Effective Risk Aversion and the Demand forSavings and Insurance

Mario J. MirandaKatie Farrin∗

April 4, 2020

Abstract

We examine the tradeoffs and complementarities that exist amongsaving, borrowing, and insurance in managing generic income risk. Wedepart from conventional static Von Neumann-Morgenstern ExpectedUtility theory by couching the insurance adoption decision in a formaldynamic framework that explicitly allows for the use of alternative fi-nancial instruments to manage risk. We find that the standard resultsof expected utility theory do not hold when agents may borrow orsave. Access to savings and borrowing “crowds out”, that is, reducesthe demand for, insurance. Agents who have access to borrowingand saving: 1) do not fully insure at actuarially favorable premiumrates; 2) are less likely to insure at loaded “market” premiums rates;and 3) may not insure at all, even if premium rates are subsidized,if they are temporarily poor. We show that the demand for insur-ance depends, not simply the curvature of the utility of consumptionfunction, but rather, more generally, on the curvature of the valuefunction, which we refer to as the “effective risk aversion. Effective

∗The authors are, respectively, Professor, Department of Agricultural, Environmentaland Development Economics, The Ohio State University, and Economist, Department ofPolicy and Evaluation, Millennium Challenge Corporation.

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risk aversion is heavily influenced by market conditions other than anindividuals primitive attitude toward risk, most notably the degree ofaccess to borrowing and savings.

Introduction

Methods for managing agricultural production risk have been the subjectof an extensive body of research that has appeared in the agricultural eco-nomics literature over the past forty years, with considerable attention beingdevoted to the optimal use of agricultural crop insurance.1 Most theoret-ical findings regarding the benefits of agricultural crop insurance employstatic Von Neumann-Morgenstern expected utility models in which an agri-cultural producer’s attitude toward risk is represented by a utility of moneyfunction whose concave curvature captures the producer’s aversion to risk.Typically, such models predict that a producer can enjoy substantial ben-efits from purchasing crop insurance, even when the premium exceeds theexpected indemnity.

It is a well-established empirical fact, however, that agricultural producersemploy crop insurance at rates that are substantially less than those pre-dicted or prescribed by static expected utility models, even when the insur-ance contracts embody subsidies(see Knight and Coble 1997). A widespreadview among agricultural risk management researchers, policy makers, and ex-tension agents is that the discrepancy between theory and practice is largelyexplained by producers’ ignorance on how best to employ crop insurance intheir production and marketing plans, prompting calls for aggressive publicoutreach and extension programs to educate agricultural producers on howto improve their financial welfare through proper use of crop insurance.

The discrepancy between theory and practice, however, may be due more toan incomplete theory of decision making under uncertainty than to flawedagricultural producer practices and perceptions. Agricultural risk manage-ment studies employing static, expected utility models are unable to ade-

1Between 1970 and 2011, the American Journal of Agricultural Economics alone pub-lished 649 articles containing “risk” in the title and 137 articles containing “risk” and“insurance” in the title.

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quately capture effective available alternatives to crop insurance for managingfarm level risk, most notably the use of saving, borrowing, and investment,which are inherently dynamic in nature. Consequently, conventional staticmodels over-state the benefits of crop insurance and cannot adequately in-form us on optimal uses of crop insurance and how its benefits might dependon a producer’s net worth, liquidity, and access to and cost of credit.

In this paper, we systematically explore how access to credit and savingsimpacts the benefits and optimal uses of agricultural crop insurance. Ouranalysis is grounded in formal theoretical models of dynamically optimizingagents that have access not only to crop insurance, but also to financial in-struments that allow the agent to build financial reserves through savings.We begin by exploring the interrelations between saving and insurance usinga two-period model for which analytic solutions exist. In the context of thismodel, we establish that the demand for and benefits of crop insurance relyheavily on a producer’s financial reserves and his/her access to credit andsavings. In particular, savings and insurance are substitutes, with the de-mand for insurance declining with interest rates on savings, and the demandfor net savings increasing with insurance premium rates. Our analysis alsoreveals that the demand for insurance is more a function of the producer’sfinancial reserves than the risk aversion embodied in his/her utility of con-sumption function, and leads us to conclude that insurance is something thepoor cannot afford and the rich do not need. None of these results emergefrom the standard static model of decision-making under uncertainty that isused almost exclusively in published theoretical and empirical work on cropinsurance demand.

We then turn to an infinite-horizon model of crop insurance demand whenproducers are able to save. This model, due to its inherent complexity, cannotbe solved analytically, but may be solved and analyzed using computationaltechniques. In the context of a fully dynamic model, the demand for cropinsurance will depend primarily on the curvature of the producer’s value func-tion, not simply the curvature of his/her utility of consumption function. Thecurvature of the value function, moreover, is endogenous to the model, beingdetermined not only by the agent’s utility of consumption but also accessto savings as indicated by the interest rates at which he/she can save. Themodel yields additional implications that do not emerge from conventionalstatic expected utility models or a two-period dynamic model. In particular,

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in a fully dynamic setting, a producer’s value function in the presence of bor-rowing and saving exhibits less curvature, and thus embodies less effectiverisk aversion, than the producer’s utility of consumption function. As such,the short- and long-run demand for crop insurance with access to savingsis less than predicted by the two-period model. The curvature of the valuefunction, moreover, can be reversed at low levels of wealth if bankruptcy andloan restructuring options exist, turning the producer into what in the con-text of static expected utility would be considered a ”risk-lover” who wouldhave no demand for crop insurance.

Our research answers questions of fundamental interest to researcher, prac-titioners, agricultural economists and policymakers interested in agriculturalrisk management not only in the developed countries, but also in the devel-oping world. Our insights can help us explain the observed low demand forcrop insurance in developed countries and shed light on the proper role foragricultural insurance in agricultural risk management. Our findings also in-form the emerging debate among development economists on whether poorfarmers in developing countries are better served by public programs thatpromote access to savings or by programs that promote access to crop orweather insurance.

Literature Review

Although the potential tradeoffs between savings and insurance have notreceived much attention from agricultural economists, the topic has been ex-amined and documented in the general theoretical economics and insuranceliteratures. Ehrlich and Becker 1972 were first to develop a theory of decisionmaking under uncertainty that allows for interaction between self-insurance,of which savings is one example, and market insurance, concluding that thetwo are substitutes. In their words: “optimal decisions about market insur-ance depend on the availability of ... other activities and should be viewedwithin the context of a more comprehensive insurance decision.” The fur-ther point out that: “apparent attitudes toward risk are dependent on marketopportunities, and real attitudes cannot easily be inferred from behavior.”

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Deaton 1991 analyzes optimal intertemporal consumption behavior of con-sumers who are able to borrow and save. He finds that when incomes exhibitlow orders of autocorrelation over time, consumes are able to significantlysmooth consumption over time by accumulating modest buffer stock of as-sets. In his model, however, the option to purchase insurance does not exist.Paxson 1992 and Morduch 1995 conclude that an individual’s ability to saveand borrow can greatly smooth consumption in the absence of alternativerisk markets. Eisenhauer 1994 considers consumer credit as a source of riskfinancing and derives explicit conditions under which it is superior to in-surance policies. He concludes that credit can be a viable alternative toinsurance for financing low-severity household risks.

Gollier 1994 examines dynamic risk management strategies and concludesthat risk-averse individuals will prefer self-insurance in the long run if the costof insurance exceeds a certain critical value, suggesting that market insuranceis most useful as a transitory strategy to protect capital and savings early inthe life cycle. Gollier 2002 develops a theoretical foundation to self-insurancethrough intertemporal diversification. He finds that in a dynamic setting,attitudes toward wealth risk and consumption risk are not the same, andself-insurance becomes an effective substitute to costly external insurance.

Most recently, Gollier 2003 analyzes a simple life-cycle model based on Deaton1991, Heaton and Lucas 1996, and Carroll 1997, in which consumers can fol-low a time-diversification (self-insurance) strategy by accumulating bufferstock wealth. He concludes that insurance would only be demanded forcatastrophic risks or by individual that face liquidity constraints, and thusthe added value of the insurance sector would be surprisingly low in such aneconomy.

Absorption of production and price shocks through some form of self-insuranceis especially common in developing countries. Deaton 1992 analyzes howfarmers in developing countries protect their living standards against fluc-tuations in income, emphasizing the role of credit markets in consumptionsmoothing. Hubbard, Skinner, and Zeldes 1995 employ a dynamic modelwith multiple sources of uncertainty to demonstrate how social insuranceprograms can discourage self-insurance and precautionary savings amongpoor households. Dercon 2002 provides a comprehensive review of recentliterature on risk-management and risk-coping strategies of poor households,

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identifies constraints on their effectiveness, and discusses policy options. Heshows that the opportunities to use assets as insurance are limited by risk,lumpiness of assets, and entry constraints, and that informal risk-sharingprovides only limited protection, leaving some of the poor exposed to verysevere negative shocks.

Kurosaki and Fafchamps 2002 also review a body of research on the typicalmechanisms of consumption smoothing, which include accumulating grain,livestock, and financial assets as a form of precautionary saving. They furtherrefer to a body of literature that analyzes efficiency of these institutions andpredicts that full insurance will not achieved despite substantial consumptionsmoothing. They also find that the households adapts production practices inresponse to consumption price risk and warn that “empirical and theoreticalwork on risk should avoid putting an exclusive emphasis on yield and outputprice risk”.

Review of Static Model

We begin by briefly reviewing the demand for insurance in the static VonNeumann-Morgenstern expected utility model, which has been the bulwarkof risk analysis in agriculture and other fields of economics.

Consider an agent endowed with predetermined initial wealth w > 0 whofaces an uncertain income y ≥ 0 and, additionally, an uncertain, but insurableloss l ≥ 0 that is independent of income, with El > 0. The agent may insureany portion x ∈ [0, 1] of the loss at a premium rate π > 0.2 That is, if theagent pays a premium xπ, he receives an indemnity xl if he experiences aloss of magnitude l. The agent chooses the coverage x that maximizes hisexpected utility of terminal wealth; that is, he solves:

max0≤x≤1

U(x)

where

U(x) ≡ Eu(w + y − (1− x)l − πx).

2In order to preclude the possibility of nonpositive terminal wealth, we further assumethat w+y > l, where y ≡ sup{y|Pr(y ≤ y) = 0} is the greatest lower bound on attainable

income and l ≡ inf{l|Pr(l ≥ 0) = 1} is the least upper bound on attainable losses.

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Here, u is the agent’s utility of wealth, which is presumed to be twice contin-uously differentiable, strictly increasing, and strictly concave. Under theseassumptions, U is twice continuously differentiable, strictly increasing, andstrictly concave, implying that the optimal coverage x is well-defined andcompletely characterized by the Karush-Kuhn-Tucker conditions:

0 ≤ x ≤ 1, x > 0 =⇒ U ′(x) ≥ 0, x < 1 =⇒ U ′(x) ≤ 0.

Theorem 1. The agent purchases coverage if, and only if,

π < π∗ ≡ Eλ(l)l.

where

λ(l) ≡ Eu′(w + y − l)Eu′(w + y − l)

.

Moreover,

π∗ > El.

Proof. The Karush-Kuhn-Tucker conditions assure that the agent purchasescoverage if, and only if,

U ′(0) = E(l − π)u′(w + y − l) > 0.

The first part of the proposition follows directly with some simple algebraicmanipulations. The second part of the proposition follows from the “covari-ance rule” (Gollier 2001, p. 94) and the facts that λ is strictly increasing, lis non-degenerate, and Eλ(l) = 1.

Theorem 1 states that agent purchases coverage if, and only if, the premiumrate π is less than the “risk adjusted” expected loss π∗, defined as the expec-tation of the loss l weighted by the relative marginal utility of wealth λ(l).The risk-adjusted expected loss π∗ exceeds the expected loss El because themarginal utility of wealth is greater for large losses than for small losses.This implies that the agent will purchase coverage even if the premium rateexceeds the expected indemnity, provided the premium rate is not excessivelyhigh.

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Theorem 2. The agent will fully insure, that is, x = 1, if, and only if, theexpected loss El equals or exceeds the premium rate π.

Proof. Since y and l are independent,

U ′(1) = E(l − π)u′(w + y − π) = (El − π)Eu′(w + y − π).

Given that U ′ is strictly decreasing, the Karush-Kuhn-Tucker conditionsguarantee that x ≥ 1 ⇐⇒ U ′(1) ≥ 0 ⇐⇒ El ≥ π.

Theorem 3. For El < π < π∗, the optimal coverage x is a continuouslydifferentiable function of the premium rate π.

Proof. If El < π < π∗, the agent will purchase coverage x > 0 such thatU ′(x) = 0. Thus, the Implicit Function Theorem guarantees that the optimalcoverage x(π) is a continuously differentiable function of the premium ratewith derivative

x′(π) =xE(l − π)u′′(w + y − (1− x)l − πx) + Eu′(w + y − (1− x)l − πx)

E(l − π)2u′′(w + y − (1− x)l − πx).

It is not possible to sign this derivative globally without further assumptions.

Note 1. At this point, I have not been able to sign this derivative globally. Ishould review Gollier for possible insights into how to do this

Theorem 4. Let V (w) denote the maximum expected utility for a givenwealth w, that is,

V (w) = max0≤x≤1

U(x,w)

where

U(x,w) ≡ Eu(w + y − (1− x)l − πx).

Then V is continuously differentiable, strictly increasing, and strictly con-cave.

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Proof. The continuous differentiability and the strict monotonicity of V fol-low directly from the Envelope Theorem, which guarantees that

V ′(w) =∂U(x,w)

∂w= Eu′(w + y − (1− x)l − πx) > 0,

where x is the optimal coverage given wealth w. To establish strict concavity,let w1 and w2 be distinct levels of wealth and let α1 and α2 be strict convexweights; furthermore, let xi be the optimal coverage for wealth wi. Then

V (α1w1 + α2w2) ≥ U(α1x1 + α2x2, α1w1 + α2w2)

= Eu(∑

i αi(wi + y − (1− xi)l − πxi)≥ E

∑i αiu(wi + y − (1− xi)l − πxi)

=∑

i αiEu(wi + y − (1− xi)l − πxi)=∑

i αiV (wi)

Since u is strictly concave, the second inequality will be an exact equalityonly if

w1 + y − (1− x1)l − πx1 = w2 + y − (1− x2)l − πx2

with probability 1, or, equivalently, only if

(w1 − w2) + (l − π)(x1 − x2) = 0

with probability 1. However, since l is not constant with probability 1, thisis possible only if x1 = x2 and w1 = w2, violating the assumption that w1

and w2 are distinct.

Basic Two-Period Model without Savings

Let us now extend the basic static Von Neumann-Morgenstern expected util-ity model to two periods, allowing the agent to purchase insurance in the firstperiod, period 0, to cover potential losses in the following period, period 1.We assume the agent has no means to save, a possibility that is examined inthe subsequent subsection.

Consider an agent endowed with predetermined wealth w faces an uncertainincome y next period and, additionally, an uncertain, but insurable loss l ≥ 0

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that is independent of income, with El > 0.3 The agent is not permittedto borrow or save, but may insure any portion x ∈ [0, 1] of the loss at apremium rate π > 0. That is, if the agent pays a premium xπ this period,he receives an indemnity xl next period if he experiences a loss of magnitudel. The agent chooses the coverage x that maximizes the sum of current anddiscounted expected future utility of consumption; that is, he solves:

max0≤x≤1

U(x)

where

U(x) ≡ u0(w − πx) + δEu1(y − (1− x)l).

Here, δ < 1 is the agent’s subjective discount factor and u0 and u1 arethe agent’s current and future utility of consumption, both of which are as-sumed to be twice continuously differentiable, strictly increasing, and strictlyconcave. Under these assumptions, U is twice continuously differentiable,strictly increasing, and strictly concave, implying that the optimal coveragex is well-defined and completely characterized by the Karush-Kuhn-Tuckerconditions:

0 ≤ x ≤ 1, x > 0 =⇒ U ′(x) ≥ 0, x < 1 =⇒ U ′(x) ≤ 0.

Theorem 5. The agent purchases coverage if, and only if,

π < π∗ ≡ Eλ(l)l.

where

λ(l) ≡ δEu′1(y − l)u′0(w)

.

Proof. The Karush-Kuhn-Tucker conditions imply that the agent purchasescoverage if, and only if,

U ′(0) = −πu′0(w − πx) + δElu′1(y − (1− x)l) > 0.

The proposition follows immediately, given some algebraic rearrangement ofterms.

3In order to preclude the possibility of nonpositive wealth in the second period, wefurther assume that y > l, where y ≡ sup{y|Pr(y ≤ y) = 0} is the greatest lower bound

on attainable income and l ≡ inf{l|Pr(l ≥ 0) = 1} is the least upper bound on attainablelosses.

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Theorem 5 states that agent purchases coverage if, and only if, the premiumrate π is less than the “risk adjusted” expected loss π∗, defined as the expecta-tion of the loss l weighted by the marginal rate of intertemporal substitutionof consumption λ(l). The risk adjusted expected loss π∗ may be greater thanor less than the expected loss El, depending on the curvatures of the utilityfunctions and the agent’s wealth w. A sufficiently rich agent (high w) willpurchase insurance, even if it is actuarially unfavorable, that is, even if thepremium exceeds the expected indemnity. Similarly, a sufficiently poor agent(low w), will not purchase insurance, even if it is actuarially favorable, thatis, even if the expected indemnity exceeds the premium.

Theorem 6. The agent will fully insure, that is, x = 1, if, and only if,

π ≤ πfull

where πfull is uniquely characterized by the nonlinear equation

πfull ≡ δEu′1(y)

u′0(w − πfull)l.

Moreover, 0 < πfull < π∗.

Proof. To prove that πfull is well-defined, let

f(π) ≡ π − δEu′1(y − l)l

u′0(w − π).

Now, f is continuous, f ′(π) > 0, f(0) < 0, and limπ→w f(π) = ∞, implyingthat f has a unique positive root on the interval (0, w), which by definitionequals πfull. Given that U ′ is strictly decreasing, the Karush-Kuhn-Tuckerconditions guarantee that x ≥ 1 if, and only if,

U ′(1) = −πu′0(w − π) + δElu′1(y − l) ≥ 0

or, equivalently, if, and only if,

f(π) ≤ 0 = f(πfull),

or, equivalently, if, and only if,

π ≤ πfull.

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Also,

πfull = Eu′1(y)

u′0(w − πfull)l < E

u′1(y − l)u′0(w)

l = π∗

since u′1(y − l) ≥ u′1(y) > 0 and 0 < u′0(w) < u′0(w − πfull)

Theorem 6, in conjunction with Theorem 5, states that an agent will fullyinsure if π ≤ πfull, will partially insure if πfull < π < π∗, and will not insureif π∗ ≤ π. If πfull < π < π∗, the agent will purchase coverage x > 0 thatsatisfies

πu′0(w − πx) = δElu′1(y − (1− x)l).

That is, at optimum coverage, the value of consumption forgone by purchas-ing a marginal unit of coverage today equals the expected present value ofconsumption provided by the additional indemnity tomorrow.

Theorem 7. For πfull < π < π∗, the optimal coverage x is a continuouslydifferentiable strictly decreasing function of the premium rate π and a con-tinuously differentiable strictly increasing function of wealth w.

Proof. For πfull < π < π∗, the optimal coverage x is characterized by thenonlinear equation U ′(x) = 0. The Implicit Function Theorem guaranteesthat over this range, the optimal coverage is a continuously differentiablefunction of the premium rate π and wealth w, with partial derivatives

∂x

∂π=

u′0(w − πx)− xπu′′0(w − πx)

π2u′′0(w − πx) + δEl2u′′1(y − (1− x)l)< 0

and

∂x

∂w=

πu′′0(w − πx)

π2u′′0(w − πx) + δEl2u′′1(y − (1− x)l)> 0.

Theorem 8. Let V (w) denote the maximum sum of current and discountedexpected future utility for a given wealth w, that is,

V (w) = max0≤x≤1

U(x,w)

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where

U(x,w) ≡ u0(w − πx) + δEu1(y − (1− x)l).

Then V is continuously differentiable, strictly increasing, and strictly con-cave.

Proof. The continuous differentiability and the strict monotonicity of V fol-low directly from the Envelope Theorem, which guarantees that

V ′(w) =∂U(x,w)

∂w= u′0(w − πx) > 0,

where x is the optimal coverage given wealth w. To prove strict concavity,begin by noting that U is jointly strictly concave in (x,w). Suppose xi isthe optimal coverage for wealth wi for i = 1, 2, where w1 6= w2; let w =α1w1 +α2w2 and x = α1x1 +α2x2 where α1 and α2 are strict convex weights.Then,

V (w) ≥ U(x,w) > α1U(x1, w1) + α2U(x2, w2) = α1V (w1) + α2V (w2).

Suppose now that loss is of the following simple form: l = L > 0 withprobability p, and equals zero otherwise. The following theorem tells us howthe optimal coverage x varies with the magnitude of the loss L, holding boththe premium rate π and the expected loss pL = K constant:

Theorem 9. If πfull < π < π∗, then an agent will cover a greater portion ofan infrequent catastrophic loss (low p, high L) than a more frequent smallerloss (higher p, lower L) with the same expectation pL and insurable at thesame premium rate π.

Proof. If πfull < π < π∗, then the agent will purchase coverage x < 1 charac-terized by U ′(x) = 0. Applying the Implicit Function Theorem allows us toconclude that

∂x

∂L=

(1− x)δpLu′′1(y − (1− x)L)

π2u′′0(w − πx) + δpL2u′′1(y − (1− x)L)> 0.

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Two-Period Model with Savings

We now extend the two-period model examined in the preceding section toallow for liquid savings.

Consider an agent endowed with predetermined wealth w faces an uncertainincome y ≥ 0 next period and, additionally, an uncertain, but insurable lossl ≥ 0 that is independent of income, with El > 0.4 The agent may save asmuch of his wealth s ≥ 0 as he pleases, earning an interest rate r > 0. Hemay also insure any portion x ≥ 0 of the uncertain loss at a premium rateπ > 0. That is, if the agent pays a premium xπ this period, he receives anindemnity xl next period if he experiences a loss of magnitude l. The agentchooses the savings s and coverage x that maximize the sum of current anddiscounted expected future utility of consumption; that is, he solves:

maxs≥0,x≥0

U(s, x)

where

U(s, x) ≡ u0(w − s− πx) + δEu1(y + (1 + r)s− (1− x)l).

Here, δ ≡ (1 + ρ)−1 where ρ > 0 is the agent’s subjective discount rate andu0 and u1 are the agent’s current and future utility of consumption, both ofwhich are assumed to be twice continuously differentiable, strictly increasing,and strictly concave.

Let

rs ≡ Eλ(1 + r)− 1

and

rx ≡ Eλl

π− 1

4In order to preclude the possibility of nonpositive wealth in the second period, wefurther assume that y > l, where y ≡ sup{y|Pr(y ≤ y) = 0} is the greatest lower bound

on attainable income and l ≡ inf{l|Pr(l ≥ 0) = 1} is the least upper bound on attainablelosses.

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denote, respectively, the expected rates of return on savings and insurancecoverage, weighted by the realized marginal rate of intertemporal substitutionof consumption

λ ≡ u′1(y + (1 + r)s− (1− x)l)

u′0(w − s− πx).

We refer to rs and rx, respectively, as the “risk-adjusted real rates of return”on savings and insurance. Forgoing a marginal unit of consumption today inorder to save is expected to yield tomorrow the current equivalent of 1 + rsunits of consumption; forgoing a marginal unit of consumption today in orderto purchase additional insurance coverage is expected to yield tomorrow thecurrent equivalent of 1 + rx units of consumption tomorrow.

Theorem 10. The optimal levels of savings s and coverage x satisfy thefollowing conditions:

rs ≤ ρ, s ≥ 0, and rs < ρ =⇒ s = 0

and

rx ≤ ρ, x ≥ 0, and rx < ρ =⇒ x = 0

Proof. The conclusion is an immediate consequence of the Karush-Kuhn-Tucker Theorem and some algebraic manipulation of the terms.

Theorem 10 states that, at an optimum, the risk-adjusted real rates of returnon savings and insurance must not exceed the discount rate, for otherwisethere would be incentive to save more or purchase more coverage. The agentwill not save or purchase insurance coverage if the associated risk-adjustedreal rate of return is less than the discount rate. If the agent saves andpurchases insurance coverage simultaneously, then the risk-adjusted real ratesof return on saving and insurance coverage are equal to the discount rate,and thus to each other.

Let

r∗ ≡ u′0(w)

δEu′1(y − l)− 1,

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and

π∗ ≡ δEu′1(y − l)lu′0(w)

.

Theorem 11. Suppose r ≤ r∗. Then the agent will not save, and will insureif, and only if, π < π∗.

Proof. Let s and x be the optimal savings and coverage, given r < r∗ andπ ≥ 0. If s > 0,

∂U

∂s= −u′0(w − s− πx) + (1 + r)δEu′1(y + (1 + r)s− (1− x)l)

< −u′0(w) + (1 + r)δEu′1(y − l)

≤ −u′0(w) + (1 + r∗)δEu′1(y − l)

= 0

in violation of the K-K-T conditions. This implies s = 0 by contradiction.Given s = 0, then for x = 0

∂U

∂x= −πu′0(w) + δEu′1(y − l)l = (π∗ − π)u′0(w)

which implies that

∂U

∂x

> 0 if π < π∗

= 0 if π = π∗

< 0 if π > π∗.

Thus, the agent will insure if, and only if, π < π∗.

Theorem 12. Suppose π ≥ π∗. Then the agent will not insure, and willsave if, and only if, r > r∗.

Proof. Let s and x be the optimal savings and coverage, given π ≥ π∗ and

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r. If x > 0,

∂U

∂x= −πu′0(w − s− πx) + δEu′1(y + (1 + r)s− (1− x)l)l

< −πu′0(w) + (1 + r)δEu′1(y − l)l

≤ −π∗u′0(w) + (1 + r)δEu′1(y − l)l

= 0

in violation of the K-K-T conditions. This implies x = 0 by contradiction.Given x = 0, then for s = 0

∂U

∂s= −u′0(w) + (1 + r)δEu′1(y − l)

which implies that

∂U

∂s

> 0 if r > r∗

= 0 if r = r∗

< 0 if r < r∗.

Thus, the agent will save a positive amount if, and only if, r > r∗.

Theorem 13. The critical interest rate r∗ below which an agent will not saveis a strictly decreasing function of wealth w; the critical premium π∗ abovewhich an agent will not insure is a strictly increasing function of wealth w.

Proof. Differentiating expressions above

∂r∗

∂w=

u′′0(w)

δEu′1(y − l)< 0.

and

∂π∗

∂w= −δu

′′0(w)Eu′1(y − l)l

(u′0(w))2> 0.

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Thus, given fixed interest rate r and premium rate π, the wealthier the agent,the more likely he is to save and insure.

Theorem 14. If r > r∗, then there exists π∗1 and π∗2, where 0 < π∗1 < π∗2 <π∗, such that

• The agent insures, but does not save if π ≤ π∗1.

• The agent saves, but does not insure if π ≥ π∗2.

• The agent both saves and insures if π∗1 < π < π∗2.

Proof. The critical premium level π∗1 is characterized jointly with the optimalcoverage x∗ at that premium by the pair of nonlinear equations

u′0(w − π∗1x∗) = (1 + r)δEu′1(y − (1− x∗)l)

π∗1u′0(w − π∗1x∗) = δEu′1(y − (1− x∗)l)l.

The critical premium level π∗2 is characterized jointly with the optimal savingss∗ at that and higher premiums by the pair of nonlinear equations

u′0(w − s∗) = (1 + r)δEu′1(y + (1 + r)s∗ − l)

π∗2u′0(w − s∗) = δEu′1(y + (1 + r)s∗ − l)l.

Figure 1 illustrates how optimal savings and insurance choices divide ther − π plane into four distinct sections. For low interest rates, below r∗, andhigh premium rates, above π∗, it optimal to neither save nor insure. In thenortheast section, where interest and premium rates are relatively high, it isoptimal to save, but not to insure. In the southwest section, where interestand premium rates are relatively low, it is optimal to insure, but not tosave. It is optimal to both save and insure only for interest-premium ratecombinations that lie between π∗1 and π∗2.

Figure 2 illustrates the demand for savings with access to insurance (blue)and without access to insurance (red). The demand for savings is not affectedby access to insurance for low interest rates, r ≤ r∗, because these interest

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π1*

π2*

r*

π*

Neither Savings

Only

Insurance

Only

Both

Interest Rate

Pre

miu

m R

ate

Savings LineInsurance Line

Figure 1: Savings and Insurance Choices, Interest-Premium Rate Plane

s*

r1* r

2* r*

Interest Rate

Sav

ings

With InsuranceWithout Insurance

Figure 2: Demand for Savings, Fixed Premium Rate

19

rates are too low to merit savings, regardless of whether insurance is available.The demand for savings is not affected by access to insurance for high interestrates, r > r∗2, because at high interest rates it is more beneficial to save thanto insure, and the agent will not insure. For intermediate interest rates,r∗ < r < r∗2, insurance “crowds out” savings, reducing it below what itwould be without insurance.

Theorem 15. If π < π∗, then there exists r∗1 and r∗2, where r∗ < r∗1 < r∗2,such that

• The agent insures, but does not save if r ≤ r∗1.

• The agent saves, but does not insure if r ≥ r∗2.

• The agent both saves and insures if r∗1 < r < r∗2.

Proof. The critical interest rate r∗1 is characterized jointly with the optimalcoverage x∗ at that and lower interests rate by the pair of nonlinear equations

u′0(w − πx∗) = (1 + r∗1)δEu′1(y − (1− x∗)l)

πu′0(w − πx∗) = δEu′1(y − (1− x∗)l)l.The critical interest rate r∗2 is characterized jointly with the optimal savingss∗ at that interest rate by the pair of nonlinear equations

u′0(w − s∗) = (1 + r∗2)δEu′1(y + (1 + r∗2)s

∗ − l)

πu′0(w − s∗) = δEu′1(y + (1 + r∗2)s∗ − l)l.

Figure 3 illustrates how optimal savings and insurance choices divide theπ − r plane into four distinct sections. For low interest rates below r∗ andhigh premium rates above π∗, it optimal to neither save nor insure. In thenortheast section, where interest and premium rates are relatively high, it isoptimal to save, but not to insure. In the southwest section, where interestand premium rates are relatively low, it is optimal to insure, but not tosave. It is optimal to both save and insure only for premium-interest ratecombinations that lie between the r∗1 and r∗2 curves.

20

r1*

r2*

r*

π*

Both Savings

Only

Neither

Insurance

Only

Premium Rate

Inte

rest

Rat

e

Savings LineInsurance Line

Figure 3: Savings and Insurance Choices, Premium-Interest Rate Plane

Figure 4 illustrates the demand for insurance with access to savings (blue)and without access to savings (red). The demand for insurance is not affectedby access to savings for high premium rates, π ≥ π∗, because these premiumrates are too high to merit purchasing coverage, regardless of whether onehas access to savings. The demand for insurance is not affected by access tosavings for low premium rates, π < π∗1, because at low premium rates rates itis more beneficial to insure than to save, and no savings is undertaken. Forintermediate premium rates, π∗1 < π < π∗, savings“crowds out” insurance,reducing it below what it would be without access to savings.

Theorem 16. Let V (w) denote the maximum sum of current and discountedexpected future utility for a given wealth w, that is,

V (w) = maxs≥0,x≥0

U(s, x, w)

where

U(s, x, w) ≡ u0(w − s− πx) + δEu1(y + (1 + r)s− (1− x)l).

Then V is continuously differentiable, strictly increasing, and strictly con-cave.

21

x*

π1* π

2* π*

Premium Rate

Cov

erag

e

With SavingsWithout Savings

Figure 4: Demand for Insurance, Fixed Interest Rate

Proof. The continuous differentiability and the strict monotonicity of V fol-low directly from the Envelope Theorem, which guarantees that

V ′(w) =∂U(s, x, w)

∂w= u′0(w − s− πx) > 0,

where s and x are, respectively, the optimal savings and coverage given wealthw. To prove strict concavity, begin by noting that U is jointly strictly concavein (s, x, w). Suppose si and xi are, respectively, the optimal savings andcoverage for wealth wi for i = 1, 2, where w1 6= w2; further suppose thatw = α1w1 + α2w2, s = α1s1 + α2s2, and x = α1x1 + α2x2 where α1 and α2

are strict convex weights. Then,

V (w) ≥ U(s, x, w) > α1U(s1, x1, w1)+α2U(s2, x2, w2) = α1V (w1)+α2V (w2).

22

Two-Period Model with Saving - Wealth Effects

For this analysis, we assume that the interest rate r > 0 and the premiumrate π > 0 are both given and examine how savings and insurance decisionsvary with wealth.

Let w∗r and w∗π be implicitly defined by

r ≡ u′0(w∗r)

δEu′1(y − l)− 1,

and

π ≡ δEu′1(y − l)lu′0(w

∗π)

.

Equivalently, let

w∗r ≡ u′−10

((1 + r)δEu′1(y − l)

),

and

w∗π ≡ u′−10

(δEu′1(y − l)l

π

).

Clearly, due to the curvature properties assumed for the utility functionu, both are well defined, provided we assume that limw→0 u

′0(w) = ∞ and

limw→∞ u′0(w) = 0, which we will assume. Moreover:

Theorem 17. Suppose w ≤ w∗r . Then the agent will not save, and willinsure if, and only if, w > w∗π.

Proof. Follows from Theorem 26.

Theorem 18. Suppose w ≤ w∗π. Then the agent will not insure, and willsave if, and only if, w > w∗r .

Proof. Follows from Theorem 27.

23

Theorem 19. The critical wealth w∗r below which an agent will not save is astrictly decreasing function of the interest rate r; the critical wealth w∗π belowwhich an agent will not insure is a strictly increasing function of the premiumrate π.

Proof. Follows from Theorem 28.

Let

π ≡ 1

1 + r

Eu′1(y − l)lEu′1(y − l)

denote the premium rate at which the agent will be indifferent to purchasinginsurance in the absence of savings. Then we have two theorems:

Theorem 20.

π

>=<

π ⇐⇒ w∗π

>=<

w∗r

Proof. Follows from Theorem 28.

Theorem 21. Optimal coverage is a strictly increasing function of wealthover the interval in which the agent insures but does not save.

Proof. Over the stated interval,

−πu′0(w − πx) + δEu′1(y − (1− x)l)l = 0.

Differentiating latter expression and rearranging:

x′(w) =πu′′0(w − πx)

δEu′′1(y − (1− x)l)l2 + π2u′′0(w − πx)> 0

Further theoretical results of interest are possible only if we place additionalrestrictions on the model. The critical assumption needed to derive theseresults appears to be the following:

24

Assumption 1. Regularity Assumption: For γ ≥ 0,

π(γ) ≡ 1

1 + r

Eu′1(y + γ − l)lEu′1(y + γ − l)

is a non-increasing function of γ.

The Regularity assumption states that the maximum premium rate the agentis willing to pay for insurance will not increase if income experiences a spread-preserving increase in its mean. More basic assumptions that guaranteeRegularity are not easy to come by.

Theorem 22. Suppose the Regularity Assumption holds. Then, if, π > π,the agent does not insure at any level of wealth.

Proof. To prove the contrapositive, suppose the agent insures for some levelof wealth. Then there is a w ≥ w∗π, such that s ≥ 0 and x = 0 are optimalfor w and

−u′0(w − s) + δ(1 + r)Eu′1(y + (1 + r)s− l) ≤ 0

−πu′0(w − s) + δEu′1(y + (1 + r)s− l)l = 0,

so that

π ≥ π((1 + r)s) =1

1 + r

Eu′1(y + (1 + r)s− l)lEu′1(y + (1 + r)s− l)

≥ π.

Theorem 23. Assume that

• the agent’s period 1 utility function u1 exhibits non-increasing absoluterisk aversion

A(w) ≡ −u′′1(w)

u′1(w)

at all levels of wealth w > 0;

25

• the loss l is binary, equaling L > 0 with probability p and 0 with prob-ability 1− p;

• income y is deterministic, and equals y with probability 1.

Then the Regularity Assumption holds.

Proof. Note that, under the given assumptions,

dπ

dγ=p(1− p)

1 + r

u′1(y + γ)u′1(y + γ − L)

Eu′1(y + γ − l)(A(y + γ)− A(y + γ − L)) ≤ 0.

Note 2. Recall that constant absolute risk aversion and constant relativerisk aversion utility functions satisfy the DARA assumption of the precedingtheorem.

What can we say about the proportion of wealth invested in insurance,πx(w)/w? The derivative of this proportion has the same sign as

wx′(w)− x(w),

which is positive if and only if

π2u′′0(w − πx)

δEu′′1(y − (1− x)l)l2 + π2u′′0(w − πx)> x(w)/w.

However, this is hard to to sign.

What can we say about the the signs of x′(w) and s′(w) when s > 0 andx > 0?

Here,

0 = −u′0(w − s− πx) + δ(1 + r)Eu′1(y + (1 + r)s− (1− x)l).

0 = −πu′0(w − s− πx) + δEu′1(y + (1 + r)s− (1− x)l)l.

26

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

Demand for Savings and Insurance − Two Period Model

Wealth

Exp

endi

ture

as

Pro

port

ion

of W

ealth

SavingsInsurance

Figure 5: Demand for Savings and Insurance as a Function of Wealth

Totally differentiating

0 = 1 + (R2Eb(l)− 1)s′ + (REb(l)l − π)x′.

0 = 1 + (REb(l)l

π− 1)s′ + (Eb(l)

l2

π− π)x′.

where

b(l) = −δu′′1(y + (1 + r)s− (1− x)l)

u′′0(w − s− πx)< 0

But this is nearly impossible to sign.

Basic Results

∂U

∂s= −u′0(w − s− πx) + δ(1 + r)Eu′1(y + (1 + r)s− (1− x)l).

∂U

∂x= −πu′0(w − s− πx) + δEu′1(y + (1 + r)s− (1− x)l)l.

27

0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

1.5

2

2.5

3Critical Interest Rate

Wealth

rsta

r

Figure 6: Critical Interest Rate

0 0.5 1 1.5 2 2.50

2

4

6

8

10

12Critical Premium Rate

Wealth

pist

ar

Figure 7: Critical Premium Rate

28

FIRST CRITICAL POINT

0 = −πu′0(w) + δEu′1(y − l)l.

provided

−u′0(w) + δ(1 + r)Eu′1(y − l) ≤ 0

which would imply

π(1 + r)Eu′1(y − l) ≤ Eu′1(y − l)l

SECOND CRITICAL POINT

0 = −πu′0(w − πx) + πδ(1 + r)Eu′1(y − (1− x)l).

0 = −πu′0(w − πx) + δEu′1(y − (1− x)l)l.

THIRD CRITICAL POINT

0 = −u′0(w − s) + δ(1 + r)Eu′1(y + (1 + r)s− l).

0 = −πu′0(w − s) + δEu′1(y + (1 + r)s− l)l.

29

so that

π(1 + r)Eu′1(y + (1 + r)s− l) = Eu′1(y + (1 + r)s− l)l.

for current parameterization, let

uL = u′(y + (1 + r)s− L)

and

u0 = u′(y + (1 + r)s)

π(1 + r)[puL + (1− p)u0] = pLuL

==========================

Infinite-Horizon Model with Savings

An infinitely-lived agent begins each period endowed with predeterminedwealth w, which he must then allocate among consumption, savings, andpurchases of insurance. The agent faces an uncertain income y ≥ 0 thefollowing period and, additionally, an uncertain, but insurable loss l ≥ 0that is independent of income, with El > 0.5 The agent may save as muchof his wealth s ≥ 0 as he pleases, earning a per-period interest rate r > 0.He may also insure any portion x ≥ 0 of the uncertain loss the followingperiod at a premium rate π > 0. That is, if the agent pays a premium xπ

5In order to preclude the possibility of nonpositive wealth in any period, we furtherassume that y > l, where y ≡ sup{y|Pr(y ≤ y) = 0} is the greatest lower bound on

attainable income and l ≡ inf{l|Pr(l ≥ 0) = 1} is the least upper bound on attainablelosses.

30

this period, he receives an indemnity xl next period if he experiences a lossof magnitude l.

The agent chooses the savings s and coverage x that maximize the sum ofcurrent and discounted expected future utility of consumption over an infinitehorizon. By Bellman’s Principle of Optimality, the agent’s value function,V (w), which denotes the maximum attainable sum of current and discountedexpected future utility of consumption given the agent’s current wealth w, ischaracterized by the functional equation

V (w) = maxs≥0,x≥0

{u(w − s− πx) + δEV (y + (1 + r)s− (1− x)l)}.

Here, δ ≡ (1 + ρ)−1 where ρ > 0 is the agent’s subjective discount rateand u is the agent’s utility of consumption, which is presumed to be twicecontinuously differentiable, strictly increasing, and strictly concave. Theagent’s consumption equals his predetermined wealth, plus borrowing, lessproduction costs, less index insurance premium payments.

Theorem 24. The agent’s Bellman functional equation possesses an uniquesolution V , which denotes the maximum sum of current and discounted ex-pected future utility for a given wealth w. The value function V is continuous,strictly increasing, and strictly concave.

Proof. Need special arguments. Not sure about differentiability. See Stokey-Lucas.

We denote by s(w) and x(w) the optimal levels of savings and insurancecoverage, respectively, given wealth w. These are the agent’s optimal policyfunctions.

For any admissible level of wealth w, let

rs(w) ≡ Eλ(w; y, l)(1 + r)− 1

and

rx(w) ≡ Eλ(w; y, l)l

π− 1

31

denote, respectively, the expected rates of return on savings and insurancecoverage, weighted by the realized marginal rate of intertemporal substitutionof consumption

λ(w; y, l) ≡ V ′(y + (1 + r)s(w)− (1− x(w))l)

u′(w − s(w)− πx(w)).

We refer to rs(w) and rx(w), respectively, as the “risk-adjusted real rates ofreturn” on savings and insurance. Forgoing a marginal unit of consumptiontoday in order to save is expected to yield tomorrow the current equivalentof 1 + rs units of consumption; forgoing a marginal unit of consumptiontoday in order to purchase additional insurance coverage is expected to yieldtomorrow the current equivalent of 1 + rx units of consumption tomorrow.

Theorem 25. For any given level of wealth w, the optimal levels of savingss(w) and coverage x(w) satisfy the following conditions:

rs(w) ≤ ρ, s(w) ≥ 0, and rs(w) < ρ =⇒ s(w) = 0

and

rx(w) ≤ ρ, x(w) ≥ 0, and rx(w) < ρ =⇒ x(w) = 0

Proof. The conclusion is an immediate consequence of the Karush-Kuhn-Tucker Theorem and some algebraic manipulation of the terms.

Theorem 25 states that, at an optimum, the risk-adjusted real rates of returnon savings and insurance must not exceed the discount rate, for otherwisethere would be incentive to save more or purchase more coverage. The agentwill not save or purchase insurance coverage if the associated risk-adjustedreal rate of return is less than the discount rate. If the agent saves andpurchases insurance coverage simultaneously, then the risk-adjusted real ratesof return on saving and insurance coverage are equal to the discount rate,and thus to each other.

For any given level of wealth w, let

r∗(w) ≡ u′(w)

δEV ′(y − l)− 1,

32

and

π∗(w) ≡ δEV ′(y − l)l

u′(w).

Theorem 26. Suppose r ≤ r∗(w). Then the agent will not save, and willinsure if, and only if, π < π∗(w).

Proof. Let s and x be the optimal savings and coverage, given level of wealthw, and suppose r < r∗(w). If s > 0,

∂U

∂s= −u′(w − s− πx) + (1 + r)δEV ′(y + (1 + r)s− (1− x)l)

< −u′(w) + (1 + r)δEV ′(y − l)

≤ −u′(w) + (1 + r∗)δEV ′(y − l)

= 0

in violation of the K-K-T conditions. This implies s = 0 by contradiction.Given s = 0, then for x = 0

∂U

∂x= −πu′(w) + δEV ′(y − l)l = (π∗ − π)u′(w)

which implies that

∂U

∂x

> 0 if π < π∗

= 0 if π = π∗

< 0 if π > π∗.

Thus, the agent will insure if, and only if, π < π∗.

Theorem 27. Suppose π ≥ π∗(w). Then the agent will not insure, and willsave if, and only if, r > r∗(w).

Proof. Let s and x be the optimal savings and coverage, given π ≥ π∗ and

33

r. If x > 0,

∂U

∂x= −πu′(w − s− πx) + δEV ′(y + (1 + r)s− (1− x)l)l

< −πu′(w) + (1 + r)δEV ′(y − l)l

≤ −π∗u′(w) + (1 + r)δEV ′(y − l)l

= 0

in violation of the K-K-T conditions. This implies x = 0 by contradiction.Given x = 0, then for s = 0

∂U

∂s= −u′(w) + (1 + r)δEV ′(y − l)

which implies that

∂U

∂s

> 0 if r > r∗

= 0 if r = r∗

< 0 if r < r∗.

Thus, the agent will save a positive amount if, and only if, r > r∗.

Theorem 28. The critical interest rate r∗(w) below which an agent will notsave is a strictly decreasing function of wealth w; the critical premium π∗(w)above which an agent will not insure is a strictly increasing function of wealthw.

Proof. Differentiating expressions above

∂r∗(w)

∂w=

u′′(w)

δEV ′(y − l)< 0.

and

∂π∗(w)

∂w= −δu

′′(w)EV ′1(y − l)l(u′(w))2

> 0.

34

Thus, the wealthier the agent, the higher must be the interest rate to inducehim to save and the lower must be the premium rate to induce him to insure.

Let w∗r and w∗π be defined by

w∗r = u′−1(

(1 + r)δEV ′(y − l)),

and

w∗π ≡ u′−1

(δEV ′(y − l)l

π

).

so that r∗(w∗r) = r and π∗(w∗π) = π. Clearly, due to the curvature propertiesassumed for the utility function u, both are well defined, provided we as-sume that limw→0 u

′(w) = ∞ and limw→∞ u′(w) = 0, which we will assume.

Moreover:

Theorem 29. Suppose w ≤ w∗r . Then the agent will not save, and willinsure if, and only if, w > w∗π.

Proof. Follows from Theorem 26.

Theorem 30. Suppose w ≤ w∗π. Then the agent will not insure, and willsave if, and only if, w > w∗r .

Proof. Follows from Theorem 27.

Theorem 31. The critical wealth w∗r below which an agent will not save is astrictly decreasing function of the interest rate r; the critical wealth w∗π belowwhich an agent will not insure is a strictly increasing function of the premiumrate π.

Proof. I am not sure that this is easy to prove, and may actually be false.See Theorem 28.

Let

π ≡ 1

1 + r

EV ′(y − l)lEV ′(y − l)

35

denote the premium rate at which the agent will be indifferent to purchasinginsurance in the absence of savings. Then we have two theorems:

Theorem 32.

π

>=<

π ⇐⇒ w∗π

>=<

w∗r

Proof. Follows from Theorem 28. Double check this.

Theorem 33. Optimal coverage is a strictly increasing function of wealthover the interval in which the agent insures but does not save.

Proof. Over the stated interval,

−πu′(w − πx) + δEV ′(y − (1− x)l)l = 0.

Differentiating latter expression and rearranging:

x′(w) =πu′′(w − πx)

δEV ′′(y − (1− x)l)l2 + π2V ′′(w − πx)> 0

Further theoretical results of interest are possible only if we place additionalrestrictions on the model. The critical assumption needed to derive theseresults appears to be the following:

Assumption 2. Regularity Assumption: For γ ≥ 0,

π(γ) ≡ 1

1 + r

EV ′(y + γ − l)lEV ′(y + γ − l)

is a non-increasing function of γ.

The Regularity assumption states that the maximum premium rate the agentis willing to pay for insurance will not increase if income experiences a spread-preserving increase in its mean. More basic assumptions that guaranteeRegularity are not easy to come by.

36

Theorem 34. Suppose the Regularity Assumption holds. Then, if, π > π,the agent does not insure at any level of wealth.

Proof. To prove the contrapositive, suppose the agent insures for some levelof wealth. Then there is a w ≥ w∗π, such that s ≥ 0 and x = 0 are optimalfor w and

−u′(w − s) + δ(1 + r)EV ′(y + (1 + r)s− l) ≤ 0

−πu′(w − s) + δEV ′(y + (1 + r)s− l)l = 0,

so that

π ≥ π((1 + r)s) =1

1 + r

EV ′(y + (1 + r)s− l)lEV ′(y + (1 + r)s− l)

≥ π.

Theorem 35. Assume that

• the agent’s value function V exhibits non-increasing absolute risk aver-sion

A(w) ≡ −V′′(w)

V ′(w)

at all levels of wealth w > 0;

• the loss l is binary, equaling L > 0 with probability p and 0 with prob-ability 1− p;

• income y is deterministic, and equals y with probability 1.

Then the Regularity Assumption holds.

Proof. Note that, under the given assumptions,

dπ

dγ=p(1− p)

1 + r

V ′(y + γ)V ′(y + γ − L)

Eu′1(y + γ − l)(A(y + γ)− A(y + γ − L)) ≤ 0.

37

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