To solve for $c(l)$, we need to express the utility function explicitly in terms of consumption $c$ and leisure $l$, given a specific utility level $\tilde{u}$, and then solve for $c$.

Restating the Setup:

We are given the utility function:

$u(c, l) = \frac{c^{1-\sigma} - 1}{1 - \sigma} + \frac{l^{1-\sigma} - 1}{1 - \sigma}$

where $c = W(1 - l)$ and $\tilde{u} = \frac{2(0.5^{1-\sigma} - 1)}{1 - \sigma}$. We need to solve the utility equation for $c(l)$ at a given $\sigma$.

Steps to Solve for $c(l)$:

1. Set the utility equation equal to $\tilde{u}$: [ \frac{(W(1 - l))^{1