Finding Pareto optimal allocations and Walrasian equilibrium allocations in the case of 3 goods

Question

We have 3 people and 3 goods

$$U_A(x,y,z)=x_Ay_Az_A^2$$

$$U_B(x,y,z)=x_B^2y_Bz_B$$

$$U_C(x,y,z)=x_Cy_C^2z_C$$

Endowments are $W_A= (1,1,1)$

$W_B= (2,1,3)$

$W_C= (1,5,1)$

I am confused due to the presence of three goods

In the case of two goods, I can find MRS=$\partial U_x/\partial U_y$

Now, how can find MRS to obtain Pareto optimal allocations and Walrasian equilibrium ?

Any help will be appreciated. Thank you.

Answer 1

Pareto Optimality:

Since preferences are convex in your case, you can find Pareto optimality in the same way. You need to solve $\text{MRS}^{A}_{v,w} = \text{MRS}^{B}_{v,w} = \text{MRS}^{C}_{v,w}$ where $\{v,w : v \neq w\} \subset \{x,y,z\}$.

The reason this works is because you can maximize $U_A$ subject to $U_B = \overline{u_B}$ and $U_C = \overline{u_C}$ and similarly for the other two agents. Since your utility functions are convex, when you set up the Lagrangian, you will find that the MRSs have to be equal to one another.

Walrasian equilibrium:

You have to solve the following (three) maximization problems: $$\max [U_P(x_P,y_P,z_P)] \text{ subject to } p_xx_P + p_y y_P + p_z z_P = p_x w_x^A + p_y w_y^A + p_z w_z^A$$

for each $P \in \{A, B, C\}$. You will be able to find $p_x : p_y : p_z$ and the allocations.

Additionally, you can verify that these points lie in the Pareto optimal set due to the first welfare theorem.

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Later edit: Your set of equations for Pareto optimality will be:

1. $\begin{aligned}\text{MRS}^A_{x,y} = \text{MRS}^B_{x,y} = \text{MRS}^C_{x,y} &\iff \frac{\frac{\partial U_A}{\partial x}}{\frac{\partial U_A}{\partial y}} = \frac{\frac{\partial U_B}{\partial x}}{\frac{\partial U_B}{\partial y}} = \frac{\frac{\partial U_C}{\partial x}}{\frac{\partial U_C}{\partial y}} \\ &\iff \frac{y_A z_A^2}{x_A z_A^2} = \frac{2x_By_Bz_B}{x_B^2z_B} = \frac{y_C^2z_C}{x_Cy_Cz_C}\end{aligned}$
2. $\begin{aligned}\text{MRS}^A_{y,z} = \text{MRS}^B_{y,z} = \text{MRS}^C_{y,z} \iff \frac{\frac{\partial U_A}{\partial y}}{\frac{\partial U_A}{\partial z}} = \frac{\frac{\partial U_B}{\partial y}}{\frac{\partial U_B}{\partial z}} = \frac{\frac{\partial U_C}{\partial y}}{\frac{\partial U_C}{\partial z}}\end{aligned}$
3. $\begin{aligned}\text{MRS}^A_{z,x} = \text{MRS}^B_{z,x} = \text{MRS}^C_{z,x} \iff \frac{\frac{\partial U_A}{\partial z}}{\frac{\partial U_A}{\partial x}} = \frac{\frac{\partial U_B}{\partial z}}{\frac{\partial U_B}{\partial x}} = \frac{\frac{\partial U_C}{\partial z}}{\frac{\partial U_C}{\partial x}}\end{aligned}$

Answer 2

To add to my previous answer (and to reply to your comment), the set of equations for Pareto optimality include the MRS equations and the feasibility equations.

MRS equations: \begin{align} \text{MRS}^A_{x,y} = \text{MRS}^B_{x,y} = \text{MRS}^C_{x,y} &\iff \frac{\frac{\partial U_A}{\partial x}}{\frac{\partial U_A}{\partial y}} = \frac{\frac{\partial U_B}{\partial x}}{\frac{\partial U_B}{\partial y}} = \frac{\frac{\partial U_C}{\partial x}}{\frac{\partial U_C}{\partial y}} \\ &\iff \frac{y_A z_A^2}{x_A z_A^2} = \frac{2x_By_Bz_B}{x_B^2z_B} = \frac{y_C^2z_C}{x_Cy_Cz_C} \tag{1} \\ \text{MRS}^A_{y,z} = \text{MRS}^B_{y,z} = \text{MRS}^C_{y,z} &\iff \frac{\frac{\partial U_A}{\partial y}}{\frac{\partial U_A}{\partial z}} = \frac{\frac{\partial U_B}{\partial y}}{\frac{\partial U_B}{\partial z}} = \frac{\frac{\partial U_C}{\partial y}}{\frac{\partial U_C}{\partial z}}\tag{2} \\ \text{MRS}^A_{z,x} = \text{MRS}^B_{z,x} = \text{MRS}^C_{z,x} &\iff \frac{\frac{\partial U_A}{\partial z}}{\frac{\partial U_A}{\partial x}} = \frac{\frac{\partial U_B}{\partial z}}{\frac{\partial U_B}{\partial x}} = \frac{\frac{\partial U_C}{\partial z}}{\frac{\partial U_C}{\partial x}} \tag{3} \end{align}

The feasibility allocations are: $$x_A + y_A + z_A = 1 + 2 + 1 = 4 \tag{4}$$ $$x_B + y_B + z_B = 1 + 1 + 5 = 7 \tag{5}$$ $$x_C + y_C + z_C = 1 + 3 + 1 = 5 \tag{6}$$

The equations $1-6$ combined will give you the final Pareto Efficient allocations.