Question

Suppose an individual \(\mathrm{U}=\mathrm{LY}-0.1 \mathrm{~L}^{2}-0.1 \mathrm{Y}^{2}\), where \(\mathrm{L}\) is the leisure and \(\mathrm{Y}\) is the income of the individual. Assuming \(\mathrm{T}\) is time available to work, derive the individual supply of work.

Answer 1

Individual's budget constraint:

$$ Y=w(T-L) $$

Substituting this in the utility function

$$ \begin{array}{l} U=L[w(T-L)]-0.1 L^{2}-0.1[w(T-L)]^{2} \\ U=w T L-L^{2}-0.1 L^{2}-0.1[w(T-L)]^{2} \end{array} $$

The FOC of maximization requires

$$ \frac{\partial U}{\partial L}=w T-2 L-0.2 L+0.2 w[w(T-L)]=0 $$

$$ \begin{array}{l} \therefore w T-2.2 L+0.2 w^{2} T-0.2 w L=0 \\ \therefore w T+0.2 w^{2} T=2.2 L+0.2 w L \\ \therefore L=\frac{w T+0.2 w^{2} T}{2.2+0.2 w} \\ \therefore N^{s}=T-L=T-\frac{w T+0.2 w^{2} T}{2.2+0.2 w} \\ \therefore N^{s}=\left(1-\frac{w+0.2 w^{2}}{2.2+0.2 w}\right) T=\left(\frac{2.2-0.8 w-0.2 w^{2}}{2.2+0.2 w}\right) T \end{array} $$

Thenm the labor supply is

$$ \therefore N^{s}=\left(\frac{2.2-0.8 w-0.2 w^{2}}{2.2+0.2 w}\right) T $$

Answer 2

Individual's budget constraint:

$Y=w(T-L)$

Substituting this in the utility function

$U=L\left [ w(T-L) \right ]-0.1L^2-0.1\left [ w(T-L) \right ]^2$

$U= wTL-L^2-0.1L^2-0.1\left [ w(T-L) \right ]^2$

The FOC of maximization requires

$\frac{\partial U}{\partial L}= wT-2L-0.2L+0.2w\left [ w(T-L) \right ]=0$

$\therefore wT-2.2L+0.2w^2T-0.2wL=0$

$\therefore wT+0.2w^2T=2.2L+0.2wL$

$\therefore L= \frac{wT+0.2w^2T}{2.2+0.2w}$

$\therefore N^s=T-L=T- \frac{wT+0.2w^2T}{2.2+0.2w}$

$\therefore N^s=\left (1- \frac{w+0.2w^2}{2.2+0.2w} \right )T=\left ( \frac{2.2-0.8w-0.2w^2}{2.2+0.2w} \right )T$

Thenm the labor supply is

$\therefore N^s=\left ( \frac{2.2-0.8w-0.2w^2}{2.2+0.2w} \right )T$