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Vasectomy bonuses and completed family size in India
Journal of Public Economics 4 (1975) 319-334. 0 North-Holland
VASECTOMY
BONUSES AND COMPLETED IN INDIA A. MARKANDYA
Publishing Company
FAMILY SIZE
and N.J. RAW
University College, London, England Received April 1973, revised version received October 1974 This paper discusses the consequences in rural India of a certain rational household decision function on the number of children born in the family when birth control opportunities are available. Households are assumed to choose whether or not to practice birth control after each child is born on the basis of maximising expected utility as a function of the consumption per head and the number of members in the family. Uncertainty arises on account of the different evaluation of children of different sexes. The model is then used to predict how completed family size would change if a government agency paid bonuses for smaller families.
1. Introduction This paper attempts to measure the impact on completed family size in India of the availability of vasectomy, when account is taken of some of the variables that are considered to be relevant. The study is divided into three parts. In the first part we calculate the natural completed family size in the absence of birth control techniques. By absence here is meant either the lack of availability or the lack of knowledge and confidence in the technique, or both. In the period 1950-1960, which is the decade from which the data for this study is taken, this situation is assumed to be approximately correct. In the second part we assume that vasectomy is widely available, with no bonuses or charges attached to it, and that the persons involved know about its availability and are making their decision concerning its use on rational grounds. The process by which such decisions are made is assumed to be a sequential one in which, at each stage, the decision-maker takes that step which gives him maximal expected utility given optimal planning at each future stage. In the third part the same decision process is assumed to operate, but now bonuses are associated with vasectomy, and may vary with the stage at which vasectomy is undertaken. In both the second and third parts the only form of contraception that is considered to be available is vasectomy. *An earlier version of this paper was contributed to the European Congress of the Econo metric Society, Budapest, 1972. We also wish to acknowledge helpful comments from D.K. La1 and W.J. Corlett, and from the referees of the Journal.
320
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and N.J. Rau, Family size in India
2. The calculation of the natural completed family size in India
The term ‘natural fertility’ is taken from Henry (1961), and is defined as the fertility which exists, or has existed, in the absence of birth control. Such fertility need not be totally physiologically determined, and may also depend on social factors, such as sexual taboos during lactation. We use the data collected by Henry on natural fertility rates for India to obtain a set of mean conditional frequencies that a woman who has had k- 1 children will conceive a kth child in the absence of contraception. To calculate these figures, a method developed by one of the present authors [Rau (1972)] is used. In applying it the starting point was the natural, age-specific legitimate fertility rates quoted by Henry for five year age groups. This series was smoothed by taking moving averages; and working from this, and taking specific account of the fact that some women of each age are post-pat-turn sterile, some totally sterile, and others die at that age, the conditional frequencies were calculated. These estimated conditional probabilities refer, however, to children born and not children surviving. To adjust these figures for surviving children-where survival is taken to mean survival beyond the length of time regarded as relevant for infant mortality (one year) -we used data on state specific mortality rates and the following independence assumption: The conditional distribution of the number of surviving children, given k children ‘ever born’ is taken to be binomial with parameters k and (1 -k), k being the infant mortality rate. This assumption may be unrealistic in view of epidemics etc., but has the advantage of making computation of the desired conditional probabilities a straightforward matter. Thus we obtained finally, estimated state specific conditional probabilities that a woman who has had k- 1 surviving children will have a kth surviving child. There are no comparable sample estimates of this data but we used our estimated frequencies to compute the mean family size for a given age distribution of women, and a given marriage age, and compared these figures with sample estimates obtained for each state in a survey conducted between July 1960 and June 1961. From a normal test of the hypothesis that the sample mean is not significantly different from the theoretical mean as calculated by US, the hypothesis is accepted by 11 of the 15 states at the one percent level. Thus we conclude that the estimated conditional probabilities are a reasonable indicator of natural fertility behaviour. 3. The utility function We assume that the ‘family decision-maker’ has a preference ordering, with respect to different levels of family size and consumption per head, which may be represented by a von Neumann-Morgenstern utility function U(c, x); here c, ‘per capita consumption’, denotes the discounted value of family lifetime consumption divided by the total number x of family members. The family
A. Markandya and N.J. Rau, Family size in India
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size variable x is taken to include mother, father and children surviving beyond the infant-mortality period. Specifically, it will be assumed that the utility function is of the form U(c, x) =
(G-x-“)(c;~-c-~) 0 ( (AY+‘+B)(c;~-c-~)
for c > cO, for c = cO, for c < cO,
(1)
where c,,, 6, E,A,B,G are constants with ~,>~,~>O,E>O,A>O,A~~+~+B>O,G~~-~. Thisutility function has the following properties :
(1) U(c, x) is defined for x 2 2, c > 0, and is continuous for all such values. (2) There exists a ‘subsistence level of consumption’ cc > 0 such that U(c, x) increases as x increases for c > co, and decreases as x increases for c < co. (3)U(c, x) is quasi-concave in c and x in the region c > co, and quasi-convex in the region c < co. (4)Partial derivatives of all orders with respect to c and x (including crosspartials) exist in the regions c # co. At c = co, partial derivatives of all orders exist with respect to x (all these derivatives being zero) but not with respect to c. (5)In each of the regions c > co, c < co, the coefficient of relative risk aversion with respect to x is equal to the (same) constant E. Also, the coefficient of relative risk aversion with respect to c is equal to the constant 6. Indifference curves for such a utility function are sketched in fig. 1. )(
/ I
2-
Fig. 1
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and NJ.
Rau, Family size in India
The use of consumption per head and family size as the only arguments of the parents’ (or father’s) utility function implies that there exists some given rule for distributing consumption among family members, given c and x. This does not seem unreasonable. For example, Sen (1966) argues, in a paper on the dual economy, that equal division of consumption among family members can be taken as a sensible rule of thumb, or as a consequence of a plausible welfaremaximising process - note, however, that the latter interpretation cannot be sustained when productivity depends on consumption : this feedback selection may have some importance at low income levels, but we ignore it throughout. The notion of subsistence consumption, denoted by c0 in (l), is ofgreat importance for our analysis. cc is characterized as that level of consumption per head above which children are ‘goods’ (aU/ax < 0 for c > cO) and below which they are ‘bad? (au/ax -C 0 for c < c,J. Such a level may be set at the survival level of consumption (which is essentially what we have done in this study) or it may be socially determined, depending, for instance, on the average level of consumption in the society where the family lives.’ A similar notion to our c0 has been employed in several studies of the welfare economics of population [see for example Meade (19541. Our specification of a direct dependence of family utility on family size is something of a departure from the approach gdopted in empirical studies of the economics of population control. Such studies [see Blomquist (1971) and Enke (1960)] regard children as ‘investment goods’ and attempt to measure the discounted surplus of their production over their consumption. But given that we are concerned in this study with ‘utility’ as it affects parents’ decisions, and given the extremely high value placed on progeny in India [Kapadia (1955)], it seems very incomplete not to take account of any direct utility children may yield to their parents. It is perhaps worth mentioning two other factors which sociologists, demographers and economists have considered important in determining family size in India. One is the son-survivorship motive [May and Heer (1968)]: for social and religious reasons parents attach a high value to having at least one son. This motive (which we ignore) should be sharply distinguished from preferences for boys over girls arising from differences in wages and hence in contributions to consumption per head; the latter effect will play a leading ‘The assumption of a fixed survival level of consumption per head as being the relevant level at which the indifference map shifts is of course contentious. It can be argued that attitudes to children are influenced by the standard of living generally enjoyed by the group concerned, so that in more wealthy areas, the level of per capita consumption below which children are a ‘bad’ is higher than in relatively poor areas. The assumption of a constant minimum level, which is equal to the survival level of consumption per person, has been retained partly because we believe that, in the context of Indian agricultural workers, variations in this level are not very large, and partly because this provides us with a point of reference from which the likely qualitative consequences of higher levels in some provinces can be assessed.
A. Markandya and N.J. Rau, Family size in India
323
role in our analysis.’ Another factor is the pension motive: parents have children, in the hope that they (parents) may be kept by them in old age [for a theoretical analysis of the economic implications of this, see Neher (1971)]. This motive may be important; certainly the relatively short life expectation of Indian peasants does not preclude a pension motive, since, for sadly obvious reasons, a life-span which is short by Western standards may nevertheless include a considerable final period of dependency. Nevertheless, the considerable survey evidence on motives for having children [Berelson et al. (1966)] does not give strong support to the hypothesis that the pension motive matters, and we shall henceforth ignore it. The utility function (1) depends on six parameters: cO, 6, E, A, B, and G. We conclude this subsection by imposing certain further restrictions on these. First we set c0 = 1. This is simply a matter of choice of units: consumption per head is measured in ‘subsistence units’. Second, we make the assumption that parents are strongly philoprogenitive in the sense that ‘being childless is as bad as starving’: the marginal utility of consumption is taken as zero for all c 2 c0 when x = 2. Taken together with the utility properties, this implies that G = 2-“. Our final restrictions concern A and B. We postulate the existence of some family size x, > 2 such that aUj& and a2U/axac exist at x = x,, c = co. Simple manipulations yield A = [.s/(.s+2)] x;(“+‘),
(2)
B = 2-“-
(3)
and [(2~+2)/(~+2)] xi&.
4. Uncertainty in decision making in the context of family planning The family decision-maker may be assumed to be uncertain about numerous items affecting c and x. Among these are: (i) the number and sex of children, (ii) infant mortality, (iii) unemployment and future prospects. In this paper we are concerned entirely with (i). We ignore (ii) by making the assumption that all infant mortality is concentrated near birth. Thus we do not ‘assume away’ infant mortality, but we do assume away uncertainty about infant mortality, by postulating that the parents know at any time which of the children born to them up to that time are ‘survivors’: when we discuss vasectomy bonuses, bonus schedules are related to the number of surviving children.
5. Earnings of children and the intertemporal decision process Given the earnings levels of the mother, father, sons and daughters, x and c may be expressed as functions of the number of sons and number of daughters. ‘One possible economic factor that has been ignored here is the fact that a daughter’s marriage involves a dowry which is a cost to the parents. It is not clear how important this is, and there is little detailed statistical data on it.
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and NJ.
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Thus, given a family with (i- 1) sons and (j- 1) daughters, we may write U(c, x) = V(i, j).
(4)
The notions of consumption per head, subsistence consumption and individual earnings should take account of the fact that all these items take place over a period of time. In this paper, we largely sidestep this problem by taking the present values of these variables at a constant discount rate, and assuming that parents make decisions on the basis of these present values. To calculate these present values, we need a time horizon and a discount rate. The mother and father’s earnings and subsistence requirements are taken from the mean age of marriage to the mean life expectancy, and the children’s earnings are counted from the age of 7 to the marriage age for their sex as part of the family earnings. The present values of all these earnings are calculated at constant discount rates of 5, 10 and 15 percent. As is well known to economists, the use of such discounted present values is a great simplification of a dynamic problem. It allows estimates of future earnings and subsistence rates to enter into the model, without taking any account of how uncertainty about them might affect individual behaviour. It also fails to represent a sequential decision-making process correctly since once a given number of children are born, a decision about the next one should really be taken with the present value of all members recalculated from the date at which the child would be expected, and these figures will differ, especially for the children already born, from the values used in their case. If, however, the children occur, in general, in the few earlier years of marriage, and there are not many of them, then such differences could be quite small. Once the children start to spread out over the married life, then this error could be substantial. Owing to the technical difficulties encountered in taking account of the order in which boys and girls are born, for this would be relevant if the fully correct procedure were to be followed, this approximation has been retained for the present. 6. The optimising method used in the computations Given the utility function and the postulate that parents maximise expected utility, we can derive formally the procedure for obtaining optimal stopping points. The reader should note that, here as elsewhere in this paper, ‘children’ means surviving children. Let the probability of the family having another child, given that there are x-2 children already, be rc, when birth control is not used, E,* when it is used. Clearly 7r,*6 7rX;when the method of birth control is vasectomy n,* = 0. We also assume that for sufficiently large x, n, = 0; let X be the smallest such x. 3 Q seems perfectly reasonable to assume that the probability for a finite number of children; we in fact took X as 21.
of conception
falls to zero
A. Markandya
and N.J. Rau, Family size in India
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Let V(i,j) be the utility obtained from stopping at (i-l) sons and G- 1) daughters, and let E(i,j) be the expected utility, given optimal planning from then on, of a family with (i- 1) sons and (j- 1) daughters. If i+j = X, then clearly E(i,j) = V(i, j). Now consider the case where i+j = x < X.
Then birth control will be adopted if and only if d(i,j) = (1/2)[E(i+l,j)+E(i,j+l)]-V(i,j)
< 9.
So, letting n’(i, j) denote the optimal probability of having another child when there are already (i- 1) sons and (j- 1) daughters, we have
=, * xx
z”(i, j) =
if d(i,j) 5 0, if d(i, j) < 0,
and E(i, j) = V(i, j) + n”(i, j) *d(i, j).
Recursing backwards from the x = X down to x = 2, we obtain E(i,j) and n’(i,j)foralli, jsuchthat2 I i+j S X. Now let q(i, j) be the probability that the family reaches the combination ‘i- 1 sons, j- 1 daughters’, and p&j) the probability that it stops there. Then thep(i,j) and q(i, j) may be derived from the z’(i, j) via the relations 40, 1) = I, q(i, 1) = (I/2)n”(i-
1, l)*q(i-
1, 1)
q(Lj) = (1/2)a”(l,j-1).q(l,j-1) q(i, j)
=
(1/2){n’(i-
1,j)*q(i-
fori=2,3 forj=
,...,
x-l,
2,3 ,...,
x-l,
l,j)+7c”(i, j- l)q(i, j- 1)) for i 2 2, j 2 2, i+j S X,
and p(i,
j)
= (1 -n”(i,j))*q(i,j)
for2
5 i+jS
X,
where z"(i, j) is understood to be zero when i+j = X. From these probabilities, the expectation and variance of family size, the number of sons and number of daughters may be computed in a straightforward fashion. 4 % can be shown that for such a decision process the expected number of boys is always equal to the expected number of girls. The variance of the number of boys and that of the number of girls are, however, quite different.
1.13 1.05 1.18 1.08 1.18 1.35 1.19 1.26 1.20 1.64 2.10 2.59 3.17
Madras Andhra Pradesh Bombay Rajastan Mysore Bihar Orissa Madhya Pradesh Uttar Pradesh Kerala West Bengal Assam Punjab
0.73 0.84 0.80 0.73 0.60 1.18 0.96 1.07 1.00 0.89 1.50 1.70 2.35
15
Girl
0.73 0.83 0.81 0.77 0.66 1.13 0.91 1.06 0.95 0.93 1.51 1.80 2.37
18 0.81 0.89 0.94 0.90 0.90 1.17 1.01 1.08 1.02 1.11 1.70 1.85 2.55
Boy
0.64 0.73 0.70 0.64 0.53 1.03 0.84 0.94 0.85 0.78 1.31 1.49 2.04
15
Girl
Ten percent
0.65 0.74 0.71 0.68 0.58 1.01 0.81 0.94 0.85 0.81 1.49 1.58 2.09
18
“Given for thirteen states, three discount rates and two marriage ages for women.
Boy
State
Five percent
0.74 0.75 0.77 0.74 0.71 0.99 0.85 0.92 0.86 1.01 1.41 1.69 2.10
Boy
0.55 0.63 0.60 0.55 0.46 0.89 0.72 0.81 0.73 0.68 1.13 1.28 1.76
15
Girl
Fifteen percent
0.56 0.64 0.61 0.58 0.49 0.89 0.71 0.82 0.74 0.70 1.16 1.36 1.81
18
Table 1 The lifetime earnings for the father, mother, boy and girl (in subsistence units).’
0.63 0.70 0.75 0.82 0.79 0.80 0.58 0.84 0.74 0.91 1.30 1.93 2.21
Mother
1.46 1.70 1.64 1.97 1.74 1.67 1.69 1.80 1.90 2.20 2.54 3.13 4.00
Father
A. Markandya and N.J. Rau, Family size in hdia
327
7. The results of the optimising behaviour with no bonuses The data and sources on which the calculations are based are given in the appendix. From these we obtain the discounted lifetime wages in subsistence units as given in table 1. Whereas the parents subsistence unit wages are independent of the discqunt rate or marriage age, the children’s are not, since a higher discount rate or an earlier marriage age imply a lower value on future earnings relative to expenditure in childhood. The states seem to fall into three groups : five low wage states where the expected child wage is always below one, five middle wage states where it is greater or less than one depending on the discount rate, and three high wage states where it is always above one. The mean parent wage is above one in all stages. The optimal numbers of boys and girls at which to undertake vasectomy were worked out for E = 6 = 0.5 and 0.8 and for values of X, (the number of members of the family at which the differentiability condition is satisfied) of six, eight and ten. It turned out that the results were very insensitive to the choice of A’,, and so all simulation results are reported for one value (X, = 6). The mean completed family size in the absence of bonuses is given in table 2 for E = 6 = 0.8. The predicted behaviour varies considerably between the three types of states distinguished. In the low wage states the size of family is always small, with the lowest low wage state (Madras) producing vasectomy with no children. In the other states in this group there are never more than two children. In the high wage states, on the other hand, the mean optimal completed family size is the same as the natural family size to one decimal place. Between the two extremes we have the middle wage states, where the behaviour is similar to the low wage states when the expected child wage is below one, and similar to the high wage states when it is above one. Surprisingly the variance of child wage has very little effect on the size of family in any group. The choice of marriage age makes a difference to the mean predicted family size only when this is close to the natural family size. For a family size of less than four children only one difference was observed in the simulations for the two different marriage ages. The results were rather insensitive as to whether E and 6 are fixed at 0.5 or 0.8. When there are differences the family size is slightly larger for the lower coefficients, as one might expect. Such differences occurred mainly when the optimal family size was small, although they only occurred in about 25 percent of such cases.
8. Vasectomy bonuses and charges
In this section the effect of vasectomy bonuses on completed family size is considered. The results reported here are of a preliminary nature, and just begin to look at this big question. Only positive bonuses have been considered so far (the question of charges had not been examined); and only a small
5.2 0.0 0.0 0.0
15 15; 5% 15; 10% 15; 15%
18 18; 5% 18; 10% 18; 15%
Natural; Optimal; Optimal; Optimal;
Natural; Optimal; Optimal; Optimal;
4.9 1.3 0.9 0.9
4.8 1.7 0.9 0.9
4.6 1.7 8:; k; 0:9
5.0 ;8 4.8 117 :9’
46 4:o
5.1
51 5’1 2.5 0:9
5.5
49 40 1’7 0.9
z 1’7 1:7
:4
49 4’9b 4’9 419
514
5.4 5.4b 54
4.7 4.7b b t:; 43b 4’3b 4’3b 413b
5.5 5.5b 55b 5:5 50 5’0b 5’Ob 5:o
‘All reported values are the mean number of children suruiuing infancy (i.e., one year of age); given for thirteen states, three discount rates and two marriage ages. bOptimal behavior involves no vasectomy.
4.8 0.0 0.0 0.0
2
Completed family size; marriage age; discount rate
State
Table 2 . The natural and optimal completed family sizes when vasectomy is available with no bonuses (a = 6 = 0.8, X, = 6)”
A. Markandya
and N.J. Rau, Family size in India
329
number of the possible ways that these bonuses could operate have been examined. Vasectomy bonuses can be regarded as a money opportunity cost to having children; so that the parent has to choose between the extra utility of the bonus and the extra expected utility of further procreation. The agency operating the bonuses has to decide at what stages to make this choice available to the parent, and at what level to fix the bonus. The decision with respect to these two variables will depend on how the agency believes it can achieve a given effect on family size at the minimum cost possible. It is important to note here that a constant bonus, irrespective of the family size, will not in general reduce family size, and may well increase it by a small amount. This is so because such a bonus is equivalent to an increase in the father’s lifetime real wage (since he can make any decision with respect to family size and still obtain the bonus); and from an inspection of the results for the no-bonus case it appears that the ‘income effect’ of an increase in the father’s wage on family size is positive. Thus, a more effective bonus scheme would probably be one which gave a bonus that depended on the number of children already in the family. In the initial calculations done here, the effects are estimated of a bonus scheme in which a father who has C children or less gets a certain bonus added to his earnings, and a father who has more than C children gets nothing. 5 The values of C are fixed at 2, 3, 4 and 5 for each bonus. This effectively places a ‘price’ on the (C+ 1)th child. The ‘prices’ that were tried were 5 and 10 percent of the father’s lifetime wage for the low wage and middle wage states, and 10, 20 and 40 percent for the high wage states. These figures were used because from a preliminary run it appeared that these were the ‘orders of magnitude’ required to achieve any impact in the various types of states. Given such a bonus scheme, it is still theoretically possible that the family size will rise in some cases - the income and substitution effect are both present on the price change, for the (C+ 1)th child onwards, and there is an ‘income’ effect alone for children up to the Cth child. There were in fact some cases where this was observed when the schemes were operated with the data available. In the low wage states, we have already observed that the family size is small (O-3 children) with zero bonuses. When bonuses are introduced at 5 and 10 percent of the lifetime wage of the father, the family size is never reduced, and in three of the states (Madras, Andhra Pradesh and Rajastan) the family size increases by one child over the zero bonus size.6 The results did not vary between the 5 and 10 percent bonus levels. For the middle wage states, we 5While pricing schemes that are more interesting or desirable from an economist’s viewpoint can be devised, it is important to bear in mind that any agency operating a pricing scheme will always be concerned with the question of administrative feasibility. In this respect the scheme outlined here is also probably a little too complicated to administer, although it might suggest some guidelines for a simpler scheme. 6This increase in the family size by one child is irrespective of the values of e and 6, or the choice of marriage age. C
4.0 1.7 0.9
4.0
A:;
4.7 1.7 1.7
152
5% 10% 15%
4.7 1.7 1.7
5.1 1.7 0.9
5.1 2.5 0.9
5% 10% 15%
1;;
2.4 0.9 0.9
4.0 09 0:9
1:; 152
4.7 1.7 1.7
2.4 1.7 0.9
5.1 1.7 1.3
2.1 0.9 0.9
5.0 3.7 1.7
5.0 4.8 1.7
5.0 4.8 1.7
5% 10% 15%
(2) c=3
4.7 1.7 1.7
2.4 1.7 0.9
5.1 1.7 1.7
2.1 0.9 0.9
5.0 2.9 1.7
(3) c=4
4.5 1.7 1.7
2.4 1.7 0.9
4.4 1.7 1.7
2.0 0.9 0.9
5.0 2.8 1.7
(4) c=5
Bonus = 5 % of father’s wage (1) c=2
No bonus
Discount rate
4.7 1.7 1.7
1.7 1.7 0.9
5.1 1.7 1.3
1.7 0.9 0.9
5.0 4.1 1.7
(la) c=2
4.7 1.7 1.7
1.7 1.7 0.9
5.1 1.7 1.7
1.7 1.3 0.9
5.0 2.3 1.7
(2a) c=3
4.6 1.7 1.7
1.7 1.7 1.3
3.3 1.7 1.7
1.7 1.7 0.9
5.0 2.3 1.7
(3a) c=4
3.9 1.7 1.7
1.7 1.7 1.7
3.3 1.7 1.7
1.7 1.7 0.9
4.7 2.3 1.7
(4a) c=5
Bonus = 10% of father’s wage
“Columns (l)-(4) and (la)-(4a) give the mean completed family size when the bonus is available to a father who undergoes vasectomy when he has already fathered C children or less. No bonus is given if he has fathered more than C children and undertakes vasectomy. The marriage age is 18.
Kerala
Uttar Pradesh
Madhya Pmoesh
Orissa
Bihar
State
Table 3 Middle wage states (.s = 6 = 0.8, X, = 6).”
A. Markandya
and N.J. Rau, Family size in India
331
present the results in table 3 for E = 6 = 0.8, marriage age of 18 years for women, and bonuses of 5 and 10 percent. The results for the other values of E and 6 and a marriage age of fifteen years are not qualitatively different. Here we observe that when the zero bonus family size is high the bonuses are effective in reducing family size in all cases but one. In general the value of C has an abrupt effect - the family reduces its size just to qualify for the bonus. This, however, is not observed in the case of Orissa and Uttar Pradesh (5 percent discount rate). When, however, the family size is low without bonuses, the results are very similar to the low wage states. In the high wage states the bonuses appear to have hardly any effect at all. A 20 percent bonus has just a small effect in West Bengal and Assam for C = 4 and C = 5 at a 15 percent rate of discount. Otherwise the optimal behaviour is one of no vasectomy, and in the case of the Punjab, bonuses of up to 40 percent have no effect. This bonus gives the percentage of the lifetime wage that would have to be paid to the father at the beginning of his married life.’ 9. Conclusions and interpretations In this study the consequences of a certain kind of rational behaviour with respect to decisions concerning family size have been investigated. When the model was run with vasectomy available freely it appeared that in the low wage states the family size was reduced sharply from the natural family size. This would suggest that for the groups studied the problem of reducing family size in these states is not one of the provision of bonuses, but of effective education concerning contraception techniques and of their widespread availability.* In the middle wage states the results were ambiguous -depending crucially on whether the discount rate chosen made the expected child wage above or below one. In the high wage states free vasectomy made virtually no impact on family size. In the middle and high wage states then there is perhaps some scope for vasectomy bonuses. When vasectomy bonuses are introduced into the model, the tentative conclusions that seem to emerge are, that in the middle wage states bonuses of 5 percent to 10 percent of the father’s married lifetime earnings would reduce the family size to two or three children, ifsuch a family size were ‘The actual payment will not of course be made at that point. This payment can be calculated by finding that figure which, when discounted at the appropriate rate of interest from the time it is made, would give the present value payment. This will of course depend on how many years it takes to achieve the desired number of children, before undertaking vasectomy. Such a procedure would be quite complicated to administer, but as a rule of thumb the actual payment could be calculated for the mean length of time that it would take to achieve the number of children at which vasectomy is undertaken. *The very fact that a distribution of family size based on natural fertility data approximates closely the observed distribution suggests that in the 1950s birth control had had no significant effect. Even as recently as 1960 very little was known by the Indian farmer of contraceptive methods. In a survey [Census of India (1961b)], 12.6 percent of families where the husband was illiterate had heard of contraceptive techniques. Given the few who knew about these methods, even fewer had practised them.
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A. Markandya and N.J. Rau, Family size in India
high in the absence of bonuses. If it were small with no bonuses, then the presence of such a bonus might actually increase family size by a small amount. This also applies to the low wage states. In the high wage states substantially larger bonuses seem to be required to have any effect. Here one might suggest the importance of reducing by legislation the role of child labour in the family context as an effective way of reducing family size.’ There are two important qualifications that should be made to these conclusions, one is with respect to the calculation of the subsistence level and the other is with respect to the valuation of the relative earnings of boys and girls. In our study we have chosen the level of consumption per head at which the utility of another child changes from negative to positive as the lowest possible subsistence level of consumption. It is conceivable that the actual level of consumption per head at which this happens is higher, especially in the middle and high wage states where agricultural workers are accustomed to higher standards of living. If this is the case, then some of the states will tend to exhibit responses to vasectomy bonuses similar to that of the low wage states, and again reinforce the conclusion that in a number of Indian states, the problem of population control is not one of the provision of bonuses but one of information and education. In the high wage states however, the unit value of consumption per head would have to increase greatly for the optimal family size to be reduced in the absence of bonuses. It seems unlikely to us that the possible interstate variations in this variable are of this magnitude, but it does seem probable that the value of subsistence consumption per head is higher in these states. If that is the case, then the bonuses tried in the simulations for these states should have rather more effect on reducing the optimal family size that is predicted here. The other likely bias in this study is the relative overvaluation of a daughter’s earnings. On account of the lack of data on the lower productivity of girls under 15, relative to boys, and on the marriage costs of daughters, and on account of the fact that there is a social preference for boys, the girls earnings, as computed, probably overvalue their utility to the parents. This implies that the expected child wage is again too high and this, along with the risk aversion of parents, should result in a smaller optimal completed family, with the bonuses required to achieve a given reduction being smaller than computed. Overall then, both factors strengthen the conclusion that in all states the optimal family size is smaller than calculated, and that when the family size is not reduced without the bonuses, then the required level of such bonuses is likely to be less than calculated here. 9vvemake no comment, however, on the ‘cost effectiveness’of this method of achieving a lower family size. Of course this remark applies equally to the resource allocation aspects of the transfer payments required to pay the bonuses. The evaluation of these factors is an important but separate question.
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This study has posed a number of problems and many remain to be solved. The data used for this study was rather scant, and it would be desirable if some time series data on earnings by sex and age were available for groups of workers. Furthermore, our knowledge of the discount rate and the coefficients of relative risk aversion among Indian peasants is almost nonexistent. As stated in the appendix, we took the value of 6 from the range of values that Fellner believed to be relevant for American data. Setting E equal to 6, quite arbitrarily, gives results that are not very sensitive to the range of values tried.” This is a field where it is felt that research is much needed.
Appendix: Data and sources Earnings Data on earnings was taken from the Agricultural Labour Survey, 1956. This provides the annual earnings from all sources, including an imputed value to the nonmonetary earnings of men, women and children employed as agricultural labourers in different states of the union. To calculate the discounted present value of the earnings of the sons and daughters, the discounted sum of the child earnings were taken from the 7th to the 15th year, and the adult male or female earnings, depending on the sex of the child, were taken from the sixteenth year to the year of marriage. The definition of the child wages from the age of seven to the age of fifteen is taken from the Agricultural Survey. The marriage ages of girls are taken at 15 and 18, as before, and the marriage ages of boys are taken at the mean age of marriage among rural males in each state of India in the 1961 Census synthetic cohorts [Census of India (1961a)l. These were the closest available figures. Sutsistence
level
The data on the subsistence level was taken from the figures calculated by Bardhan (1970). He took Dr. W.R. Aykroyd’s definition of an ‘adequate’ diet and priced it at 1960-61 rural retail prices. This figure was then scaled up by the ratio of food to nonfood expenditure for an adult to obtain the subsistence level of an adult. We took this value for the adult subsistence level and adjusted it to 1956 prices to obtain the figure relevant for our calculations. The child’s subsistence level was taken as a fraction of that for the adult [see Bardhan (1970)], but excluding nonfood expenditure. We felt that in a family context much of such expenditure is of an overhead nature and therefore is unlikely to influence marginal decisions concerning children. “‘In some sensitivity runs we took various combinations of E and 6 between 0.5 and 0.8 for representative states. The use of different values of E and 6 in this range did not appear to alter the results qualitatively. However, as mentioned earlier, values outside this range could always lead to either very small families or very large ones.
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The coejicienrs
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of relative risk aversion
There appears to be little empirical evidence on the coefficient of relative risk aversion with respect to consumption (1+6 in our notation), even for developed countries. Fellner (1967, p. 68) uses data on food consumption in the United States to derive ‘Fisher-Frisch elasticities of the marginal utility of income’ of between 1.5 and 1.8, and identifies this parameter with the coefficient of relative risk aversion. As a result of reading this paper, and of various experiments with higher values leading to very low numbers of children, we decided to use the values 6 = 0.5 and 0.8 in our simulations. For obvious reasons we had even less of an idea of what seemed ‘possible’ values for E. Faute de mieux we decided to experiment with values of E in the same range of those of 6 - higher values again yielding insensitive and absurd results. Infant mortality The data on infant mortality was taken from the National Sample Survey, fourteenth round: July 1958 to June 1959, no. 76. These were the closest available figures and provided state- and sex-specific infant mortality rates for rural areas. References Agricultural Labour in India, 1957, Report of the second enquiry, Vols. I-XIV. Berelson, B.B., 1966, Kap studies in fertility, in: B.B. Berelson et al., eds., Family planning and population programmes (University of Chicago, Chicago), 655-668. Bhardhan, P.K., 1970, On the minimum level of living and the rural poor, Indian Economic Review V (new series), 129-136. Blomquist, A.G., 1971, Foreign aid population growth and gains from birth control, Journal of Development Studies 8, 5-22. Census of India, 1961a, Tables on mean age at marriage, males and females - 1961 synthetic cohorts (rural figures). Census of India, 1961b, National sample survey no. 116, 16th round, Tables with notes on family planning. Enke, S., 1960, The gains to India from population control, Review of Economics and Statistics 42, 175-188. Fellner, W., 1967, Operational utility: The theoretical background and a measurement, in: W. Fellner et al., eds., Ten economic studies in the tradition of Irving Fisher (Wiley, New York). Henry, L., 1961, Some data on national fertility, Eugenics Quarterly 8, 81-91. Henry, L., 1963, La feconditt naturelle, observation-theorie-r&hats, in: International Population Conference proceedings, vol. 2 (London), 97-108. Kapadia, K.M., 1955, Marriage and family in India (Oxford University Press, Bombay). May, D.A. and D.M. Hear, 1968, Son survivorship motivation and family size in India: A computer simulation, Population Studies 22, 199-210. Meade, J., 1955, Trade and welfare (Oxford University Press, London), 80-101. Neher, P.A., 1971, Peasants procreation and pensions, American Economic Review 61, 380-389. Rau, N.J., 1972, A method of inferring the probability distribution of family size from natural fertility data, mimeo. (University College, London). Sen, A.K., 1966, Peasants and dualism with or without surplus labour, Journal of Political Economy 74,425-450. Stiglitz, J., 1969, Behaviour towards risk with many commodities, Econometrica 37, 660-667.
VASECTOMY
BONUSES AND COMPLETED IN INDIA A. MARKANDYA
Publishing Company
FAMILY SIZE
and N.J. RAW
University College, London, England Received April 1973, revised version received October 1974 This paper discusses the consequences in rural India of a certain rational household decision function on the number of children born in the family when birth control opportunities are available. Households are assumed to choose whether or not to practice birth control after each child is born on the basis of maximising expected utility as a function of the consumption per head and the number of members in the family. Uncertainty arises on account of the different evaluation of children of different sexes. The model is then used to predict how completed family size would change if a government agency paid bonuses for smaller families.
1. Introduction This paper attempts to measure the impact on completed family size in India of the availability of vasectomy, when account is taken of some of the variables that are considered to be relevant. The study is divided into three parts. In the first part we calculate the natural completed family size in the absence of birth control techniques. By absence here is meant either the lack of availability or the lack of knowledge and confidence in the technique, or both. In the period 1950-1960, which is the decade from which the data for this study is taken, this situation is assumed to be approximately correct. In the second part we assume that vasectomy is widely available, with no bonuses or charges attached to it, and that the persons involved know about its availability and are making their decision concerning its use on rational grounds. The process by which such decisions are made is assumed to be a sequential one in which, at each stage, the decision-maker takes that step which gives him maximal expected utility given optimal planning at each future stage. In the third part the same decision process is assumed to operate, but now bonuses are associated with vasectomy, and may vary with the stage at which vasectomy is undertaken. In both the second and third parts the only form of contraception that is considered to be available is vasectomy. *An earlier version of this paper was contributed to the European Congress of the Econo metric Society, Budapest, 1972. We also wish to acknowledge helpful comments from D.K. La1 and W.J. Corlett, and from the referees of the Journal.
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2. The calculation of the natural completed family size in India
The term ‘natural fertility’ is taken from Henry (1961), and is defined as the fertility which exists, or has existed, in the absence of birth control. Such fertility need not be totally physiologically determined, and may also depend on social factors, such as sexual taboos during lactation. We use the data collected by Henry on natural fertility rates for India to obtain a set of mean conditional frequencies that a woman who has had k- 1 children will conceive a kth child in the absence of contraception. To calculate these figures, a method developed by one of the present authors [Rau (1972)] is used. In applying it the starting point was the natural, age-specific legitimate fertility rates quoted by Henry for five year age groups. This series was smoothed by taking moving averages; and working from this, and taking specific account of the fact that some women of each age are post-pat-turn sterile, some totally sterile, and others die at that age, the conditional frequencies were calculated. These estimated conditional probabilities refer, however, to children born and not children surviving. To adjust these figures for surviving children-where survival is taken to mean survival beyond the length of time regarded as relevant for infant mortality (one year) -we used data on state specific mortality rates and the following independence assumption: The conditional distribution of the number of surviving children, given k children ‘ever born’ is taken to be binomial with parameters k and (1 -k), k being the infant mortality rate. This assumption may be unrealistic in view of epidemics etc., but has the advantage of making computation of the desired conditional probabilities a straightforward matter. Thus we obtained finally, estimated state specific conditional probabilities that a woman who has had k- 1 surviving children will have a kth surviving child. There are no comparable sample estimates of this data but we used our estimated frequencies to compute the mean family size for a given age distribution of women, and a given marriage age, and compared these figures with sample estimates obtained for each state in a survey conducted between July 1960 and June 1961. From a normal test of the hypothesis that the sample mean is not significantly different from the theoretical mean as calculated by US, the hypothesis is accepted by 11 of the 15 states at the one percent level. Thus we conclude that the estimated conditional probabilities are a reasonable indicator of natural fertility behaviour. 3. The utility function We assume that the ‘family decision-maker’ has a preference ordering, with respect to different levels of family size and consumption per head, which may be represented by a von Neumann-Morgenstern utility function U(c, x); here c, ‘per capita consumption’, denotes the discounted value of family lifetime consumption divided by the total number x of family members. The family
A. Markandya and N.J. Rau, Family size in India
321
size variable x is taken to include mother, father and children surviving beyond the infant-mortality period. Specifically, it will be assumed that the utility function is of the form U(c, x) =
(G-x-“)(c;~-c-~) 0 ( (AY+‘+B)(c;~-c-~)
for c > cO, for c = cO, for c < cO,
(1)
where c,,, 6, E,A,B,G are constants with ~,>~,~>O,E>O,A>O,A~~+~+B>O,G~~-~. Thisutility function has the following properties :
(1) U(c, x) is defined for x 2 2, c > 0, and is continuous for all such values. (2) There exists a ‘subsistence level of consumption’ cc > 0 such that U(c, x) increases as x increases for c > co, and decreases as x increases for c < co. (3)U(c, x) is quasi-concave in c and x in the region c > co, and quasi-convex in the region c < co. (4)Partial derivatives of all orders with respect to c and x (including crosspartials) exist in the regions c # co. At c = co, partial derivatives of all orders exist with respect to x (all these derivatives being zero) but not with respect to c. (5)In each of the regions c > co, c < co, the coefficient of relative risk aversion with respect to x is equal to the (same) constant E. Also, the coefficient of relative risk aversion with respect to c is equal to the constant 6. Indifference curves for such a utility function are sketched in fig. 1. )(
/ I
2-
Fig. 1
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The use of consumption per head and family size as the only arguments of the parents’ (or father’s) utility function implies that there exists some given rule for distributing consumption among family members, given c and x. This does not seem unreasonable. For example, Sen (1966) argues, in a paper on the dual economy, that equal division of consumption among family members can be taken as a sensible rule of thumb, or as a consequence of a plausible welfaremaximising process - note, however, that the latter interpretation cannot be sustained when productivity depends on consumption : this feedback selection may have some importance at low income levels, but we ignore it throughout. The notion of subsistence consumption, denoted by c0 in (l), is ofgreat importance for our analysis. cc is characterized as that level of consumption per head above which children are ‘goods’ (aU/ax < 0 for c > cO) and below which they are ‘bad? (au/ax -C 0 for c < c,J. Such a level may be set at the survival level of consumption (which is essentially what we have done in this study) or it may be socially determined, depending, for instance, on the average level of consumption in the society where the family lives.’ A similar notion to our c0 has been employed in several studies of the welfare economics of population [see for example Meade (19541. Our specification of a direct dependence of family utility on family size is something of a departure from the approach gdopted in empirical studies of the economics of population control. Such studies [see Blomquist (1971) and Enke (1960)] regard children as ‘investment goods’ and attempt to measure the discounted surplus of their production over their consumption. But given that we are concerned in this study with ‘utility’ as it affects parents’ decisions, and given the extremely high value placed on progeny in India [Kapadia (1955)], it seems very incomplete not to take account of any direct utility children may yield to their parents. It is perhaps worth mentioning two other factors which sociologists, demographers and economists have considered important in determining family size in India. One is the son-survivorship motive [May and Heer (1968)]: for social and religious reasons parents attach a high value to having at least one son. This motive (which we ignore) should be sharply distinguished from preferences for boys over girls arising from differences in wages and hence in contributions to consumption per head; the latter effect will play a leading ‘The assumption of a fixed survival level of consumption per head as being the relevant level at which the indifference map shifts is of course contentious. It can be argued that attitudes to children are influenced by the standard of living generally enjoyed by the group concerned, so that in more wealthy areas, the level of per capita consumption below which children are a ‘bad’ is higher than in relatively poor areas. The assumption of a constant minimum level, which is equal to the survival level of consumption per person, has been retained partly because we believe that, in the context of Indian agricultural workers, variations in this level are not very large, and partly because this provides us with a point of reference from which the likely qualitative consequences of higher levels in some provinces can be assessed.
A. Markandya and N.J. Rau, Family size in India
323
role in our analysis.’ Another factor is the pension motive: parents have children, in the hope that they (parents) may be kept by them in old age [for a theoretical analysis of the economic implications of this, see Neher (1971)]. This motive may be important; certainly the relatively short life expectation of Indian peasants does not preclude a pension motive, since, for sadly obvious reasons, a life-span which is short by Western standards may nevertheless include a considerable final period of dependency. Nevertheless, the considerable survey evidence on motives for having children [Berelson et al. (1966)] does not give strong support to the hypothesis that the pension motive matters, and we shall henceforth ignore it. The utility function (1) depends on six parameters: cO, 6, E, A, B, and G. We conclude this subsection by imposing certain further restrictions on these. First we set c0 = 1. This is simply a matter of choice of units: consumption per head is measured in ‘subsistence units’. Second, we make the assumption that parents are strongly philoprogenitive in the sense that ‘being childless is as bad as starving’: the marginal utility of consumption is taken as zero for all c 2 c0 when x = 2. Taken together with the utility properties, this implies that G = 2-“. Our final restrictions concern A and B. We postulate the existence of some family size x, > 2 such that aUj& and a2U/axac exist at x = x,, c = co. Simple manipulations yield A = [.s/(.s+2)] x;(“+‘),
(2)
B = 2-“-
(3)
and [(2~+2)/(~+2)] xi&.
4. Uncertainty in decision making in the context of family planning The family decision-maker may be assumed to be uncertain about numerous items affecting c and x. Among these are: (i) the number and sex of children, (ii) infant mortality, (iii) unemployment and future prospects. In this paper we are concerned entirely with (i). We ignore (ii) by making the assumption that all infant mortality is concentrated near birth. Thus we do not ‘assume away’ infant mortality, but we do assume away uncertainty about infant mortality, by postulating that the parents know at any time which of the children born to them up to that time are ‘survivors’: when we discuss vasectomy bonuses, bonus schedules are related to the number of surviving children.
5. Earnings of children and the intertemporal decision process Given the earnings levels of the mother, father, sons and daughters, x and c may be expressed as functions of the number of sons and number of daughters. ‘One possible economic factor that has been ignored here is the fact that a daughter’s marriage involves a dowry which is a cost to the parents. It is not clear how important this is, and there is little detailed statistical data on it.
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Thus, given a family with (i- 1) sons and (j- 1) daughters, we may write U(c, x) = V(i, j).
(4)
The notions of consumption per head, subsistence consumption and individual earnings should take account of the fact that all these items take place over a period of time. In this paper, we largely sidestep this problem by taking the present values of these variables at a constant discount rate, and assuming that parents make decisions on the basis of these present values. To calculate these present values, we need a time horizon and a discount rate. The mother and father’s earnings and subsistence requirements are taken from the mean age of marriage to the mean life expectancy, and the children’s earnings are counted from the age of 7 to the marriage age for their sex as part of the family earnings. The present values of all these earnings are calculated at constant discount rates of 5, 10 and 15 percent. As is well known to economists, the use of such discounted present values is a great simplification of a dynamic problem. It allows estimates of future earnings and subsistence rates to enter into the model, without taking any account of how uncertainty about them might affect individual behaviour. It also fails to represent a sequential decision-making process correctly since once a given number of children are born, a decision about the next one should really be taken with the present value of all members recalculated from the date at which the child would be expected, and these figures will differ, especially for the children already born, from the values used in their case. If, however, the children occur, in general, in the few earlier years of marriage, and there are not many of them, then such differences could be quite small. Once the children start to spread out over the married life, then this error could be substantial. Owing to the technical difficulties encountered in taking account of the order in which boys and girls are born, for this would be relevant if the fully correct procedure were to be followed, this approximation has been retained for the present. 6. The optimising method used in the computations Given the utility function and the postulate that parents maximise expected utility, we can derive formally the procedure for obtaining optimal stopping points. The reader should note that, here as elsewhere in this paper, ‘children’ means surviving children. Let the probability of the family having another child, given that there are x-2 children already, be rc, when birth control is not used, E,* when it is used. Clearly 7r,*6 7rX;when the method of birth control is vasectomy n,* = 0. We also assume that for sufficiently large x, n, = 0; let X be the smallest such x. 3 Q seems perfectly reasonable to assume that the probability for a finite number of children; we in fact took X as 21.
of conception
falls to zero
A. Markandya
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325
Let V(i,j) be the utility obtained from stopping at (i-l) sons and G- 1) daughters, and let E(i,j) be the expected utility, given optimal planning from then on, of a family with (i- 1) sons and (j- 1) daughters. If i+j = X, then clearly E(i,j) = V(i, j). Now consider the case where i+j = x < X.
Then birth control will be adopted if and only if d(i,j) = (1/2)[E(i+l,j)+E(i,j+l)]-V(i,j)
< 9.
So, letting n’(i, j) denote the optimal probability of having another child when there are already (i- 1) sons and (j- 1) daughters, we have
=, * xx
z”(i, j) =
if d(i,j) 5 0, if d(i, j) < 0,
and E(i, j) = V(i, j) + n”(i, j) *d(i, j).
Recursing backwards from the x = X down to x = 2, we obtain E(i,j) and n’(i,j)foralli, jsuchthat2 I i+j S X. Now let q(i, j) be the probability that the family reaches the combination ‘i- 1 sons, j- 1 daughters’, and p&j) the probability that it stops there. Then thep(i,j) and q(i, j) may be derived from the z’(i, j) via the relations 40, 1) = I, q(i, 1) = (I/2)n”(i-
1, l)*q(i-
1, 1)
q(Lj) = (1/2)a”(l,j-1).q(l,j-1) q(i, j)
=
(1/2){n’(i-
1,j)*q(i-
fori=2,3 forj=
,...,
x-l,
2,3 ,...,
x-l,
l,j)+7c”(i, j- l)q(i, j- 1)) for i 2 2, j 2 2, i+j S X,
and p(i,
j)
= (1 -n”(i,j))*q(i,j)
for2
5 i+jS
X,
where z"(i, j) is understood to be zero when i+j = X. From these probabilities, the expectation and variance of family size, the number of sons and number of daughters may be computed in a straightforward fashion. 4 % can be shown that for such a decision process the expected number of boys is always equal to the expected number of girls. The variance of the number of boys and that of the number of girls are, however, quite different.
1.13 1.05 1.18 1.08 1.18 1.35 1.19 1.26 1.20 1.64 2.10 2.59 3.17
Madras Andhra Pradesh Bombay Rajastan Mysore Bihar Orissa Madhya Pradesh Uttar Pradesh Kerala West Bengal Assam Punjab
0.73 0.84 0.80 0.73 0.60 1.18 0.96 1.07 1.00 0.89 1.50 1.70 2.35
15
Girl
0.73 0.83 0.81 0.77 0.66 1.13 0.91 1.06 0.95 0.93 1.51 1.80 2.37
18 0.81 0.89 0.94 0.90 0.90 1.17 1.01 1.08 1.02 1.11 1.70 1.85 2.55
Boy
0.64 0.73 0.70 0.64 0.53 1.03 0.84 0.94 0.85 0.78 1.31 1.49 2.04
15
Girl
Ten percent
0.65 0.74 0.71 0.68 0.58 1.01 0.81 0.94 0.85 0.81 1.49 1.58 2.09
18
“Given for thirteen states, three discount rates and two marriage ages for women.
Boy
State
Five percent
0.74 0.75 0.77 0.74 0.71 0.99 0.85 0.92 0.86 1.01 1.41 1.69 2.10
Boy
0.55 0.63 0.60 0.55 0.46 0.89 0.72 0.81 0.73 0.68 1.13 1.28 1.76
15
Girl
Fifteen percent
0.56 0.64 0.61 0.58 0.49 0.89 0.71 0.82 0.74 0.70 1.16 1.36 1.81
18
Table 1 The lifetime earnings for the father, mother, boy and girl (in subsistence units).’
0.63 0.70 0.75 0.82 0.79 0.80 0.58 0.84 0.74 0.91 1.30 1.93 2.21
Mother
1.46 1.70 1.64 1.97 1.74 1.67 1.69 1.80 1.90 2.20 2.54 3.13 4.00
Father
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327
7. The results of the optimising behaviour with no bonuses The data and sources on which the calculations are based are given in the appendix. From these we obtain the discounted lifetime wages in subsistence units as given in table 1. Whereas the parents subsistence unit wages are independent of the discqunt rate or marriage age, the children’s are not, since a higher discount rate or an earlier marriage age imply a lower value on future earnings relative to expenditure in childhood. The states seem to fall into three groups : five low wage states where the expected child wage is always below one, five middle wage states where it is greater or less than one depending on the discount rate, and three high wage states where it is always above one. The mean parent wage is above one in all stages. The optimal numbers of boys and girls at which to undertake vasectomy were worked out for E = 6 = 0.5 and 0.8 and for values of X, (the number of members of the family at which the differentiability condition is satisfied) of six, eight and ten. It turned out that the results were very insensitive to the choice of A’,, and so all simulation results are reported for one value (X, = 6). The mean completed family size in the absence of bonuses is given in table 2 for E = 6 = 0.8. The predicted behaviour varies considerably between the three types of states distinguished. In the low wage states the size of family is always small, with the lowest low wage state (Madras) producing vasectomy with no children. In the other states in this group there are never more than two children. In the high wage states, on the other hand, the mean optimal completed family size is the same as the natural family size to one decimal place. Between the two extremes we have the middle wage states, where the behaviour is similar to the low wage states when the expected child wage is below one, and similar to the high wage states when it is above one. Surprisingly the variance of child wage has very little effect on the size of family in any group. The choice of marriage age makes a difference to the mean predicted family size only when this is close to the natural family size. For a family size of less than four children only one difference was observed in the simulations for the two different marriage ages. The results were rather insensitive as to whether E and 6 are fixed at 0.5 or 0.8. When there are differences the family size is slightly larger for the lower coefficients, as one might expect. Such differences occurred mainly when the optimal family size was small, although they only occurred in about 25 percent of such cases.
8. Vasectomy bonuses and charges
In this section the effect of vasectomy bonuses on completed family size is considered. The results reported here are of a preliminary nature, and just begin to look at this big question. Only positive bonuses have been considered so far (the question of charges had not been examined); and only a small
5.2 0.0 0.0 0.0
15 15; 5% 15; 10% 15; 15%
18 18; 5% 18; 10% 18; 15%
Natural; Optimal; Optimal; Optimal;
Natural; Optimal; Optimal; Optimal;
4.9 1.3 0.9 0.9
4.8 1.7 0.9 0.9
4.6 1.7 8:; k; 0:9
5.0 ;8 4.8 117 :9’
46 4:o
5.1
51 5’1 2.5 0:9
5.5
49 40 1’7 0.9
z 1’7 1:7
:4
49 4’9b 4’9 419
514
5.4 5.4b 54
4.7 4.7b b t:; 43b 4’3b 4’3b 413b
5.5 5.5b 55b 5:5 50 5’0b 5’Ob 5:o
‘All reported values are the mean number of children suruiuing infancy (i.e., one year of age); given for thirteen states, three discount rates and two marriage ages. bOptimal behavior involves no vasectomy.
4.8 0.0 0.0 0.0
2
Completed family size; marriage age; discount rate
State
Table 2 . The natural and optimal completed family sizes when vasectomy is available with no bonuses (a = 6 = 0.8, X, = 6)”
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number of the possible ways that these bonuses could operate have been examined. Vasectomy bonuses can be regarded as a money opportunity cost to having children; so that the parent has to choose between the extra utility of the bonus and the extra expected utility of further procreation. The agency operating the bonuses has to decide at what stages to make this choice available to the parent, and at what level to fix the bonus. The decision with respect to these two variables will depend on how the agency believes it can achieve a given effect on family size at the minimum cost possible. It is important to note here that a constant bonus, irrespective of the family size, will not in general reduce family size, and may well increase it by a small amount. This is so because such a bonus is equivalent to an increase in the father’s lifetime real wage (since he can make any decision with respect to family size and still obtain the bonus); and from an inspection of the results for the no-bonus case it appears that the ‘income effect’ of an increase in the father’s wage on family size is positive. Thus, a more effective bonus scheme would probably be one which gave a bonus that depended on the number of children already in the family. In the initial calculations done here, the effects are estimated of a bonus scheme in which a father who has C children or less gets a certain bonus added to his earnings, and a father who has more than C children gets nothing. 5 The values of C are fixed at 2, 3, 4 and 5 for each bonus. This effectively places a ‘price’ on the (C+ 1)th child. The ‘prices’ that were tried were 5 and 10 percent of the father’s lifetime wage for the low wage and middle wage states, and 10, 20 and 40 percent for the high wage states. These figures were used because from a preliminary run it appeared that these were the ‘orders of magnitude’ required to achieve any impact in the various types of states. Given such a bonus scheme, it is still theoretically possible that the family size will rise in some cases - the income and substitution effect are both present on the price change, for the (C+ 1)th child onwards, and there is an ‘income’ effect alone for children up to the Cth child. There were in fact some cases where this was observed when the schemes were operated with the data available. In the low wage states, we have already observed that the family size is small (O-3 children) with zero bonuses. When bonuses are introduced at 5 and 10 percent of the lifetime wage of the father, the family size is never reduced, and in three of the states (Madras, Andhra Pradesh and Rajastan) the family size increases by one child over the zero bonus size.6 The results did not vary between the 5 and 10 percent bonus levels. For the middle wage states, we 5While pricing schemes that are more interesting or desirable from an economist’s viewpoint can be devised, it is important to bear in mind that any agency operating a pricing scheme will always be concerned with the question of administrative feasibility. In this respect the scheme outlined here is also probably a little too complicated to administer, although it might suggest some guidelines for a simpler scheme. 6This increase in the family size by one child is irrespective of the values of e and 6, or the choice of marriage age. C
4.0 1.7 0.9
4.0
A:;
4.7 1.7 1.7
152
5% 10% 15%
4.7 1.7 1.7
5.1 1.7 0.9
5.1 2.5 0.9
5% 10% 15%
1;;
2.4 0.9 0.9
4.0 09 0:9
1:; 152
4.7 1.7 1.7
2.4 1.7 0.9
5.1 1.7 1.3
2.1 0.9 0.9
5.0 3.7 1.7
5.0 4.8 1.7
5.0 4.8 1.7
5% 10% 15%
(2) c=3
4.7 1.7 1.7
2.4 1.7 0.9
5.1 1.7 1.7
2.1 0.9 0.9
5.0 2.9 1.7
(3) c=4
4.5 1.7 1.7
2.4 1.7 0.9
4.4 1.7 1.7
2.0 0.9 0.9
5.0 2.8 1.7
(4) c=5
Bonus = 5 % of father’s wage (1) c=2
No bonus
Discount rate
4.7 1.7 1.7
1.7 1.7 0.9
5.1 1.7 1.3
1.7 0.9 0.9
5.0 4.1 1.7
(la) c=2
4.7 1.7 1.7
1.7 1.7 0.9
5.1 1.7 1.7
1.7 1.3 0.9
5.0 2.3 1.7
(2a) c=3
4.6 1.7 1.7
1.7 1.7 1.3
3.3 1.7 1.7
1.7 1.7 0.9
5.0 2.3 1.7
(3a) c=4
3.9 1.7 1.7
1.7 1.7 1.7
3.3 1.7 1.7
1.7 1.7 0.9
4.7 2.3 1.7
(4a) c=5
Bonus = 10% of father’s wage
“Columns (l)-(4) and (la)-(4a) give the mean completed family size when the bonus is available to a father who undergoes vasectomy when he has already fathered C children or less. No bonus is given if he has fathered more than C children and undertakes vasectomy. The marriage age is 18.
Kerala
Uttar Pradesh
Madhya Pmoesh
Orissa
Bihar
State
Table 3 Middle wage states (.s = 6 = 0.8, X, = 6).”
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present the results in table 3 for E = 6 = 0.8, marriage age of 18 years for women, and bonuses of 5 and 10 percent. The results for the other values of E and 6 and a marriage age of fifteen years are not qualitatively different. Here we observe that when the zero bonus family size is high the bonuses are effective in reducing family size in all cases but one. In general the value of C has an abrupt effect - the family reduces its size just to qualify for the bonus. This, however, is not observed in the case of Orissa and Uttar Pradesh (5 percent discount rate). When, however, the family size is low without bonuses, the results are very similar to the low wage states. In the high wage states the bonuses appear to have hardly any effect at all. A 20 percent bonus has just a small effect in West Bengal and Assam for C = 4 and C = 5 at a 15 percent rate of discount. Otherwise the optimal behaviour is one of no vasectomy, and in the case of the Punjab, bonuses of up to 40 percent have no effect. This bonus gives the percentage of the lifetime wage that would have to be paid to the father at the beginning of his married life.’ 9. Conclusions and interpretations In this study the consequences of a certain kind of rational behaviour with respect to decisions concerning family size have been investigated. When the model was run with vasectomy available freely it appeared that in the low wage states the family size was reduced sharply from the natural family size. This would suggest that for the groups studied the problem of reducing family size in these states is not one of the provision of bonuses, but of effective education concerning contraception techniques and of their widespread availability.* In the middle wage states the results were ambiguous -depending crucially on whether the discount rate chosen made the expected child wage above or below one. In the high wage states free vasectomy made virtually no impact on family size. In the middle and high wage states then there is perhaps some scope for vasectomy bonuses. When vasectomy bonuses are introduced into the model, the tentative conclusions that seem to emerge are, that in the middle wage states bonuses of 5 percent to 10 percent of the father’s married lifetime earnings would reduce the family size to two or three children, ifsuch a family size were ‘The actual payment will not of course be made at that point. This payment can be calculated by finding that figure which, when discounted at the appropriate rate of interest from the time it is made, would give the present value payment. This will of course depend on how many years it takes to achieve the desired number of children, before undertaking vasectomy. Such a procedure would be quite complicated to administer, but as a rule of thumb the actual payment could be calculated for the mean length of time that it would take to achieve the number of children at which vasectomy is undertaken. *The very fact that a distribution of family size based on natural fertility data approximates closely the observed distribution suggests that in the 1950s birth control had had no significant effect. Even as recently as 1960 very little was known by the Indian farmer of contraceptive methods. In a survey [Census of India (1961b)], 12.6 percent of families where the husband was illiterate had heard of contraceptive techniques. Given the few who knew about these methods, even fewer had practised them.
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A. Markandya and N.J. Rau, Family size in India
high in the absence of bonuses. If it were small with no bonuses, then the presence of such a bonus might actually increase family size by a small amount. This also applies to the low wage states. In the high wage states substantially larger bonuses seem to be required to have any effect. Here one might suggest the importance of reducing by legislation the role of child labour in the family context as an effective way of reducing family size.’ There are two important qualifications that should be made to these conclusions, one is with respect to the calculation of the subsistence level and the other is with respect to the valuation of the relative earnings of boys and girls. In our study we have chosen the level of consumption per head at which the utility of another child changes from negative to positive as the lowest possible subsistence level of consumption. It is conceivable that the actual level of consumption per head at which this happens is higher, especially in the middle and high wage states where agricultural workers are accustomed to higher standards of living. If this is the case, then some of the states will tend to exhibit responses to vasectomy bonuses similar to that of the low wage states, and again reinforce the conclusion that in a number of Indian states, the problem of population control is not one of the provision of bonuses but one of information and education. In the high wage states however, the unit value of consumption per head would have to increase greatly for the optimal family size to be reduced in the absence of bonuses. It seems unlikely to us that the possible interstate variations in this variable are of this magnitude, but it does seem probable that the value of subsistence consumption per head is higher in these states. If that is the case, then the bonuses tried in the simulations for these states should have rather more effect on reducing the optimal family size that is predicted here. The other likely bias in this study is the relative overvaluation of a daughter’s earnings. On account of the lack of data on the lower productivity of girls under 15, relative to boys, and on the marriage costs of daughters, and on account of the fact that there is a social preference for boys, the girls earnings, as computed, probably overvalue their utility to the parents. This implies that the expected child wage is again too high and this, along with the risk aversion of parents, should result in a smaller optimal completed family, with the bonuses required to achieve a given reduction being smaller than computed. Overall then, both factors strengthen the conclusion that in all states the optimal family size is smaller than calculated, and that when the family size is not reduced without the bonuses, then the required level of such bonuses is likely to be less than calculated here. 9vvemake no comment, however, on the ‘cost effectiveness’of this method of achieving a lower family size. Of course this remark applies equally to the resource allocation aspects of the transfer payments required to pay the bonuses. The evaluation of these factors is an important but separate question.
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This study has posed a number of problems and many remain to be solved. The data used for this study was rather scant, and it would be desirable if some time series data on earnings by sex and age were available for groups of workers. Furthermore, our knowledge of the discount rate and the coefficients of relative risk aversion among Indian peasants is almost nonexistent. As stated in the appendix, we took the value of 6 from the range of values that Fellner believed to be relevant for American data. Setting E equal to 6, quite arbitrarily, gives results that are not very sensitive to the range of values tried.” This is a field where it is felt that research is much needed.
Appendix: Data and sources Earnings Data on earnings was taken from the Agricultural Labour Survey, 1956. This provides the annual earnings from all sources, including an imputed value to the nonmonetary earnings of men, women and children employed as agricultural labourers in different states of the union. To calculate the discounted present value of the earnings of the sons and daughters, the discounted sum of the child earnings were taken from the 7th to the 15th year, and the adult male or female earnings, depending on the sex of the child, were taken from the sixteenth year to the year of marriage. The definition of the child wages from the age of seven to the age of fifteen is taken from the Agricultural Survey. The marriage ages of girls are taken at 15 and 18, as before, and the marriage ages of boys are taken at the mean age of marriage among rural males in each state of India in the 1961 Census synthetic cohorts [Census of India (1961a)l. These were the closest available figures. Sutsistence
level
The data on the subsistence level was taken from the figures calculated by Bardhan (1970). He took Dr. W.R. Aykroyd’s definition of an ‘adequate’ diet and priced it at 1960-61 rural retail prices. This figure was then scaled up by the ratio of food to nonfood expenditure for an adult to obtain the subsistence level of an adult. We took this value for the adult subsistence level and adjusted it to 1956 prices to obtain the figure relevant for our calculations. The child’s subsistence level was taken as a fraction of that for the adult [see Bardhan (1970)], but excluding nonfood expenditure. We felt that in a family context much of such expenditure is of an overhead nature and therefore is unlikely to influence marginal decisions concerning children. “‘In some sensitivity runs we took various combinations of E and 6 between 0.5 and 0.8 for representative states. The use of different values of E and 6 in this range did not appear to alter the results qualitatively. However, as mentioned earlier, values outside this range could always lead to either very small families or very large ones.
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The coejicienrs
A. Markandya
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of relative risk aversion
There appears to be little empirical evidence on the coefficient of relative risk aversion with respect to consumption (1+6 in our notation), even for developed countries. Fellner (1967, p. 68) uses data on food consumption in the United States to derive ‘Fisher-Frisch elasticities of the marginal utility of income’ of between 1.5 and 1.8, and identifies this parameter with the coefficient of relative risk aversion. As a result of reading this paper, and of various experiments with higher values leading to very low numbers of children, we decided to use the values 6 = 0.5 and 0.8 in our simulations. For obvious reasons we had even less of an idea of what seemed ‘possible’ values for E. Faute de mieux we decided to experiment with values of E in the same range of those of 6 - higher values again yielding insensitive and absurd results. Infant mortality The data on infant mortality was taken from the National Sample Survey, fourteenth round: July 1958 to June 1959, no. 76. These were the closest available figures and provided state- and sex-specific infant mortality rates for rural areas. References Agricultural Labour in India, 1957, Report of the second enquiry, Vols. I-XIV. Berelson, B.B., 1966, Kap studies in fertility, in: B.B. Berelson et al., eds., Family planning and population programmes (University of Chicago, Chicago), 655-668. Bhardhan, P.K., 1970, On the minimum level of living and the rural poor, Indian Economic Review V (new series), 129-136. Blomquist, A.G., 1971, Foreign aid population growth and gains from birth control, Journal of Development Studies 8, 5-22. Census of India, 1961a, Tables on mean age at marriage, males and females - 1961 synthetic cohorts (rural figures). Census of India, 1961b, National sample survey no. 116, 16th round, Tables with notes on family planning. Enke, S., 1960, The gains to India from population control, Review of Economics and Statistics 42, 175-188. Fellner, W., 1967, Operational utility: The theoretical background and a measurement, in: W. Fellner et al., eds., Ten economic studies in the tradition of Irving Fisher (Wiley, New York). Henry, L., 1961, Some data on national fertility, Eugenics Quarterly 8, 81-91. Henry, L., 1963, La feconditt naturelle, observation-theorie-r&hats, in: International Population Conference proceedings, vol. 2 (London), 97-108. Kapadia, K.M., 1955, Marriage and family in India (Oxford University Press, Bombay). May, D.A. and D.M. Hear, 1968, Son survivorship motivation and family size in India: A computer simulation, Population Studies 22, 199-210. Meade, J., 1955, Trade and welfare (Oxford University Press, London), 80-101. Neher, P.A., 1971, Peasants procreation and pensions, American Economic Review 61, 380-389. Rau, N.J., 1972, A method of inferring the probability distribution of family size from natural fertility data, mimeo. (University College, London). Sen, A.K., 1966, Peasants and dualism with or without surplus labour, Journal of Political Economy 74,425-450. Stiglitz, J., 1969, Behaviour towards risk with many commodities, Econometrica 37, 660-667.