Problem Breakdown:

The problem is an optimization with the objective function:

$\max_{y_1, y_2} \, \left( y_1 - k y_1^2 + y_2 - k y_2^2 \right)$ subject to: $y_1 + y_2 \leq Y,$ where $k > 0$ and $Y > \frac{1}{k}$.

(a) Solution to the Optimization Problem:

The Lagrangian for this problem is:

$\mathcal{L}(y_1, y_2, \lambda) = \left( y_1 - k y_1^2 + y_2 - k y_2^2 \right) + \lambda (Y - y_1 - y_2),$ where $\lambda \geq 0$ is the Lagrange multiplier.

Step 1: First-order conditions (FOCs)

Take the partial derivatives of the Lagrangian with respect to $y_1$, $y_2$, and $\lambda$:

1. $\frac{\partial \mathcal{L}}{\partial y_1} = 1 - 2k y_1 - \lambda = 0$,
2. $\frac{\partial \mathcal{L}}{\partial y_2} = 1 - 2k y_2 - \lambda = 0$,
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = Y - y_1 - y_2 = 0$.

Step 2: Solve the system of equations

From the first two conditions: $\lambda = 1 - 2k y_1, \quad \lambda = 1 - 2k y_2.$ Equating these gives: $y_1 = y_2.$

Substitute $y_1 = y_2$ into the constraint $y_1 + y_2 \leq Y$: $2y_1 \leq Y \quad \implies \quad y_1 = y_2 = \frac{Y}{2}.$

Step 3: Check the optimality condition

The candidate solution $y_1 = \frac{Y}{2}$ and $y_2 = \frac{Y}{2}$ must satisfy the constraint $y_1 + y_2 = Y$, which it does. Thus, the solution is:

$y_1^* = y_2^* = \frac{Y}{2}.$

Step 4: Objective function value at the optimum

The maximum value of the objective function is: $f(y_1^*, y_2^*) = \left(\frac{Y}{2} - k \left(\frac{Y}{2}\right)^2\right) + \left(\frac{Y}{2} - k \left(\frac{Y}{2}\right)^2\right),$ which simplifies to: $f(y_1^*, y_2^*) = Y - k\frac{Y^2}{2}.$

(b) Does the constraint bind at the optimum?

The constraint binds if $y_1 + y_2 = Y$. From the solution, $y_1^* + y_2^* = \frac{Y}{2} + \frac{Y}{2} = Y$. Hence, the constraint binds at the optimum.

Interpretation:

At the optimal solution, both $y_1$ and $y_2$ split the available budget $Y$ equally to maximize the objective function. The constraint ensures that the resources $Y$ are fully utilized, reflecting an efficient allocation.

Do you want further clarifications or details on any step? Here are 5 related questions to expand this solution:

1. How does the value of $k$ influence the optimal allocation of $y_1$ and $y_2$?
2. What happens if $Y < \frac{1}{k}$? Does the solution remain feasible?
3. Can you explain the role of the Lagrange multiplier $\lambda$ in this problem?
4. How would the solution change if the constraint was $y_1 + y_2 < Y$ (strict inequality)?
5. What would be the optimal solution if $k = 0$ (no quadratic penalty)?

Tip: Always check whether the constraint binds by evaluating the slack variable $\lambda$. If $\lambda > 0$, the constraint