I don't know if this is the right forum to ask these questions (if so, tell me where to ask them). I have my microeconomics exam soon and I would like to know if I solved a couple of exercises correctly: Consider a barter economy with two people ($A$ and $B$) and two goods ($x$ and $y$). In the initial setup, person $A$ has one unit of good $x$ and three units of good $y$, while person $B$ has three units of good $x$ and one unit of good $y$. The utility function of Person A is $U_A(x_A, y_A) = x_A·y_A$, that of Person B is $U_B(x_B, y_B) = x_B+y_B$, where $x_i$ and $y_i$ respectively are the amount of good consumed by Person $i=A, B$
a) Initial equipment:
i) Draw the initial equipment in the Edgeworth box given above. Label this point ω.
ii) First consider the two consumers separately. Calculate the marginal rate of substitution for consumers A and B.
iii) In the Edgeworth box given above, sketch the consumer indifference curves achieved in the initial equipment. Denote the indifference curve of consumer $A$ by $I_A$ and that of consumer $B$ by $I_B$. Is the initial equipment Pareto optimal? Explain your answer.
for ii) the marginal rate is
$MR_A(x_A,y_A)= \frac{y_a}{x_a}$ so $MR_A(1,3)=3$
$MR_B(x_B,y_B)= 1$ so $MR_B(3,1)=1$0
For i) and iii) here the Edgeworth box:
For iii) the initial point is not pareto optimal since every point in the green marked area would be a better point.