Microeconomic Analysis of the Leisure Time of Managers

Abstrakt: Managers will be constrained in their choice of time structure by the budged available to them over a given time period. The utility function can be shown by drawing a set of indifference curves for a representative manager. When the managerґs indifference curves are superimposed upon this budged constraint, we can identify a unique utility maximising combination of income and leisure time. There are two implications of the increase in the wage rate: the substitution and income effects.When leisure time is a normal good, then we may state that the income and substitution effects will work in opposite directions. For managers on low hourly wages, the substitution effect will outweigh the income effect and an increase in the wage rate will lead to an increase in the number of hours work. At high wage rates, it is more likely that an increase in wage rates will reduce the overall number of hour worked. In this case, the income effect is sufficiently large to outweigh the substitution effect.

The framework used in the analysis can reflect some special cases of the income/leisure time choice which confront manager in real life. 1) In reality, the hours that eached manager works is determine in advance and offered on a “take it or leave it” basis. The resultes is that many managers do not have the opportunity to maximize their utility. 2) If leisure is an inferior good, both the substitution and income effect work in the same direction, a wage increase unambiguously causes the hours worked to rise. 3) There are many real situations in which the leisure/work decision is influes by the unearned income. Manager consumes more leisure if the leisure is a normal good; he consumes fewer hours of leisure if manager views leisure es an inferior good and special situation is when the number of hours he works is unaffectes by the extra unearned income.

For the purpose of this analysis, it will be assumed the managers can be characterised by a utility function containing just two variables. The first is income, assumed to be derived from paid employment (referred to as market work). Although working is likely to cause a degree of disutility arising from, for example, the monotony of the tasks involved and constrain it imposes on lifestyle, the income that is generated can be used to provide positive utility. It can be converted not only into a variety of commodities, ranging from the basic necessities of the life such as food and shelter, but also into a recreation activities such as tourism, playing sport or videogames and gambling. Obviously the higher the income that is available to manager, the grater the spending options there are. The second variable we shall include in the managerґs utility function is leisure time itself. This can also be assumed to be desirable since it provides the opportunity to enjoy the commodities which can be afforded. Thus we may define the utility function as U = f(I,H), where U denotes utility, I represents income and H denotes leisure time.

Utility maximising combination of income and leisure time

This can be shown diagrammatically by drawing a set of indifference curves for a representative manager. In figure 1 income I is represented on the vertical axis while increments of leisure time, (H) are measured along the horizontal axis. However, because the analysis is concerned with measuring the availability of leisure time within a set period of time, for example a day or a week, the x axis will contain an upper bound. For the purposes of this analysis, it will be assumed that this limit is set at 24 hours.

It can therefore be seen that a move from left to right along the horizontal axis simultaneously represents an increase in leisure time and reduction in working time since both activities are being met from the fixed stock of available time. This limit of time available also means that the set of indifference curves which have been inserted on figure 1 cannot extend beyond the 24-hour limit. Since both income and leisure are desirable commodities, the indifference curves are negative sloped – a gain in one of the commodities must be met by a loss in the other if constant utility is to be maintained. Their convexity to the origin indicates a diminishing marginal rate of substitution between the variables. In other words, as the relative abundance of either income or leisure time, the lower the value we place on additional units of it.

Figure 1 Utility maximising combination of income and leisure time

At this point it should be reemphasis that managers have distinctive utility function. In terms of this analysis, they will differ in their relative valuation of leisure time and income. Some of them, for example, will place a high premium on the amount of leisure time they have at their disposal such that at any given level of utility a large reduction in income is compensated for a comparatively small increment in leisure time. Conversely, there are managers who place a relatively higher value upon their income who can be compensated for a large reduction in their leisure time by a seemingly small increase in income.

Managers will be constrained in their choice of time structure by the budged available to them over a given time period. Let us construct the budget constraint. At one extreme, the greater amount of leisure a manager can consume is if no work is done at all, namely the entire 24-hour period. At the other hand, there is a limit upon the amount of income which can be earned from any given occupation. In theory, the most which can be earned is the hourly wage multiplied by 24. In the practice this is obviously impractical since no one could work effectively for 24 hours a day. Nevertheless, the greater the hourly the wage rate of a given occupation, the steeper the budget constraint facing the manager (for any given calibration of the graph) and the large the number of attainable leisure time/income combination.

When the managerґs indifference curves are superimposed upon this budged constraint, we can see that there is unique utility maximising leisure/income combination. This is identified in the figure 1 as point O. This can be seen that at the utility maximising point the manager will work 24 - H₁ hours, leaving 0 - H₁ hours as leisure time. The income from this employment is I₁.

Income and substitution effects

The next stage of the analysis will focus upon what will happen if the manager faces a change in the hourly wage rate. In terms of the framework set out above, such change will alter the slope of the budget constraint, so that it will be flatter for a reduction in the hourly rate and steeper if there is an increase. Let us look how a utility-maximising manager will react to an increase in the hourly wage. It is possible to isolate two effects. First, any change in the real wage will have a impact upon the opportunity cost of leisure time. In this case, the opportunity cost will increase since it is now more expensive to take an extra hour of leisure due to the higher forgone earnings. Economists refer to this as the substitution effect.

The second implication of the increase in the wage rate is that it adds to the combinations of leisure time and income which are available. In this case there is no a priori hypothesis as to how the manager will respond to these enhanced opportunities. It will depend upon the manager utility function. However, it would not be unreasonable to expect the manager to take the advantage of the increase in the wage rate not only to add to his or her daily stream of income but also to increase the amount of leisure time available each day. This is referred to as the income effect, reflecting the fact that the manager is now better off. It must be stressed once again, however, that whereas we can predict the direction of the substitution effect, the income effect is dependent upon the preferences of manager. The sum of the substitution effect and income effect is known as the total effect. The income and the substitution effect are demonstrated diagrammatically in the figure 2.

The increase in the hourly rate of pay is denoted by the upward pivoting of the budget constraint. This provides the opportunity for the manager concerned to experience a higher level of utility, reflected in the new point of tangency B. Thus the manager in question has been made better off than before since in this example, he or she has not only been able to expand the amount of available leisure time but also to increase income. For this manager, this point of tangency provides more utility than any other income/leisure combination on the new budget constraint. The total effect of this change is therefore the shift from A to B in figure 2.

Figure 2a Income and substitution effects - decrease in the number of hours work

We can highlight the substitution effect brought about by this increase in income by adding an artificial budget constraint to the diagram which reflects the new wage rate (and hence price of the leisure time) but is drawn tangential to the old level of utility, characterized by indifference curve IC₁. This identifies the pure substitution effect since any gains in utility from the extra income and leisure time has been suppressed. By making the manager neither better nor worse off than before, we are isolating the manager response to the pure change in relative prices. Thus, we may define the move from the point A to point P in figure 2 as the substitution effect and this reflects the fact that the manager will consume less of the more expensive commodity (namely leisure) and undertake more work.

We are now in a position to identify the second of the two changes brought about by the change in the wage rate, namely the income effect. It should be recalled that this focuses upon the fact that the change in the wage rate has widened the managerґs attainable set. We can therefore compare points Q and P in terms of the change in the consumption of income and leisure time which has been brought about by change in the real income of the manager. The magnitude of increase in real income is mirrored in the distance between the artificial budget constraint and the new budget constrain. The fact that we have now suppressed the change in relative prices is reflected in these two lines being drawn parallel to each other.

When the income and substitution effects are combined, we get the total effect and this is the change in optimum from A to B. In terms of changes along the horizontal axis of figure 2a, the substitution effect is the move from H₁ to Hґ, while the income effect is move from Hґ to H₂. In this example it can be seen that income effect outweighs the substitution effect such that the manager chooses to work fewer hours than before.

Figure 2b Income and substitution effects - increase in the number of hours work

Provided it is reasonable to assume that leisure time is a normal good, then we may state that the income and substitution effects will work in opposite directions. However there is no theoretical basis to guide us as to which effect will dominate the other in the final outcome, but we may expect that much will depend upon the wage rate. For managers on low hourly wages, it is likely that, although leisure is perceived as being normal, the substitution effect will outweigh the income effect. This will result in an increase in the wage rate leading to an increase in the number of hours work. This possibility is demonstrated in figure 2b, where the substitution effect leads to a reduction in leisure time from H₁ to Hґ and the income effect leads to a reduction in leisure time from Hґ to H₂. At this wage rates, it is more likely that an increase in wage rates will reduce the overall number of hour worked. This is because the manager concerned would already be earning sufficient money to improve the overall quality of life by adding to the available leisure time. In this case, the income effect is not only positive (as expected for the normal good) but is also sufficiently large to outweigh the substitution effect.

Manager in real life

Take it or leave it

Somebody may feel that the framework used in this analysis is somewhat abstract and does not really reflect the choice of the manager in real life. For example, time is here simply broken down into leisure time and work time whereas in reality time is also used for eating, sleeping and housework. Furthermore, there is also the possibility of unearned income, arising from accumulated savings or from state benefit. These possibilities should not be seen as a problem. Extremely complex budget constraints can be calculated to take account of the even greater interaction between taxes and welfare benefits, however, the inclusion of such moments would merely serve to complicate the basic analysis.

A point which should be considered, however, is that most managers do not have an infinitely free choice with respect to the number of hours they work during the basic working week (working day). Usually, the hours that each manager works is determine in advance and offered on a “take it or leave it” basis. Therefore many managers do not have the opportunity to maximize their utility. In other words, they are constrained to consume an amount of leisure time which lies on a lower indifference curve that they would prefer to be on. Thus the more realistic outcome is the one depicted in figure 3.

Figure 3 Take it or leave it

In this case, it can be seen that the contractual obligation imposed by the employer force the manager to consume H₁ units of leisure time. These contracts with the optimal amount of leisure time for this manager H₂. In other words, the manager only receives the utility associated with indiference curve IC₁, rather than IC₂ which is just tangential to the time budget line.

Leisure is an inferior good

If leisure is an inferior good, both the substitution and income effect work in the same direction, and hours of leisure definitely fall. So if leisure is an inferior good, a wage increase unambiguously causes the hours worked to rise.

Manager optimum with unearned income

There are many real situations in which the leisure/work decision depended on the unearned income. How does this no-strings-attached income affect the number of hours he wants to work? Does his utility increase?

We shall start at the point of manager optimum without unearned income. As manager receives the unearned income, it affects his budget constraint. The extra income causes a parallel upward shift of budget constraint. His new budget constrain has the same slope as before because his wage does not change. The extra income cannot buy manager more time, of course, so new budget constraint cannot extend to the right of the time constraint. It can be shown that the relative position of the new original optimum depends on manager tastes. Manager consumes more leisure if the leisure is a normal good; he consumes fewer hours of leisure if manager views leisure is an inferior good and special situation is when the number of hours he works is unaffected by the extra unearned income.

Manager supply curve

Using consumer theory, we can derive the daily demand curve for leisure of the manager, which is the time spent on activities other than work. By subtraction the demand curve for leisure from 24 hours, we obtain the manager supply curve, which shows how the number of hours worked by manager varies with the wage. Depending on whether leisure is an inferior good or a normal good, and whether the substitution effect of the change of wage rate outweighs the income effect, the supply curve of manager may be upward sloping or backward bending.

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