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Ricardo
Ricardo
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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.

To add to my previous answer, 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.

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.

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Ricardo
Ricardo
  • 23
  • 2

To add to my previous answer, the set of equations for Pareto optimality will be given byinclude 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.

To add to my previous answer, the set of equations for Pareto optimality will be given by 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.

To add to my previous answer, 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.

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Ricardo
Ricardo
  • 23
  • 2

To add to my previous answer, the set of equations for Pareto optimality will be given by 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.