Timo Boppart and Per Krussell (BK) did a really interesting paper last year on the past and future of work hours. More people should read it and I want to understand it, so here is some simple summary notes.

Graph: work hours down since 1870

Since the 1800’s, work hours per person has decreased in rich countries, which might mean that the income effect dominate the substitution effect in the long run, given some common neoclassical assumption.

A usual assumption is that the two effects cancel each other, and that labor supply therefor is stable in the long run. Theoretically this can be described in many ways. BP point out that King-Plosser-Rebelo (1998) show that balanced growth with constant hours worked in this (standard neoclassical growth) setting only holds if the utility function can be written as

$u\left(c,h\right)=\frac{\left(c v\left(h\right)\right)^{1-\theta}-1}{1-\theta}$ for $\sigma\neq 1$,  and

$u\left(c,h\right)=\log\left(c\right)+\log v\left(h\right)$ for $\sigma=1$.

where c and h is consumption and hours worked. $\sigma$ is intertemporal elasticity of substitution.

BP present a utility function where growth of consumption and negative growth of work hours is constant in the long run along the balanced growth path. The income effect thus dominates. They argue that, given certain assumptions, a utility function in this setting must have the following form:

$u\left(c,h\right) =\frac{\sigma\left(cb\left(hc^{\frac{v}{1-v}}\right)\right)^{\frac{1-\sigma}{\sigma}}}{1-\sigma}$

for $\sigma \neq 1$, and for $\sigma =1$:

$u\left(c,h\right)=\log\left(c\right)+\log\left(b\left(hc^{\frac{v}{1-v}}\right)\right)$

where b is an twice continuously differentiable function, $\sigma$ is risk aversion, and v is preferences over consumption and leisure, assumed to be over 0 (v=0 is the standard case with constant work hours). Then the choice between work and leisure in each time period, defined by the marginal rate of substitution, is

$\frac{u_{2}\left(c,h\right)}{u_{1}\left(c,h\right)}=c^{1/(1-v)}b_{1}\left(hc^{v/(1-v)}\right)$

The term $hc^{v/(1-v)}$ will be constant in the long run, meaning that with higher income, work hours decrease. So if this is a reasonably correct description of how people really behave in the long run with respect to leisure and consumption, we will most probably work even shorter and/or fewer days in the future, if labor productivity and real wages continues to grow.