19.2 Wages and the Budget Constraint

To model the leisure-consumption tradeoff, let’s assume that “good 1” is “leisure time” (denoted $R$ for “relaxation” to distinguish it from $L$ for labor) and “good 2” is “money” (denoted $C$ for “consumption”). (Note: Some other people use the abbreviation $Le$ for leisure; there’s no global convention.)

We’ll consider the problem of a worker with a job which pays some amount $w$ per hour, and who can choose how many hours to work (“labor,” denoted $L$). They also have a nonwage income of $M$ (perhaps savings, perhaps money from family or some other source). Therefore, if they work for $L$ hours, their total amount of money for consumption is \(C = M + wL\) This worker is endowed with 24 hours of time in the day, which they can split between leisure ($R$) and labor ($L$): \(L + R = 24\) Combining these two equations allows us to write a budget line in terms of the two “goods,” leisure and consumption: \(\begin{aligned} C &= M + w(24 - R)\\ wR + C &= 24w + M \end{aligned}\) Notice that this is just an endowment budget constraint \(p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2\) if we interpret the variables as follows: \(\begin{aligned} x_1 &= R & \text{ (leisure time)}\\ x_2 &= C & \text{ (consumption)}\\ e_1 &= 24 & \text{ (available time)}\\ e_2 &= M & \text{ (nonwage income)} \end{aligned}\) The “price” of time is the wage rate $w$: that is, each hour the worker chooses to spend in leisure rather than working, they’re giving up $w$ dollars of consumption. (Alternatively, you could think of $w$ as the price at which they can sell their leisure time.)

Notice as well that the right-hand side of the budget constraint is the “liquidation value” of the endowment: that is, if the worker chose to work for all 24 hours, the maximum amount of money they could earn would be $M + 24w$.

Putting this all together, we can see that the worker’s budget constraint extends from the endowment point at $(24, M)$ to the point $(0, M + 24w)$, with a slope of $-w$. In other words, the labor market allows this person to convert their time endowment into money at a wage rate $w$ dollars per hour: