Suppose I spend my daily budget of 18 non-working hours on a mixture of sleeping hours and other leisure . My utility is given by . How many hours do I sleep, given that I select my optimal bundle?

Answer 1

To solve your problem and determine how many hours you sleep given your utility function, we need to clarify a few things:

1. Utility Function: You mentioned that your utility is given by a specific function, but you did not provide the exact form of that function. The shape of the utility function will impact your optimal choice between sleeping and other leisure activities.

2. Constraints: You also stated that you have a total of 18 non-working hours, which implies that your time is limited. This creates a constraint for your sleep hours and leisure hours.

The problem typically involves formulating a utility maximization scenario with a budget constraint, which often takes the following general form:

• Let \( S \) represent the number of hours you sleep.
• Let \( L \) represent the number of leisure hours you engage in other activities.

Given the constraint: \[ S + L = 18 \]

And if you have a utility function \( U(S, L) \), your goal is to maximize \( U(S, L) \) subject to this constraint.

The solution typically involves using the method of Lagrange multipliers, or you may derive an optimal choice directly if the utility function is simple (like Cobb-Douglas).

Steps to Solve:

1. Define the utility function: Specify the form of your utility function \( U(S, L) \).

2. Formulate the Lagrangian: Set up your optimization problem: \[ \mathcal{L}(S, L, \lambda) = U(S, L) + \lambda(18 - S - L) \] where \( \lambda \) is the Lagrange multiplier.

3. Take the first-order conditions: Differentiate the Lagrangian with respect to \( S \), \( L \), and \( \lambda \), and set the derivatives equal to zero.

4. Solve the system of equations: This will give you the optimal amounts of \( S \) and \( L \).

Without the specific form of your utility function or any additional details, it’s impossible to provide a specific numerical answer for your optimal sleeping hours. Please provide that information if you want a more detailed analysis!