# How to derive labor supply function

Suppose a worker has the utility function $U = L^\alpha C ^{(1-\alpha)}$ where $L$ describes leisure hours and $C$ is a consumption good. The wage rate is W and non-labor income is $100. Assume that the price of consumption is$1. Derive the labour supply curve assuming that the maximum hours that can be worked is 24.

First, we should describe the workers budget constraint. The worker has non-labor income of \$100 plus the wage earnings for each hour ($H$) they work, which constitutes all of their income. If we assume that they spend all their income on the consumption good, then they will have the budget constraint

$WH + 100 = 1C$

As the utility function is a function of leisure and consumption, we can replace the hours in the budget constraint with leisure using our knowledge that workers have 24 hours that they split between leisure and labor such that:

$24 = L + H$

giving

$H = 24 - L$

Therefore, the budget constraint can be expressed as:

$W(24 - L) + 100 = C$

or

$100 + 24W = WL + C$

The second term on the left-hand side $24W$ can be conceptualized as if the worker sells all of their possible hours for work and then purchases them back as leisure.

When deriving the labor supply curve, we start by actually finding the leisure demand curve. First we equate the marginal product divided by the marginal cost for leisure and the consumption good such that:

$\frac{MU_L}{MC_L} =\frac{MU_C}{MC_C}$

where $MU_L$ is the derivative of the utility function with respect leisure and same for consumption. This equation gives:

$\frac{\alpha L^\alpha C ^{(1-\alpha)} }{W*L} =\frac{(1-\alpha) L^\alpha C ^{(1-\alpha)}}{1C}$

Note: expressing the $MU_L$ as $\frac{L^\alpha C ^{(1-\alpha)}}{L}$ makes it convenient to simplify. Hopefully it is obvious to the reader that $\alpha L^{\alpha-1} C ^{(1-\alpha)} = \alpha \frac{L^\alpha C ^{(1-\alpha)}}{L}$

Solving for consumption yields

$C = \frac{1-\alpha}{\alpha}W*L$

Substituting this back into the budget constraint gives

$24W + 100 = WL +\frac{1-\alpha}{\alpha}WL$

Rearranging for L gives the leisure demand function:

$L = \alpha (24 +\frac{100}{W})$

To find the labor supply curve, we replace $L$ with $24 - H$ such that:

$24 - H = \alpha (24 +\frac{100}{W})$

Which can be solved for $H$ as

$H = (1-\alpha)24 -\alpha\frac{100}{W}$