I have to characterize the Pareto optimal allocation of the following problem:
Consider two-agents-two-goods economy. The preferences of the agents are given by the following utility functions:
$\upsilon_1(x_{1,1},x_{2,1})=x_{1,1}-0.25x^2_{2,1}$,
$\upsilon_2(x_{1,2},x_{2,2})=x_{1,2}+x^{0.5}_{2,2}$
where $x_{j,i}$ denotes the consumption of good j by agent i and agents have the following endowements $e_1=(0,8)$ and $e_2=(16,8)$.
I know that since the second good for the first consumer is a "bad good" in a Pareto optimal allocation $x_{2,1}$ will be equal to zero.
My question is if there is a way to obtain this results mathematically?
With positive consumption, the marginal utility of $x_2$ is negative for the first consumer, and positive for the second consumer, $$\frac{\partial u_{1}}{\partial x_2}<0 \text{ and } \frac{\partial u_{2}}{\partial x_2}>0$$. The Pareto optimal allocations $(a,0),(b,16)$, where $a+b=16$.