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

Learn how to solve for the consumption function c(l) in a utility maximization problem involving a consumption-leisure tradeoff. This problem applies key economic concepts such as utility functions, optimization, and first-order conditions. Suitable for undergraduate economics students.

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