The image presents a utility maximization problem that involves the consumption function and the marginal utility conditions for an intertemporal consumption model. Here’s a breakdown of the key components:

1. Marginal Utility Conditions:

   • Equation $(a')$: $U'(C_t) = \beta (1 + r_t) E_t (U'(C_{t+1}))$
   • Equation $(a'')$: $U'(C_t) = \beta E_t ((1 + X_t) U''(C_{t+1}))$

   These equations represent the first-order conditions for utility maximization in a dynamic setting, where future consumption depends on expected utility and the interest rate $r_t$ or another factor $X_t$.

2. Utility Function:

   • The function $U(C) = aC - bC^2$ is given, where $a$ and $b$ are constants.
   • This is a quadratic utility function, suggesting diminishing marginal returns to consumption as $C$ increases.

3. Condition for $\beta (1 + r_t) = 1$:

   • This condition implies that the discount factor adjusted by the interest rate equals 1, which might indicate a situation of time consistency in consumption decisions.

4. Objective:

   • The question asks how consumption in period $t+1$ (i.e., $C_{t+1}$) relates to consumption in period $t$ (i.e., $C_t$).

To solve this, we need to use the given utility function and the marginal utility condition $(a')$ to express $C_{t+1}$ in terms of $C_t$ under the assumption that $\beta (1 + r_t) = 1$.

Solution Steps:

1. Compute the Marginal Utility $U'(C)$: Given $U(C) = aC - bC^2$, $U'(C) = a - 2bC.$

2. Substitute $U'(C_t)$ and $U'(C_{t+1})$ into $(a')$: Using the condition $(a')$: $a - 2bC_t = \beta (1 + r_t) E_t(a - 2bC_{t+1}).$ Given $\beta (1 + r_t) = 1$, this simplifies to: $a - 2bC_t = E_t(a - 2bC_{t+1}).$

3. Expectations and Relationship: Solving this equation further will reveal the functional form of $C_{t+1}$ in terms of $C_t$, factoring in expectations about future consumption.

Would you like me to proceed with detailed calculations or analysis on how $C_{t+1}$ explicitly depends on $C_t$? Here are some further related questions you might be interested in:

1. How does the discount factor $\beta$ affect intertemporal consumption choices?
2. What happens if $\beta (1 + r_t) \neq 1$?
3. Can we derive a closed-form solution for $C_{t+1}$ in terms of $C_t$?
4. What role does the quadratic utility function play in shaping the consumption path?
5. How would changes in parameters $a$ and $b$ affect the optimal consumption levels?

Tip: Always check the stability conditions in dynamic optimization problems to understand long-term behavior in consumption models.