Solution

1. 1

   Marginal rates of substitution between the public good $y$ and the private good are

2. 2
   $$\text{MRS}^A_{y,x} = \frac{\partial u_A/\partial y}{\partial u_A/\partial x_A} = 1, \qquad \text{MRS}^B_{y,x} = \frac{\partial u_B/\partial y}{\partial u_B/\partial x_B} = \tfrac{1}{2}.$$
3. 3

   The marginal rate of transformation between $y$ and the private good, from

4. 4

   $y + m(x_A + x_B) = 1$, is $\text{MRT}_{y,x} = \tfrac{1}{m}$ (units of private

5. 5

   good required per additional unit of public good).

6. 6

   The Samuelson condition for Pareto optimality with a public good is

7. 7
   $$\text{MRS}^A_{y,x} + \text{MRS}^B_{y,x} \;=\; \text{MRT}_{y,x},$$
8. 8

   which yields

9. 9
   $$1 + \tfrac{1}{2} = \tfrac{1}{m} \quad\Longrightarrow\quad m = \tfrac{2}{3}.$$
10. 10

    This value is not among the options listed. Given the printed answer choices

11. 11

    $\{4/5, 4/7, 4/9, 4/11\}$, the question as transcribed is internally inconsistent

12. 12

    with the standard Samuelson criterion, so the problem statement (in particular

13. 13

    the utility specifications or the production technology) is flagged for review.

Answer structure / marking notes

Content note