19.3 Optimal Choice of Labor Supply

Having established the budget line, establishing the optimal choice is just the same as in any consumer theory problem: we maximize the utility function subject to the budget constraint. This will allow us to find the labor supply curve for the worker: that is, the optimal amount of labor supplied at each wage rate, given their preferences over time and money, and the amount of nonwage income they have.

For example, suppose we have a Cobb-Douglas utility function representing preferences over leisure ($R$) and consumption ($C$) of \(u(R,C) = R^\alpha C^{1 - \alpha}\) with the associated MRS \(MRS = {\alpha \over {1-\alpha}}\times {C \over R}\) and the budget constraint \(wR + C = 24w + M\) The “price ratio” in this case is $w/1 = w$: that is, you can earn $w$ dollars per hour, and each dollar buys one unit of consumption. Setting the MRS equal to the price ratio gives us \(C = w \times \frac{1-\alpha}{\alpha} \times R\) Plugging this into our budget constraint and solving for the optimal $R$ gives us the optimal quantity of leisure: \(\begin{aligned} wR + w \times \frac{1-\alpha}{\alpha} \times R &= 24w + M\\ wR &= \alpha(24w + M)\\ R^\star &= \alpha\left(24 + {M \over w}\right) \end{aligned}\) This is the gross demand for leisure, but what we’re looking to find is the net supply of labor. Since they start out with 24 hours of leisure and want to consume $R^\star$ hours of leisure, their labor supply is \(\begin{aligned} L^\star(w) &= 24 - R^\star(w)\\ &= 24 - \alpha\left(24 + {M \over w}\right)\\ &= (1-\alpha)24 - {\alpha M \over w} \end{aligned}\) Visually, we can see this in the diagram below. The optimal “bundle” is $X$; the labor supply is the horizontal difference between that point and the endowment at 24 hours of labor:

Note that we assume in this model that you can only sell your time, not buy additional time (notwithstanding the enterprising student who once submitted an exam question involving Hermione Granger and a time turner). This also implies that there might be conditions under which you choose not to sell any of your time: specifically, when you either have a lot of nonwage income and/or face a very low wage.

Finally, the “net supply of good 1” in this case is the labor supply curve for the individual: