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In a pure-exchange economy with two individuals, A and B, and two goods, X and Y, the determination of Pareto optimal allocations requires finding all allocations of goods between the two individuals such that it is impossible to make one individual better off without making the other worse off. The total endowments of goods in the economy are fixed: $X_T = X_A + X_B = 5 + 3 = 8$ units of good X, and $Y_T = Y_A + Y_B = 3 + 4 = 7$ units of good Y. Individual A has a utility function $U_A = X_A Y_A^2$, and individual B has a utility function $U_B = X_B^2 Y_B$. To characterize the Pareto optimal allocations, we use the concept of marginal rates of substitution (MRS). The marginal utilities of individual A with respect to goods X and Y are: $$ MU_{X_A} = \frac{\partial U_A}{\partial X_A} = Y_A^2 $$ $$ MU_{Y_A} = \frac{\partial U_A}{\partial Y_A} = 2 X_A Y_A $$ Thus, the marginal rate of substitution of good X for good Y for individual A is: $$ MRS_A = \frac{MU_{X_A}}{MU_{Y_A}} = \frac{Y_A^2}{2 X_A Y_A} = \frac{Y_A}{2 X_A} ________ ______ _____ ___ _______ ________ __________ ______ __________ ________ ________.
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