# Labor-leisure tradeoff with linear in consumption utility function

The labor-leisure tradeoff is the tradeoff between working more hours and earning a wage for an extra hour versus the extra benefit received for consuming an extra hour of leisure.

The labor-leisure tradeoff can be used to determine the optimal labor supply by an individual. For example, consider a consumer with the following utility function:

$U = C - \frac{1}{2}(H)^2$

where C is the level of consumption and H is the labour supplied. This is called a linear in consumption utility function because the marginal utility for consuming an extra unit of consumption is always 1. We can also observe that the marginal disutility from working an extra hour increases as the amount of labour supplied increases.

Suppose that this particular worker only receives wage income and does not save any income. His/her budget constraint would be:

$wH = C$

It is also assumed that the price of consumption is 1 in this case. If we substitute out the consumption function from our utility function we can re-write our utility function as such:

$U = wH - \frac{1}{2}(H)^2$

We can find the optimal labour supply now by taking the derivative of U wrt to H, such that:

$\frac{dU}{dH} = w - H= 0$

Which implies that in equilibrium $w = H$. To see the labour leisure tradeoff, we note that the consumers time constraint for a day is:

$24 = L + H$

where L is the amount of leisure that a worker enjoys. Rearranging and substituting out for H, we find:

$L = 24 - w$

Thus there is a 1-to-1 negative relationship between leisure and wage. As the wage rate increases, the consumer will consume less leisure and work more.