The question in the image is related to Kreuzer-Ramsey economics, focusing on the utility function and optimal allocation of consumption and labor. The utility function and production technology are provided. Let's break down the tasks from the image:

1. Optimize the Kreuzer-Ramsey problem and write down the Lagrangian function.
   The utility function $u(c, 1-l)$ and production technology $y = Zl^\beta$ are given. To solve this, you'll need to:

   • Formulate the optimization problem by maximizing the given utility subject to the production constraint.
   • Write the Lagrangian considering both utility and production functions.

2. Find the Euler equation.
   The Euler equation is derived from the Lagrangian's first-order conditions, which give the intertemporal consumption-labor trade-off.

3. Using the Euler equation, find the relationship between the marginal utility of leisure and consumption.
   Here, you need to use the Euler equation and derive expressions connecting the optimal allocation between consumption $c$ and leisure $1 - l$, using the parameter $\alpha$.

4. Analyze the behavior of the utility function and technology when $\alpha \to 1$.
   This part requires examining what happens to the system when $\alpha$, a parameter in the utility function, approaches 1.

5. If the central planner prefers producing public goods (G) instead of private goods (C), redefine the optimization and solve for the new allocations.
   This question asks for a modification in the problem where the planner optimizes for public good provision instead of private consumption.

Would you like to proceed with solving these in detail, or would you like a specific part broken down first?