The Time Allocation Budget Line

Let’s return to the example of Clive, who likes leisure but also likes having money to spend. We now assume that Clive has a total of 80 hours per week that he could spend either working or enjoying as leisure time. (The remaining hours in his week, we assume, are taken up with necessary activities, mainly sleeping.) Let’s also assume, initially, that his hourly wage rate is $10.

A time allocation budget line shows an individual’s trade-off between consumption of leisure and the income that allows consumption of marketed goods.

His consumption possibilities are defined by the time allocation budget line in Figure 19A-1, a budget line that shows Clive’s trade-offs between consumption of leisure and income. Hours of leisure per week are measured on the horizontal axis, and the money he earns from working is measured on the vertical axis.

The horizontal intercept, point X, is at 80 hours: if Clive didn’t work at all, he would have 80 hours of leisure per week but would not earn any money. The vertical intercept, point Y, is at $800: if Clive worked all the time, he would earn $800 per week.

Why can we use a budget line to describe Clive’s time allocation choice? The budget lines found in Chapter 10 and its appendix represent the trade-offs facing consumers deciding how to allocate their income among different goods. Here, instead of asking how Clive allocates his income, we ask how he allocates his time. But the principles underlying the allocation of income and the allocation of time are the same: each involves allocating a fixed amount of a resource (80 hours of time in this case) with a constant trade-off (Clive must forgo $10 for each additional hour of leisure). So using a budget line is just as appropriate for time allocation as it is for income allocation.

The Time Allocation Budget Line Clive’s time allocation budget line shows his trade-off between work, which pays a wage rate of $10 per hour, and leisure. At point X he allocates all his time, 80 hours, to leisure but has no income. At point Y he allocates all his time to work, earning $800, but consumes no leisure. His hourly wage rate of $10, the opportunity cost of an hour of leisure, is equal to minus the slope of the time allocation budget line. We have assumed that point A, at 40 hours of leisure and $400 in income, is Clive’s optimal time allocation choice. It obeys the optimal time allocation rule: the additional utility Clive gets from one more hour of leisure must equal the additional utility he gets from the goods he can purchase with one hour’s wages.

As in the case of ordinary budget lines, opportunity cost plays a key role. The opportunity cost of an hour of leisure is what Clive must forgo by working one less hour—$10 in income. This opportunity cost is, of course, Clive’s hourly wage rate and is equal to minus the slope of his time allocation budget line. You can verify this by noting that the slope is equal to minus the vertical intercept, point Y divided by the horizontal intercept, point X—that is, −$800/(80 hours) = −$10 per hour.

The optimal time allocation rule says that an individual should allocate time so that the marginal utility gained from the income earned from an additional hour worked is equal to the marginal utility of an additional hour of leisure.

To maximize his utility, Clive must choose the optimal point on the time allocation budget line in Figure 19A-1. In Chapter 10 we saw that a consumer who allocates spending to maximize utility finds the point on the budget line that satisfies the utility-maximizing principle of marginal analysis: the marginal utility per dollar spent on two goods must be equal. Although Clive’s choice involves allocating time rather than money, the same principles apply.

Since Clive “spends” time rather than money, the counterpart of the utility-maximizing principle of marginal analysis is the optimal time allocation rule: the marginal utility Clive gets from the extra money earned from an additional hour spent working must equal the marginal utility of an additional hour of leisure.