El aprendizaje activo

We will now turn to the two types o f regimes discussed in section 3.3 and analyse the conditions that determine the existence and size o f a critical wage above which people choose to fully recuperate. Under regime 1 people choose to fully recuperate and even

take some post-recuperation leisure time, whereas under regime 2 people choose to work part o f the recuperation time or exactly recuperate (regime la discussed previously, where the whole period is spent in leisure and none in labour, is ruled out by our assumption o f a positive net minimum consumption requirement).^^

Regime 1: In order to analyse the amount o f leisure chosen by those who at least

recuperate fully we can use equation (3.5) or (3.8) to substitute for and ^'(7^. From

equation (3.18) the expression for leisure is derived:

I = ^Lz l . (3.19)

yw^ + w

( a X

where y - --- and (z > 0 is the net minimum consumption. We know from the

\ l - a )

analysis in section 3.3 that the wage group that will choose to at least fully recuperate fulfils the condition o ï L ^ s r. Hence, we focus on the case where there is a critical wage, w , for which L ( w ) = L =sr, and L (w)> L for all wages above the critical wage ( V w > w ). L , 0 <L <T, is ûiQ critical value o f leisure above which people choose to fully recuperate.

Differentiating equation 3.19 with respect to wage we obtain:

Æ ^ + (3.20)

Leisure demand is hence strictly increasing in wage as long as we assume <7^1.

From equation (3.19) we observe that if the income obtained by working the entire period exactly equals the net minimum consumption requirement, then no leisure is taken

(i.e. L=0 when w T = a \ whereas when wages tend towards infinity leisure time tends

towards total time (i.e. L T os w oo). Hence there always exists a unique w for

which L ( w ) = L.

The budget constraint given in equation (3.13) (c ^ w C>(L) - a) implies that total consumption (consumption, c, plus the minimum consumption requirement, m) in the type 1 regime must be smaller or equal to unearned income, i.e. c + m < ÿ . This again will imply that the net minimum consumption requirement is equal to, or less than Q (a < 0). Given a fixed level o f required minimum consumption, this type o f regime is therefore only possible for high levels o f unearned income, ÿ . Certainly, the amount of unearned income has to be equal to or larger than the required minimum consumption, i.e. y ^ m . An interesting extension to the model would be to make unearned income a function o f income saved from previous periods. In that case, and assuming that the current wage rate is a function o f previous wage rate, the wage rate would not be irrelevant for this type o f regime.

In the Cobb-Douglas case, where <7 = 7, we see that once again no leisure is taken

when earning the existence wage (i.e. L = 0 when w T = a \ whereas when wages tend

towards infinity leisure time now tends towards an amount less than total time (i.e. L

T / ( I + y ) 2iS w~> 06). In order to ensure that there is a critical wage w for which L ( \ v ) ^ L, it is in this case necessary to assume that L < T/(1 + y).

Analytical expressions for the critical wage could be derived in at least two cases,

the first being the Cobb-Douglas case where cr= 7, and the second is where cr= 0.5, and

the results can be found in appendix 3.C.

In order to give an idea as to what happens to the critical wage for various values o f the elasticity o f substitution, three numerical examples w ill now be analysed. In each

case we assume that a=7. Example 1 is defined by a=0.5, s=0.5, and r=0.3, example 2

by a=0.2, s=0.5, and r=0.3, and example 3 by a=0.5, s=0.5, and r=0.8. The parameter values are substituted into equation (3.19), and different wage rates are simulated for the

different values o f <7 in order to find the critical wage, w, that equalises the two sides.

When comparing examples 1 and 2 in table 3.1 we observe that the lower crthe

lower the critical wage above which people choose to fully recuperate if a ^ 1-a, whereas

in the opposite case the opposite occurs. Furthermore, when comparing examples 1 and 3 it is clear that a longer duration o f the recuperation period requires a higher critical wage rate (i.e. the wage rate that separates those who fully recuperate from those who do not).

A wage o f 1.43 allows the individual to exactly survive in examples 1 and 2 when working the entire period, given that the minimum consumption requirement on wage income is assumed to be 1 in this example.^^ Hence, with an elasticity o f substitution o f e.g. 0.8 individuals with a wage rate more than 1.24 times (1.77/1.43) as high as the minimum (net) consumption requirement will choose enough leisure time to at least recuperate fully in the case o f example 1.

Regime 2: We can calculate the range o f wages for which individuals would choose to exactly recuperate for the first illness case, i.e. & (!)= P (1- k s r /(I-r). It was shown that for this sickness case the total effect on efficient working time from working during illness would not be positive, and hence no one would work during recuperation. The existence wage is then given by the net minimum consumption requirement divided by the maximum obtainable efficient labour time;

See the footnote under table 3.1 for the calculation o f this existence wage.

^min = a / ( l - r) (3.21)

Hence, at any wage between the existence wage, Wmmy and the critical wage, w , the

individuals will exactly recuperate. In examples 1 and 2 in table 3.1 the survival wage for case 1 has been illustrated.

In the second illness case, where = ) , we found the

1 — r

following expression for maximum working time for given amounts o f leisure:

0 (L ) = s r - L + ( l- r ) c ^ ^ ^ for 0 <L < s r (3.22)

Substituting for x^O in equation (3.22) gives us the following existence wage:

Wmm=^a/s<W (3.23)

Hence, at any wage between the existence wage and the critical wage individuals will

choose an amount o f efficient leisure between 0 < s r . The exact amount can only be

derived numerically after having assumed different parameter values. In example 3 table 3.1 the survival wage for case 2 has been illustrated.

3.5 Summary

This chapter has considered the choice an individual faces on how to spend her sickness time, given that this choice affects the post-recuperation productivity o f labour, as well as the utility derived from leisure (productivity o f leisure) during that period. The analysis finds that if any leisure is taken in the post-recuperation phase then recuperation is spent entirely in leisure (recuperating). The reason for this is that recuperating during the recuperation phase has the additional advantage o f increasing the productivity in both work and leisure in the post-recuperation phase. Hence, when the individual chooses to take out the equivalent o f the recuperation time or less in leisure she will only take it during the recuperation time. How much o f this time she chooses to work will, however, depend on the productivity-decreasing effect o f the illness, as well as on the wage level, and the minimum consumption requirement.

An implication o f the above finding is that the more leisure an individual takes, the more likely she is to stay at home during recuperation. Given a CES utility function and an empirically supported elasticity o f substitution between leisure and consumption, and assuming that the minimum consumption constraint on wage income net o f unearned income is non-negative, there will be a tendency for an increase in the wage to increase the demand for leisure. In this case the analysis suggests that people with low wages

(below a certain level determined by preferences and the budget constraint) w ill be working during sickness time, which in turn will imply a lower productivity in their post­ recuperation phase. The adverse health effect from a given illness is therefore influenced by the choice made on how to spend ones time, and will be larger for low-wage individuals than higher-wage individuals.

The income and welfare implications o f this finding, as well as its implication for traditional illness measurements are discussed in chapter 4.

Table 3. 1: Critical wage (i.e. a wage rate above this wage yields post-recuperation leisure) for different values of a (a=l).

a

Example 1 Example 2 Examples

a=0.5, s=0.5, r=0.3 a=0.2, s=0.5, r=0.3 a=0.5, s=0.5, r=0.8

(case 1: s <1-r) W (case 1: s <1-r) w (case 3: s >1-r) w 0.10 1.43 1.65 1.43 1.62 2.00 7.44 0.20 1.43 1.67 1.43 1.61 2.00 8.03 0.30 1.43 1.68 1.43 1.59 2.00 8.85 0.40 1.43 1.69 1.43 1.58 2.00 10.03 0.50 1.43 1.71 1.43 1.56 2.00 11.90 0.60 1.43 1.73 1.43 1.55 2.00 15.26 0.70 1.43 1.75 1.43 1.54 2.00 22.91 0.80 1.43 1.77 1.43 1.53 2.00 52.69 0.90 1.43 1.79 1.43 1.52 2.00 1073 1.00 1.43 1.82 1.43 1.51 2.00 -

Source: Own calculations.

* A wage rate o f one (w = /) allows a healthy individual to exactly survive when working the entire period (normalised to one), given that the minimum consumption requirement on wage income is assumed to be 1 (a=7). In example 1, the parameter values fulfil a necessary condition for being a sickness case 1: 5 < 1-r.

With a case 1, no additional efficient time can be gained by working during recuperation, hence, maximum

efficient working time is the exact post-recuperation time. Survival wage in this case is therefore Wmm = a/[l-r]

(=1/0.7=1.43). In example 3, the parameter values fulfil a necessary condition for being a case 3: s > 1-r. With

a case 3, additional efficient time can be gained by working during recuperation, hence, maximum efficient working time is total real time (=1) multiplied by the productivity during sickness (s). Survival wage in this