# Index insurance, probabilistic climate forecasts, and production.

ABSTRACT

Index insurance and probabilistic seasonal forecasts are becoming available in developing countries to help farmers manage climate risks in production. Although these tools are intimately related, work has not been done to formalize the connections between them. We investigate the relationship between the tools through a model of input choice under uncertainty, forecasts, and insurance. While it is possible for forecasts to undermine insurance, we find that when contracts are appropriately designed, there are important synergies between forecasts, insurance, and effective input use. Used together, these tools overcome barriers preventing the use of imperfect information in production decision making.

INTRODUCTION

Climate-related risks have profound impacts on agricultural producers around the world. (1) These effects are particularly important in developing countries, where agriculture makes a significant contribution to gross domestic product (World Bank, 2001) and insurance markets are underdeveloped or nonexistent. Recently, microinsurance and probabilistic seasonal forecasts have become available to help farmers manage climate risks in production. Although these tools are intimately related, work has not been done to formalize the fundamental connections between them. (2)

It is well known that climate risk can keep low-income households in poverty traps (see a review by Barnett, Barrett, and Skees, 2007). Lack of assets and risk exposure may lead households to forego activities with high returns, perpetuating their poverty. In response to this challenge, innovative insurance pilots to help farmers in developing countries have been gaining increasing interest (Hellmuth et al., 2009). (3) Recent studies analyze the potential of insurance in helping to escape poverty traps (Kovacevic and Pflug, Forthcoming).

Work in the agricultural economics literature has examined the relationship between insurance and input usage for both farm-level insurance (e.g., Ramaswami, 1993; Babcock and Hennessy, 1996) and index insurance (Chambers and Quiggin, 2000; Mahul, 2001). Focusing on the relationship between insurance and input use, this literature does not address interactions between insurance, climate forecast, and input decisions. Trade-offs between basis risk and moral hazard afforded by index-based products have been analyzed from an insurance company standpoint (Doherty and Richter, 2002).

Seasonal climate is predictable in many regions of the world (Goddard et al., 2001), with the El Nino Southern Oscillation (ENSO) linked to variations of seasonal precipitation (Ropelewski and Halpert, 1987). Work in agricultural and climate science has modeled the impact of probabilistic climate forecasts on production decisions (e.g., Hansen, 2002) but has, with few exceptions (Mjelde, Thompson, and Nixon, 1996; Cabrera, Letson, and Podesta, 2005), ignored the impact of insurance. (4)

Although yields and production practices are impacted by ENSO phases, this relationship is not built into existing insurance products. Insurance implementers acknowledge that climate forecasts may undermine the financial soundness of a product by providing opportunities for intertemporal adverse selection. The advocated strategies are to finalize insurance transactions contracts months ahead of time, before the forecast is informative (see, e.g., Hess and Syroka, 2005; World Bank, 2005), or allow insurance premiums to reflect forecast information (Skees, Hazell, and Miranda, 1999). Implementation of these strategies is undermined by the lack of a clear conceptual understanding about the interaction between seasonal forecast, insurance, and production decisions. Potential opportunities to exploit synergies may be being missed. Since the reinsurance industry uses probabilistic seasonal climate forecasts in pricing (Hellmuth et al., 2009), (5) it is important to understand forecasts and pricing from the perspective of the insurance company that must develop a product that: (1) results in products that provide value to farmers, unlocking consumer demand; (2) reflects fluctuating reinsurance costs; and (3) cannot be undermined through strategic use of forecast information. As microinsurance projects advance, the use of ENSO in these products is being increasingly explored, increasing the urgency for formalizing the basic features connecting insurance, production decisions, and forecasts. Existing analyses of insurance and forecast focus on pricing while ignoring the production value of the interaction can lead to misguided decision making as these production benefits are critical in driving insurance demand.

Since the fundamental relationship between insurance, input use, and seasonal climate forecasts has not been addressed, we propose to fill this gap by explicitly modeling input use and index insurance demand given probabilistic seasonal climate forecasts. While seasonal climate forecasts, insurance contracts, and input use choices can each be used to mitigate uncertainty and risk, they each play a different role and have potential synergies or unanticipated impacts when used together.

We investigate the relationship just described through the simplest possible model of input choice under uncertainty, forecasts, and index insurance. Our goal is to formalize the fundamental features of the relationship between insurance, forecasts, and production in a highly stylized model, in order to represent the key features in the most transparent way. Although it is not central to our problem, we include features in our model that allow for a basic discussion of the impacts of basis risk because it is of so much interest in the literature on new index-based insurance products (Golden, Wang, and Yang, 2007). In this line, the inclusion of the idiosyncratic shock allows the modeling of basis risk. The way in which the shock is introduced, linearly in this case to follow the literature and simplify the analysis, will provide only a subset of the results possible. Work focusing more on complex basis risk modeling would provide deeper insights into these issues.

We find that if contracts are appropriately designed there are important synergies between forecasts and insurance and effective input use. Insurance allows the farmer to map a probabilistic forecast into a much more deterministic payout, allowing the farmer to commit to production choices that take advantage of forecast information that is too noisy to utilize without risk protection. We also find that the presence of skillful probabilistic forecasts may affect the demand for insurance as well as its effectiveness as a risk-reducing tool. In our stylized treatment, basis risk attenuates impacts but does not lead to findings that are fundamentally different.

We begin by presenting the base framework that will be used throughout the article. Probabilistic seasonal climate forecasts and index insurance with their well-known impacts on production decisions are then introduced individually in order to provide benchmarks for our findings. Next, we combine the instruments (forecasts and insurance) and analyze their joint interactions with production practices. The last section provides concluding remarks and proposes some avenues for future research.

PRELIMINARIES AND BASE MODEL

Consider first a competitive farmer with a single crop with yield (y) dependent on the level of a controllable input (N, may be thought of as nitrogen, an improved seed, or the level of technology used), a systemic weather shock (r, hereafter rainfall) affecting all farmers in the area, and an idiosyncratic production shock ([epsilon]) as follows:

y = f(N, r) + [epsilon]. (1)

A special case of this yield function was used by Mahul (2001). (6) For simplicity, let rainfall take only two values, [r.sub.g] and [r.sub.b] (denoting good and poor or bad growing condition, respectively). It is assumed that [f.sub.N](N,r) = [partial derivative]f(N,r)/[partial derivative]N > 0, and [f.sub.NN](N,r) = [[partial derivative].sup.2]f(N,r)/[partial derivative][N.sup.2] [less than or equal to] 0. We assume further that f(N, [r.sub.g]) > f(N, [r.sub.b]), and [f.sub.N](N, [r.sub.g]) > [f.sub.N](N, [r.sub.b]) for all N. (7)

The value of the random variables is learned after the input has been applied. In the sections to follow, the systemic shock r is the variable on which the forecast provides information and on which the index insurance is written. The farmer knows that the climatological (historical) probability of observing r = [r.sub.g] is [[omega].sub.g]. The expected value of the idiosyncratic shock, which by definition is independent of the systemic shock, is assumed to be zero, and its variance is given by [sigma]2[epsilon]. Both the price of the controllable input ([p.sub.N]) and the price of the output (p) are assumed to be nonrandom, and the latter price is normalized to I without loss of generality. (8) Conditional on the idiosyncratic shock, and defining [[pi].sup.0] (N, r) = f(N, r) - [p.sub.N]N, profits for the farmer are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The farmer is assumed risk averse with a Bernoulli utility function given by u([pi]), with u''([pi]) < 0 < u'([pi]). If the farmer's choice is on the level of the input to apply, the farmer's problem is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](3)

where as in Mahul the indirect utility function [??](*) is [??]([[pi].sup.0]) = Eu([[pi].sup.0] + [epsilon]) for all [[pi].sup.0]. Kihlstrom, Romer, and Williams (1981) show the indirect utility function is increasing and concave in [[pi].sup.0]. It is well known that producers will self-insure in this context by selecting a level of inputs that reduce the magnitude of a loss when one occurs. The level of inputs selected will be lower than that of a risk-neutral producer. For future

reference notice that, in the absence of insurance or seasonal forecasts, the expected level, and variability of profits are given by

E([pi]) = E(E([pi]|[epsilon])) = [[omega].sub.g][[pi].sup.0]([N.sup.*],[r.sub.g]) + (1 - [[omega].sub.g])[[pi].sup.0]([N.sup.*],[r.sub.b]) (4a)

and

Var([pi]) = E(Var([pi]|[epsilon])) + Var(E([pi]|[epsilon])) = [[omega].sub.g](1 - [[omega].sub.g])([[pi].sup.0]([N.sup.*],[r.sub.g]) - [[pi].sup.0][([N.sup.*],[r.sub.b])).sup.2] + [[sigma].sup.2.sub.[epsilon]]. (4b)

FORECASTS WITHOUT INSURANCE

Suppose now that a skillful probabilistic seasonal climate forecast is available before decisions over the input are made. The forecast indicates the future state of the world (high or low rainfall) for the coming season and an associated uncertainty given by the probability that the forecast is incorrect. If high levels of rainfall are forecasted, there is probability cog I g that rainfall is actually high. Thus, forecast error is climate risk updated by the information in the forecast. A forecast is skillful if 1 > [[omega].sub.g|g] > [[omega].sub.g], and 1 > [[omega].sub.b|b] > [[omega].sub.b]. (9) Assume that the forecast is unbiased; that is, the frequency with which a high rainfall forecast is issued ([m.sub.g]) equals that of high rainfall years ([[omega].sub.g]).

In this situation, the decision of the farmer will depend on the forecast received. If a good year is forecasted, the decision of the farmer is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The [N.sup.*g], the optimal amount of input use when a good year is forecasted, will depend on the skill of the forecast. Again, the farmer self-insures against the uncertainty in the forecast.

INSURANCE WITHOUT A FORECAST

Suppose now that instead of a forecast, insurance (I) is available to farmers, and they must decide how much of it to buy at a price r per unit. To include a simple representation of basis, we assume that the insurance is available for the systemic shock (r) but not for the idiosyncratic shock ([epsilon]). In this case, the objective function and first-order conditions (at an interior solution) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

If the insurance is actuarially fair (r = 1 - [[omega].sub.g]), one obtains the standard result that the risk-averse farmer insures fully against the systematic risk; that is, [I.sup.*] = [[pi].sup.0] ([N.sup.*], [r.sub.g]) [[pi].sup.0]([N.sup.*],[r.sub.b]) = f([N.sup.*],[r.sub.g])- f([N.sup.*],[r.sub.b]), (10) and a higher level of inputs (the risk-neutral level) is utilized. The farmer's expected level and variability of profits when actuarially fair insurance is available are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9a)

and

Var([[pi].sub.I]) = E(Var([[pi].sub.I]|[epsilon])) + Var(E([[pi].sub.I]|[epsilon])) = [[sigma].sup.2][epsilon]. (9b)

Although profit variability is reduced, Equation (9b) indicates that some basis risk remains for farmers even when they are fully insured against the systemic shock.

COMBINING THE FORECAST WITH INSURANCE The impacts of the interaction between forecasts and insurance depend critically on the timing of the forecast who has little flexibility, who must commit to exogenously determined production practices prior to the forecast information and insurance decision. We then model the case in which a farmer simultaneously makes insurance and production decisions after the forecast is available.

Effects of a Skillful Forecast on Insurance Purchases With Fixed N

Consider a farmer who is constrained to commit to production decisions before insurance becomes available but after a skillful forecast has been issued. If skillful seasonal forecasts are released before the closing date for the insurance purchase, the premium rates must be modified to reflect the climate information available for the insurance to be financially sustainable, and the problem is state contingent. When a good year is forecasted, the actuarially fair rate becomes lower (from [tau] = [[omega].sub.b] to [[tau].sub.1] = [[omega].sub.b|g]) and the farmer's problem and first-order condition (for an interior solution) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[??]'([[pi].sup.0](N,[r.sub.g]) - [[tau].sub.1]I)/[??]'([[pi].sup.0](N,[r.sub.b]) - (1 - [[tau].sub.1])I = [[omega].sub.b|g]/[[omega].sub.g|g] (1 - [[tau].sub.1])/[[tau].sub.1]. (11)

Since the insurance is actuarially fair, the farmer will insure fully against the systemic

risk setting [I.sup.*] = [[pi].sup.0] (N,[r.sub.g]) - [[pi].sup.0] (N,[r.sub.b]) = [[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]. Furthermore, if inputs cannot be changed, the insurance purchase depends neither on whether the forecast is for a good or bad year nor on its skill. Hence, if a bad year is forecasted, and the premium rates reflect it (defining [[tau].sub.2] = [[omega].sub.b|b] > [[omega].sub.b] > [[tau].sub.1]), an analogous problem can be solved and the farmer will insure fully against the systemic risk.

When a forecast for a good year is issued, profits equal [[pi].sup.*.sub.g]|[epsilon]. = [[pi].sup.0.sub.g] - [[tau].sub.1]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]) + [epsilon] across realizations of the insured variable and thus E([[pi].sup.*.sub.g]) = [[pi].sup.0.sub.g] - [[tau].sub.1]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b] and Var([[pi].sup.*.sub.g]) = [[sigma].sup.2.sub.[epsilon]]. If the forecast is for a poor year, profits equal [[pi].sup.*.sub.b]|[epsilon]. = [[pi].sup.0.sub.g] - [[tau].sub.2]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]) + [epsilon] across realization of r, expected profits are E([[pi].sup.*.sub.g]) = [[pi].sup.0.sub.g] - [[tau].sub.2]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]), and Var([[pi].sup.*.sub.b]) = [[sigma].sup.2.sub.[epsilon]]. Since the forecast is unbiased and the insurance is actuarially fair, we have E([[pi].sup.*]|[epsilon])= [[omega].sub.g][[pi].sup.0.sub.g] + (1 - [[omega].sub.g])[[pi].sup.0.sub.b] + [epsilon, and E([[pi].sup.*]) = [[omega].sub.g][[pi].sup.0.sub.g] + (1 - [[omega].sub.g])[[pi].sup.0.sub.b].

Notice that expected profits change across realizations of the forecast. The difference is given by E([[pi].sup.*.sub.g] - E([[pi].sup.*.sub.b] = ([[tau].sub.2]) - ([[tau].sub.1]) ([[tau].sup.0.sub.g] - [[tau].sup.0.sub.b]]). Since the insurance is actuarially fair, [[tau].sub.2] = [[omega].sub.b|b] and [[tau].sub.1] = [[omega].sub.b|g], indicating that as the skill of the forecast increases, so does the difference in expected profits across forecasts. The resulting profit variability is

Var([pi]*) + E(Var([[pi].sup.*]|[epsilon])) + (Var(E([pi].sup.*]|[epsilon])) = [[omega].sub.g](1 - [[omega].sub.g])[([[tau].sub.2] - [[tau].sub.1])[([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b])].sup.2] + [[sigma].sup.2.sub.[epsilon]]. (12)

Equation (12) indicates that the existence of a skillful forecast that is issued before purchases of the insurance are made increases the variability of profits when compared to the no-forecast situation. In the absence of the forecast (or when the forecast has no skill), we have [tau] = [[tau].sub.1] = [[tau].sub.2] and thus the farmer will only face the idiosyncratic risk (compare with Equation (9b)). As a skillful forecast is introduced, the difference [[tau].sub.2] - [[tau].sub.1] increases, undermining the effectiveness of the insurance to provide protection against the insurable risk. In the limit, with a perfect forecast, we have [[tau].sub.1] = [[omega].sub.b|g] = 0 and [[tau].sub.2] = [[omega].sub.b|b] = 1 yielding the same variance of profits as the uninsured case (Equation (4b)) for a fixed N.

In summary, since the forecast is assumed to be unbiased and available to both parties, the ex ante expected profit in this and the variability of that profit increases when the forecast is available. Hence, the presence of a forecast undermines the effectiveness of the insurance as a risk-mitigation mechanism in this situation and reduces welfare.

Choice of Both Insurance and Input Purchases in the Presence of a Skillful Forecast

To analyze the full interaction between the risk management tools and production decisions, we now allow farmers to choose both the level of the controllable input and the insurance purchase after observing the skillful forecast. Finally, the systemic and idiosyncratic shocks are observed. Since the forecast is released before farmers make their decisions, we have a state-contingent problem. The objective and first-order conditions when a good year is forecasted are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Since the insurance is actuarially fair, we know that [I.sup.*] = [[pi].sup.0]([N.sup.*g],[r.sub.g]) - [[pi].sup.0]([N.sup.*g],[r.sub.b]) = f([N.sup.*g], [r.sub.g]) - f([N.sup.*g], [r.sub.b]). Using this result, the first-order conditions evaluated at the optimum are written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Equation (16) indicates that in the presence of a state-dependent, actuarially fair insurance, the risk-neutral solution is replicated. Although the existence of the idiosyncratic risk imposes utility penalties, the presence of market inputs applied will maximize expected profits.

To investigate how the farmer's decisions are affected by the skill of the forecast, we need to sign [[partial derivative]I.sup.*g]/[[partial derivative][[omega].sub.g|g], and [[partial derivative]N.sup.*g]/ [partial derivative][[omega].sub.g|g]. Comparative statics on the system given by (16) and (17) indicate that

[[partial derivative]I.sup.*g]/[[partial derivative][[omega].sub.g|g] = ([f.sub.N]([N.sup.*g],[r.sub.g]) - [f.sub.N]([N.sup.*g],[r.sub.b]))[[partial derivative]N.sup.*g]/[[partial derivative][omega].sub.g|g]. (18)

Since we assumed that the marginal productivity of N is higher in good years, the partial derivatives in Equation (18) have the same sign. The effect of the skill of the forecast on the optimal nitrogen application is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where H is the determinant of the Hessian of the problem (positive by second-order sufficient conditions (SOSC) for a maximum). E ([??]"([[pi].sup.*]|g)) is negative by SOSC. The second term in the numerator is positive by technology assumptions, and thus [[partial derivative]N.sup.*g]/[[partial derivative][omega].sub.g|g] > 0 and [[partial derivative]I.sup.*g]/[[partial derivative][[omega].sub.g|g]> 0. Analogous analysis and previous results indicate that the farmer will purchase less insurance and use less inputs when the forecast is for a poor year.

The previous comparative statics exercise reveals that, counter to intuition, when the skillful forecast indicates a good (poor) year is likely, the farmer will purchase more (less) of an insurance of actuarially fair price. The expected change in overall insurance purchases brought about by a forecast of increasing skill depends on the relative adjustment induced by each kind of forecast (good versus poor growing conditions) and the natural frequency of each event, protecting the farmer from climate risk, it protects the farmer from forecast error.

The farmer is able to remove uncertainty from forecast error and improve utility by operating at the expected profit-maximizing input level instead of self-insuring with less aggressive changes in input. When a good year is forecast, the farmer can intensify to the expected profit-maximizing level, and when a bad year is forecast the farmer can prevent losses through the efficient level of input reduction while still maintaining inputs at a level that maximizes expected profits by taking into account the chance that a good year may still occur. Ex ante expected profits when both insurance and forecast are allowed to interact with the farmer's input decisions are given by

E([[pi].sup.*]) = [[omega].sub.g]([[pi].sup.0]([N.sup.*g],[r.sub.g]) - [[tau].sub.1][I.sup.*g]) + (1 - [[omega].sub.g])([pi]0([N.sup.*b],[r.sub.g]) - [[tau].sub.2][I.sup.*b]) (20a)

Var([[pi].sup.*]) = [[omega].sub.g](1 - [[omega].sub.g])[(E([[pi].sup.*g]) - E([[pi].sup.*b])).sup.2] + [[sigma].sup.2.sub.[epsilon]] (20b)

where we used the assumptions that the insurance is actuarially fair and that the forecast is unbiased ([m.sub.g] = [[omega].sub.g]). E ([[pi].sup.*.sub.i]) denotes expected profits for an i = g, b forecast. The actuarially fair insurance will lead farmers to maximize expected profits, and the skill of the forecast allows farmers to make better-informed decisions. Thus, expected profits increase when both the insurance and a skillful forecast are available. However, the introduction of a forecast comes at the cost of increasing profit variability. If the forecast has no skill, the farmer will not adjust input usage, and thus expected profits are invariant to the information released. In this situation, the insurance is able to remove the systemic risk (first term in Equation (20b)). If the skill of the forecast creates a wedge between expected profits obtained under different forecasts, the ex ante variance of profits increases and the effectiveness of the insurance to manage variability is reduced. Counter to intuition, the variability of profits when both risk management tools are available can be higher than when none is available. This can be seen by comparing Equations (20b) and (4b). Whenever expected profits under different forecasts differ more than the profit difference in the base case, variability will be increased.

In this case, with a perfect forecast, there is no role for insurance, while insurance is completely relied upon when the forecast has no skill. The difference is that the forecast directly allows improved input application that leads to increased yields and increased profits, while the insurance does not directly increase profits but allows the farmer to behave less conservatively. Thus, with insurance and a forecast, the farmer can have increased variability because of the potential to produce more in good years. However, to the extent that bad years are perfectly forecast, the farmer must face the full brunt of the drought, albeit with full information for optimal input use.

Since insurance plays different roles when priced using climatology or the forecast, it may be worthwhile to offer both pre- and postforecast policies: preforecast to protect against climatology and postforecast to protect against forecast error. The relative value of the pre- versus postforecast depends on the skill of the forecast and the farmer's flexibility in making changes in order to use effectively the forecast information in production to increase profits in good years and reduce damages in bad years. Future work should analyze the potential value of this risk management strategy.

CONCLUSIONS

Risk-driven barriers to development and innovations in financial markets have fed a renewed interest in the search that allows farmers to intensify their operations and invest in higher returns but in riskier activities. This is touted as key in helping farmers in developing countries escape poverty traps.

A substantial effort has been devoted to the study of the interaction between insurance and input decisions. Work has also explored the relationship between climate forecasts and input usage. Since previous literature has said little about the interaction between insurance, in particular index insurance, and climate forecasts, we have formalized and studied the basic relationship between forecasts, insurance, and production decisions through a theoretical model.

Climate scientists have made remarkable progress at forecasting rainfall and temperature deviations from long situations with the capacity to threaten the effectiveness and survival of existing index insurance mechanisms to alleviate poverty.

We find fundamental interactions between insurance and probabilistic climate forecasts. Insurance (in the absence of moral hazard effects) will induce farmers to use more of a risk-increasing input. The presence of a skillful probabilistic find that if an actuarially fair insurance is available, and the farmer's profits are not sufficiently responsive to the input mix, the introduction of a climate forecast harms the farmer if the premiums reflect the forecast (even if they are actuarially fair). Hence, a necessary condition for farmers to prefer a state-contingent, commercially viable insurance product is that farmers can increase their profits by taking the forecast information into account. Perhaps surprisingly, we find that forecast information may induce farmers to buy more insurance even as it reduces risk. The intuition is that the forecast may widen the wedge between optimized profits among states of the world. Although basis risk is an important issue in the treatment of index insurance, instead of having fundamental implications to the basic relationship between the forecast and insurance, we find that our straightforward representations of basis risk simply attenuate our findings. Nevertheless, it is likely that future work on this topic would be of value.

Since insurance priced using climatological probabilities protects against the climate and insurance priced on forecast probabilities protects against forecast error, farmer preferences for climatological- versus forecast-based insurance mirror the value of the forecast information in production. It is likely that both products could be useful, particularly when farms are heterogeneous, especially in the rates at which they are willing to trade expected levels by variability in profits Insurance prices may communicate forecast information when farmers do not have direct access to the forecast. Studies exploring the potential of insurance prices as aggregators of forecast information would be valuable.

Implementation of forecast-contingent insurance policies will require nontrivial innovation, as current insurance regulations and financing methodologies are not necessarily well suited to quickly fluctuating premiums, value at risk, and market size. Because an insurance policy typically does not include the option for resale at a market price, the pricing of information cannot directly rely on market movements. For insurance, it is likely that information pricing will be explicitly engineered into the products offered. Future work addressing these issues may be worthwhile.

Since insurance providers must typically reinsure their risks, the forecast-dependent price fluctuations of global weather derivative markets will lead to variations in reinsurance costs that must somehow be managed. Retail products that adjust based on the forecast could be one alternative that insurers have to address this problem. Future work will need to address both the technical issues of appropriately translating forecast information into an unbiased insurance and the financial and implementation issues of how to build a product that can be marketed and financed by an insurance company, that meets the demands of clients, and that falls within the allowable legal framework of insurance. One ENSO-based strategy might be to charge a nonvarying premium for a base liability calculated for an unfavorable ENSO phase and to increase the liability covered at no cost when the forecast is favorable. These changes might be financed by the insurer through purchases of ENSO derivatives or related products.

REFERENCES

Babcock, B. A., and D. A. Hennessy, 1996, Input Demand Under Yield and Revenue Insurance, American Journal of Agricultural Economics, 78: 416-427.

Barnett, B. J., C. B. Barrett, and J. R. Skees, 2007, Poverty Traps and Index-Based Risk Transfer Products, University of Georgia, Department of Agricultural and Applied Economics, Working paper.

Cabrera, V. E., D. Letson, and G. Podesta, 2005, The Value of Climate Information When Farm Programs Matter, Southeast Climate Consortium Technical Report Series SECC-05-004, Gainesville, FL.

Chambers, R. G., and J. Quiggin, 2000, Optimal Producer Behavior in the Presence of Area-Yield Crop Insurance, American Journal of Agricultural Economics, 84: 320-334.

Chang, C. C., S. K. Lin, and M. T. Yu, 2011, Valuation of Catastrophe Equity Puts With Markov-Modulated Poisson Processes, Journal of Risk and Insurance, 78: 447-473.

Doherty, N. A., and A. Richter, 2002, Moral Hazard, Basis Risk, and Gap Insurance, Journal of Risk and Insurance, 69: 9-24.

Goddard, L., S. J. Mason, S. E. Zebiak, C. F. Ropelewski, R. Basher, and M. A. Cane, 2001, Current Approaches to Seasonal-to-Interannual Climate Predictions, International Journal of Climatology, 21: 1111-1152.

Golden, L. L., M. Wang, and C. Yang, 2007, Handling Weather Related Risks Through the Financial Markets: Considerations of Credit Risk, Basis Risk, and Hedging, Journal of Risk and Insurance, 74:319-346

Hansen, J. W., 2002, Applying Seasonal Climate Prediction to Agricultural Production, Agricultural Systems, 74: 305-307.

Hellmuth, M. E., D. E. Osgood, U. Hess, A. Moorhead, and H. Bhojwani (eds.), 2009, Index Insurance and Climate Risk: Prospects for Development and Disaster Management (New York: Climate and Society No. 2. International Research Institute for Climate and Society (IRI) Columbia University).

Hess, U., and J. Syroka, 2005, Weather-Based Insurance in Southern Africa: The Case of Malawi, The World Bank Agricultural and Rural Development Discussion paper No. 13.

Just, R. E., and R. D. Pope, 2002, A Comprehensive Analysis of the Role of Risk in U.S. Agriculture (Boston: Kluwer Academic Publishers).

Kihlstrom, R., D. Romer, and S. Williams, 1981, Risk Aversion With Random Initial Wealth, Econometrica, 49 (July): 911-920.

Kovacevic, R. M., and G. Ch. Pflug, Forthcoming, Does Insurance Help Escape the Poverty Trap? A Ruin Theoretic Approach, Journal of Risk and Insurance. DOI: 10.1111/j.1539-6975.2010.01396.x.

Mahul, O., 2001, Optimal Insurance Against Climatic Experience, American Journal of Agricultural Economics, 83: 593-604.

Mjelde, J. W., T. N. Thompson, and C. J. Nixon, 1996, Government Institutional Effects on the Value of Seasonal Climate Forecast. American Journal of Agricultural Economics, 78:175-188

Moschini, G., and D. A. Hennessy, 2001, Uncertainty, Risk Aversion, and Risk Management for Agricultural Producers, in: B. Gardner and G. Rausser, eds., Handbook of Agricultural Economics, Volume 1 (Amsterdam: Elsevier), pp. 87-153.

Ramaswami, B., 1993, Supply Response to Agricultural Insurance: Risk Reduction and Moral Hazard Effects, American Journal of Agricultural Economics, 75: 914-925.

Ropelewski, C., and M. Halpert, 1987, Global and Regional Scale Precipitation Patterns Associated With the E1 Nino/Southern Oscillation, Monthly Weather Review, 115(8): 1606-1626.

Skees, J., P. Hazell, and M. Miranda, 1999, New Approaches to Crop Yield Insurance in Developing Countries, International Food and Policy Research Institute, Environment and Production Technology Division Discussion paper No. 55.

World Bank, 2001, World Development Report 2000/2001: Attacking Poverty (New York: Oxford University Press).

World Bank, 2005, Managing Agricultural Production Risk: Innovations in Developing Countries, Agricultural and Rural Development Department Report No. 32727GLB.

(1) Since there is a large body of literature on the role of risk in agriculture (see, e.g., Just and Pope, 2002; Moschini and Hennessy, 2001), it is worthwhile to note that there are several sources of risk that are relevant from the farmer's perspective, including production, price, technological, and policy uncertainties. Since our focus is on climate risks, a case of production uncertainty, we will assume that prices are nonrandom.

(2) One type of insurance that is becoming available is index insurance, which is similar to a weather derivative because payouts are calculated based on a weather-based index. It is important to understand the relationship between insurance and forecasts in this context because insurance does not include an option common to weather derivatives, the option to perform repeatedly marginal transactions in a dynamic market. Therefore, instead of relying

on market-based updates for optimal use of information, mechanisms to incorporate the information must be built directly into the retail contracts.

(3) One example is the index insurance for groundnut and maize farmers in Malawi (Hess and Syroka, 2005). In this case, the insurance provides the risk protection required for lenders to be willing to provide the credit farmers need to be able to adopt yield- and quality-increasing seeds.

(4) The two exceptions just mentioned analyze the impact of several government programs (including traditional yield insurance) on the value of seasonal forecast information. Based on numerical simulations of specific situations, these studies motivate the need for work that derives the fundamental relationships explaining their results.

(5) Longer-term climate forecasts have been recently investigated for the pricing of catastrophe equity puts (Chang, Lin, and Yu, 2011).

(6) The function used by Mahul (2001) is y = g(N)r + h(N) + [epsilon].

(7) The latter inequality reflects the assumption that the marginal response to an input is higher when other factors (e.g., rain) are not limiting. The assumption could be relaxed and the direction of the inequality reversed. In this case, the signs of the comparative statics presented in the following sections should also be reversed.

(8) In essence, we are assuming that the production region is small, in the sense that production shortfalls in this area do not "upset" world (and local) markets. Results will likely be attenuated if correlations between yields and prices are present. Malawi provides examples for both, export-oriented crops such as groundnuts and tobacco for which the assumption holds and local crops such as maize in which output affects prices (Hellmuth et al., 2009).

(9) Improved climate models can only provide a conditional distribution of yields. An inherent level of uncertainty is maintained as long as [[omega].sub.i|i] < 1 holds. Only in the case of a perfect forecast ([[omega].sub.i|i] = 1, for i = b,g), will all the systemic uncertainty be removed. Idiosyncratic variability still remains in this model.

(10) This result is analogous to Proposition 2 in Mahul (2001) with independent risks, where the trigger for the insurance is the maximum value of the weather variable, and the slope of the indemnity function with respect to the index equals its marginal productivity (given an input decision).

Miguel A. Carriquiry is at the Center for Agricultural and Rural Development, Iowa State University. Daniel E. Osgood is at the International Research Institute for Climate and Society, Columbia University. Osgood can be contacted via e-mail: deo@iri.columbia.edu.

DOI: 10.1111/j.1539-6975.2011.01422.x

Index insurance and probabilistic seasonal forecasts are becoming available in developing countries to help farmers manage climate risks in production. Although these tools are intimately related, work has not been done to formalize the connections between them. We investigate the relationship between the tools through a model of input choice under uncertainty, forecasts, and insurance. While it is possible for forecasts to undermine insurance, we find that when contracts are appropriately designed, there are important synergies between forecasts, insurance, and effective input use. Used together, these tools overcome barriers preventing the use of imperfect information in production decision making.

INTRODUCTION

Climate-related risks have profound impacts on agricultural producers around the world. (1) These effects are particularly important in developing countries, where agriculture makes a significant contribution to gross domestic product (World Bank, 2001) and insurance markets are underdeveloped or nonexistent. Recently, microinsurance and probabilistic seasonal forecasts have become available to help farmers manage climate risks in production. Although these tools are intimately related, work has not been done to formalize the fundamental connections between them. (2)

It is well known that climate risk can keep low-income households in poverty traps (see a review by Barnett, Barrett, and Skees, 2007). Lack of assets and risk exposure may lead households to forego activities with high returns, perpetuating their poverty. In response to this challenge, innovative insurance pilots to help farmers in developing countries have been gaining increasing interest (Hellmuth et al., 2009). (3) Recent studies analyze the potential of insurance in helping to escape poverty traps (Kovacevic and Pflug, Forthcoming).

Work in the agricultural economics literature has examined the relationship between insurance and input usage for both farm-level insurance (e.g., Ramaswami, 1993; Babcock and Hennessy, 1996) and index insurance (Chambers and Quiggin, 2000; Mahul, 2001). Focusing on the relationship between insurance and input use, this literature does not address interactions between insurance, climate forecast, and input decisions. Trade-offs between basis risk and moral hazard afforded by index-based products have been analyzed from an insurance company standpoint (Doherty and Richter, 2002).

Seasonal climate is predictable in many regions of the world (Goddard et al., 2001), with the El Nino Southern Oscillation (ENSO) linked to variations of seasonal precipitation (Ropelewski and Halpert, 1987). Work in agricultural and climate science has modeled the impact of probabilistic climate forecasts on production decisions (e.g., Hansen, 2002) but has, with few exceptions (Mjelde, Thompson, and Nixon, 1996; Cabrera, Letson, and Podesta, 2005), ignored the impact of insurance. (4)

Although yields and production practices are impacted by ENSO phases, this relationship is not built into existing insurance products. Insurance implementers acknowledge that climate forecasts may undermine the financial soundness of a product by providing opportunities for intertemporal adverse selection. The advocated strategies are to finalize insurance transactions contracts months ahead of time, before the forecast is informative (see, e.g., Hess and Syroka, 2005; World Bank, 2005), or allow insurance premiums to reflect forecast information (Skees, Hazell, and Miranda, 1999). Implementation of these strategies is undermined by the lack of a clear conceptual understanding about the interaction between seasonal forecast, insurance, and production decisions. Potential opportunities to exploit synergies may be being missed. Since the reinsurance industry uses probabilistic seasonal climate forecasts in pricing (Hellmuth et al., 2009), (5) it is important to understand forecasts and pricing from the perspective of the insurance company that must develop a product that: (1) results in products that provide value to farmers, unlocking consumer demand; (2) reflects fluctuating reinsurance costs; and (3) cannot be undermined through strategic use of forecast information. As microinsurance projects advance, the use of ENSO in these products is being increasingly explored, increasing the urgency for formalizing the basic features connecting insurance, production decisions, and forecasts. Existing analyses of insurance and forecast focus on pricing while ignoring the production value of the interaction can lead to misguided decision making as these production benefits are critical in driving insurance demand.

Since the fundamental relationship between insurance, input use, and seasonal climate forecasts has not been addressed, we propose to fill this gap by explicitly modeling input use and index insurance demand given probabilistic seasonal climate forecasts. While seasonal climate forecasts, insurance contracts, and input use choices can each be used to mitigate uncertainty and risk, they each play a different role and have potential synergies or unanticipated impacts when used together.

We investigate the relationship just described through the simplest possible model of input choice under uncertainty, forecasts, and index insurance. Our goal is to formalize the fundamental features of the relationship between insurance, forecasts, and production in a highly stylized model, in order to represent the key features in the most transparent way. Although it is not central to our problem, we include features in our model that allow for a basic discussion of the impacts of basis risk because it is of so much interest in the literature on new index-based insurance products (Golden, Wang, and Yang, 2007). In this line, the inclusion of the idiosyncratic shock allows the modeling of basis risk. The way in which the shock is introduced, linearly in this case to follow the literature and simplify the analysis, will provide only a subset of the results possible. Work focusing more on complex basis risk modeling would provide deeper insights into these issues.

We find that if contracts are appropriately designed there are important synergies between forecasts and insurance and effective input use. Insurance allows the farmer to map a probabilistic forecast into a much more deterministic payout, allowing the farmer to commit to production choices that take advantage of forecast information that is too noisy to utilize without risk protection. We also find that the presence of skillful probabilistic forecasts may affect the demand for insurance as well as its effectiveness as a risk-reducing tool. In our stylized treatment, basis risk attenuates impacts but does not lead to findings that are fundamentally different.

We begin by presenting the base framework that will be used throughout the article. Probabilistic seasonal climate forecasts and index insurance with their well-known impacts on production decisions are then introduced individually in order to provide benchmarks for our findings. Next, we combine the instruments (forecasts and insurance) and analyze their joint interactions with production practices. The last section provides concluding remarks and proposes some avenues for future research.

PRELIMINARIES AND BASE MODEL

Consider first a competitive farmer with a single crop with yield (y) dependent on the level of a controllable input (N, may be thought of as nitrogen, an improved seed, or the level of technology used), a systemic weather shock (r, hereafter rainfall) affecting all farmers in the area, and an idiosyncratic production shock ([epsilon]) as follows:

y = f(N, r) + [epsilon]. (1)

A special case of this yield function was used by Mahul (2001). (6) For simplicity, let rainfall take only two values, [r.sub.g] and [r.sub.b] (denoting good and poor or bad growing condition, respectively). It is assumed that [f.sub.N](N,r) = [partial derivative]f(N,r)/[partial derivative]N > 0, and [f.sub.NN](N,r) = [[partial derivative].sup.2]f(N,r)/[partial derivative][N.sup.2] [less than or equal to] 0. We assume further that f(N, [r.sub.g]) > f(N, [r.sub.b]), and [f.sub.N](N, [r.sub.g]) > [f.sub.N](N, [r.sub.b]) for all N. (7)

The value of the random variables is learned after the input has been applied. In the sections to follow, the systemic shock r is the variable on which the forecast provides information and on which the index insurance is written. The farmer knows that the climatological (historical) probability of observing r = [r.sub.g] is [[omega].sub.g]. The expected value of the idiosyncratic shock, which by definition is independent of the systemic shock, is assumed to be zero, and its variance is given by [sigma]2[epsilon]. Both the price of the controllable input ([p.sub.N]) and the price of the output (p) are assumed to be nonrandom, and the latter price is normalized to I without loss of generality. (8) Conditional on the idiosyncratic shock, and defining [[pi].sup.0] (N, r) = f(N, r) - [p.sub.N]N, profits for the farmer are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The farmer is assumed risk averse with a Bernoulli utility function given by u([pi]), with u''([pi]) < 0 < u'([pi]). If the farmer's choice is on the level of the input to apply, the farmer's problem is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](3)

where as in Mahul the indirect utility function [??](*) is [??]([[pi].sup.0]) = Eu([[pi].sup.0] + [epsilon]) for all [[pi].sup.0]. Kihlstrom, Romer, and Williams (1981) show the indirect utility function is increasing and concave in [[pi].sup.0]. It is well known that producers will self-insure in this context by selecting a level of inputs that reduce the magnitude of a loss when one occurs. The level of inputs selected will be lower than that of a risk-neutral producer. For future

reference notice that, in the absence of insurance or seasonal forecasts, the expected level, and variability of profits are given by

E([pi]) = E(E([pi]|[epsilon])) = [[omega].sub.g][[pi].sup.0]([N.sup.*],[r.sub.g]) + (1 - [[omega].sub.g])[[pi].sup.0]([N.sup.*],[r.sub.b]) (4a)

and

Var([pi]) = E(Var([pi]|[epsilon])) + Var(E([pi]|[epsilon])) = [[omega].sub.g](1 - [[omega].sub.g])([[pi].sup.0]([N.sup.*],[r.sub.g]) - [[pi].sup.0][([N.sup.*],[r.sub.b])).sup.2] + [[sigma].sup.2.sub.[epsilon]]. (4b)

FORECASTS WITHOUT INSURANCE

Suppose now that a skillful probabilistic seasonal climate forecast is available before decisions over the input are made. The forecast indicates the future state of the world (high or low rainfall) for the coming season and an associated uncertainty given by the probability that the forecast is incorrect. If high levels of rainfall are forecasted, there is probability cog I g that rainfall is actually high. Thus, forecast error is climate risk updated by the information in the forecast. A forecast is skillful if 1 > [[omega].sub.g|g] > [[omega].sub.g], and 1 > [[omega].sub.b|b] > [[omega].sub.b]. (9) Assume that the forecast is unbiased; that is, the frequency with which a high rainfall forecast is issued ([m.sub.g]) equals that of high rainfall years ([[omega].sub.g]).

In this situation, the decision of the farmer will depend on the forecast received. If a good year is forecasted, the decision of the farmer is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The [N.sup.*g], the optimal amount of input use when a good year is forecasted, will depend on the skill of the forecast. Again, the farmer self-insures against the uncertainty in the forecast.

INSURANCE WITHOUT A FORECAST

Suppose now that instead of a forecast, insurance (I) is available to farmers, and they must decide how much of it to buy at a price r per unit. To include a simple representation of basis, we assume that the insurance is available for the systemic shock (r) but not for the idiosyncratic shock ([epsilon]). In this case, the objective function and first-order conditions (at an interior solution) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

If the insurance is actuarially fair (r = 1 - [[omega].sub.g]), one obtains the standard result that the risk-averse farmer insures fully against the systematic risk; that is, [I.sup.*] = [[pi].sup.0] ([N.sup.*], [r.sub.g]) [[pi].sup.0]([N.sup.*],[r.sub.b]) = f([N.sup.*],[r.sub.g])- f([N.sup.*],[r.sub.b]), (10) and a higher level of inputs (the risk-neutral level) is utilized. The farmer's expected level and variability of profits when actuarially fair insurance is available are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9a)

and

Var([[pi].sub.I]) = E(Var([[pi].sub.I]|[epsilon])) + Var(E([[pi].sub.I]|[epsilon])) = [[sigma].sup.2][epsilon]. (9b)

Although profit variability is reduced, Equation (9b) indicates that some basis risk remains for farmers even when they are fully insured against the systemic shock.

COMBINING THE FORECAST WITH INSURANCE The impacts of the interaction between forecasts and insurance depend critically on the timing of the forecast who has little flexibility, who must commit to exogenously determined production practices prior to the forecast information and insurance decision. We then model the case in which a farmer simultaneously makes insurance and production decisions after the forecast is available.

Effects of a Skillful Forecast on Insurance Purchases With Fixed N

Consider a farmer who is constrained to commit to production decisions before insurance becomes available but after a skillful forecast has been issued. If skillful seasonal forecasts are released before the closing date for the insurance purchase, the premium rates must be modified to reflect the climate information available for the insurance to be financially sustainable, and the problem is state contingent. When a good year is forecasted, the actuarially fair rate becomes lower (from [tau] = [[omega].sub.b] to [[tau].sub.1] = [[omega].sub.b|g]) and the farmer's problem and first-order condition (for an interior solution) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[??]'([[pi].sup.0](N,[r.sub.g]) - [[tau].sub.1]I)/[??]'([[pi].sup.0](N,[r.sub.b]) - (1 - [[tau].sub.1])I = [[omega].sub.b|g]/[[omega].sub.g|g] (1 - [[tau].sub.1])/[[tau].sub.1]. (11)

Since the insurance is actuarially fair, the farmer will insure fully against the systemic

risk setting [I.sup.*] = [[pi].sup.0] (N,[r.sub.g]) - [[pi].sup.0] (N,[r.sub.b]) = [[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]. Furthermore, if inputs cannot be changed, the insurance purchase depends neither on whether the forecast is for a good or bad year nor on its skill. Hence, if a bad year is forecasted, and the premium rates reflect it (defining [[tau].sub.2] = [[omega].sub.b|b] > [[omega].sub.b] > [[tau].sub.1]), an analogous problem can be solved and the farmer will insure fully against the systemic risk.

When a forecast for a good year is issued, profits equal [[pi].sup.*.sub.g]|[epsilon]. = [[pi].sup.0.sub.g] - [[tau].sub.1]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]) + [epsilon] across realizations of the insured variable and thus E([[pi].sup.*.sub.g]) = [[pi].sup.0.sub.g] - [[tau].sub.1]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b] and Var([[pi].sup.*.sub.g]) = [[sigma].sup.2.sub.[epsilon]]. If the forecast is for a poor year, profits equal [[pi].sup.*.sub.b]|[epsilon]. = [[pi].sup.0.sub.g] - [[tau].sub.2]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]) + [epsilon] across realization of r, expected profits are E([[pi].sup.*.sub.g]) = [[pi].sup.0.sub.g] - [[tau].sub.2]([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b]), and Var([[pi].sup.*.sub.b]) = [[sigma].sup.2.sub.[epsilon]]. Since the forecast is unbiased and the insurance is actuarially fair, we have E([[pi].sup.*]|[epsilon])= [[omega].sub.g][[pi].sup.0.sub.g] + (1 - [[omega].sub.g])[[pi].sup.0.sub.b] + [epsilon, and E([[pi].sup.*]) = [[omega].sub.g][[pi].sup.0.sub.g] + (1 - [[omega].sub.g])[[pi].sup.0.sub.b].

Notice that expected profits change across realizations of the forecast. The difference is given by E([[pi].sup.*.sub.g] - E([[pi].sup.*.sub.b] = ([[tau].sub.2]) - ([[tau].sub.1]) ([[tau].sup.0.sub.g] - [[tau].sup.0.sub.b]]). Since the insurance is actuarially fair, [[tau].sub.2] = [[omega].sub.b|b] and [[tau].sub.1] = [[omega].sub.b|g], indicating that as the skill of the forecast increases, so does the difference in expected profits across forecasts. The resulting profit variability is

Var([pi]*) + E(Var([[pi].sup.*]|[epsilon])) + (Var(E([pi].sup.*]|[epsilon])) = [[omega].sub.g](1 - [[omega].sub.g])[([[tau].sub.2] - [[tau].sub.1])[([[pi].sup.0.sub.g] - [[pi].sup.0.sub.b])].sup.2] + [[sigma].sup.2.sub.[epsilon]]. (12)

Equation (12) indicates that the existence of a skillful forecast that is issued before purchases of the insurance are made increases the variability of profits when compared to the no-forecast situation. In the absence of the forecast (or when the forecast has no skill), we have [tau] = [[tau].sub.1] = [[tau].sub.2] and thus the farmer will only face the idiosyncratic risk (compare with Equation (9b)). As a skillful forecast is introduced, the difference [[tau].sub.2] - [[tau].sub.1] increases, undermining the effectiveness of the insurance to provide protection against the insurable risk. In the limit, with a perfect forecast, we have [[tau].sub.1] = [[omega].sub.b|g] = 0 and [[tau].sub.2] = [[omega].sub.b|b] = 1 yielding the same variance of profits as the uninsured case (Equation (4b)) for a fixed N.

In summary, since the forecast is assumed to be unbiased and available to both parties, the ex ante expected profit in this and the variability of that profit increases when the forecast is available. Hence, the presence of a forecast undermines the effectiveness of the insurance as a risk-mitigation mechanism in this situation and reduces welfare.

Choice of Both Insurance and Input Purchases in the Presence of a Skillful Forecast

To analyze the full interaction between the risk management tools and production decisions, we now allow farmers to choose both the level of the controllable input and the insurance purchase after observing the skillful forecast. Finally, the systemic and idiosyncratic shocks are observed. Since the forecast is released before farmers make their decisions, we have a state-contingent problem. The objective and first-order conditions when a good year is forecasted are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Since the insurance is actuarially fair, we know that [I.sup.*] = [[pi].sup.0]([N.sup.*g],[r.sub.g]) - [[pi].sup.0]([N.sup.*g],[r.sub.b]) = f([N.sup.*g], [r.sub.g]) - f([N.sup.*g], [r.sub.b]). Using this result, the first-order conditions evaluated at the optimum are written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Equation (16) indicates that in the presence of a state-dependent, actuarially fair insurance, the risk-neutral solution is replicated. Although the existence of the idiosyncratic risk imposes utility penalties, the presence of market inputs applied will maximize expected profits.

To investigate how the farmer's decisions are affected by the skill of the forecast, we need to sign [[partial derivative]I.sup.*g]/[[partial derivative][[omega].sub.g|g], and [[partial derivative]N.sup.*g]/ [partial derivative][[omega].sub.g|g]. Comparative statics on the system given by (16) and (17) indicate that

[[partial derivative]I.sup.*g]/[[partial derivative][[omega].sub.g|g] = ([f.sub.N]([N.sup.*g],[r.sub.g]) - [f.sub.N]([N.sup.*g],[r.sub.b]))[[partial derivative]N.sup.*g]/[[partial derivative][omega].sub.g|g]. (18)

Since we assumed that the marginal productivity of N is higher in good years, the partial derivatives in Equation (18) have the same sign. The effect of the skill of the forecast on the optimal nitrogen application is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where H is the determinant of the Hessian of the problem (positive by second-order sufficient conditions (SOSC) for a maximum). E ([??]"([[pi].sup.*]|g)) is negative by SOSC. The second term in the numerator is positive by technology assumptions, and thus [[partial derivative]N.sup.*g]/[[partial derivative][omega].sub.g|g] > 0 and [[partial derivative]I.sup.*g]/[[partial derivative][[omega].sub.g|g]> 0. Analogous analysis and previous results indicate that the farmer will purchase less insurance and use less inputs when the forecast is for a poor year.

The previous comparative statics exercise reveals that, counter to intuition, when the skillful forecast indicates a good (poor) year is likely, the farmer will purchase more (less) of an insurance of actuarially fair price. The expected change in overall insurance purchases brought about by a forecast of increasing skill depends on the relative adjustment induced by each kind of forecast (good versus poor growing conditions) and the natural frequency of each event, protecting the farmer from climate risk, it protects the farmer from forecast error.

The farmer is able to remove uncertainty from forecast error and improve utility by operating at the expected profit-maximizing input level instead of self-insuring with less aggressive changes in input. When a good year is forecast, the farmer can intensify to the expected profit-maximizing level, and when a bad year is forecast the farmer can prevent losses through the efficient level of input reduction while still maintaining inputs at a level that maximizes expected profits by taking into account the chance that a good year may still occur. Ex ante expected profits when both insurance and forecast are allowed to interact with the farmer's input decisions are given by

E([[pi].sup.*]) = [[omega].sub.g]([[pi].sup.0]([N.sup.*g],[r.sub.g]) - [[tau].sub.1][I.sup.*g]) + (1 - [[omega].sub.g])([pi]0([N.sup.*b],[r.sub.g]) - [[tau].sub.2][I.sup.*b]) (20a)

Var([[pi].sup.*]) = [[omega].sub.g](1 - [[omega].sub.g])[(E([[pi].sup.*g]) - E([[pi].sup.*b])).sup.2] + [[sigma].sup.2.sub.[epsilon]] (20b)

where we used the assumptions that the insurance is actuarially fair and that the forecast is unbiased ([m.sub.g] = [[omega].sub.g]). E ([[pi].sup.*.sub.i]) denotes expected profits for an i = g, b forecast. The actuarially fair insurance will lead farmers to maximize expected profits, and the skill of the forecast allows farmers to make better-informed decisions. Thus, expected profits increase when both the insurance and a skillful forecast are available. However, the introduction of a forecast comes at the cost of increasing profit variability. If the forecast has no skill, the farmer will not adjust input usage, and thus expected profits are invariant to the information released. In this situation, the insurance is able to remove the systemic risk (first term in Equation (20b)). If the skill of the forecast creates a wedge between expected profits obtained under different forecasts, the ex ante variance of profits increases and the effectiveness of the insurance to manage variability is reduced. Counter to intuition, the variability of profits when both risk management tools are available can be higher than when none is available. This can be seen by comparing Equations (20b) and (4b). Whenever expected profits under different forecasts differ more than the profit difference in the base case, variability will be increased.

In this case, with a perfect forecast, there is no role for insurance, while insurance is completely relied upon when the forecast has no skill. The difference is that the forecast directly allows improved input application that leads to increased yields and increased profits, while the insurance does not directly increase profits but allows the farmer to behave less conservatively. Thus, with insurance and a forecast, the farmer can have increased variability because of the potential to produce more in good years. However, to the extent that bad years are perfectly forecast, the farmer must face the full brunt of the drought, albeit with full information for optimal input use.

Since insurance plays different roles when priced using climatology or the forecast, it may be worthwhile to offer both pre- and postforecast policies: preforecast to protect against climatology and postforecast to protect against forecast error. The relative value of the pre- versus postforecast depends on the skill of the forecast and the farmer's flexibility in making changes in order to use effectively the forecast information in production to increase profits in good years and reduce damages in bad years. Future work should analyze the potential value of this risk management strategy.

CONCLUSIONS

Risk-driven barriers to development and innovations in financial markets have fed a renewed interest in the search that allows farmers to intensify their operations and invest in higher returns but in riskier activities. This is touted as key in helping farmers in developing countries escape poverty traps.

A substantial effort has been devoted to the study of the interaction between insurance and input decisions. Work has also explored the relationship between climate forecasts and input usage. Since previous literature has said little about the interaction between insurance, in particular index insurance, and climate forecasts, we have formalized and studied the basic relationship between forecasts, insurance, and production decisions through a theoretical model.

Climate scientists have made remarkable progress at forecasting rainfall and temperature deviations from long situations with the capacity to threaten the effectiveness and survival of existing index insurance mechanisms to alleviate poverty.

We find fundamental interactions between insurance and probabilistic climate forecasts. Insurance (in the absence of moral hazard effects) will induce farmers to use more of a risk-increasing input. The presence of a skillful probabilistic find that if an actuarially fair insurance is available, and the farmer's profits are not sufficiently responsive to the input mix, the introduction of a climate forecast harms the farmer if the premiums reflect the forecast (even if they are actuarially fair). Hence, a necessary condition for farmers to prefer a state-contingent, commercially viable insurance product is that farmers can increase their profits by taking the forecast information into account. Perhaps surprisingly, we find that forecast information may induce farmers to buy more insurance even as it reduces risk. The intuition is that the forecast may widen the wedge between optimized profits among states of the world. Although basis risk is an important issue in the treatment of index insurance, instead of having fundamental implications to the basic relationship between the forecast and insurance, we find that our straightforward representations of basis risk simply attenuate our findings. Nevertheless, it is likely that future work on this topic would be of value.

Since insurance priced using climatological probabilities protects against the climate and insurance priced on forecast probabilities protects against forecast error, farmer preferences for climatological- versus forecast-based insurance mirror the value of the forecast information in production. It is likely that both products could be useful, particularly when farms are heterogeneous, especially in the rates at which they are willing to trade expected levels by variability in profits Insurance prices may communicate forecast information when farmers do not have direct access to the forecast. Studies exploring the potential of insurance prices as aggregators of forecast information would be valuable.

Implementation of forecast-contingent insurance policies will require nontrivial innovation, as current insurance regulations and financing methodologies are not necessarily well suited to quickly fluctuating premiums, value at risk, and market size. Because an insurance policy typically does not include the option for resale at a market price, the pricing of information cannot directly rely on market movements. For insurance, it is likely that information pricing will be explicitly engineered into the products offered. Future work addressing these issues may be worthwhile.

Since insurance providers must typically reinsure their risks, the forecast-dependent price fluctuations of global weather derivative markets will lead to variations in reinsurance costs that must somehow be managed. Retail products that adjust based on the forecast could be one alternative that insurers have to address this problem. Future work will need to address both the technical issues of appropriately translating forecast information into an unbiased insurance and the financial and implementation issues of how to build a product that can be marketed and financed by an insurance company, that meets the demands of clients, and that falls within the allowable legal framework of insurance. One ENSO-based strategy might be to charge a nonvarying premium for a base liability calculated for an unfavorable ENSO phase and to increase the liability covered at no cost when the forecast is favorable. These changes might be financed by the insurer through purchases of ENSO derivatives or related products.

REFERENCES

Babcock, B. A., and D. A. Hennessy, 1996, Input Demand Under Yield and Revenue Insurance, American Journal of Agricultural Economics, 78: 416-427.

Barnett, B. J., C. B. Barrett, and J. R. Skees, 2007, Poverty Traps and Index-Based Risk Transfer Products, University of Georgia, Department of Agricultural and Applied Economics, Working paper.

Cabrera, V. E., D. Letson, and G. Podesta, 2005, The Value of Climate Information When Farm Programs Matter, Southeast Climate Consortium Technical Report Series SECC-05-004, Gainesville, FL.

Chambers, R. G., and J. Quiggin, 2000, Optimal Producer Behavior in the Presence of Area-Yield Crop Insurance, American Journal of Agricultural Economics, 84: 320-334.

Chang, C. C., S. K. Lin, and M. T. Yu, 2011, Valuation of Catastrophe Equity Puts With Markov-Modulated Poisson Processes, Journal of Risk and Insurance, 78: 447-473.

Doherty, N. A., and A. Richter, 2002, Moral Hazard, Basis Risk, and Gap Insurance, Journal of Risk and Insurance, 69: 9-24.

Goddard, L., S. J. Mason, S. E. Zebiak, C. F. Ropelewski, R. Basher, and M. A. Cane, 2001, Current Approaches to Seasonal-to-Interannual Climate Predictions, International Journal of Climatology, 21: 1111-1152.

Golden, L. L., M. Wang, and C. Yang, 2007, Handling Weather Related Risks Through the Financial Markets: Considerations of Credit Risk, Basis Risk, and Hedging, Journal of Risk and Insurance, 74:319-346

Hansen, J. W., 2002, Applying Seasonal Climate Prediction to Agricultural Production, Agricultural Systems, 74: 305-307.

Hellmuth, M. E., D. E. Osgood, U. Hess, A. Moorhead, and H. Bhojwani (eds.), 2009, Index Insurance and Climate Risk: Prospects for Development and Disaster Management (New York: Climate and Society No. 2. International Research Institute for Climate and Society (IRI) Columbia University).

Hess, U., and J. Syroka, 2005, Weather-Based Insurance in Southern Africa: The Case of Malawi, The World Bank Agricultural and Rural Development Discussion paper No. 13.

Just, R. E., and R. D. Pope, 2002, A Comprehensive Analysis of the Role of Risk in U.S. Agriculture (Boston: Kluwer Academic Publishers).

Kihlstrom, R., D. Romer, and S. Williams, 1981, Risk Aversion With Random Initial Wealth, Econometrica, 49 (July): 911-920.

Kovacevic, R. M., and G. Ch. Pflug, Forthcoming, Does Insurance Help Escape the Poverty Trap? A Ruin Theoretic Approach, Journal of Risk and Insurance. DOI: 10.1111/j.1539-6975.2010.01396.x.

Mahul, O., 2001, Optimal Insurance Against Climatic Experience, American Journal of Agricultural Economics, 83: 593-604.

Mjelde, J. W., T. N. Thompson, and C. J. Nixon, 1996, Government Institutional Effects on the Value of Seasonal Climate Forecast. American Journal of Agricultural Economics, 78:175-188

Moschini, G., and D. A. Hennessy, 2001, Uncertainty, Risk Aversion, and Risk Management for Agricultural Producers, in: B. Gardner and G. Rausser, eds., Handbook of Agricultural Economics, Volume 1 (Amsterdam: Elsevier), pp. 87-153.

Ramaswami, B., 1993, Supply Response to Agricultural Insurance: Risk Reduction and Moral Hazard Effects, American Journal of Agricultural Economics, 75: 914-925.

Ropelewski, C., and M. Halpert, 1987, Global and Regional Scale Precipitation Patterns Associated With the E1 Nino/Southern Oscillation, Monthly Weather Review, 115(8): 1606-1626.

Skees, J., P. Hazell, and M. Miranda, 1999, New Approaches to Crop Yield Insurance in Developing Countries, International Food and Policy Research Institute, Environment and Production Technology Division Discussion paper No. 55.

World Bank, 2001, World Development Report 2000/2001: Attacking Poverty (New York: Oxford University Press).

World Bank, 2005, Managing Agricultural Production Risk: Innovations in Developing Countries, Agricultural and Rural Development Department Report No. 32727GLB.

(1) Since there is a large body of literature on the role of risk in agriculture (see, e.g., Just and Pope, 2002; Moschini and Hennessy, 2001), it is worthwhile to note that there are several sources of risk that are relevant from the farmer's perspective, including production, price, technological, and policy uncertainties. Since our focus is on climate risks, a case of production uncertainty, we will assume that prices are nonrandom.

(2) One type of insurance that is becoming available is index insurance, which is similar to a weather derivative because payouts are calculated based on a weather-based index. It is important to understand the relationship between insurance and forecasts in this context because insurance does not include an option common to weather derivatives, the option to perform repeatedly marginal transactions in a dynamic market. Therefore, instead of relying

on market-based updates for optimal use of information, mechanisms to incorporate the information must be built directly into the retail contracts.

(3) One example is the index insurance for groundnut and maize farmers in Malawi (Hess and Syroka, 2005). In this case, the insurance provides the risk protection required for lenders to be willing to provide the credit farmers need to be able to adopt yield- and quality-increasing seeds.

(4) The two exceptions just mentioned analyze the impact of several government programs (including traditional yield insurance) on the value of seasonal forecast information. Based on numerical simulations of specific situations, these studies motivate the need for work that derives the fundamental relationships explaining their results.

(5) Longer-term climate forecasts have been recently investigated for the pricing of catastrophe equity puts (Chang, Lin, and Yu, 2011).

(6) The function used by Mahul (2001) is y = g(N)r + h(N) + [epsilon].

(7) The latter inequality reflects the assumption that the marginal response to an input is higher when other factors (e.g., rain) are not limiting. The assumption could be relaxed and the direction of the inequality reversed. In this case, the signs of the comparative statics presented in the following sections should also be reversed.

(8) In essence, we are assuming that the production region is small, in the sense that production shortfalls in this area do not "upset" world (and local) markets. Results will likely be attenuated if correlations between yields and prices are present. Malawi provides examples for both, export-oriented crops such as groundnuts and tobacco for which the assumption holds and local crops such as maize in which output affects prices (Hellmuth et al., 2009).

(9) Improved climate models can only provide a conditional distribution of yields. An inherent level of uncertainty is maintained as long as [[omega].sub.i|i] < 1 holds. Only in the case of a perfect forecast ([[omega].sub.i|i] = 1, for i = b,g), will all the systemic uncertainty be removed. Idiosyncratic variability still remains in this model.

(10) This result is analogous to Proposition 2 in Mahul (2001) with independent risks, where the trigger for the insurance is the maximum value of the weather variable, and the slope of the indemnity function with respect to the index equals its marginal productivity (given an input decision).

Miguel A. Carriquiry is at the Center for Agricultural and Rural Development, Iowa State University. Daniel E. Osgood is at the International Research Institute for Climate and Society, Columbia University. Osgood can be contacted via e-mail: deo@iri.columbia.edu.

DOI: 10.1111/j.1539-6975.2011.01422.x

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Author: | Carriquiry, Miguel A.; Osgood, Daniel E. |
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Publication: | Journal of Risk and Insurance |

Geographic Code: | 1USA |

Date: | Mar 1, 2012 |

Words: | 5831 |

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