THE IMPACT OF INTERLINKED INDEX INSURANCE AND CREDIT CONTRACTS ON FINANCIAL MARKET DEEPENING AND SMALL FARM PRODUCTIVITY

Transcription

1 THE IMPACT OF INTERLINKED INDEX INSURANCE AND CREDIT CONTRACTS ON FINANCIAL MARKET DEEPENING AND SMALL FARM PRODUCTIVITY Micael R. Carter Lan Ceng Alexander Sarris University of California, Davis University of California, Davis University of Atens Abstract. Tis paper explores te relationsip between credit and index insurance market development using a teoretical model in wic small farm ouseolds ave te option to eiter (i) adopt a capital-intensive tecnology tat is risky but ig-yielding, or (ii) self-insure by adopting a traditional low-input and low-yielding tecnology. We sow tat neiter market is likely to develop in isolation from te oter, and tat uptake of improved tecnology will be low absent efforts to link credit and insurance. Te failure of index insurance markets to independently develop is not per se due to te existence of basis risk or to its expense, as self-insurance strategies are similarly caracterized by basis risk and are costly to te ouseold as tey reduce mean incomes. However, we sow tat te interlinkage of credit and index insurance contracts can allow bot markets to develop because te interlinked contract is more likely to stocastically dominate self-insurance. Te analysis also sows tat te way interlinkage will work depends fundamentally on te nature of te credit market and te degree to wic lenders are able to demand and seize collateral in te event of loan default. Tis interplay between collateral and te nature of credit-insurance interlinkage as direct and important implications for te design of programs to simultaneously boost small farm productivity and deepen rural financial markets. Keywords: Index insurance, Credit rationing, Interlinkage, Tecnology adoption Department of Agricultural and Resource Economics, mrcarter@ucdavis.edu. Department of Agricultural and Resource Economics, lanceng@ucdavis.edu. Department of Economics, alekosar@otenet.gr.

2 1. Introduction Te correlated risks and information asymmetries tat typify many low-income, small-older agricultural economies can keep rural financial markets (credit and insurance market) tin or even absent. 1 Te costs of tese tin markets are obvious and well documented, but te solution is far less clear. An earlier generation of efforts to employ conventional agricultural insurance to address te risk needs of te small farm sector failed under te weigt of transactions costs, adverse selection and moral azard (Hazell (1992); Barrett et al. (2008); Rotscild and Stiglitz (1976)). Wile it is tempting to declare te problem unsolvable, te pernicious role tat risk plays in te construction and perpetuation of rural poverty demands furter efforts in tis area. Enabled by tecnological advances in remote sensing and meteorological data, novel forms of agricultural index insurance would appear to offer a solution to tis problem of tin financial markets twinned wit low small farm productivity. Unlike conventional insurance wose indemnities is determined by individual outcomes, index insurance indemnifies insured farmers based on an index tat is correlated wit individual outcomes but is not influenced by individual beavior. Despite teir advantages in overcoming moral azard and adverse selection, agricultural index insurance contracts ave met wit sometimes indifferent demand and low uptake by te intended beneficiary populations. Wile tere can be multiple explanations for low uptake rates, tis paper argues on teoretical grounds tat uptake and impacts will be iger wen index insurance is interlinked wit credit contracts. Put simply, our argument is tat eiter market, credit or insurance, in isolation is likely to be tin or slow to develop in small-older agriculture. Wen contracts are interlinked, te gains in market deepening and productivity growt are likely to be iger. We sow tat impacts of interlinkage are somewat subtle, and differ across different types of economic environments. Interlinking index insurance wit credit is far different from simply bundling te two contracts togeter. Giné and Yang (2009) empirically estimate te impact of bundled contracts on take-up of credit under borrowers limited liability using a field experiment in Malawi. Farmers in te treatment group were offered bundled loan contracts, wile tose in control group were offered stand-alone loan contracts. Te result sows tat te take-up rate of bundled loan contracts was 13% lower tan tat of stand-alone loans. Miranda and Gonzalez-Vega (2010) build a model in a similar context wit limited liability loans. Teir simulation sows tat loans bundled wit index insurance raise loan default rates and reduce lenders expected profits. Tey attribute te poor 1 See Binswanger and Rosenzweig (1986) for a persuasive, if somewat informal discussion of tese points. 1

3 performance of bundled index insurance to ig basis risk or loading costs, concluding tat index insurance does not ave muc value for individual farmers. However, tese two studies overlook te positive externality generated by borrowers purcases of index insurance on lenders and do not endogenize loan contracts terms. Wen borrowers purcase index insurance under limited liability, index insurance not only reduces borrowers risks but also protects lenders by reducing default rates. Interlinked index insurance contracts internalize tis externality by allowing loan contract terms to respond to insurance contract, tus increasing te value of index insurance to individual farmers. Our model suggests tat interlinked contracts outperform bot non-interlinked and stand-alone loan contracts, especially in a low collateral environment, wic induce ig take-up rates of financial products, ig productivity tecnology and raise farmers welfare level. Tere is a tendency to explain low uptake rates of index insurance by inappropriate analogy to developed country experience, or by general statements tat basis risk or loading costs are too ig (Smit and Watts, 2009; Miranda and Gonzalez-Vega, 2010). However, te fact is tat te self-insurance strategies employed by small farmers expose farmers to significant basis risk and are actuarially unfair, wit ig implicit loadings and premium. Te question is tus not weter or not tere is basis risk under index insurance, but weter farmers welfare under index insurance can stocastically dominate tat under self-insurance. Asking te question tis way motivates te searc for ways to combine index insurance wit adoption of iger-yielding, but riskier tecnologies. Tat is, index insurance will more likely stocastically dominate self-insurance if it is a non-zero sum proposition tat simultaneously allows an increase in expected income even as it reduces risk exposure. We ere explore ways in wic tis migt appen troug te interlinkage of index insurance wit credit contracts. To do tis, Section 2 presents a stylized model of te tecnology and contracts potentially available to producer-consumer ouseolds in a low-income agricultural economy. We sow tat index insurance contracts can be represented as a mean preserving squeeze of te stocastic distribution determining output, and te agricultural credit supply is determined by lenders exposure to covariant default risks. Section 3 ten explores ouseolds demand for tecnology, credit and insurance facing tree insurance scemes associated wit ig-yielding tecnology: 1) no formal insurance, were loan contracts provide implicit insurance wen loans are not fully collateralized, 2) noninterlinked or bundled index insurance and credit, 3) interlinked index insurance and credit. We analyze eac sceme in two stylized environments: one caracterized by ig levels of collateral suc 2

4 as in Latin America and anoter caracterized by low levels of collateral suc as in Africa and Cina. Te introduction of non-interlinked insurance substantially improves te demand for ig-yielding tecnology and financial products wen te collateral level is ig, but as small positive or even negative impacts on ouseolds demand wen te collateral level decreases. Tis is consistent wit studies by Giné and Yang (2009) and Miranda and Gonzalez-Vega (2010). In contrast, interlinked index insurance significantly boosts ouseolds demand for ig-yielding tecnology in bot low and ig collateral environments. Section 4 analyzes te impacts of different insurance scemes wen te credit market reaces equilibrium. We sow tat te demand for new tecnology and financial products under no formal insurance and non-interlinked insurance scemes discussed in Section 3, will be coked off by te increased price of credit. However, wen insurance is interlinked wit credit, lenders are willing to provide any amount of agricultural loans at a fixed low price, so tat any expansion of credit demand will be satisfied. Section 4.2 analyzes te eterogeneous impact of interlinked contracts. Section 5 concludes wit te main findings. 2. Environment, Tecnology and Financial Contracts Tis section lays out a stylized model of te risk-averse and small farm ouseold. Wile igly simplified, te model captures te key elements of te small farm problem tat are relevant to te problem at and, including te self-insurance options available to te ouseold. Agricultural production is influenced by bot covariant and idiosyncratic socks. Against tis backdrop we define te set of potential financial contracts tat could be offered by competitive lenders and insurance firms Risks and Houseold Production. Small farm ouseolds are assumed to ave access to two tecnologies, a traditional tecnology wit low, but stable returns, and a iger yielding, but riskier tecnology tat requires substantial use of purcased inputs. Bot tecnologies are subject to idiosyncratic (or specific) socks, θ s and covariant socks, θ c 2. We assume a multiplicative structure and write te output of low-yielding tecnology as: (2.1) y l = θg l were θ =(θ c + θ s ) wit support [0, θ], probability distribution function denoted f(θ), cumulative distribution function denoted F (θ) and E(θ) =1. Te net income from low yielding tecnology is 2 Te simulation uses 75% of covariant risk over te total risk. 3

5 denoted as ρ l = y l. Similarly, we write te output of te ig-yielding tecnology as (2.2) y = θg (K) were K is te amount of purcased inputs required. Te superiority of ig-yielding tecnology is te iger expected net return, g (K) K > g l. We furter assume capital K can only be financed by borrowing from te rural credit market. Denote te loan contract as l < K, r, χ >, were r is te contractual interest rate and χ is te collateral required (Section 2.3 gives details on te determination of contract terms). Net returns to te ouseold under tis loan contract are as follows: θg (K) (1 + r)k, if θ> θ (2.3) ρ = χ, oterwise were θ = (1+r)K χ g (K) is te tresold level of te sock suc tat te value of te collateral plus te output produced just equals te value required for full loan repayment. Note tat tis specification sarply assumes tat te ouseold retains no income (or collateral assets) until after te loan is repaid. Assuming te separability between ouseold s consumption and production, ouseold consumption is given by c t = ρ t +W +B, t =, l, were W is te ouseold s consumable and collateralizable wealt and B is te risk-free income from off-farming activities. Te lowest consumption can be under te ig-yielding tecnology is c = W + B χ. Figure 5 sows ouseold consumption as a function of te stocastic factor under te two tecnologies. Te dased line represents te low tecnology. Te solid and dotted lines represent te ig tecnology in a ig and low collateral environment respectively. As collateral level decreases, te consumption floor rises and lenders bear more down-side risk. Assume te ouseold is risk-averse and as a conventional concave utility function, u(c). For purposes of later numerical analysis tat we will use to illustrate various propositions, we assume tat te utility function exibits Constant Relative Risk Aversion (CRRA). Houseolds coose between low-yielding and ig-yielding tecnology to maximize expected utility. Te population of te economy is distributed following te joint probability distribution function (ψ, W ), were ψ is te Arrow-Pratt measure of relative risk aversion. 4

6 Figure 2 demonstrates te effectiveness of self-insurance by adopting low yielding tecnology, using te numerical specification detailed in Appendix A. Te black solid line and te blue dased line depict te CDFs of ouseold consumption under ig tecnology financed by fully-collateralized loans and low tecnology respectively. We see tat low tecnology substantially reduces te probability of low outcomes. However, tis self-insurance strategy is far from perfect. First, it is actuarially unfair, reducing expected ouseold income (and consumption) by te difference in expected incomes between te two tecnologies (23% of expected income reduction in te numerical parameters used to generate te figure). Second, compared to an idealized contract tat sielded te ouseold against any consumption losses any time wen te ig yielding tecnology results in production less tan its expected value (illustrated in Figure 2by te pink das-dotted line) 3, self-insurance exposes te ouseold to residual or basis risk as tere is still a substantial probability of consumption well below te expected level. As can be seen, tis idealized contract stocastically dominates self-insurance. Wile te index insurance contracts discussed in te next section are clearly not going to dominate tis type of idealized contract eiter, te relevant question is weter tey can dominate self-insurance given te basis risk and implicit loadings associated wit it Index Insurance Contracts. Unlike conventional agricultural insurance tat pays off based on individual outcomes (y t in our notation), an index insurance contract pays off based on direct observation of te covariant sock (θ c ) or on average yields (θ c g t ) 4. To keep matters simple, we will assume tat te index insurance contract is based directly on θ c. We denote te insurance contract for tecnology t as I t < ˆθ c, t,z t,β t >, were ˆθ c is te strike point for te contract, t is indemnity normalized by g t, z t is te normalized actuarially fair premium and β t is te normalized loading or markup of te insurance as a percentage of z t. To simplify te matematical analysis te following teoretical structure assumes te insurance contract is actuarially fair, but te simulation results are based on a 30% of loading costs. Te indemnity is defined by t (θ c ) = 1( ˆθ c >θ c )(ˆθ c θ c ) 5. 3 Te contract illustrated in Figure 2 emulates a multi-peril contract tat restores farm income to its average level any time an idiosyncratic or covariant sock occurs. Subtracted from consumption is te actuarially fair premium for suc coverage. Suc contracts typically do not exist for te small farm sector because of moral azard, adverse selection and ig transaction costs. 4 As discussed by many autors, index insurance avoids te moral azard, transactions costs and adverse selection problems tat render conventional agricultural insurance unsustainable. 5 Tis implicitly assumes tat te farmers can coose te optimal insurance coverage level to reduce te basis risk, so tat insurance contract always reduces farmers risks 5

7 Under te actuarially fair insurance contract, gross returns to te farm ouseold are given by: (2.4) y I = (θ c + θ s )g t +(ˆθ c θ c )g t z t g t =(ˆθ c + θ s z t )g t, if ˆθ c >θ c. t (θ c + θ s )g t z t g t, =(θ c + θ s z t )g t oterwise were z t = E[1( ˆθ c >θ c )(ˆθ c θ c )]. Note tat tis specifications assumes tat te ouseold first receives indemnity and pays insurance premium, and ten applies te net income to repaying outstanding debt before bolstering its consumption. As a prelude to later analysis, define θ I = θ + s(θ), were s(θ) = 1( ˆθ c >θ c )( ˆθ c θ c ) z t. Ten yt I can be written as yt I = θ I g t. Tis indicates tat index insurance contract essentially transforms te multiplicative sock from θ to θ I, were θ I represents te net output sock. Since z t is te actuarially fair premium, E[θ I ]=E[θ] = 1. Denote te PDF and CDF of θ I as f I (θ) and F I (θ). We proof tat θ I is a mean preserving squeeze of θ and te following two properties old (see matematical proof in Appendix B): (2.5) θ 0 [F (θ) F I (θ)]dθ =0 (2.6) y 0 [F (θ) F I (θ)]dθ 0 0 y< θ. Figure 5illustrates te PDF of θ and θ I Credit Supply. We assume tat te credit markets are competitive and tat banks are willing (at te margin) to offer an agricultural loan tat provides an expected return equal to an opportunity cost of funds, π a. In te first part of tis section, we assume π a is exogenously given and examine te nature of marginal loan offers and te impact of insurance on loan contract terms. In te second part, we endogenize π a and explore te impact of index insurance on aggregate supply of agricultural credit. Te Iso-expected profit contract locus. Under a standard agricultural loan contract l(k, r, χ), lender s profits, π, are given by: (1 + r)k, if θ > θ (2.7) π = χ + θg (K), oterwise 6.

8 , and lenders expected profits are given by: θ (2.8) E(π) = [1 F ( θ)]rk + (χ + θg (K) K)f(θ)dθ 0. Since lender s profits π are concave in te random variable θ, expected profits E(π) decrease wen te variance of θ increases olding oters constant. Assume π a is exogenously given. Using te implicit function teorem, we can caracterize te iso-expected profits contract locus as tose combinations of interest rates and collateral requirements tat just yield expected returns equal to π a (E(π) = π a ). Figure 5 depicts te iso-expected profit lines wit and witout index insurance. Wen borrowers are not insured, te locus as a negative slope r χ = F ( θ) (1 F ( θ))k < 0. Tis is because as collateral level declines, lenders ave to raise interest rates to compensate te loss from iger default rates. Wen borrowers are insured, lenders face net output sock θ I and te locus becomes muc less steeper wit te slope satisfying F ( θ) (1 F ( θ))k < F I ( θ) (1 F I ( θ))k < 0. Tis is because even toug low collateral level exposes lenders to iger risks, index insurance compensates te loss so tat lenders can maintain te level of interest rates 6. In general, we would not expect a lender to be genuinely indifferent between te different points on te iso-expected profit loci. As explored by a number of papers, iger collateral and lower interest rate contracts diminis incentives for morally azardous beavior and adverse selection by lenders (for a recent treatment, see Boucer et al. (2008)). In te analysis ere, we ignore borrower eterogeneity tat migt generate adverse selection (e.g., differences in borrower onesty or individual level eterogeneity in te structure of risk). We also ignore potential sources of morally azardous beavior (e.g., credit diversion as in Carter (1988) or non-contractible effort as in Boucer et al. (2008)). Instead, we follow Stiglitz and Weiss (1981)and simply assume tat lenders demand tat borrowers present a minimum amount of collateral in order to leverage of a loan of size K. Wile tese assumption are somewat artificial, tey allows us to focus on te impact of index insurance in two distinctive, but empirically important environments. Te first of tese migt be considered to be representative of areas of Latin America were agricultural land is individually titled and potentially can be seized in te event of loan default. In tese environments, we will assume tat lenders require a collateral of value χ, as sown in Figure 5. In oter areas were agricultural land ownersip is less individualized and less securely titled (e.g., in Cina and many 6 Te slope of te iso-profit locus under insurance is flatter tan under no insurance but it is still downward sloping, since uninsured idiosyncratic risks can cause default 7

9 parts of Africa), we will assume tat te te loan package requires a lower amount of collateral, denoted χ l in Figure 5. Wile we could impose additional structure on te model to endogenize collateral levels, our goal ere is to sow tat index insurance and its interaction wit small farm productivity and financial markets will exibit subtle differences across tese two types of stylized agricultural economies. Aggregate credit supply to te agricultural sector. Te analysis in te prior section considered te conditions of competitive loan supply taking te lender s overall loan portfolio as given. Wen loan repayment is subject to purely idiosyncratic socks, te lender s overall portfolio will be selfinsuring. However, a portfolio of agricultural loans will not be purely self-insuring as a negative covariant sock (e.g., a drougt) could trigger a large scale episode of default. To explore tis issue furter, tis section examines te lender s portfolio decisions on aggregate credit supply. We assume tat in te sort-run, te lender as sufficient loanable funds to extend n loans of size K. Te lender can extend type a agricultural loans, or type b loans, wic we assume to be risk free (or subject only to idiosyncratic socks and terefore self-insuring). Te lender s gross rate of return from te portfolio of n loans, G, can be written as: (2.9) G = na i=1 π i( π a )+n b π. n. If G falls below a critical tresold level π, te lender faces a penalty function, P (G), wic reduces te lender s net portfolio returns. Te net portfolio returns, N, is written as: G,if G> π (2.10) N = G P (G) = G P (G), oterwise, were P, P 0 Te penalty for low gross portfolio return occurs for several reasons. First, wen te lender realizes too low a gross return on te loan portfolio, it runs afoul of reserve and oter regulatory requirements. Second, wen te gross portfolio return is too low, te lender as to sell a large amount of collateral to repay depositors, wic drives down te price of collateral and lenders net return. Next, low gross return from te portfolio forces te lender to borrow from te money market and pay for ig interest rates. Lastly, from a political economy perspective, te lender understands tat a massive 8

10 default, driven by a drougt or oter unfavorable event, will likely trigger a political economy reaction wit te government tempted to mandate at least partial default forgiveness. 7 Now assuming te expected net portfolio return π is exogenous in a competitive market, lenders ave to satisfy te participation constraint in wic E(N) π. Given tis constraint, lenders ave to adjust π a as te composition of te portfolio canges. Let F G and f G denote te CDF and PDF of G. Taking te expected value of N, te Lender s Participation Constraint (LPC) can be written as: (2.11) π + n π a n ( π a π) 0 P (G)f G (G)dG π (LPC), were te integral term is te expected penalty. Using implicit function teorem, te expected return of agricultural loan can be written as a function of te quantity of agricultural loans, π a = π a (n a ). Te function π a (n a ) represents an aggregate supply curve of agricultural loans, wic is sown in Figure 5. It sows tat in absence of formal insurance and low collateral environment, π a as to increase dramatically above π to maintain te participation constraint as te sare of agricultural loans rises in te lending portfolio (te black solid line). Tis is because low collateral requirement exposes lenders to large covariant risk and increases te probability of paying penalty. Te penalty policy P implies an increasing marginal cost of unit agricultural lending wen lenders are exposed to covariant risks. Te introduction of formal insurance reduces te cost of credit and tus flattens te supply curve (te black dased line). Tus, wen loans are insured, π a keeps constant at a low level of π as te number of agricultural loans increases. As te collateral level increases, uninsured supply curves (te red and blue solid lines ) sift down and insured supply curves (te red and blue dased lines) keep equal to π. In Figure 5 all insured supply curves and te uninsured supply curve in a low collateral environment overlap togeter on te flat straigt line of π a = π. Insurance isolates te rate of return of agricultural loans from te impact of te collateral level. (Appendix C provides a matematical proof of te sape of te function π a = π a (n a ) and te determinant factors.) Given a fixed level of χ, te aggregate supply function can also be written in terms of interest rates as r = r(n a χ, f I (θ),f(θ),p), wic is sown in Figure 6. Similar wit Figure 5, wen index insurance is not available, te lower te collateral level, te steeper te rise of interest rates. Wen 7 Following te 1998 El Nino event, te Peruvian government instituted a financial rescue tat instructed agricultural lenders to forgive outstanding debt (see Trivelli ). 9

11 borrowers are insured, all te supply curves become straigt flat lines and lenders would supply as many loans as tey could at fixed interest rates. Te level of interest rates under insurance is sligtly different wen collateral level canges. Te interest rates under low collateral (te black dased line) is iger tan tat under medium and ig collateral (te red and blue dased line) because low collateral exposes lenders to more idiosyncratic risks. 3. Demand for credit, insurance and tecnology Under Alternative Insurance Scemes Tis section analyzes farmers optimal coices of tecnology and financial contracts under alternative insurance scemes. In Section3.1 we simulate te CDFs of ouseold consumption under te four projects: te fallback project of low-yielding tecnology (self-insurance), ig-yielding tecnology associated wit stand-alone loan contracts (implicit insurance), ig-yielding tecnology associated wit non-interlinked and interlinked index insurance. Ten we analyze farmers coices between te low-yielding and ig-yielding tecnology in eac of te tree insurance scemes by comparing te expected utility function. In Section 3.2, we sow tat wen ig-yielding tecnology is associated wit stand-alone loan contract, farmers are likely to be risk-rationed, eiter because lenders carge ig interest rates wen collateral level is low or because farmers ave to bear substantial default risks wen collateral level is ig. In bot cases, farmers are likely to coose low tecnology rater tan ig tecnology. In Section 3.3 te ig-yielding tecnology is associated wit non-interlinked index insurance and credit contracts, were index insurance is introduced as an independent market. We sow tat in low collateral environments, te impact of non-interlinked index insurance on uptakes of ig tecnology is adverse or minimal due to te existing implicit insurance provided by loan contracts. In contrast, in ig collateral environments, non-interlinked insurance substantially improves ouseold welfare by crowding in demand for credit and ig tecnology. In Section 3.4, we examine te impact of contractual interlinkage in wic loans and insurance are interlinked as a single contract (i.e., because loans are linked, lenders know wen a loan is or is not secured by an insurance contract) and interest rates are endogenously determined by borrowers purcase of insurance. We sow tat even in low collateral environment, interlinked insurance increases uptakes of credit and ig tecnology by inducing lender to lower interest rates and increase credit supply. It sould be stressed tat te analysis is predicated on te simultaneous existence of an improved, capital-dependent tecnology. 10

12 3.1. Stocastic dominance of interlinked index insurance. Figure 5 draws te CDFs of consumption under te four projects tat are discussed below wen collateral level is low. Te implicit insurance associated wit ig tecnology (te black solid line) is ard to compete wit te selfinsurance associated wit low tecnology (te blue dased line), since implicit insurance as iger probability of low consumption tan self-insurance. Te introduction of te non-interlinked index insurance (te green dased line) makes a small improvement over te implicit insurance. But igly risk-averse farmers would still prefer self-insurance to non-interlinked insurance, because limited liability reduces te value of non-interlinked index insurance. However, wen index insurance is interlinked wit loan contracts, te CDF sifts forward from te green dased line to te red dased line, so tat te interlinked insurance is very likely to stocastically dominate self-insurance Absent formal insurance. Tis section compares farmers expected utility between low tecnology and ig tecnology wen formal insurance is not available. Under te specification in Section 2, ouseolds expected utility under low tecnology, V l, and ig tecnology, V, are given by: (3.1) V l = u(θg l + W + B)f(θ)dθ θ 0 θ (3.2) V = F ( θ)u(c)+ θ u(θg (1 + r)k + W + B)f(θ)dθ.. Te ouseold will coose ig tecnology and demand for credit if l = V V l > 0. Using te expressions above, we rewrite l as: (3.3) l = F ( θ)u(w + B χ) + ˇθ θ 0 u(θg l + W + B)f(θ)dθ [u(θg (1 + r)k + W + B) u(θg l + W + B)]f(θ)dθ θ + θ [u(θg (1 + r)k + W + B) u(θg l + W + B)]f(θ)dθ ˇθ 11

13 were ˇθ satisfies ˇθg (1 + r)k = ˇθg l, wic is te tresold of θ wen te net output of te ig-yielding tecnology is equal to te output of te low-yielding tecnology. Since ˇθ = (1+r)K ˇθ > g g l, (1+r)K χ θ(= g ). Te first and te second term in square brackets of equation 3.3 are bot strictly negative (even wen χ =0), representing te risks farmers ave to bear wen bad socks occur. Te first term represents te risks implicitly insured by loan contracts, wile te second term represents te risks tat are not covered by loan contracts. Te tird term is positive, representing te gain from te ig-yielding tecnology wen good socks appen 8. In te ig collateral environment, te lower bound of income c = W + B χ is low and θ is close to zero. Tis means te coverage of te implicit insurance is small and te sum of te first two terms, te total risks farmers ave to bear, is negatively big. Risk-averse farmers are likely to ave a negative l and coose low tecnology. Tis is called risk rationing by Boucer et al. (2008). Te risk-rationed are tose ouseolds wo escewed te risk of borrowing and instead self-insured teir liveliood by coosing te low income activity. In te low collateral environment, c rises and θ expands te implicit insurance. Te risks farmers ave to bear are reduced. However, te implicit insurance is far from being perfect. First, te sum of te first two terms is still negative even wen χ =0, meaning farmers still ave to bear risks wen bad sock appens. Second and more importantly, since production risks, especially covariant risks, are passed to lenders under low collateral level, lenders ask for iger expected rate of return and raise interest rates as discussed in Section 2.3. As iger interest rates make te tird term positively small, risk-averse farmers are still likely to be risk rationed. Terefore, in absence of formal insurance, te demand for ig tecnology and credit are likely to be low in bot low and ig collateral environments. Figure 8 depicts certainty equivalent (CE) of te four projects for a representative farmer wose CRRA coefficient is equal to 2. Under te specification in Appendix A, te CE of ig tecnology witout formal insurance (te black solid line) is lower or only sligtly iger tan te CE of low tecnology (te blue dased line). Tis is because ig tecnology is too risky wen collateral level is ig and te implicit insurance is too costly tat eats up te expected profit of ig tecnology wen collateral level is low. Te rest of te paper focuses on suc scenario were l < 0 due to risk rationing. Te next two subsections will analyze ow index insurance reduces risk rationing in different collateral environments. 8 Te ig-yielding tecnology is defined suc tat g (1 + r)k > g l at any interest rates offered by lenders. Terefore, price-rationing is excluded. 12

14 3.3. Non-interlinked index insurance contracts. Tis section compares farmers expected utility between low-yielding tecnology and ig-yielding tecnology tat is associated wit noninterlinked index insurance and credit contract. Te non-interlinked index insurance is merely bundled wit loan contracts, were loan contracts terms are independent of index insurance. Before deriving expected utility, we first examine ow actuarially fair non-interlinked contracts cange farmers expected consumption level. Te difference of expected consumption from taking igyielding tecnology between wit non-interlinked index insurance and witout formal insurance is equal to θ (3.4) E(c I ) E(c )=χ[f ( θ) F I ( θ)] [θg (K) (1 + r)k][f(θ) f I (θ)]dθ θ Integrating by parts and using te properties of mean-preserving spread (equation 2.5 and 2.6 ), te above expression reduces to: (3.5) E(c I ) E(c )=g θ 0 [F I (θ) F (θ)]dθ 0 wic is negative wen loans are not fully collateralized and equal to zero wen loans are fully collateralized. It indicates tat te lower te collateral level, te lower te farmers expected consumption under non-interlinked contracts. Farmers optimal coices over ig and low yielding tecnology are based on V I and V l, wic represent te value function of ig-yielding tecnology under non-interlinked insurance and lowyielding tecnology respectively. Weter farmers will adopt ig tecnology depends on te sign of V I V l, wic can be written as (3.6) I l = V I V l =(V I V )+(V V l ) = I + l were l < 0, and te sign of I is ambiguous. Since loan contract terms are isolated from index insurance under non-interlinked contracts, interest rates can be written as r = r(χ, n a,f(θ),p). Ten we ave 13

15 θ (3.7) V I = U(c)F I ( θ)+ and θ (3.8) I = V I V = U(c)[F I ( θ) F ( θ)] + θ U[θg (1 + r)k + W + B]f I (θ)dθ θ U[θg (1 + r)k + W + B](f I (θ) f(θ))dθ After integrating by part of I twice, we ave (3.9) I = U (c)g θ 0 θ [F I (θ) F (θ)]dθ + [ θ θ 0 (F I (y) F (y))dy]u g 2 dθ Since θ is a mean-preserving spread of θ I, te first term of equation 3.9 is non-positive and te second term is non-negative. As can be seen from equation (3.5), te first term of I represents te cange of expected utility due to te cange in expected consumption. Te second term represents te cange of expected utility due to te cange in consumption fluctuation. Tis indicates tat risk neutral farmers for wo te first term is non-positive and te second term is zero, would always prefer te implicit insurance rater tan te non-interlinked insurance. Tis is consistent wit te conclusion drawn from equation 3.5. As for te risk-averse farmers, te sign and magnitude of I can be determined by collateral requirement. If fully collateralized wit χ = (1+r)K and θ =0, te first term of equation 3.9 srinks to zero and I is positive, indicating tat farmers will be willing to buy non-interlinked insurance, since non-interlinked contracts maintain expected income level and reduce risks. Non-interlinked index insurance crowd in risk-rationed and raises uptakes of financial products and ig-yielding tecnology. If χ =0and θ > 0, I decreases and is more likely to be negative, indicating farmers will not be willing to buy non-interlinked insurance, since tey are already insured by loan contract and insurance premium lowers te expected income. Intuitively, under a low collateral environment, te lender bears most of te risk. Insurance is valuable to lenders by transferring te risk from te lender to te insurance provider, but yields no benefit to te ouseold wo noneteless pays for insurance premium. In contrast, under a ig collateral environment, te ouseold wo bears nearly all te risk, enjoys te gains from te insurance. Te dased line in Figure 8 illustrates te 14

16 certainty equivalent of V I as a percentage of tat of low tecnology. We see tat te CE under non-interlinked insurance is even lower tan tat under implicit insurance ( I is negative) wen collateral is low, wic is consistent wit te empirical evidence from Giné and Yang (2009). As collateral rises, te CE of te non-interlinked becomes iger tan implicit insurance, and finally iger tan self-insurance of low tecnology. Tis indicates tat non-interlinked insurance can only solve risk rationing in a ig collateral environment Interlinked insurance contracts. Tis section compares farmers expected utility between low tecnology and ig tecnology associated wit interlinked index insurance and credit contracts. Farmers coice of tecnology is based on te value functions of te two projects, denoted as V II V l respectively. Farmers make decisions based on te sign of V II as and V l, wic can be disaggregated (3.10) II l = V II V l =(V II V I )+(V I V )+(V V l ) = II I + I + l were l < 0 and I factors influencing te sign of II I. increases in χ as sown in te above section. Te rest of tis section explores As sown in Section 2.3, ouseolds purcase of index insurance influences lenders expected return. Under interlinked contracts, interest rates te lender offers are endogenously determined by index insurance contract, wic can be written as r I = r m (χ, n a,f I (θ)). Because θ is a function of interest rates, te critical point of θ wen default occurs is denoted as θ I = θ(r I ) under interlinked insurance. Ten V II becomes θ (3.11) V II = U(c)F I ( θ I )+ θ I U[θg (1 + r I )K + W + B]f I (θ)dθ Te difference of expected utility between interlinked and non-interlinked insurance, II I written as 15 can be

17 (3.12) II I = V II θ V I = (U[θg (1 + r I )K + W + B] U[θg (1 + r)k + W + B])f I (θ)dθ θ θ + (U[θg (1 + r I )K + W + B] U(c)) f I (θ)dθ θ I As Figure 6 sows, wen χ< (1 + r)k, r I <r, and θ I < θ. Tus II is positive wen loans I are not fully collateralized, and is equal to zero wen loans are fully collateralized. II I is always non-negative and decreasing in χ. Tis means te interlinked insurance is always at least as good as non-interlinked insurance for farmers. Interlinkage will tus are able to crowd-in more credit demand by lowering interest rates for farmers wo purcase index insurance. Since I is increasing in χ, interlinked insurance as advantages over non-interlinked insurance especially in a low collateral environment. Te dotted line in Figure 8 denotes te CE of V II of V I decreases., wic always lies above te CE. Te gap between te dased and te dotted line (representing II) increases as collateral I Combining II, I I, l togeter, te dotted solid line in Figure 8 demonstrates II l. Te Certainty Equivalent (CE) of te interlinked ig tecnology for a typical ouseold is almost constant around 1.5% more tan tat of low tecnology regardless of te cange in collateral level. Tis can be explained using te mean and variance of te income from interlinked ig tecnology. Since interlinked insurance always brings te cost of unit credit back to a constant level π as sown in Figure 5, farmers expected income from interlinked contracts is equal to (3.13) E(c II )=E(y ) (1 + π)k + W + B wic is independent of collateral level. Te income variance satisfies (3.14) V ar(c II )= V ar(y )V ar(θ I ) V ar(θ) 16

18 wic is also independent of collateral level. Since te expected utility is mainly determined by te first and second order of income, te above two equations indicate tat te CE under interlinked insurance mainly depends on te productivity of te tecnology and risk structure (basis risk), but is not influenced by te caracteristics of te te credit market suc as collateral level and numbers of agricultural loans, wic ave significant impacts on te performance of non-interlinked contracts. 4. Farm Productivity and te financial Market development in te Equilibrium of te te credit market Tis section analyzes farm productivity and te credit market development wen te credit market reaces an equilibrium 9. In absence of interlinked insurance or fully collateralization, te aggregate supply curve of credit sown in Figure 6 is uprising and tus any increase in demand discussed in Section 3 will raise interest rates tat coke off te increased expansion. Wen insurance and credit are interlinked, te credit supply curve is flattened at a constant level and tus te increased demand induced by insurance will not be coked off by interest rates. In oter words, wile non-interlinked contracts only sift credit demand curve, interlinked contracts sift bot curves of credit demand and credit supply and tus induce a ig uptake rate of ig tecnology and financial products. Te second part of tis section analyzes te eterogeneous impact of index insurance and sows tat interlinked contracts can crowd in igly risk-averse and poor smallolders, wo are excluded from te credit market wen interlinked contracts are not available te credit market Equilibrium. According to Section 2.3, we can write te aggregate supply of agricultural loans, n s a, as a function of te price r conditional on collateral level, te distribution function of θ, purcase of insurance and penalty function: (4.1) n s a = n s a(r χ, f(θ), f I (θ), P) According to Section 3, aggregate effective demand of agricultural loans, n d a, is a function of r conditional on collateral, te distribution of θ, purcase of insurance and te distribution of population on risk preference and wealt: 9 Since te model does not consider imperfect information problems of moral azard and adverse selection, interest rates do not affect te riskiness of lenders return, and tus tere exists an equilibrium interest rates tat equates credit demand and supply. 17

19 (4.2) n d a = n d a(r χ, f(θ), f I (θ), (ψ, W )). Because te population is eterogeneous in ψ and W, te demand function is downward sloping in r. Te te credit market reaces an equilibrium wen demand equals supply, (4.3) n s a = n d a = n a Te quantity of agricultural loans and interest rates at te equilibrium, n a and r, vary on te different insurance scemes associated wit ig tecnology: no formal insurance, non-interlinked and interlinked index insurance. Figure 9 sows te supply and demand curve of credit under te tree insurance scemes in different collateral environments. Point A, B, C represent te equilibrium allocations under no formal insurance, non-interlinked and interlinked index insurance respectively. Te orizontal axis, n a%, represents te percentage of farmers wo obtain an agricultural loan and adopt ig-yielding tecnology. Te rest of population, 1 n a%, use traditional low-yielding tecnology. In a low collateral environment (te first grap in Figure 9 ), demand and supply curve witout formal insurance interact at a point wit ig price and relatively low quantity. Wen loans are insured wit non-interlinked contract, supply curve keeps uncanged and demand curve sifts to te left due to te implicit insurance provided by low collateral. Te lower demand curve drives down bot r and n a, as sown by te black arrow from point A to B. Tis coincides wit te empirical observation tat bundled loan contracts reduce uptake rates of loans by Giné and Yang (2009). Wen te two financial contracts are interlinked, te demand curve sifts in te same way as te one under te non-interlinked, and te supply curve sifts down and becomes flat. Te sifting credit supply curve generates an equilibrium wit lower r and iger n a, as sown by te red arrow from point B to C. In a medium collateral environment, non-interlinked contracts increase credit demand and improves te equilibrium towards a iger n a but also drives up te price. Interlinked contracts increase n a furter by lowering supply curve. Finally in te ig collateral environment, non-interlinked contracts induce a big expansion of te demand so tat almost te wole population obtain credit. Since te lender is fully insured by te ig collateral, te interlinked contract performs as well as te non-interlinked. 18

20 4.2. Te eterogeneous impact of index insurance. Te impact of insurance on individuals varies as teir risk preference and wealt cange. Te empirical evidence from Giné and Yang (2009) sows tat wealt indicators ave positive (altoug not significant) impact and risk aversion as negative impact on uptakes of non-interlinked contracts. Figure 10 sows simulated critical levels of risk aversion coefficient and wealt (ψ,w ) wen te credit market reaces an equilibrium, below wic ouseolds adopt te ig tecnology and above wic tey do not. In addition to risk rationing, tis figure also considers te quantity-rationed wo cannot borrow wen teir wealt level is lower tan te collateral requirement. Te solid lines represent no formal insurance, te dased lines non-interlinked contracts, and te das-dotted lines interlinked contracts. Te red lines denote low collateral environment and te green lines denote ig collateral environment. In a low collateral environment, igly risk-averse and poor farmers in te nortwest corner of Figure 10 are risk-rationed out of te te credit market wen formal insurance is not available. Non-interlinked index insurance worsens te risk-rationing and expand te rationing area to te souteast. Tis is because te implicit insurance provided by loan contracts renders formal insurance effectively an increases in te interest rate on te loan (Giné and Yang, 2009). However, te introduction of interlinked index insurance reduces risk rationing, moving te boundary towards te nortwest so muc tat igly risk-averse and poor farmers are willing to borrow from te te credit market and adopt ig-yielding tecnology. In a ig collateral environment, two types of credit rationing occur wen formal insurance is not available. First, poor farmers on te left side of te green vertical line are quantity-rationed out because of lack of wealt to put as collateral. Second, among tose farmers wo are eligible to apply for a loan (on te rigt side of te green vertical line), igly risk-averse farmers are risk-rationed because tey fear te loss of collateral. Te introduction of non-interlinked insurance contracts reduces risk rationing and crowd in all te eligible farmers into te te credit market. Because lenders are protected by ig collateral, interlinked contracts benefit farmers and performs as well as non-interlinked contracts. 5. Conclusion Covariant risks associated wit agricultural activities can amper development of te rural credit market and tus prevent poor smallolder farmers from escaping poverty. On te oter and, despite its advantage of addressing imperfect information problems (moral azard and adverse 19

21 selection), novel index insurance experiences low uptake rates in te field. Our model suggests tat te two financial markets, credit and insurance, ave to interlink wit eac oter and associate wit income-enancing tecnologies, in order to acieve market deepening and productivity growt. Wile te uptake of novel index insurance contracts as at times been disappointing, te simple explanation tat uptake is slow because index insurance contracts are not actuarially fair and ave basis risk overlooks te fact tat small farmers in low income economies typically self-insure using mecanisms at tat are costly (actuarially unfair) and expose te farmer to significant basis risk. Tese inefficient forms of self-insurance tus leave ample space in wic tey can be stocastically dominated by formal index insurance contracts. Te analysis ere sows tat tis kind of stocastic domination is most likely to occur wen index insurance is combined wit te introduction of improved tecnologies and credit contract. Te analysis as compared tree insurance scemes associated wit te ig-yielding tecnology: no formal insurance (implicit insurance provided by loan contracts), bundled or non-interlinked index insurance were loan contract terms are independent of index insurance, and interlinked index insurance were loan contract terms are endogenously determined by index insurance. Te model and te simulation results sow tat te impact of index insurance differs wen collateral environment canges. In a low collateral environment, interlinked contracts crowd in poor and igly risk-averse farmers, wo are excluded from te financial market wen absent of formal insurance or under non-interlinked contracts. Non-interlinked contracts ave small or even negative impacts on uptakes of financial products and ig-yielding tecnology. In a ig collateral environment, interlinked and non-interlinked contracts perform equally well in reducing risk-rationing. Wile subject to empirical confirmation, tese teoretically grounded observations ave significant implications for te design of efforts to promote bot small farm productivity growt and rural financial market deepening. References Bank, World, World Development Report 2008: Agriculture for Development, New York: Oxford University Press, Barnett, B. J., C. B. Barrett, and J. R. Skees, Poverty Traps and Index-Based Risk Transfer Products, World Development, 2008, 36 (10), Barrett, Cristoper B., Displaced Distortions: Financial Market Failures and Seemingly Inefficient Resource Allocation, ttp:// , Micael R. Carter, and Munenobu Ikegami, Poverty Traps and Social Protection, Binswanger, Hans P. and Mark R. Rosenzweig, Beavioural and material determinants of 20

22 production relations in agriculture, Journal of Development Studies, 1986, 22 (3), Boucer, Stepen and Caterine Guirkinger, Risk, Wealt, and Sectoral Coice in Rural Credit Markets, Social Science Researc Network Working Paper Series, October 2007., Micael R. Carter, and Caterine Guirkinger, Risk Rationing and Wealt Effects in Credit Markets: Teory and Implications for Agricultural Development, Social Science Researc Network Working Paper Series, April Braverman, Avisay and Josep E. Stiglitz, Sarecropping and te Interlinking of Agrarian Markets, Te American Economic Review, 1982, 72 (4). Carter, M., Equilibrium credit rationing of small farm agriculture, Journal of Development Economics, February 1988, 28 (1), Carter, Micael, Cristoper B. Barrett, and Andrew Mude, A Productive Safety Net for Nortern Kenya s Arid and Semi-Arid Lands: Te HSNP+ Program, Carter, Micael R., Environment, Tecnology and te Social Articulation of Risk in West African Agriculture, Economic Development and Cultural Cange, 1997, 45 (2), Giné, Xavier and Dean Yang, Insurance, credit, and tecnology adoption: Field experimental evidencefrom Malawi, Journal of Development Economics, May 2009, 89 (1), Hazell, Peter B. R., Te appropriate role of agricultural insurance in developing countries, Journal of International Development, 1992, 4 (6), Lybbert, Travis J., Cristoper B. Barrett, Solomon Desta, and D. Layne Coppock, Stocastic wealt dynamics and risk management among a poor population, Te Economic Journal, 2004, 114 (498), Miranda, Mario J. and Claudio Gonzalez-Vega, Systemic Risk, Index Insurance, and Optimal Management of Agricultural Loan Portfolios in Developing Countries, American Journal of Agricultural Economics, December 2010, 92 (6). Rotscild, M. and J. Stiglitz, Equilibrium in Competitive Insurance Markets: An essay on te Economics of Imperfect Information, Te Quarterly Journal of Economics, November 1976, 90 (4), Smit, Vince and Myles Watts, Index Based Agricultural Insurance in Developing Countries: Feasibility, Scalability and Sustainability, Stiglitz, Josep E. and Andrew Weiss, Credit Rationing in Markets wit Imperfect Information, Te American Economic Review, 1981, 71 (3), Bank (2007); Barnett et al. (2008); Barrett (2006); Barrett et al. (2008); Binswanger and Rosenzweig (1986); Boucer and Guirkinger (2007); Boucer et al. (2008); Braverman and Stiglitz (1982); Carter (1997); Carter et al. (2008); Giné and Yang (2009); Hazell (1992); Lybbert et al. (2004); Rotscild and Stiglitz (1976); Smit and Watts (2009); Stiglitz and Weiss (1981) 21