Chris Elbers
email: celbers@feweb.vu.nl

Jan Willem Gunning

Lei Pan

Insurance and Rural Welfare: What Can Panel Data Tell Us?

Assessing the scope for insurance in rural communities usually requires a structural model of household behavior under risk. One of the few empirical applications of such models is the study by Rosenzweig and Wolpin (1993) who conclude that Indian farmers in the ICRISAT villages would not benefit from the introduction of formal weather insurance. In this paper we investigate how models such as theirs can be estimated from panel data on production and assets. We show that if assets can take only a limited number of values the coefficients of the model cannot be estimated with reasonable precision. We also show that this can affect the conclusion that insurance would not be welfare improving.

Introduction

Insecurity is a key aspect of poverty. There now is substantial evidence that exposure to risk (in the absence of well-functioning financial markets) is often reflected in very large volatility of a household's consumption over time (e.g. Baulch and Hoddinott, 2000;Dercon, 2005). There still is little clarity on the policy implications of such churning. For example, Rosenzweig and Wolpin (1993) argue that rural households need micro credit, but Dercon (2005) favors formal insurance arrangements. While many insurance schemes are now being piloted (including rainfall insurance in Ethiopia and option contracts on coffee prices in several African countries) very little is known of their costs and benefits relative to existing risk coping institutions. To assess new policy initiatives in this area we need to know more about their welfare effects.

In choosing risk-coping strategies households face, of course, a trade-off between the mean and the volatility of their income. For example, a household can reduce the volatility of its income through crop diversification, but it will thereby lower its mean income by foregoing the gains from specialization. The household thereby pays an implicit risk premium and this must be taken into account in the evaluation of policy interventions, such as micro credit, price stabilization or formal insurance.

The evaluation should assess not only the change in volatility as a result of the intervention, but also the change in the (implicit) premium.

Ideally, the scope for insurance can be evaluated by comparing villages with and without insurance where treatment villages have been selected randomly and the floor under their consumption level. They found that the introduction of actuarially fair insurance (AFI) would not be welfare improving: households were already sufficiently protected through informal insurance. (Strictly speaking RW could draw no such conclusion. In their analysis a household that would switch from the informal arrangement to AFI would still have to pay the implicit premium of the informal arrangement. Clearly, this biases the comparison in favor of the status quo.) [START_REF] Rosenzweig | Wealth, Weather Risk and the Composition and Profitability of Agricultural Investments[END_REF] using the same data reached the opposite conclusion: the poorer ICRISAT farmers would benefit greatly from the introduction of insurance. Average profits in the bottom wealth quartile would increase by about one third for a reduction in weather risk of one standard deviation. However, their conclusion was not based on a dynamic model: a household's total wealth was taken as given. [START_REF] Elbers | Growth and Risk: Methodology and Micro Evidence[END_REF] estimated a structural model for smallholder households in Zimbabwe. They found a massive effect of the introduction of AFI: on average in their sample households would accumulate a capital stock (cattle) twice as large (over a 50-year period) as in the absence of insurance (when consumption smoothing is the only risk coping strategy available to them). As in RW this model is estimated on the basis of production and capital stock data but in this study the capital stock is a continuous variable.

The use of a discrete concept of the capital stock is intuitively appealing: cattle ownership is likely to be less subject to measurement error than, say, consumption data; the number of cows or bullocks is a discrete variable which is easily remembered. However, this advantage may well come at very high econometric costs, as we show in this paper. The question we investigate is whether the RW research agenda is feasible in the sense that robust conclusions about the welfare effects of insurance can be derived from a structural model estimated from production and capital stock data if the capital stock can take only a limited number of values. Our approach is to specify a simplified version of the RW model; to use the model as data generating mechanism; to find out how accurately we can recover the underlying behavioral parameters by estimation, given the values of all other parameters and given the true model specification; and, finally, to use the findings to assess the robustness of the policy conclusion on the desirability of the introduction of insurance.

The structure of the paper is as follows. In the next section we show under what conditions the use of limited dependent variables in this class of models may lead to large standard errors. In section 3 we specify the model and use simulation experiments to derive the distribution of the structural coefficients. We find, as expected, very large errors. It turns out that in this class of models the RW conclusion as to the welfare effects of insurance is not robust. Section 4 concludes. to the optimization problem can be characterized by an investment function , with

F

0 0 1 ˆ)) ( ( ˆk k k f k t t = = +
We will assume that the solution is unique. Note that the optimization problem is stationary, implying that does not depend on t .

There are several ways to extend the decision problem to a stochastic framework.

0 0 0 = = a w i t a k = +1 ˆ, if 1 ) ( + < i t i w k f w
Since the asset levels i a are integer-valued they do not carry local information about the behavioral parameters; the parameters must therefore be determined from the threshold values i w . The number of threshold values therefore determines the number of parameters of the behavioral process that can be estimated.

As an example take the case where k settles on a steady state value of 2 and all households have initial values 0 k below this number. Then the data will at best allow the researcher to determine the thresholds 1 w and 2 w . 4 Consequently only two parameters of the behavioral process can (normally) be estimated. Whether heterogeneity can indeed be used to solve the estimation problem is an open question. 6 RW in fact fix the value of one of the three behavioral parameters, the discount factor. Further, as we will show for a simplified version of their model, but with similar parameter values, heterogeneity introduced by pump ownership does little to improve the situation since conditionally on pump-less threshold values, behavioral parameters have almost no effect on pump-inclusive thresholds. Also, age-based heterogeneity has very little effect on investment behavior except for households with elderly heads; this gives such households inordinate weight in the estimation procedure.

We conclude that estimation of behavioral parameters underlying an accumulation process is inherently difficult if the asset involved can take (or is observed to take) only a few discrete values. The problem can be solved by putting additional constraints on the parameters, but any counterfactual analyses based on the estimated parameters must then be checked for robustness against such constraints. We also suggest that the identification problem can be solved if the asset becomes continuous.

The model

In the simplified RW model each agent solves:

= T o t t t c c E 1 )] 0 , [max
= = + + + ) ( ) ( 1 1 (1) 0 k is given ) ( 1 t t w k = + (2) b t b t b b t t p c w p c w p c c p w w 2 2 , 2 / 1 / 0 , 1 / 0 , 0 ) ( min min min min + > + < + < = (3)
where the agent's instantaneous utility function is characterised by the parameter Note from (1) and (3) that the insurance arrangement will pay out only to households without cattle: a household with min c c < will have to sell its cattle before it is entitled to consumption support since c is defined as the difference between wealth at hand ( w ) which includes the cattle asset and the investment.

The optimal policy function will depend on and P . Given these two parameters the function will involve two threshold values for wealth at hand, 1 w , 2 w such that in arrangement. It would be more natural to specify different parameters for these roles so that the welfare effect of a change in the protection offered by the insurance scheme can be analyzed.

F

Simulations set-up

We impose the following values (based on the RW estimates) for the three structural coefficients:

1469 min = c , 95 . 0 = , 964 . 0 = . We set 992 = k p , 6338 = p p , 0 0 = , 326 1 = , 1800 2 =
, 1795 = p (based on RW estimates and the value used in their paper). The distribution of is normal with zero mean and 2293 = .

Note that this is very high relative to the value of min c . The observation error is also normally distributed with zero mean and standard deviation 427 =

. The maximum plan horizon 0 T is 70 periods. We solve the investment thresholds w by using backward recursion which is discussed in Rust (1994) as the main solution method for finite-horizon models. With these parameter values the investment thresholds w are virtually insensitive to age for all but the oldest households (see Figure 1). We have therefore eliminated age heterogeneity, imposing = 0 T .

For these parameter values we find thresholds 2461 1 = w and 3453 2 = w for 8 There are two possible cases for the order of the thresholds. Besides the case used in this paper, the only possible case is the one mentioned in footnote 4. Since we do observe households with one bullock in the data, we exclude this case in this paper. given the values of all other parameters and the true underlying model specification.

Maximum Likelihood Estimation

Define

1 1 = t D if 1 = t k
, and zero otherwise. Similarly for

1 2 = t D if 2 = t k
. With this notation the likelihood contribution of a household is proportional to

t= 0 9 [Pr(k t +1 = 0 | wt ,k t ,P t ) 1 D 1,t +1 D 2,t +1 × Pr(k t +1 = 1 | wt ,k t ,P t ) D 1,t +1 × Pr(k t +1 = 2 | wt ,k t ,P t ) D 2,t
) ( ) 0 , , | 0 Pr( 0 1 1 = = = + t t t t t w P k w k (4)
Note that the likelihood depends only indirectly, through the threshold values w , on the parameters . Maximizing the likelihood therefore involves computing the threshold values as a function of . To reduce the computational burden we have calculated the thresholds on a grid of parameters . Linear interpolation13 of these threshold values on the grid values of is then used to approximate the threshold values for non-grid values of . The interpolated values turn out to be highly accurate approximations to the exact threshold values. We then substitute the interpolated threshold values

) ( ˆ w
in the likelihood function. For instance, in equation ( 2) we get

) ) ( ( ) 0 , , | 0 r( P ˆ0 1 1 = = = + t t t t t w P k w k
It is now straightforward to maximize the likelihood with respect to . In all cases we have used the true parameter values (

95 . 0 = , 964 . 0 = , 1469 min = c
) as initial point for the optimizing algorithm. Moreover, we have restricted parameters to the bounds imposed of the grid of parameters for which the true thresholds have been computed, i.e. intervals ) (0.59,0.99 for , 89) (0.899,0.9

for and (0,2997) for min c .

The bounds have been chosen to make the thresholds estimated precisely. Since 12 Note that

t k t p t t k p P k w = + ) ( ~.

Results

With the above choice of model and parameters it turns out that the threshold values 2 , 1 w are virtually insensitive to parameters and . Hence it is impossible to estimate these parameters with any accuracy. Given the typical size of household shocks and only 30 households, the influence of and on the likelihood function is almost absent. We find that tends to settle on one of the bounds and tends to stay very close to its initial value. What are the implications for the robustness of the conclusion that actuarially fair insurance would not raise welfare? We investigate this by calculating W , the net increase in welfare (i.e. the expected value of discounted utility) as a result of replacing the informal arrangement supporting min c with AFI. 14 We do this for each of six household types, defined by whether they have (initially) a pump and whether they start with 0, 1 or 2 bullocks. Table 1 shows W for each of these household types and for various values of in the 0.99) (0.59, range, and putting all other parameters to their true value. 15 The values shown in the Table are calculated under the assumption that no household will acquire a pump: the first three household types remain pumpless throughout.

The Table shows ) the policy conclusion is thereby fixed as well.

Conclusion

There is a renewed interest in insurance mechanisms to assist rural households in risk coping. It is rarely possible to evaluate such interventions through (quasi)

experimental evaluation methods. In principle estimating a structural model of household behavior under risk (using panel data) is a viable alternative. Rosenzweig and Wolpin (1993) applied this approach to the ICRISAT data and found that insurance would not be welfare improving. In this paper we have investigated the robustness of this policy conclusion. We have considered a class of models of household behavior under risk where assets can take only a small number of values. This severely restricts the scope for estimating structural coefficients. In the RW case heterogeneity could solve this problem but we have shown (for a simplified version of their model) that in small samples two of the coefficients cannot be estimated with reasonable precision in spite of this heterogeneity. Since the policy conclusion on the desirability of introducing formal insurance is sensitive to the value of these coefficients, the conclusion is probably not robust. such models is the study by Rosenzweig and Wolpin (1993) who conclude that Indian farmers in the ICRISAT villages would not benefit from the introduction of formal weather insurance. In this paper we investigate how models such as theirs can be estimated from panel data on production and assets. We show that if assets can take only a limited number of values the coefficients of the model cannot be estimated with reasonable precision. We also show that this can affect the conclusion that insurance would not be welfare improving. African countries) very little is known of their costs and benefits relative to existing risk coping institutions. To assess new policy initiatives in this area we need to know more about their welfare effects.

In choosing risk-coping strategies households face, of course, a trade-off between the mean and the volatility of their income. For example, a household can reduce the volatility of its income through crop diversification, but it will thereby lower its mean income by foregoing the gains from specialization. The household thereby pays an implicit risk premium and this must be taken into account in the evaluation of policy interventions, such as micro credit, price stabilization or formal insurance.

The evaluation should assess not only the change in volatility as a result of the intervention, but also the change in the (implicit) premium. Households can also use consumption smoothing by accumulating or decumulating assets to cope with Here the implicit premium is the loss of production as a result of using assets for this purpose.

Ideally, the scope for insurance can be evaluated by comparing villages with and without insurance where treatment villages have been selected randomly and the distribution of shocks is known. Alternatively, if locations differ in risk exposure this can be exploited as a natural experiment to infer how behavior would change under insurance. Clearly, the villages must then be similar in all other respects.

Matching or differencing offers some scope for relaxing this requirement but only if all relevant variables are observed. In practice unobserved heterogeneity is likely to be a major problem. In that case there is no alternative to estimating a structural model, using panel data. Such a model can be used to derive how agents would behave if insurance were available, even if no agent had been observed in that situation. Obviously, this requires estimation of the model's structural parameters.

An important advantage of this procedure is that the researcher does not need to know the distribution of the shocks to which the households are exposed;

estimates of the parameters characterizing that distribution will be generated as part of the estimation procedure. Lucas (2003) suggested on the basis of a back-of-envelope calculation that insurance could not have a substantial effect on growth. However, he considered a situation with much less risk than is common in many developing countries.

Unfortunately, for developing countries there are few empirical studies.

A notable exception is the famous paper by Rosenzweig and Wolpin (1993) (henceforth RW). They estimated a structural model using the ICRISAT data on average in their sample households would accumulate a capital stock (cattle) twice as large (over a 50-year period) as in the absence of insurance (when consumption smoothing is the only risk coping strategy available to them). As in RW this model is estimated on the basis of production and capital stock data but in this study the capital stock is a continuous variable.

In many rural economies the key asset of a household is livestock, e.g. one or two bullocks. The use of a discrete concept of the capital stock is therefore a natural choice. In addition, it is well-known that households recall their cattle ownership to the optimization problem can be characterized by an investment function , with

0 0 1 ˆ)) ( ( ˆk k k f k t t = = +
We will assume that the solution is unique. Note that the optimization problem is stationary, implying that does not depend on t .

There are several ways to extend the decision problem to a stochastic framework.

One possibility is to allow for random shocks in ) (k f and to maximize the expected and corresponding asset levels i a so that

0 0 0 = = a w i t a k = +1 ˆ, if 1 ) ( + < i t i w k f w
Since the asset levels i a are integer-valued they do not carry local information about the behavioral parameters; the parameters must therefore be determined from the threshold values i w . The number of threshold values therefore determines the number of parameters of the behavioral process that can be estimated.

As an example take the case where k settles on a steady state value of 2 and all households have initial values 0 k below this number. Then the data will at best allow the researcher to determine the thresholds 1 w and 2 w . 5 Consequently only two parameters of the behavioral process can (normally) be estimated.

More information can be obtained if there is more heterogeneity between observed cases. For instance, if the time horizon of the optimization is finite (say, until the death of the agent) and agents differ in age, then the investment function and the threshold values i w become age-dependent. Also, the production function

) (k f
could differ across households, again leading to multiplication of observable threshold values. If such heterogeneity affects some (but not all) of the parameters it can be exploited in pooled estimation.

5 It is possible that We conclude that estimation of behavioral parameters underlying an accumulation process is inherently difficult if the asset involved can take only a few discrete values. The problem can be solved by putting additional constraints on the parameters, but any counterfactual analyses based on the estimated parameters must then be checked for robustness against such constraints. We also suggest that the identification problem can be solved if the asset becomes continuous.

6 Rust (1994) provides detailed discussion of the identification of dynamic discrete choice models, presenting sufficient conditions under which the model can be identified and a simple and general estimation theory is available. One of the restrictions required in his discussion is that shocks enter into the utility function additively, which is violated in the RW model. As Rust (1994) mentions, no general estimation theory exists for this case. For our simulations we use a simplified version of the RW model. 7 We first describe the model and then discuss the simulations.

The model

In the simplified RW model each agent solves:

= T o t t t c c E 1 )] 0 , [max( max 1 min subject to: t t k t p t t t t p t k t k p P k w P P p k p c + + + = = + + + ) ( ) ( 1 1 (1) 0 k is given ) ( 1 t t w k = + (2) t k k t k k t t w p c p c w p c c p w w + + < + < = 2 2 , 2 / 1 / 0 , 1 / 0 , 0 ) ( min min min min (3)
where the agent's instantaneous utility function is characterised by the parameter 0 > ( 1 ), is the discount factor, min c is a minimum consumption level supported by an informal insurance arrangement, 8 c is the level of consumption 7 In this model consumption smoothing is the only coping mechanism. [START_REF] Maitra | Is Consumption Smooth at the Cost of Volatile Leisure? An Investigation of Rural India[END_REF] finds evidence of another coping mechanism, changes in labour supply. 8 In this RW specification min c has unfortunately a dual role: it is the level of consumption below which a household cannot survive and also the level guaranteed by the informal insurance arrangement. It would be more natural to specify different parameters for these roles so that the welfare effect of a change in the protection offered by the insurance scheme can be analyzed. Note from (1) and (3) that the insurance arrangement will pay out only to households without cattle: a household with min c c < will have to sell its cattle before it is entitled to consumption support since c is defined as the difference between wealth at hand ( w ) which includes the cattle asset and the investment.

F

The optimal policy function will depend on and P . Given these two parameters the function will involve two threshold values for wealth at hand, 1 w , 2 w such that in each period the household chooses

1 = k for 2 1 w w w < , 2 = k for 2 w w and 0 = k otherwise 9 .
9 There are two possible cases for the order of the thresholds. Besides the case used in this paper, the only possible case is the one mentioned in footnote 4. Since we do observe households with one bullock in the data, we exclude this case in this paper.

Simulations set-up

We impose the following values (based on the RW estimates) for the three Note that this is very high relative to the value of min c . The observation error is also normally distributed with zero mean and standard deviation 427 =

. The maximum plan horizon 0 T is 70 periods. We solve the investment thresholds w by using backward recursion which is discussed in Rust (1994) as the main solution method for finite-horizon models. With these parameter values the investment thresholds w are virtually insensitive to age for all but the oldest households.

We given the values of all other parameters and the true underlying model specification.

Maximum Likelihood Estimation

Define

1 1 = t D if 1 = t k
, and zero otherwise. Similarly for

1 2 = t D if 2 = t k
. With this notation the likelihood contribution of a household is proportional to

t= 0 9 [Pr(k t +1 = 0 | ˜ w t ,k t ,P t ) 1 D 1,t +1 D 2,t +1 × Pr(k t +1 = 1 | ˜ w t ,k t ,P t ) D 1,t +1 × Pr(k t +1 = 2 | ˜ w t ,k t ,P t ) D 2,t +1 ]

Results

With the above choice of model and parameters it turns out that the threshold values 2 , 1 w are virtually insensitive to parameters and . Hence it is impossible to estimate these parameters with any accuracy. We find that tends to settle on one of the bounds and tends to stay very close to its initial value. Figure 1 experimental evaluation methods. In principle estimating a structural model of household behavior under risk (using panel data) is a viable alternative. Rosenzweig and Wolpin (1993) applied this approach to the ICRISAT data and found that insurance would not be welfare improving. In this paper we have investigated the robustness of this policy conclusion. We have considered a class of models of household behavior under risk where assets can take only a small number of values. This severely restricts the scope for estimating structural coefficients. In the RW case heterogeneity could solve this problem but we have shown (for a simplified version of their model) that in small samples two of the coefficients cannot be estimated with reasonable precision in spite of this heterogeneity. Since the policy conclusion on the desirability of introducing formal insurance is sensitive to the value of these coefficients, the conclusion is probably not robust. This does not mean that we cannot use panel data to assess the scope for insurance. Rather, it implies that if asset data indeed take only a small number of values (relative to the number of parameters to be estimated) then estimation

  in an informal insurance arrangement which established a

  methods. They assume a finite decision horizon, leading to heterogeneity in asset holding policies across decision makers of different age.Irreversibility of the installation of a pump leads to a shift in the production function for pump owners 5 .

  we use a simplified version of the RW model. We first describe the model and then discuss the simulations.

  , and min c should be close to their true values if they could be estimated from the model the choice of different bounds do not make difference to the estimation of the parameters.

Figure

  hand, min c can be estimated fairly accurately. Figure3displays the sampling distribution of the estimator for min c . The mean and standard deviation determined from 100 simulations are 1444.7 and 80.3.Recall that the heterogeneity in pump ownership raises the number of thresholds above the number of parameters so that the problem discussed in section 2 of having to recover three coefficients from two threshold values does not arise.However, it turns out that the extra pair of thresholds do not convey sufficient additional information: as may be seen from Figures 4 and 5 the thresholds for pump-owning households are almost insensitive to the behavioral parameters given the corresponding thresholds for the other households. Hence the second convey little extra information on the parameters. In fact, we find that the thresholds are almost exclusively determined by the value of min c , so that the combined figures trace essentially a one-dimensional sub-set in the space of threshold values. This is why only a single parameter can be estimated with reasonable precision.

Figure

  Figure 1: Investment thresholds 1 _ w (below) and 2 _ w (above) for households without a pump.

1

  See e.g.Tanner (1997) for US evidence andElbers et al. (forthcoming) for Zimbabwe evidence. Indian villages. The model describes investment behavior under risk where investment in bullocks is the key decision variable. Bullock ownership can take three values: 0, 1 or 2. The model is estimated from data on production and bullock ownership, but not on consumption. RW stressed that the villagers in their sample participated in an informal insurance arrangement which established a floor under their consumption level. They found that the introduction of actuarially fair insurance (AFI) would not be welfare improving: households were already sufficiently protected through informal insurance.[START_REF] Rosenzweig | Wealth, Weather Risk and the Composition and Profitability of Agricultural Investments[END_REF] using the same data reached the opposite conclusion: the poorer ICRISAT farmers would benefit greatly from the introduction of insurance. Average profits in the bottom wealth quartile would increase by about one third for a reduction in weather risk of one standard deviation. However, their conclusion was not based on a dynamic model: a household's total wealth was taken as given. Elbers, Gunning and Kinsey (forthcoming) estimate a structural model for smallholder households in Zimbabwe. They found a massive effect of the introduction of AFI:

  methods. They assume a finite decision horizon, leading to heterogeneity in asset holding policies across decision makers of different age.Irreversibility of the installation of a pump leads to a shift in the production function for pump owners.Whether heterogeneity can indeed be used to solve the estimation problem is an open question.6 RW in fact fix the value of one of the three behavioral parameters, the discount factor. Further, as we will show for a simplified version of their model, but with similar parameter values, heterogeneity introduced by pump ownership does little to improve the situation since conditionally on pump-less threshold values, behavioral parameters have almost no effect on pump-inclusive thresholds. Also, age-based heterogeneity has very little effect on investment behavior except for households with elderly heads; this gives such households inordinate weight in the estimation procedure.

  RW estimates and the value used in their paper). The distribution of is normal with zero mean and 2293 = .

  , and min c should be close to their true values if they could be estimated from the model the choice of different bounds do not make difference to the estimation of the parameters.

Figure 1 :

 1 Figure 1: Sampling Distribution of Behavioral Parameters ( , )

Page 12 of 50 Editorial

  The set-up of the simulations is as follows. Each of thirty households is endowed with exogenous values 0 k and t P . Next we generate a series of eight 10 shocks t and measurement errors t for each household. The shocks are independent across households. 11 The measurement errors are applied to the true values t w to generate the 'observed' values t w ~. The dataset now consists of a vector

						Submitted Manuscript
	households without a pump and again	1 = P w	2461	and	w	2 = P	3453	for households
	owning a pump. 9
	{ k	1	, w 1	,...,	k	F o r } , | , 0 t s s P k w for each household. The maximum likelihood estimates of
	P the three behavioral parameters	, ( min	,	)
						e e r
							R e v i e w
							12

Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK c = are derived for this data set (with all other parameters set at their true values). We then generate a new data set and a new set of estimates of . By repeating this procedure of data generation and estimation many times we generate the sampling distribution of the -estimators,

Table 1 :

 1 relative to the number of parameters to be estimated) then estimation requires heterogeneity which (unlike the heterogeneity allowed for by RW) leads to independent variation in threshold values. The procedure we have described can easily establish whether this condition is satisfied. Another way to solve the problem is by treating asset variable(s) continuous, e.g., by us ing livestock (an aggregate of cattle, goats, sheep etc.) rather than bullocks as the asset variable. Welfare Effect of Insurance

	of values (Submitted Manuscript
	F o
	r
	P
	e e r
	R e v i e w
	18

This does not mean that we cannot use panel data to assess the scope for insurance. Rather, it implies that if asset data indeed take only a small number Page 19 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Page 21 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Page 22 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o
	r
	P
	e e r
	R e v i e w
	22

Page 30 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  An implication is that observing the number of cattle is likely to be less subject to measurement error than, say, consumption. However, this advantage may well come at very high econometric costs, as we show in this paper. The question we investigate is whether the RW research agenda is feasible in the sense that robust conclusions about the welfare effects of insurance can be derived from a structural model estimated from production and capital stock data if the capital stock can take only a limited number of values. Our approach is to specify a simplified version of the RW model; to use the model as data generating mechanism; to find out how accurately we can recover the underlying behavioral parameters by estimation, given the values of all other parameters and given the true model specification; and, finally, to use the findings to assess the robustness of the policy conclusion on the desirability of the introduction of insurance.

	Submitted Manuscript
	F o 0 , t k o 2 Limited Dependent Variables t c Consider the following deterministic intertemporal optimization problem = o t t t k c c u t t ) ( max , subject to the constraints quite accurately. F ) ( 1 = + + t t t k f k c
	r r
	P e e r P where c is consumption, k the capital stock, 0 k given, e r e and a discount factor. The usual interpretation of the model is that output ) (c u the instantaneous utility function
	R e v i e w can be consumed or used as input into next period's production. Note the R absence of a depreciation term ) ( t k f k ) 1 ( : to get a more compact notation the e production function ) (k f represents the real value of output plus the after-v i w e production value of the asset. A solution t k ˆ, ,... 1 = t
	4 5

The structure of the paper is as follows. In the next section we show under what conditions the use of limited dependent variables in this class of models may lead to large standard errors. In section 3 we specify the model and use simulation experiments to derive the distribution of the structural coefficients. We find, as expected, very large errors. It turns out that in this class of models the RW conclusion as to the welfare effects of insurance is not robust. Section 4 concludes.

Page 32 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  because no more than n values of k are observed. In the stochastic case (with

					Submitted Manuscript
	value of summed discounted utility with respect to investment policies. If the
	shocks are serially independent the optimal policy is again an investment function ) (k f subject to shocks) output is a continuous variable while k is discrete. The
	of the form investment function is now a step function characterized by threshold values i )) ( ( k f . w
	In the deterministic case the functions	f	(	)	and	) ( can be identified from a
	sufficient number of different 0 k observations and subsequent observations on the
	capital stock and output. (Note that there is no need to observe consumption since
	F o it follows from 1 t t k c = + (subject to shocks) and t f + k ˆis sufficient to recover the behavioral parameters of the ) ( t k .) The question arises whether observation of ) ( t k f F o
					r r
	P e e r process, namely the discount factor P affirmative: by integrating the Euler conditions for an optimal accumulation path we and the parameters of ) (c u . The answer is can normally recover as well as the utility function (over the relevant part of its domain and up to an affine transformation). 2 Knowledge of the behavioral e e r
	R e v i e w parameters allows us to study counterfactual situations, such as the introduction of insurance. 3 Now suppose that t k is restricted to a limited number ( n ) of integer values. 4 Recovering the behavioral parameters of the model now becomes problematic. For R e v i e w
	example, in the deterministic case since there is a one to one mapping from t k to
	t k ˆand	f	( t k	)	can only take n numbers, if the number of behavioral parameters
	exceeds n they cannot be recovered, irrespective of the number of observations
					6 7

  takes the values 0 , 1 , 2 for = k 0, 1, 2 respectively; this picks up the effect of bullock ownership on income. The productivity of pumps is captured by p . The age of the household head (which determines the time remaining until time T ) is denoted by . In this formulation investment in pumps (which is irreversible) is exogenous (contrary to the RW formulation). Agents without a pump do not expect to acquire one. Expected utility is maximized with respect to the

	before any support from the insurance arrangement, w is wealth at hand (available
	for consumption and investment), k is the capital stock (constrained to take the
	values 0, 1 or 2) with price k p , P is a dummy variable indicating whether the
	household owns a pump (with price p p ) and is an income shock (a draw from a
	distribution known by the agent). Income shocks are independent over time. The
	o r P e e r (k policy function function ) .
	R e v i e w
	10

Page 37 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	Like RW we assume that 0 k , , P and the capital stock t k are observed without
	error, but that instead of t w the researcher observes	t ~, where t is t t w w + =
	measurement error, i.i.d. and independent of household shocks t .
	F o
	r
	P
	e e r
	R e v i e w
	11

  have therefore eliminated age heterogeneity, imposing =

	The set-up of the simulations is as follows. Each of thirty households is endowed
	with exogenous values 0 k and t P . Next we generate a series of eight 11 shocks t
	and measurement errors t for each household. The shocks are independent
	across households. 12 The measurement errors are applied to the true values t w to
	generate the 'observed' values t w ~. The dataset now consists of a vector
	{ k	1	, w 1	,...,	k	s	, w	s	|	k	0	,	t P	}	for each household. The maximum likelihood estimates of
	the three behavioral parameters F o r other parameters set at their true values). We then generate a new data set and a ) , , ( min c = are derived for this data set (with all
	new set of estimates of . By repeating this procedure of data generation and P r e e estimation many times we generate the sampling distribution of the -estimators,
																R e v i e w
																0 T	. For these
	parameter values we find thresholds	1 = w	2461	and	w	2 =	3453	for households
	without a pump and again	1 = P w	2461	and	w	2 = P	3453	for households owning a
	pump. 10
																12

  In the Table we show a particular distribution. Under this distribution the change in aggregate welfare depends on the value of the discount factor. For low values of (i.e. a high discount rate) the net effect is negative, for values of 0.79 or higher it is positive: AFI would be accepted. There is a renewed interest in insurance mechanisms to assist rural households in risk coping. It is rarely possible to evaluate such interventions through (quasi)

	the combined figures trace essentially a one-dimensional sub-set in the space of arrangement. For richer households this advantage of the min c arrangement is
	threshold values. This is why only a single parameter can be estimated with offset by a larger difference between min c and mean consumption: the informal
	reasonable precision. arrangement gives them little downward protection. As a result, households with
	What are the implications for the robustness of the conclusion that actuarially fair pumps or with two bullocks would switch to AFI. Clearly, the aggregate effect
	insurance would not raise welfare? We investigate this by calculating W , the net depends on the distribution of the population over household types and on the 4 Conclusion
	is a sampling distribution, based on 100 c can be estimated fairly accurately. The mean and standard typical scatter plot of the joint ) , ( simulations. On the other hand, min deviation determined from 100 simulations are 1444.7 and 80.3. Recall that the heterogeneity in pump ownership raises the number of thresholds above the number of parameters so that the problem discussed in section 2 of having to recover three coefficients from two threshold values does not arise. However, it turns out that the extra pair of thresholds do not convey sufficient F o r P e e r R e v i e w increase in welfare (i.e. the expected value of discounted utility) as a result of replacing the informal arrangement supporting min c with AFI. 15 We do this for each of six household types, defined by whether they have (initially) a pump and whether they start with 0, 1 or 2 bullocks. Table 1 shows W for each of these household types and for various values of in the 0.99) (0.59, range, and putting all other parameters to their true value. The Table shows that the first two types of households (those who initially have no pump and at most one bullock) would not be willing to give up the informal insurance arrangement (which guarantees a consumption level min c ) in exchange for AFI. The reason is simple: under AFI they would no longer enjoy the positive income shocks which they experience under the (asymmetric) informal insurance F o r P e e r R e v i e w F o r P e e r discount factor. switching to insurance then welfare gains might be positive even for the first two R e v i e w
	additional information: as may be seen from Figures 2 and 3 the thresholds for types of households.
	pump-owning households are almost insensitive to the behavioral parameters Since cannot be estimated with precision, Table 1 implies that any policy
	given the corresponding thresholds for the other households. Hence the second conclusion on the welfare effect of introducing formal insurance will not be robust.
	pair of thresholds convey little extra information on the parameters. In fact, we find If is estimated (in a relatively small sample) it may easily settle on one of the two
	that the thresholds are almost exclusively determined by the value of min c , so that
	15 16

16

The values shown in the Table

are

calculated under the assumption that no household will acquire a pump: the first three household types remain pumpless throughout. This example underestimates the case for AFI, for two reasons. First, as noted above, under AFI the implicit premium of the informal insurance would no longer have to be paid but this is not taken into account. Secondly, we have treated pump investment as exogenous. In the RW world pump investment is endogenous. A pumpless household might receive (at some future date) a positive shock large enough to enable it to buy a pump. At that stage insurance would become attractive. Therefore, if households were not forced to adopt insurance now or stay with the informal arrangement forever, but were instead offered the option of

Page 44 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  18 requires heterogeneity which (unlike the heterogeneity allowed for by RW) leads to independent variation in threshold values. The procedure we have described can easily establish whether this condition is satisfied. An alternative is to treat assets as continuous variables, e.g ., by using livestock (an aggregate of cattle, goats, sheep etc.) rather than bullocks as the capital stock. Rosenzweig, Mark R. and Hans Binswanger (1993) Wealth, Weather Risk and the Composition and Profitability of Agricultural Investments, Economic Journal, vol. 103, pp. 56-78. Rosenzweig, Mark R. and Kenneth Wolpin (1993) Credit Market Constraints, Consumption Smoothing, and the Accumulation of Durable Production Assets in Low-Income Countries: Investment in Bullocks in India, Journal of PoliticalEconomy, vol.101, pp. 223-244. Rust, John (1994) Structural estimation of Markov decision processes, in Handbook ofEconometrics, vol. 4, ed. by R. Engle and D. McFadden, Amsterdam: Applied Economics, vol. 29,

	Submitted Manuscript P o F r e e r R e v i e w Submitted Manuscript F o r P e e r R e v i e w F o r P e e r R e v i e w North Holland. Tanner, Evan (1997) Shifts in US Savings: Long-run Asset Accumulation versus F o r P e e r Consumption Smoothing, Submitted Manuscript R e v i e w
	19 20 21

Page 45 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Page 47 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 :

 1 Welfare Effect of Insurance

			Submitted Manuscript			Page 48 of 50
	Endowment
	initial k	0	1	2	0	1	2
	pump	0	0	0	1	1	1
	discount							weighted
	factor							effect
	.590	-25.8	-35.6	27.5	26.0	17.7	8.5	loss
	.615	-27.8	-37.6	30.4	27.0	18.6	9.4	loss
	.640	-30.0	-40.0	33.8	28.1	19.7	10.4	loss
	.665	-32.5	-42.7	37.7	29.4	20.9	11.6	loss
	.690	-35.5	-45.8	42.3	30.8	22.3	13.0	loss
	.715	-39.0	-49.5	47.8	32.4	23.9	14.7	loss
	.740	-43.2	-53.9	54.5	34.2	25.8	16.7	loss
	.765	-48.4	-59.2	62.6	36.3	28.0	19.1	loss
	.790	-54.7	-65.7	72.8	38.9	30.8	22.2	gain
	.815	-62.8	-74.1	86.0	42.1	34.2	26.2	gain
	.840	-73.5	-85.0	103.4	46.2	38.7	31.5	gain
	.865	-88.2	-99.8	127.4	51.5	44.8	38.8	gain
	.890	-109.6 -121.5 162.7	59.1	53.5	49.6	gain
	.915	-143.5 -155.6 218.7	70.8	67.1	66.7	gain
	.940	-203.7 -216.0 318.6	91.1	91.2	97.3	gain
	.965	-328.8 -341.4 526.8	132.8	141.1	161.6	gain
	.990	-640.5 -653.4 1047.0 236.6	266.2	323.6	gain
	weights	1.8	1	1	1	1	1

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

We do not measure the effect of irreversibility in this paper but only keep this assumption to be in line with RW.

Rust (1994) provides detailed discussion of the identification of dynamic discrete choice models, presenting sufficient conditions under which the model can be identified and a simple and general estimation theory is available. One of the restrictions required in his discussion is that shocks enter into the utility function additively, which is violated in the RW model. As Rust (1994) mentions, no general estimation theory exists for this case.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Note that the threshold values are insensitive to pump ownership.

Eight is also the number of periods in RW. RW do not mention the number of households, but the ICRISAT data set contains 30 medium-size households with two or more observations. RW use the data from this medium-size group of households.

This is, of course, a simplification; in practice there will be covariance, e.g. in the case of weather shocks.

The interpolation method used is a trivariate version of the "Four Point Formula" inAbramowitz and Stegun (1972, p. 882).

Recall that this measure is biased since (as RW recognise) the cost of the informal arrangement would be reflected in a premium which would not have to be paid under AFI. Since RW do not know this cost (which is reflected in

Elbers and Gunning (2002).

From a positive economics perspective this is the very reason for writing accumulation as an optimization problem instead of being satisfied with a purely descriptive function .

This is similar to the situation studied by RW where asset ' k ' is in fact a vector of stocks: bullocks, pump and calves of various ages, all integer valued. The number of bullocks can take only three values: 0, 1 and 2.

Eight is also the number of periods in RW. RW do not mention the number of households, but the ICRISAT data set contains 30 medium-size households with two or more observations. RW use the data from this medium-size group of households.

This is, of course, a simplification; in practice there will be covariance, e.g. in the case of weather shocks.

Recall that this measure is biased since (as RW recognise) the cost of the informal arrangement would be reflected in a premium which would not have to be paid under AFI. Since RW do not know this cost (which is reflected in

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript