Labor supply choices

The choice of a person to supply his own labor services can be thought as a choice with endowment. It is recommended to read the lecture on choices with endowment before reading this one. The consumer can, in this context, choose to work more and consume more or rest more and consume less.

THE BUDGET CONSTRAINT

Even if the idea of the budget constraint remains the same of the normal choices with endowment, we have to adapt that concept to the topic of labor supply. If the person we are analysing chooses to work, he will receive a wage each hour that we are going to call ω. The number of hours worked is indicated by L. If that person has another source of income that is received whether the he works or not he is said to have a nonlabour income and we indicate it as M. The price of consumption is labelled as p and the amount consumed is indicated with C. Therefore the budget constraint can be written as:

pC = M + ωL

the value of what you consume must be equal to the value of what you have thanks to non labor and labor income

we can rewrite the budget constraint as:

pC - ωL = M

if we denote as the maximum amount of hours that the person can work (24h) and if we add the same quantity to both sides we get:

pC - ωL + ωḸ= M + ωL which becomes pC + ω (Ḹ– L) = M + ωL

the maximum amount the person can consume without working is defined as: Ć= M/p; it represents the person’s endowment of consumption. We can rewrite M as M = pĆ. The budget constraint becomes:

pC + ω (Ḹ– L) = pC + ωL

(Ḹ – L) is the difference between the total hours the person can work and the hours he really works. Therefore it represents the number of hours that person does not work and therefore the amounts of hours he rests. The budget constraint becomes in the end:

pC + ωR = pĆ + ωḸ

the maximum amount of hours the person can work is the same as the maximum amount he can rest Ṝ. Therefore we can rewrite the expression as:

pC + ωR = pĆ + ωṜ

the wage represents therefore the price of labor but also the price of leisure. This happens because, when a person decides not to work, he is giving up what he could have earned by working. pC + ωḸ represents the consumer’s full income: the maximum value the consumer can have. Graphically, the budget line passes through the endowment as always but has a slope of - ω/p. Notice that, on the right of the endowment, the line does not exist; this is because we indicated with Ṝ the maximum amount of hours a person can rest or, alternatively, the maximum amount of hours a person can spend during his day. Therefore, no matter what the person will do, he can not increase that time. Once again the optimal solution occurs where the marginal rate of substitution (which comes from the isoquant representing the consumer’s preferences) is equal to the slope of the budget line and therefore when the isoquant and the budget constraint are tangent.

WHAT HAPPENS WHEN THE NON LABOR INCOME OR WAGE INCREASE

Suppose a person wins the lottery and therefore its non labor income increases massively, his demand for labor would increase as well because leisure is considered a normal good (the more you earn the more of it you will demand). If instead, is the wage that increases, not only the consumer’s income will rise but also the cost of doing nothing increases: you give up more money when you decide not to work and therefore leisure becomes relatively more expensive. This pushes the request for leisure to a lower level because of the substitution effect. Obviously this is not the only factor we have to consider because, by earning more money per hour, the person could work less and at the same time earn the same amount he earned before the wage increase. The final result depends on which one between the substitution and the endowment income effect is stronger. This final result can be expressed as:

When the wage rate is low, the substitution effect is larger and therefore the demand for leisure becomes lower. To understand why consider that, when a person does not earn much, a small increase in the wage rate will better up his lifestyle. When the wage rate is high, increasing it more will not make such a high difference because the person considered is rich yet and therefore, for the rich, is better to work less, keep the same conditions of life and rest more. This relationship is depicted graphically by a curve that bends backward when the wage rate is high.