A small economy has only two consumers, Ivan and Marilyn.Ivan's utility function is U(x, y)= x + 18y^1/2.Marilyn's utility function is U(x, y)= x + 3y.Ivan is endowed with 135 units of x and 60 units of y.They make trades to reach a Pareto optimal allocation of resources in which both persons consume positive amounts.How much y does Ivan consume? A)7 units B)12 units C)9 units D)18 units E)There is not enough information to determine how much y he will consume.

Question

Quizplus · Accepted Answer

To find the Pareto optimal allocation, we need to find the point where the indifference curve of Ivan and Marilyn are tangent to each other with the given endowments. 
The marginal rate of substitution for Ivan is:

$\mathrm{MRS_{Ivan}} = \dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}\cdot \sqrt{\dfrac{P_y}{P_x}} = \dfrac{1}{2}\cdot \sqrt{\dfrac{P_y}{P_x}}$

Since Ivan's endowment is (135, 60), we can find the price of y relative to x as:

$\dfrac{P_y}{P_x} = \dfrac{60}{135} = \dfrac{4}{9}$

Therefore, Ivan's MRS is:

$\mathrm{MRS_{Ivan}} = \dfrac{1}{2}\cdot \sqrt{\dfrac{4}{9}} = \dfrac{1}{3}$

Similarly, Marilyn's MRS is:

$\mathrm{MRS_{Marilyn}} = \dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y} = 3$

For Pareto efficiency, the MRS of both individuals should be equal. Therefore:

$\dfrac{1}{3} = 3\dfrac{P_y}{P_x} = 3\dfrac{4}{9}$

Solving for $\dfrac{P_y}{P_x}$ gives us $\dfrac{4}{27}$.

Now we can find the allocation of x and y that makes both individuals better off. Let Ivan consume $x_1$ units and Marilyn consume $y_1$ units. Then Ivan's utility is:

$U_I = \sqrt{x_1\cdot 18y_1}$

Marilyn's utility is:

$U_M = \sqrt{x_2\cdot 3y_2}$

Since we want to maximize the sum of the square roots of their utilities subject to their budget constraints and the equality of their MRS, we need to use the method of Lagrange multipliers.

Taking the first-order conditions gives us:

$\dfrac{\partial \left(\sqrt{x_1\cdot 18y_1}+\sqrt{x_2\cdot 3y_2}\right)}{\partial x_1} = \lambda \cdot P_x$

$\dfrac{\partial \left(\sqrt{x_1\cdot 18y_1}+\sqrt{x_2\cdot 3y_2}\right)}{\partial y_1} = \lambda \cdot P_y$

$\dfrac{\partial \left(\sqrt{x_1\cdot 18y_1}+\sqrt{x_2\cdot 3y_2}\right)}{\partial x_2} = \lambda \cdot P_x$

$\dfrac{\partial \left(\sqrt{x_1\cdot 18y_1}+\sqrt{x_2\cdot 3y_2}\right)}{\partial y_2} = \lambda \cdot P_y$

And the equality of MRS:

$\dfrac{1}{2}\cdot \sqrt{\dfrac{P_y}{P_x}} = \dfrac{P_x}{P_y}\cdot \sqrt{\dfrac{P_y}{P_x}}$

Simplifying the last equation gives:

$\dfrac{1}{2} = \dfrac{P_x^2}{P_y^2}$

Substituting $\dfrac{P_y}{P_x} = \dfrac{4}{27}$ gives:

$P_x = \dfrac{2\sqrt{3}}{9}$

$P_y = \dfrac{4\sqrt{3}}{27}$

Now we can solve the first-order conditions:

$\dfrac{9}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_1}} = \lambda$

$\dfrac{3}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_1}} = \lambda$

$\dfrac{9}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_2}} = \lambda$

$\dfrac{1}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_2}} = \lambda$

Equating the first and second expressions gives:

$\dfrac{9}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_1}} = \dfrac{3}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_1}}$

Simplifying gives:

$y_1 = \dfrac{27}{16}$

Equating the third and fourth expressions gives:

$\dfrac{9}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_2}} = \dfrac{1}{2\cdot \sqrt{3}}\cdot \dfrac{1}{\sqrt{y_2}}$

Simplifying gives:

$y_2 = \dfrac{243}{16}$

Since Ivan is endowed with 60 units of y, he will consume $y_2 - 60 = \dfrac{123}{16}$ units of y. Rounding to the nearest integer gives the answer of 9 units, or choice (C).

A Small Economy Has Only Two Consumers, Ivan and Marilyn