Pareto-optimal allocations in the Edgeworth box

If agent A has utility function $u(x_1, x_2)$ and agent B has utility function $v(x_1, x_2)$, what are the equation(s) that characterize the Pareto-efficient allocation of goods in the Edgeworth box? Assume that the initial endowment of agent A is $(\omega_A^1, \omega_A^2)$ and the initial endowment of agent B is $(\omega_B^1, \omega_B^2)$.

I don't know why this got down-voted. I'll post my attempt anyway...

$MRS_A(x_A^1, x_A^2)=MRS_B(x_B^1, x_B^2)$

$\displaystyle\left.\frac{\partial{u}/\partial{x_1}}{\partial{u}/\partial{x_2}}\right|_{(x_A^1, x_A^2)}=\left.\frac{\partial{v}/\partial{x_1}}{\partial{v}/\partial {x_2}}\right|_{(x_B^1, x_B^2)}$

In addition, $x_A^1+x_B^1=\omega_A^1+\omega_B^1$

and $x_A^2+x_B^2=\omega_A^2+\omega_B^2$.

These 3 equations in 4 unknowns determine the contract curve.

• How is this an attempt? Did you derive these equations or did you find them in a book? – Giskard Aug 1 '18 at 19:23
• Also, you seem to be missing the individual rationality conditions if you want to the define the contract curve. – Giskard Aug 1 '18 at 19:25
• @denesp My textbook says that the consumers will have to have the same MRS between each of the two goods. I just wrote that down as an equation. I don't know what you mean by individual rationality conditions. – Thomas Aug 1 '18 at 19:54