Answer

Solution

1. 1

   Compute $A$'s marginal rate of substitution at the endowment $(x_A, y_A) = (0, y)$:

2. 2
   $$\frac{\partial u_A}{\partial x_A} = 1 - m(x_A - y_A), \qquad \frac{\partial u_A}{\partial y_A} = 1 + m(x_A - y_A).$$
3. 3

   At $(0,y)$:

4. 4
   $$\text{MRS}^A_{x,y} = \frac{\partial u_A/\partial x_A}{\partial u_A/\partial y_A} = \frac{1 + my}{1 - my}.$$
5. 5

   Agent $B$'s utility is linear with $\text{MRS}^B_{x,y} = 1$.

6. 6

   For $A$ to voluntarily transfer $Y$ to $B$ in exchange for $X$ (so that both

7. 7

   agents are made strictly better off), $A$ must value $X$ relative to $Y$ more

8. 8

   than $B$ does at the endowment, i.e.

9. 9
   $$\text{MRS}^A_{x,y} > \text{MRS}^B_{x,y} \;=\; 1.$$
10. 10

    Assuming $1 - my > 0$ (so that marginal utilities are positive at the endowment),

11. 11
    $$\frac{1+my}{1-my} > 1 \quad\Longleftrightarrow\quad 1 + my > 1 - my \quad\Longleftrightarrow\quad my > 0.$$
12. 12

    Since the endowment $y > 0$, this reduces to $m > 0$.

Answer structure / marking notes

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