 5. Assume that 2 e w 2 0. An increase in the A&O subsidy rate, s d, increases 22

23 the monopsony symmetric equilibrium compensation rate where w m = Θ ( D ) 2 e m 0, BT where T 2Θ ( ) ( ) 2 ( ) ( ) 2 ( D + RΘ2 D 3 2 D B 1 α e m B 2 1 α e m e 1 ) D B 0, m2 B e m e m B 2 C A e wm Θ 2 D m2 (1 α) e 0, and R (α wm ) m2 (1 α) + s d 0. For suiciently low compensation rates wm 0 and wm α 0. In Proposition 5 the sign o wm is determined over the entire space o exogenous policy variables (values o s, α, and s d ) conditional on 2 e 0. The signs o wm w 2 and wm α potentially change over this space, but we show in the proo o Proposition 5 that or a suiciently small compensation rate, say w m 0, then it must be that wm 0 and wm α 0. Note that 0 w m α + (1 α)s d, so that insurance company proits are non-negative or any w m in this range. I w m = α + (1 α)s d, then the collusive monopsony wage is the same as the competitive wage rate, w c. As w m decreases, the compensation rate or insurance agents alls and the returns or insurance companies increase. This argument suggests that insurance companies in competitive markets beneit rom binding compensation rate ceilings. Proposition 5 shows how the exogenous policy variables aect the monopsony compensation rate. We now examine the total eect o s, α, and s d on agent eort levels in a monopsony setting. The total change in eort level includes both the direct eect and the indirect eect through changes in the compensation rate. Proposition 6. Assume that 2 e w 2 0. An increase in the administration and operations subsidy rate, s d, increases the monopsony symmetric equilibrium agent eort level where de m ds d = Θ2 BT ( ) 3 D 0. (1 α) e m For suiciently low compensation rates, an increase in s lowers the eort level ( dem ds 0) and an increase in α increases the eort level ( dem dα 0). 23

24 As with changes in the equilibrium compensation rate, it is possible that dem ds and dem dα change signs as the compensation rate increases, but we cannot show that this is the case without highly restrictive assumptions about crop insurance demand and insurance agent costs. We now combine the results o Propositions 5 and 6 to discuss the eects o changes in s and α. An increase in s lowers the premium rate that armers pay or insurance and increases the quantity demanded o insurance. Since a lower premium rate has the same eect on the quantity demanded as a greater agent eort level, insurance companies respond to the increase in s by lowering the compensation rate ( wm eort level ( dem ds to increase their proits. 0) which lowers an agent s 0). The decrease in agent compensation rates allows insurance companies There is a similar story or an increase in α. An increase in α increases the premium rate armers pay or insurance and decreases the quantity o insurance they demand. Insurance companies respond by increasing the compensation rate ( wm α o agents ( em α 0). 0) to increase the eort level 7 Analysis This section examines the impact o marginal policy changes and changes in crop prices. First, we evaluate marginal changes in policy conditional on no prior government policy. This analysis allows us to evaluate the marginal eects o an increase in s, α, or s d separately and make direct comparisons between the competitive and collusive models. Next, we consider increases in demand or insurance due to changes in agricultural crop prices. This analysis is relevant as crop prices have increased over time and because as crop prices increase the amount o crop insurance armers purchase also increases. 24

25 7.1 Marginal Policy Changes The model includes three exogenous policy variables: s, α, and s d. In Sections 5 and 6 we described how marginal changes in these policy variables change the equilibrium agent compensation rate (w ) and agent eort level (e ). The ocus there was on the direction o each change. Here the ocus is on the magnitude o the changes. The relative magnitudes o these changes are equally as important since while a policy maker may ind it useul to know that an increase in s d will increase equilibrium agent eort, it is also useul to know by how much agent eort will increase. In addition to changes in agent eort and compensation rate, we extend our analysis to consider changes in the market quantity o crop insurance and the associated changes in the costs/beneits due to a marginal change in policy. This analysis is relevant in a case where policy makers consider a change in policy to increase the quantity o insurance armers buy but are also interested in the cost/beneit tradeos associated with that change. Our analysis also provides a ramework to evaluate the choice o policy instruments. While it is important to understand the change in the market quantity o insurance and the change in costs/beneits due to a marginal change in s, α, or s d, it is also important to be able to evaluate whether a policy goal is more easily achieved with a marginal change in s, α, or s d. We consider marginal changes in each policy variable or both the competitive and collusive monopsony symmetric equilibrium. In order to make direct comparisons across policy variables and equilibrium types, we determine the eect associated with a change in s, α, and s d conditional on the baseline case o no government intervention. With no prior government intervention, s = α = s d = 0, and the equilibrium compensation rate and eort are zero. Additionally, the premium rate is p =, and the symmetric equilibrium quantity o crop insurance is Q = KΘD(, 0). Conditioning on no prior government intervention allows us to compare the magnitude o the dierent policy eects as a change in α and a change in s may have dierent eects conditional on the values o α, s, and s d. One drawback to this analysis is that our results do 25