the welfare bias from omitting climatic variability in economic studies of global warming

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Ž . JOURNAL OF ENVIRONMENTAL ECONOMICS AND MANAGEMENT 33, 221]239 1997 ARTICLE NO. EE971000 The Welfare Bias from Omitting Climatic Variability in Economic Studies of Global Warming 1 Michael G. Dalton Department of Economics, Stanford Uni ¤ ersity, Stanford, California 94305-6072 Received June 12, 1996; revised November 18, 1996 This article analyzes the welfare effects of climatic variability from global warming in a stochastic economic growth model and shows that these may be significant. An empirical analysis indicates that the effects of climate change with variability are greater than the corresponding effects without it. Effects with variability are also shown to be more sensitive to variations in the rate of climate change. Q 1997 Academic Press 1. INTRODUCTION The threat of global warming is a major environmental problem facing the world today. Scientists warn that the accumulation of greenhouse gases in the atmo- sphere, such as carbon dioxide, will cause significant changes in climate. The literature on this topic is extensive but a great deal of uncertainty remains. Most studies in this literature focus on changes in mean temperature and precipitation. However, other properties of climate, in particular its variability, are also likely to be important. For example, the corn harvest in the United States, the world’s leading producer of this crop, was reduced by nearly 40% during the drought of wx 1988 5 . Several recent studies have stressed the importance of climatic variability w x 13, 20, 11 , but they do not examine its welfare implications. On the other hand, studies in the economics literature on global warming usually w x analyze welfare, many in an agricultural context 10, 14, 15, 21 , but most, including w x Nordhaus’ 18, 19 influential DICE analysis, ignore the effects of climatic variabil- wx ity. However, some steps have been taken to address this. Kaiser et al. 9 study the effects of climate change, including those related to risk, on farm-level decisions in w x a microeconomic framework. Mendelsohn et al. 16, 22 include variables that measure climatic variability in a cross section regression analysis to estimate its effects on land values and find that these are significant. Both sets of studies 1 I am grateful to Lawrence Goulder, Edward Green, James Jordan, Antonio Merlo, Herb Mohring, Stephen Schneider, Gunther Schulze, G. David Tilman, and two anonymous referees for many helpful ¨ comments, conversations, and suggestions. I also thank Greg Spodin of the Minnesota Department of Natural Resources for providing the historical climate data used in the paper. This research received support from the National Science Foundation. 221 0095-0696r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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Ž .JOURNAL OF ENVIRONMENTAL ECONOMICS AND MANAGEMENT 33, 221]239 1997ARTICLE NO. EE971000

The Welfare Bias from Omitting Climatic Variability inEconomic Studies of Global Warming1

Michael G. Dalton

Department of Economics, Stanford Uni ersity, Stanford, California 94305-6072

Received June 12, 1996; revised November 18, 1996

This article analyzes the welfare effects of climatic variability from global warming in astochastic economic growth model and shows that these may be significant. An empiricalanalysis indicates that the effects of climate change with variability are greater than thecorresponding effects without it. Effects with variability are also shown to be more sensitiveto variations in the rate of climate change. Q 1997 Academic Press

1. INTRODUCTION

The threat of global warming is a major environmental problem facing the worldtoday. Scientists warn that the accumulation of greenhouse gases in the atmo-sphere, such as carbon dioxide, will cause significant changes in climate. Theliterature on this topic is extensive but a great deal of uncertainty remains. Moststudies in this literature focus on changes in mean temperature and precipitation.However, other properties of climate, in particular its variability, are also likely tobe important. For example, the corn harvest in the United States, the world’sleading producer of this crop, was reduced by nearly 40% during the drought of

w x1988 5 . Several recent studies have stressed the importance of climatic variabilityw x13, 20, 11 , but they do not examine its welfare implications.

On the other hand, studies in the economics literature on global warming usuallyw xanalyze welfare, many in an agricultural context 10, 14, 15, 21 , but most, including

w xNordhaus’ 18, 19 influential DICE analysis, ignore the effects of climatic variabil-w xity. However, some steps have been taken to address this. Kaiser et al. 9 study the

effects of climate change, including those related to risk, on farm-level decisions inw xa microeconomic framework. Mendelsohn et al. 16, 22 include variables that

measure climatic variability in a cross section regression analysis to estimate itseffects on land values and find that these are significant. Both sets of studies

1I am grateful to Lawrence Goulder, Edward Green, James Jordan, Antonio Merlo, Herb Mohring,Stephen Schneider, Gunther Schulze, G. David Tilman, and two anonymous referees for many helpful¨comments, conversations, and suggestions. I also thank Greg Spodin of the Minnesota Department ofNatural Resources for providing the historical climate data used in the paper. This research receivedsupport from the National Science Foundation.

2210095-0696r97 $25.00

Copyright Q 1997 by Academic PressAll rights of reproduction in any form reserved.

MICHAEL G. DALTON222

abstract from investment, storage, and other dynamic decisions and rely on partialequilibrium analysis, distinguishing them from our work.2

We analyze climatic variability in a dynamic general equilibrium setting using aone sector stochastic growth model.3 Our agents fully account for the uncertaintyin the economic environment; so, our model provides a consistent framework foranalyzing decision making in this setting. First, we analyze the structural relation-ships between welfare and the model’s parameters that describe climate. Theseparameters characterize the mean and variance of precipitation and temperatureand the correlation between them. We find that the relationships between welfareand the climatic means are similar to those between welfare and variability.Therefore, abstracting from variability may seriously underestimate the impacts ofclimate change. To assess this, we empirically analyze these relationships.

Our empirical analysis focuses on the welfare effects from a gradually changingclimate on the economy for a range of scenarios found in the recent literature onglobal warming. Since it is expected to be among the most vulnerable industries, wefollow the other studies that include variability and focus our empirical work onagriculture.4 The parameters in our model that characterize agricultural produc-tion are estimated using a crop growth simulator and historical climate data. Wesimulate the growth of only one crop, corn, but this is less restrictive than itappears. In a general sense, we are really concerned about capturing the biophysi-cal relationships of crop growth in an economic model of production with climatechange. Corn production is a good representation of the biophysical relationshipsand of the diversity of uses for agricultural products, making it the best simple way

2 w xKaiser et al. 9 focus on crop switching and other responses available to farmers. They include themeans and variances of precipitation and temperature but abstract from the correlation between them.

w x w xMendelsohn et al. 16, 22 extend their empirical Ricardian approach from 15 that focused on landvalues along a climatic gradient to include variables that measure diurnal and interannual variability inprecipitation and temperature. Their results show that climatic variability is statistically and qualita-tively significant. In general, they find that increases in interannual variability are harmful whileincreases in diurnal variability are beneficial. Like Kaiser et al., they do not analyze any cross-effectsbetween precipitation and temperature, including correlation.

3 w xIn real business cycle models, climatic variability is often cited as a source of productivity shocks 2 .Our model supports this interpretation but is more general than the usual specification found in thereal business cycle literature since it allows the distribution functions that characterize the randomshocks to be nonstationary to include climate change.

4Of course, the effect of climate change on other industries is also important. Because of theflexibility of the economic growth framework, it is straightforward to extend our model to includedifferent industries. However, empirical work to support such an extension does not currently exist andis beyond the scope of this study. For our analysis to apply to the whole economy, it is sufficient that thestructure of the shocks in different industries be similar to those of agriculture. To keep the focus of thepaper on agriculture, we restrict the range of adaptation by producers for our empirical analysis. The

w xpaper by Mendelsohn et al. 15 shows the importance of adaptation in assessing the impact of climatechange. They measure the full range of adaptation to climate by regressing land values in the UnitedStates onto local climatic variables such as temperature and precipitation. The empirical component ofour study falls into the broad category that they call ‘‘the production function approach.’’ This approachtends to overestimate the negative effects from climate change since it abstracts from many forms ofadaptation. However, they note that this approach provides a ‘‘useful baseline for estimating the impact

Ž .of climate change on farming’’ p. 753 and stress the complementarity between the two methodologies.

WELFARE AND CLIMATIC VARIABILITY 223

of incorporating these features into our economic framework.5 Our empiricalanalysis shows that the effects of climate change on agriculture are greater withvariability than without it.6 The effects with variability are also more sensitive tovariations in the rate of climate change. For example, the welfare effect fromreducing rainfall is significantly greater when the correlation between precipitationand temperature is included in the model. Though not as striking, welfare effectswith variability are also more sensitive to variations in the rates of change of otherclimatic parameters such as mean temperature.

The paper proceeds as follows. Section 1 illustrates the welfare bias fromomitting climatic variability. Section 2 presents an analytical model to study therelationship between climatic variability and welfare. Section 3 gives an empiricalassessment of the welfare effects from global warming. Sections 4 and 5 discuss ourresults and future research.

2. THE BIAS FROM OMITTING VARIABILITY

The possibility of a welfare bias from ignoring climatic variability is illustrated inFig. 1. There we consider the welfare effects of climate change on the value of aneconomy. The case illustrated in the figure is special: we assume that welfare is at aŽ .local maximum for the benchmark climate. However, this is not the only possiblecase. In the following sections, we generalize the climate model that is pictured inFig. 1 and explore different cases. In some of these, climate change is welfareimproving.7 To motivate the case of a climatic maximum, think of an agriculturalsystem as having adapted to its present climate. Climate change forces adaptationto a new set of conditions and if this is sufficiently costly welfare decreases.

To understand the figure, suppose climate is represented by a single variable, forexample temperature, that exhibits a natural variability which we assume israndom. Most studies of global warming use a summary description of climatebased on mean values. However, it is often important to know how close a randomvariable usually is to this value. This is certainly true for temperature. Consider a

5Corn is an extremely important crop: in the United States it is number one by area planted andglobally it is among the top five. It can tolerate diverse climatic and geographical conditions; it also hasmany end uses, including consumption by people and feed for livestock. We use the crop growth model

w xCERES-Maize 7 to simulate the relationship between climate and yield. A whole family of thesemodels is available to simulate the growth of other crops, such as wheat and soy beans. In reference tothe issue of adaptation discussed in footnote 3, the structure of these models is similar but the

w xperformance of CERES-Maize has been shown to be superior to the others 27 . We do consider alimited type of adaptation by allowing choice of crop variety but abstract from others such as fertilizerapplication, irrigation, and switching crops or activities.

6Our concept of variability allows correlation between temperature and precipitation. Evidence inour climate data shows significant negative correlation between precipitation and temperature in thesummer growing season. This implies that it is more often dry when it is hot, the conditions moststressful on plants. Negative correlation between these variables is a general climatic property ofcontinental interiors. For the whole world, the situation is just the opposite: warmer temperatures imply

Ž .a wetter climate. I thank Stephen Schneider personal communication for clarifying this point.7 Ž .In general, the welfare effects of climate change positive or negative depend on parameter values

that are described in the theory section of the paper. Estimates of these are given in the empiricalsection. We analyze the impact of climate change on two different crop populations. For one of these,the parameter estimates imply that some amount of warming is actually beneficial. It is straightforwardto draw a different set of isovalue curves such that welfare increases with movement along the ray OAin Fig. 1.

MICHAEL G. DALTON224

FIG. 1. The bias from omitting variability.

stable climate that is warm everyday and an unstable one that is extremely hot oneday and bitterly cold the next. These two patterns may have the same averagetemperature, but they are likely to have very different consequences for agricul-tural production. In fact, some researchers have suggested that the variability oftemperature is more important to agriculture than its mean value. To comparethese different climates, we measure welfare by a value function that depends onboth the mean and variance of climate.

In the figure, the horizontal axis measures a climatic mean m and the verticalaxis measures its variance s 2. We assume that the origin O represents the currentvalues of mean and variance, m and s 2. Welfare decreases for small changes in0 0

Ž 2 .the mean or increases in the variance. The value function V m, s is representedby its contour or isovalue curves: these represent indifference curves over the meanand variance of climate. We have drawn two of these indifference curves, V and1V . With the assumption that the initial climate is optimal, indifference curves2closer to the origin have higher value, illustrated by the arrow pointing toward theorigin on V . In particular, the value of V is greater than that of V .2 1 2

Models that abstract from climatic variability analyze changes in climatic meansonly. The evolution of climate change in these models is assumed to move alongthe horizontal axis, represented by the ray Om. For illustration, suppose thatclimate shifts from m to a new mean m from global warming. Models that ignore0 1

WELFARE AND CLIMATIC VARIABILITY 225

variability predict that welfare decreases from its initial value of V at the origin to0the value of V , the isovalue curve that intersects the horizontal axis at m .1 1Suppose that climatic variability also changes and that the relationship betweenmean and variance is given by the ray OA. Then, the shift of the mean to m is1accompanied by a shift in the variance to s 2. The value of the economy after this1

Ž 2 .shift of climate is V , the isovalue curve that goes through the point m , s . Since2 1 1the value of V is less than V , ignoring climatic variability in this example2 1underestimates the welfare effect from global warming by the amount V y V .1 2This difference can be translated into a climatic mean m that is associated with2the initial level of variance s 2; in this case, m is strictly greater than m .0 2 1

3. THE MODEL

In this section, we present an analytical model that generalizes Fig. 1. We usethe traditional stochastic growth framework to describe the economy. There is arepresentative household that has access to a constant returns production technol-ogy that uses capital and labor to produce a good that can be consumed orinvested. The productivity of capital and labor is subject to a random shock thathas a distribution which depends on climate.

3.1. Consumption

Consumption at time t, C G 0, is measured as per capita consumption of realtoutput by a representative household. Time is indexed in discrete units over thenon-negative integers by t. Since output is stochastic, consumption is representedby a sequence of random variables C , t G 0, and utility is measured by thetexpected value:

` 1yjC y 1ttE b N . 1Ž .Ý tž /1 y jts0

The variable N G 0 denotes the population level at time t. For convenience, wetnormalize the size of the population to unity. Including population growth andtechnological change is straightforward but does not affect our results; for clarity,we abstract from these. The parameter 0 - b - 1 is the discount factor whichmeasures patience or time preference of the household. The parameter j is theconstant rate of relative risk aversion. As j ª 1, the utility function converges tothe natural log of consumption. We use this fact below to derive analyticalsolutions for the model.8

3.2. Production

In addition to being consumed, real output can be invested in a stock of physicalcapital, measured by K G 0. Investment decisions are made at the end of date t,tbefore the climate shock at t q 1 is observed, and there is no intervening

8 w xEmpirically, it has been difficult to distinguish j and b 12 . Setting j s 1 is a way of normalizingthe relationship between these parameters.

MICHAEL G. DALTON226

Ž .uncertainty during storage, etc. . Then, the resource constraint describing feasibleallocations is

Y s C q K y 1 y d K . 2Ž . Ž .t t tq1 t

The parameter d is the depreciation rate of the physical capital stock and Y istoutput at date t. To simplify the analysis, we assume that the depreciation rate isequal to one so that the capital stock completely depreciates after being used.9

Production at date t is the product of a Cobb]Douglas production function thatuses inputs of capital K and labor N .10 Production is subject to a climate shockt t

Ž .W that is related to the level of production by a measurable function l ? which istdefined as

Y s l W K aN 1ya . 3Ž . Ž .t t t t

For the rest of the paper, we abstract from issues related to the supply of labor andassume that it is fixed at a single unit at each t. The W index climate and aretassumed to be a sequence of independently distributed random variables, each withmean m and variance s 2.11 Since we do not consider the control of globalt twarming in this paper, we assume that these parameters are exogenously deter-mined.

Ž .The general form of l ? is based on a single plant’s physiological development inw xcontinuous time 25 and is defined by

hŽW .tl W s e . 4Ž . Ž .t

˜Ž . Ž .The function h ? is assumed to be continuous hence measurable ; we appeal tothe Stone]Weierstrass Theorem and approximate this function with a polynomialof the form

Mmh W s r y g W . 5Ž . Ž .ÝM t m t

ms1

9A heuristic interpretation of the capital stock is that of seed for an annual crop. For d - 1,consumption increases; so, the welfare effects from climate change are smaller because of diminishingmarginal utility. Therefore, assuming d s 1 gives an upper bound on welfare effects which is consistentwith other assumptions that we make. Of course, this restriction biases our empirical results. Unfortu-nately, allowing d - 1 precludes closed form solutions. Numerical techniques for solving the stochastic

w xgrowth model in the case are well known 8 , but implementing one of these is beyond the scope of thispaper. An alternative, suggested by an anonymous referee, is to compensate for this restriction by usingother parameters in the model; this is discussed below.

10Our model economy is very abstract and our definition of capital includes all reproducible factors,the government sector, and land. We could separate inputs of energy, to endogenize global warming.Since this warming is mainly an external effect, assuming a large number of producers would create aprocess of climatic change that is exogenous to a representative decision maker’s optimization problem.Since we do not consider the control of greenhouse warming in this paper, we simplify the presentationby abstracting from this component of the problem.

11An index that we use for empirical work is water deficit. It is the difference between potentialŽ . Ževapotranspiration the amount of moisture a plant could use and actual evapotranspiration the

.amount that is actually available . Water deficit captures the interacting effects from both temperatureand moisture availability. The independence assumption is consistent with spectral analysis of ourclimate data which supports a white noise description of the W at the 95% confidence level based ont

w xFisher’s test 4, p. 284 . Independence can be relaxed using, for example, a Markov description of the W .tThis case precludes analytical solutions and numerical techniques, such as Judd’s projection methoddiscussed above, must be used to characterize solutions to the model. We anticipate that this type ofcorrelation, the nature of which is quite different that the bivariate correlation that we describe below,may be very important. This is a topic of ongoing research.

WELFARE AND CLIMATIC VARIABILITY 227

The parameters g describe the sensitivity of production to climate and r charac-mterizes maximum yield for given inputs of capital and labor.12 For illustration,suppose for the moment that each W G 0 and that each g G 0 with at least onet m

Ž . Ž .of the latter being strictly positive. Then for W equal to zero ideal conditions , l ?tis at its maximum value er; for large values of W , it is close to zero. In general,teach g can be positive or negative and the case of each being strictly positivemcorresponds to the situation in Fig. 1 where the initial climate is optimal. If any ofthe g are negative, then the initial climate need not be optimal; that is, them

Ž .maximum of l ? may not be W s 0. In this case, r does not correspond to themaximum yield and climate change may be beneficial. We focus on the quadraticcase with M s 2.13

Ž .We extend the form of 5 to a multivariate setting that includes a stochasticprocess for precipitation P and one for temperature T ; the climate index W ist t tassumed to be a function of these underlying processes. We focus on an analog of

Ž . 14the quadratic version of 5 :

T T 2t t

h P , T s r q a q b . 6Ž . Ž .t t PP' tt

In general, the random variables P , T , and W are assumed to be defined on thet t tsame probability space but have different distributions. To make this formulationoperational, we assume a joint distribution for temperature and precipitation ateach t defined by the probability density:

ln r2n r2y1f P , T s PŽ .

2 2'G nr2 2pf 1 y rŽ . Ž .2y1 T y t T y t'= exp 2lP q y 2 r 2lP . 7Ž .2 ž / ž /ž /ž /f f2 1 y rŽ .

Ž .G ? is the usual gamma function. The parameters l and n characterize themarginal distribution of precipitation, t and f characterize temperature, and rcharacterizes correlation between them. If precipitation and temperature areuncorrelated, that is r s 0, then P has a gamma distribution with mean nr2l and

12 For much of this paper, we assume that these parameters are exogenous to the decision maker’sproblem. A more general specification allows them to be determined endogenously. This couldrepresent adaptation by the model’s agents to changes in the parameters describing climate. Forexample, an important way for farmers to adapt to climate is by choosing among different crop varietiesand the values of r and the g depend on this choice. We consider this type of adaptation below for am

w xfinite number of crop varieties. Smulders 23 analyzes a stochastic growth model that allows agents toto trade off between an investment’s expected return and its variability with a continuous relationshipbetween these quantities.

13 This is the simplest case for which the variance of climate matters for welfare analysis. Empiricalresults from higher order cases are not significantly different than those from the quadratic case. This isdiscussed further below.

14 Ž . M N n m r2More generally, we consider the family h P , T s Ý Ý a T rP . The speci-M , N t t ms0 ns0 m , n t tŽ . Ž .fication in 6 is a member of this family. The remarks made above about the quadratic version of 5

are also relevant here. Using the more general form for analysis improves the fit of the model but doesnot significantly affect our welfare results.

MICHAEL G. DALTON228

variance nr2l2 and T has a normal distribution with mean t and variance f 2. Infact, the marginal distribution of P is gamma with the same parameters even if ris not zero; however, the marginal density of T is not normal in this case.

Ž .There are several issues relevant to the specification in 7 . First, its form isrelated to the bivariate normal distribution. We are considering seasonal aggre-gates of temperature and precipitation, and these aggregates can be thought of asthe accumulated sum of many daily trials. Therefore, appealing generally to aCentral Limit Theorem provides some justification for the normality assumptionsof both temperature and precipitation. Moreover, the gamma specification ensuresthe non-negativity of P. This is crucial for the full mass of our agents beliefs to beon the same set as that of the underlying process generating climate data, which isreasonable to avoid biasing the model in the direction of impossible events.15

Furthermore, the covariance of precipitation and temperature is important. Thereis evidence of significant negative correlation between P and T in our climatedata. This correlation drives our empirical results and we show that neglecting itcan substantially underestimate the effects of global warming.

3.3. Equilibrium

In this section, we maximize the expected utility of a representative householdsubject to the production constraints described above to study the welfare effectsof climate change. The allocation that solves this maximization, or planner’s,problem coincides with that of a competitive equilibrium for the decentralizedeconomy.16 The planner’s problem is

`tMax E b ln CŽ .Ý tž /� 4C , Kt tq1 ts0

8Ž .2ryg W yg W a1 t 2 ts.t. C q K s e K ,t tq1 t

2K given, W ; independent m , s .Ž .0 t t t

We characterize solutions to this problem by a value function and a policyfunction. Since the production shocks are independent over time, the capital stockis the only state variable. Because it allows us to derive analytical solutions to themodel, we focus on the case with j and d equal to one. For this case, investing a

w xconstant fraction of output is the optimal savings rule 24 . This fraction depends

15 We can, of course, violate this criteria with the excuse of approximation. Then, there is a direct linkbetween the model’s accuracy and its welfare results. We avoid this difficulty by maintaining thenon-negativity constraint for precipitation. Of course, the same difficulty exists for the temperatureprocess, which ignores the lower bound of absolute zero. Obviously, we think including the constraintfor precipitation is more important.

16 The full social optimum can be studied by endogenizing the climate change process with an energyinput as described in footnote 10. Here we are concerned with the ‘‘restricted’’ social planner’s problemwhere the climate change is exogenously determined. It is straightforward to show that the outcome ofthe restricted planner’s problem coincides with the market equilibrium for the ‘‘competitive’’ case with alarge number of myopic agents and is thus the appropriate object of study here.

WELFARE AND CLIMATIC VARIABILITY 229

on the parameters of the model describing technology and preferences but not onthe climate parameters m and s 2:17

t t

K s aberyg1Wtyg 2W 2t K a . 9Ž .tq1 t

The value function pictured in Fig. 1 is described by the following propositionwhich is proved in the Appendix.

PROPOSITION 1. Let the W be represented by a stationary process with parameterst2 Ž .m and s . Then for the ¨alue function V K that corresponds to the maximizationŽ .problem in 8 , there are functions A and B such that

V K s A a , b , K q B a , b r y g m y g s 2 q m2 . 10Ž . Ž . Ž . Ž .Ž .Ž .1 2

Ž .The first term on the right-hand side of 10 depends on the discount factor b ,the elasticity of output with respect to capital a , and the initial level of the capitalstock K. The second term depends on the climate parameters m and s . Changesin these parameters can raise or lower the value of the economy depending onvalues of g and g .1 2

The bias illustrated in Fig. 1 can be quantified by computing the differencebetween the value function before and after climate change. A literal interpreta-tion of this procedure, however, is that of a discrete, unexpected shift in climate.This raises questions about when and how this jump occurs. Because of discount-ing, the model’s welfare predictions are sensitive to these choices. To avoid thisdifficulty, we instead impose gradual, expected trends in the climate parametersthat are consistent with the current predictions from the scientific literature on thistopic. These trends are described by anticipated changes in temperature andprecipitation. To utilize these predictions, we use the distributional assumptions

Ž .above about the P and T and their relationship with the W given by 6 .t t tSince we have only limited information on the predicted dynamic paths of the

stochastic parameters that describe climate, we construct dynamic paths betweenthese points by linearly interpolating.18 The value function of this case is character-ized by the following proposition which is proved in the Appendix.

17 The policy rule is invariant to changes in climate. This version of the model exhibits a type ofcertainty equivalence which is similar to that exhibited by the better known model with quadratic utilityand linear constraints. By certainty equivalence, we mean that there is a deterministic analog of thestochastic model that has identical welfare properties. Since in this case the parameters that describethe model’s stochastic features affect its welfare properties but not policy, this version is appropriate forwelfare analysis only. To derive policy conclusions, a different version of the model must be used. Thereare many ways of doing this: one is to consider different values of j or d ; another is to allow correlationof climate over time; yet another is to endogenize the choice of r and the g by allowing adaptationmand adding more crops or crop varieties to the model. We do the last in a simple way below.

18 Typically, scenarios in the literature on global warming give only the predicted change inŽtemperature and precipitation for some reference event for example, the doubling of atmospheric

.carbon dioxide from its preindustrial level . Of course, nonlinear interpolations are also possible, butthe linear one is simplest for analysis. Also note that we are treating each scenario as given; that is, as ifit occurs with certainty, and we compare different scenarios directly. It would be straightforward tointroduce a probability distribution over scenarios to get around the apparent inconsistency of themodel’s agents having more information than its designers. This, however, would necessarily introducean additional subjective component into the model. Moreover, we have only a limited number of

Ž .scenarios less than a half dozen to compare and are interested only in welfare effects and not inoptimal policy responses; therefore, it is prudent to rank the different scenarios according to best andworst cases.

MICHAEL G. DALTON230

PROPOSITION 2. Let the ratios T r P be independent o¨er time, the P and T be't t t tŽ . Ž .jointly distributed according to 7 , and the t , f , n , l , r be linear interpolationst t t t t

Ž . Ž .between t , f , n , l , r at date t and t , f , n , l , r at date t . Then for the0 0 0 0 0 0 1 1 1 1 1 1Ž . Ž . Ž .¨alue function V K that corresponds to the maximization problem in 8 with 6

Ž .instead of 5 , there are functions A, D and G such that

V K s A a , b , K q D a , b , a, b , r , t , f , n , l , rŽ . Ž . Ž .0 0 0 0 0

y G a , b , a, b , t , t , f , f , n , n , l , l , r , r , t , t . 11Ž . Ž .0 1 0 1 0 1 0 1 0 1 0 1

The first term of this value function depends on the initial level of the capitalstock; the second and third terms are more relevant to our focus on climatechange. The second term represents the baseline value of the benchmark climate.The third term represents the effect on value from climate change. To assess themagnitude of this effect, we need estimates for all parameters in G, the topic ofthe next section. The following proposition links the value function in proposition 1to that in proposition 2 and is proved in the Appendix.

PROPOSITION 3. The ¨alue functions defined by propositions 1 and 2 are related by'the change of ¨ariable W s Tr P and

G n y 1 r2Ž .Ž . ' 'm s l t q 2l fr , 12Ž .G nr2Ž .

22 22lf 1 y r 2 G n y 1 r2Ž .Ž .Ž .2 2s s q lt y . 13Ž .ž /ž /n y 2 n y 2 G nr2Ž .

This proposition shows that the mean and variance of the climate index W arerelated in a very nonlinear way to the parameters of the underlying processes thatgenerate temperature and precipitation. The mean of the climate index m and itsvariance s 2 depend on the temperature parameters t and f, the precipitationparameters n and l, and the correlation parameter r. Therefore social welfare,measured by the value function, also depends on these parameters. In general awetter climate, represented by a larger value of n , decreases both m and s 2. Awarmer climate, represented by a larger value of t , increases both as long as n issufficiently large, which is within the range that is empirically relevant. The effectsfrom changes in f and r on m depend on the sign of r. Increases in f raise s 2

while increases in the absolute value of r decrease it. Next, we present estimatesfor all of the parameters in the model and relate predictions about global warmingto changes in those that characterize climate t , f, n , l, and r.

4. EMPIRICAL ANALYSIS

Table I presents four predictions of future climate from the recent literature onglobal warming. It gives projected changes in precipitation and temperature fromtheir current levels to those associated with a doubling of atmospheric carbon over

w xpreindustrial levels. The first scenario, which is used by Mendelsohn et al. 15 , is aw xforecast by the Intergovernmental Panel on Climate Change 6 . The remaining

Ž .scenarios are predictions from the following general circulation models GCMs for

WELFARE AND CLIMATIC VARIABILITY 231

TABLE IClimate Change Senarios

IPCC GISS GFDL OSU

Ž .Change in precipitation mmrday 8% 0.3 y0.2 0.1Ž .Change in temperature degrees Celsius 3.0 3.6 5.0 3.5

the summer growing season in the United States: Goddard Institute of SpaceŽ . Ž .Studies GISS , Geophysical Fluid Dynamics Laboratory GFDL , and Oregon

Ž .State University OSU . For a description of these models and their predictions,w xsee Gates 5 . It is reported there that the conditions for which these predictions

are valid occur in 50 years. The GFDL scenario, with higher temperatures and adecrease in precipitation, is the most severe of those in Table I. To derive an upperbound on welfare effects, we focus on this scenario in our analysis.

Estimates of some of the model’s parameters and their projected values withclimate change for the GFDL scenario are given in Table II. We consider twovalues for the discount factor b. The lower value of 0.95 is from a standard real

w xbusiness cycle model by Cooley and Prescott 2 , which has a structure similar toours, that is calibrated with data from the U.S. economy. The higher value of 0.985

w xis from Cline 1 , who argues that this choice is appropriate for economic studies ofglobal warming due to the long time horizons associated with climate change. The

w xvalue used by Nordhaus 17]19 in his DICE model is in the middle of our range.For the elasticity of output with respect to capital a , we analyze a broad range

of values to include various definitions of capital and to compensate for ourrestriction on the depreciation rate. For comparison, the value calibrated byCooley and Prescott using a ‘‘broad’’ definition of capital that includes thegovernment sector and inventories is 0.4. A survey of agricultural production

w xfunctions 26 reports values of a , interpreted in this context to include all inputsexcept labor, between 0.45 and 0.7. Larger values of a are relevant to our studybecause increasing the depreciation rate from its ‘‘true’’ value, which we havedone, is at least partially balanced by increasing a . However, we measure theeffects of global warming by the percentage change in the present value of theeconomy from its benchmark. For the family of parameter values that we consider,the absolute value of this change is largest for a between 0.5 and 0.55 and isapproximately symmetric around these values. This symmetry implies that focusing

TABLE IISome Parameter Values

Parameter Estimate

a 0.500, 0.999b 0.950, 0.985t 2867.830, 3440.600f 148.927, 148.990n 22.605, 19.683l 0.031, 0.029r y0.581, y0.582

MICHAEL G. DALTON232

analysis on the range above 0.5 is done without loss of generality since results fromthe range below 0.5 are approximately the mirror image of those above it.

Two values are reported for each of the climate parameters in Table II: the firstvalue is a benchmark derived from historical climate data and the second is aprojection based on the GFDL scenario. The benchmark estimates are functions ofthe sample statistics from a time series of precipitation and temperature data fromcentral Minnesota.19 The projected values of the climate parameters are obtainedby scaling the precipitation and temperature data up by the factors given for theGFDL scenario in Table I and following the same procedure used to derive thebenchmark estimates.

Ž .Estimates for a, b, and r in 6 are obtained by simulating harvests using thew xcrop growth model CERES-Maize 7 and the historical climate data. We consider

two different crop populations. The first, population S, consists of a single varietythat is now commonly planted in central Minnesota. The second, population A,consists of all varieties present in the CERES-Maize genetic library. We allow asimple form of adaptation by giving farmers perfect foresight for the cominggrowing season at the time of planting, but there is uncertainty when investmentdecisions are made, allowing them to choose the highest yielding variety for eachseason.20 The estimates of a, b, and r are the coefficients from a log-quadraticleast squares approximation of the climate]yield relationship embodied inCERES-Maize. An appropriate measure of fit for this approximation is R2. For thequadratic model, R2 is only about 0.35 for population S but is significantly higherfor population A.21 The estimated coefficients imply that the quadratic approxima-tion for population A is a convex parabola that opens upward and for population Sis a concave parabola that opens downward. In the former case, if W is initiallyclose to zero then increasing it lowers yields. In the latter case, increasing W raisesyields, showing that some amount of climate change could benefit population S.

An important aspect of climatic variability is the time scale. Our analysis stressesinterannual variability and we use a single growing season as a fundamental unit ofmeasurement. This corresponds to a crop life cycle with a single stage. However,

19 The data consists of daily maximum and minimum temperatures and precipitation. Samplestatistics from the climate data and their relation to the climate parameters are given in the Appendix.We experimented with other methods of estimation, including maximum likelihood, and derivedconfidence intervals based on our distributional assumptions. Our results are insensitive to variationwithin these intervals at a 99.9% level of confidence. We consider dryness at the beginning of a growingseason to have second order effects and do not explicitly account for it in this study. The estimates froma single location immediately introduces a problem of geographical scale. Minnesota is at the northernboundary of the agricultural region in the United States and may not be representative of the changesthat will take place at other locations. This motivates further empirical analysis. Including several

Žlocations with uncorrelated weather patterns would decrease aggregate variability in the limit, a law of.large numbers applies and aggregate variance goes to zero , perhaps lowering the predicted welfare

effects of climate change.20 More generally, there would be uncertainty about the weather in the coming growing season too.

This induces a more interesting planting decision with a portfolio choice component involving distribu-tions over the different varieties. This more general, and significantly more complex, problem is beyond

w xthe scope of this paper. See Dalton 3 for a more detailed discussion.21As discussed in the theory section of the paper, we can improve the fit of the crop growth

component of the model by using a higher order approximation. For example, a fourth orderapproximation has an R2 equal to about 0.64 for population S. However, welfare results from the higher

Ž .order models we analyzed a fourth order approximation of the form in footnote 14 are notsignificantly different than those from the quadratic case.

WELFARE AND CLIMATIC VARIABILITY 233

TABLE IIIEstimates of Crop Parameters

Parameter Population A Population S

y3 y4a y6.43 = 10 9.88 = 10y6 y6b 1.22 = 10 y8.91 = 10

r 10.47 9.112R 0.35 0.52

the sensitivity of a crop to climate does vary over its life cycle, making the effects ofintra-annual variability also important. To incorporate this, CERES-Maize parti-tions a crop’s life cycle into five different stages. Based on results from themultistage model, we double the negative coefficients in Table III and includeresults from doing so as an upper bound on sensitivity from intra-annual variability.As part of our analysis, we also halve these coefficients. A plausible interpretationof this is the effect of carbon dioxide fertilization on crop productivity. The welfareresults from this case and others are given in the next section.

5. RESULTS

Welfare effects for a range of settings based on the GFDL climate changescenario are given in Tables IV and V.22 With respect to Fig. 1, these differentsettings correspond to different values after climate change for the mean of theclimate index m and its variance s 2. Omitting variability restricts welfare analysis

Ž .to movements along the horizontal m axis in the figure. Including variabilityallows a more general evolution of climate, for example along the ray OA. Our

Ž . Ž .welfare measure is the ratio Gr A q D of the functions in 11 . Since the sumA q D is the baseline value of the economy with no climate change and G is the

TABLE IVŽ Ž ..Loss of Value Gr A q D for Population S

GFDL Dt Dn Dl Da Db

Cline’s bWith variability 1.17 0.64, 2.46 0.91, 1.88 1.28, 0.95 1.15, 1.09 0.27, 3.01Without variability 1.15 0.60, 2.49 0.92, 1.76 1.26, 0.94 1.13, 1.07 0.27, 2.97

Cooley and Prescott’s bWith variability 0.34 0.19, 0.71 0.26, 0.54 0.37, 0.28 0.34, 0.32 0.08, 0.87Without variability 0.33 0.17, 0.72 0.27, 0.51 0.36, 0.27 0.33, 0.31 0.08, 0.86

Ž .Note. All values in percentages % .

22 We normalize the size of the initial capital stock to unity. The implications of this assumption areclear from an inspection of the value function derived in the Appendix. A single term in the derivation

Ž . Ž . Ž .of A ? depends on capital: a ln K r 1 y ab . With a single unit of capital as the initial value, thisterm equals zero; for larger values, this term equals f 6.8, 13.6, 20.4, and 27.2 for starting capital stocks

3 6 9 12 Ž . 2 Ž .of 10 , 10 , 10 , and 10 , respectively. The remaining terms in A ? are on the order of 10 and in D ?

on the order of 103. Some simple algebra implies that the gross value of the implied bias in our welfareresults is then strictly less than 10y3.

MICHAEL G. DALTON234

TABLE VŽ Ž ..Loss of Value Gr A q D for Population A

GFDL Dt Dn Dl Da Da

Cline’s bWith variability 3.52 2.08, 6.38 2.87, 5.08 3.79, 2.96 3.47, 3.29 1.57, 8.05Without variability 3.51 2.01, 6.50 2.93, 4.91 3.79, 2.95 3.47, 3.29 1.57, 8.03

Cooley and Prescott’s bWith variability 1.02 0.60, 1.84 0.83, 1.47 1.10, 0.86 1.01, 0.97 0.45, 2.33Without variability 1.02 0.58, 1.88 0.85, 1.42 1.10, 0.85 1.01, 0.96 0.45, 2.32

Ž .Note. All values in percentages % .

absolute loss from global warming, this ratio is the percentage loss in value fromglobal warming.23

The percentage loss in value from climate change is given for both Cline’s andCooley and Prescott’s discount factor for two different structural versions of themodel. Values in rows with ¨ariability have the effects of climatic variabilityincluding correlation between precipitation and temperature. Values in rows with-out ¨ariability have the constraint that both f and r are equal to zero. This casecorresponds to the predictions of models that abstract from the effects of climaticvariability.24 Reference values appear in column one and the results of sensitivityanalysis are presented in the other columns. The reference case consists of climatechange from its benchmark description to that associated with the GFDL scenarioand, because of the empirical estimates and symmetry discussed in the previoussection, a s 0.5. Columns two, three, and four give values from varying the rate ofchange of the temperature parameter t and the precipitation parameters n and l,respectively. Variations in f and r are not presented as they are not significantlyaffected by the scaling procedure that generates our climate projections. The firstvalue in each of these columns corresponds to half the anticipated change for theparameter with the GFDL scenario and the second value to double the anticipatedchange. To cover the range of empirical estimates and to compensate for ourrestriction on the depreciation rate, column five gives values from increasing theelasticity of output with respect to capital a . The first value in this columncorresponds to a s 0.75 and the second value to 0.999. Recall that by symmetry,these results also apply to a in the range from 0.001 to 0.5. Column six gives valuesfrom varying the negative coefficient from the climate]yield approximations sum-marized in Table III. The first value in this column corresponds to half thereference value of the negative coefficient, representing decreased climatic sensi-

23 Ž .We summarize with this measure because it is given in percentage unitless terms and because it isw xconvenient in terms of the model’s structure. Nordhaus 19 summarizes the welfare effects from global

warming by measuring the change in the present value of consumption. Our measure is similar, but weŽuse the log of consumption instead reflecting our choice of utility function which is the same as

.Nordhaus and our model’s stochastic structure. These two measures can be compared. For a rangerelevant to our analysis, the percentage change of log-consumption is strictly greater than thepercentage change of consumption. To compare with Nordhaus, Table 1 in his paper reports the impactof climate change on the present value of consumption to be a 0.76% reduction relative to a policy of nocontrols and a 0.72% reduction relative to his optimal policy.

24 Ž .Analyzing the case with no covariability r s 0 but not f produces results almost identical to thecase with no variability.

WELFARE AND CLIMATIC VARIABILITY 235

tivity, and the second value is from doubling the reference value, increasingsensitivity. A plausible interpretation of the first value is decreased sensitivity dueto carbon dioxide fertilization and of the second value is increased sensitivity dueto intra-annual variability.

The loss of value from climate change for population S and the GFDL scenarioas a reference is predicted to be more than 1% for Cline’s discount factor andabout one-third of that for Cooley and Prescott’s.25 These predictions are within

Žthe standard range reported in economic studies of global warming see footnote.23 and at the lower end of studies that use empirical techniques similar to ours

w x10, 21 . The effect on value is about 2% greater with variability than without it.Aside from the large change in t and the small change in n , the model withvariability is also more sensitive to variation in the rates of climate change. This ismost noticeable for the large change in n : losses are about 7% greater withvariability than without it. Increasing a from 0.5 to 0.75 decreases the effect ofclimate change by about 2%, and increasing it from 0.75 to 0.999 decreases it byabout 5%.

Results for population A are given in Table V. The values in this table are largerthan those in Table IV; for example, the loss of value associated with Cline’sdiscount factor is about 3.5% compared with 1.17% for population S. This issurprising since population A allows some adaptation. It occurs because of theperfect foresight assumption and the fact that crop varieties with higher yields aremore sensitive to climate.26 This suggests that developed agricultural systems, withtheir reliance on higher yielding varieties, may actually be more sensitive to climatechange than systems that rely on varieties with lower but more stable yields. Asdiscussed in the previous section, the maximum or envelope of the many varietiespresent in population A is best approximated by a convex quadratic. This implies arelatively large reduction of yields for even small changes in climate. The analysiswith population A provides insight but may not be realistic because of the perfectforesight assumption. It measures the impact of climate change relative to abenchmark with yields that are too large compared to those that would be observedwith more plausible planting strategies. Like population S, population A is moresensitive to climate change with variability, but less so: losses with the large changein n are only about 3.5% compared with 7% above. Since population A allowssome adaptation, it makes sense that this population is relatively less sensitive tovariations in the rate of climate change.

6. CONCLUSIONS

This article examines the implications of climatic variability for welfare analysesof global warming. Our stochastic growth model demonstrates a clear relationship

25 The case with Cooley and Prescott’s discount factor and population S is the most optimistic ofw x Žthose in our empirical analysis. The results of Mendelsohn et al.’s 15 analysis is relevant here recall

.the remarks in footnote 3 in the Introduction . It is entirely possible that allowing a wider scope ofadaptation in our analysis, for example, to include different crops or industries, would completelyoutweigh the negative effects from global warming and could even result in net benefits.

26 Estimates of a, b, and r for several single varieties present in the CERES-Maize genetic libraryshow that there is a trade off between r and crop sensitivity to climate. That is, larger values of r areaccompanied by a negative coefficient of a or b that is larger in absolute value, suggesting that higheryielding varieties are more sensitive to climate.

MICHAEL G. DALTON236

between climatic variability and welfare. Our results show that abstracting fromclimatic variability may seriously bias predictions about the welfare effects fromglobal warming, possibly underestimating the costs of climate change. For the typeof cost]benefit analysis that has recently been used to analyze optimal controlstrategies, underestimating these costs leads to policy recommendations with toolittle control of warming.

To assess the magnitude of this bias, we do an empirical analysis. Our resultsyield several general insights. First, our analysis shows that losses from climatechange are predicted to be greater with variability than without it. Second, themodel with variability is more sensitive to variations in the rate of climate change.Third, we show that the impact of global warming on welfare depends on the scopeof adaptation and find that, surprisingly, systems that adapt can actually beaffected more than others that do not.

Further research is needed to include crop switching and other modes ofadaptation and to broaden the geographical range of the analysis. In addition,more research is needed to include other types of climatic variability, such as theseverity and frequency of storms and floods.

APPENDIX A: THE VALUE FUNCTION

Ž .The value function corresponding to the maximization problem in 8 is obtainedŽ . Ž . Ž 2 .by substituting 9 into the objective in 8 . Let e equal E g W q g W for the0 1 2

benchmark climate and e be the value of this expectation for the climate1associated with a doubling of atmospheric carbon dioxide over preindustrial levels.Interpolating the climatic trends implies that the final form of the value function is

ln 1 y ab a ln ab 1 1Ž . Ž .V K s q yŽ .0 ž /1 y b 1 y a 1 y b 1 y ab

a ln K r y e 1 aŽ .0 0q q yž /1 y ab 1 y a 1 y b 1 y ab

e y e b a a 11 0y q y . 14Ž .2 ž /ž /1 y a t y t 1 y a 1 y ab 1 y bŽ . Ž . 1 y bŽ .1 0

2 'Ž . Ž .Computing e and e requires evaluating E W and E W where W s Tr P .0 1The first two moments of W are calculated by integrating with respect to the

Ž .density in 7 . Let c be the constant term on that function; then, arranging terms,integrating with respect to a normal density, and integrating by parts implies

G n y 1 r2Ž .Ž . ' 'E W s l t q 2l fr , 15Ž . Ž .G nr2Ž .

2l2 2 2 2E W s t q f 1 y rŽ . Ž .Ž .

n y 2

'2 2 G n y 1 r2Ž .Ž . 2q ltfr q 2l fr . 16Ž . Ž .G nr2Ž .

WELFARE AND CLIMATIC VARIABILITY 237

APPENDIX B: THE CLIMATE DATA AND PARAMETER ESTIMATES

The climate data is a time series of daily maximum and minimum temperaturesŽand precipitation from 1890 through 1994 in the Twin Cities area Minneapolis and

.St. Paul of Minnesota and consists of daily precipitation amounts and maximumand minimum temperatures. The daily data during a 120 day summer growingseason is used to simulate corn harvests for each of the years in the data set.Sample statistics of the climate data are used to derive estimates of the climateparameters and are defined in terms of aggregate daily precipitation and aggregatedaily average temperatures over the growing season. The aggregate data set ismeasured in millimeters of precipitation per growing season and in degrees Celsiusper growing season. Benchmark statistics and statistics corresponding to each ofthe climate projections given in Table I are given in Table VI. The benchmarkstatistics are calculated using the aggregate sample data and statistics for each ofthe projected scenarios are obtained by scaling the daily climate data by theappropriate factors found in Table I. The statistics in this table are used toconstruct estimates for each of the climate parameters in the model from thefollowing equations which are derived by the same steps of integration laid outabove:

nE P s , 17Ž . Ž .

2l

'2 G n q 1 r2Ž .Ž .E T s t q fr , 18Ž . Ž .

G nr2Ž .n

V P s , 19Ž . Ž .22l

2G n q 1 r2Ž .Ž .22 2V T s f 1 y r q n fr y 2 fr , 20Ž . Ž . Ž .Ž . ž /G nr2Ž .

G n q 1 r2 frŽ .Ž .COV P , T s . 21Ž . Ž .' l2 G nr2Ž .

TABLE VISample Statistics

Scenario P T S S SP T P T

Benchmark 358.94 2460.58 106.77 135.60 y6466.76IPCC 387.66 2820.58 115.31 135.60 y6984.10GFDL 334.94 3060.58 } } }

GISS 394.94 2892.58 } } }

OSU 370.94 2880.58 } } }

Note. Dashed values equal the benchmark.

MICHAEL G. DALTON238

APPENDIX C: NOMENCLATURE

m climatic means 2 climatic varianceC consumption at date ttN population at date ttb discount factorj rate of relative risk aversionK capital stock at date ttd depreciation rate of capitalY output at date tta elasticity of output with respect to capitalW stochastic climate shock at date ttr intercept of relation between yield and climateg sensitivity coefficients of relation between yield and climatemP precipitation at date ttT temperature at date tta, b sensitivity coefficients of relation between yield, temperature and precipita-

tionn , l parameters of the marginal distribution for precipitationt measure of mean for temperaturef 2 measure of variance for temperaturer measure of covariance between temperature and precipitation

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