Consider an exchange economy with consumers A and B. A has an endowment of 10 units of x and 50
units of y, while B has 40 units of x and 40 units of y. A has utility function u A(x,y) = x Ay A and B has a utility
function u B(x,y) = x By B.
c. Find the Pareto efficient allocation that gives B the same utility as he gets at his endowment.

In this exchange economy problem, we aim to find the **Pareto-efficient allocation** of goods \( x \) and \( y \) between two consumers (A and B) that ensures B gets the same utility as in his initial endowment.

### Step-by-Step Approach

#### 1. **Endowments and Utilities**:
- **A's endowment**: \( (x_A^e = 10, y_A^e = 50) \)
- **B's endowment**: \( (x_B^e = 40, y_B^e = 40) \)
- **A's utility function**: \( u_A(x_A, y_A) = x_A \cdot y_A \)
- **B's utility function**: \( u_B(x_B, y_B) = x_B \cdot y_B \)

#### 2. **Initial Utility of Consumer B**:
To find the initial utility that B gets from their endowment:

\[
u_B^e = x_B^e \cdot y_B^e = 40 \cdot 40 = 1600
\]

So, consumer B has an initial utility of \( u_B = 1600 \).

#### 3. **Pareto Efficiency Condition**:
In a Pareto-efficient allocation, the **Marginal Rate of Substitution (MRS)** for both consumers must be equal.

The MRS for consumer A is:

\[
\text{MRS}_A = \frac{\partial u_A / \partial x_A}{\partial u_A / \partial y_A} = \frac{y_A}{x_A}
\]

The MRS for consumer B is:

\[
\text{MRS}_B = \frac{\partial u_B / \partial x_B}{\partial u_B / \partial y_B} = \frac{y_B}{x_B}
\]

Equating the two:

\[
\frac{y_A}{x_A} = \frac{y_B}{x_B}
\]

This implies:

\[
\frac{y_A}{x_A} = \frac{y_B}{x_B} \quad \Rightarrow \quad y_A \cdot x_B = y_B \cdot x_A \quad \text{(1)}
\]

#### 4. **Resource Feasibility Condition**:
The total endowment of goods is fixed. Therefore, the allocations must satisfy the following feasibility conditions:

\[
x_A + x_B = 10 + 40 = 50 \quad \text{(total endowment of } x \text{)}
\]
\[
y_A + y_B = 50 + 40 = 90 \quad \text{(total endowment of } y \text{)}
\]

#### 5. **Solve for the Allocation**:
We now solve for the allocations \( x_A \), \( x_B \), \( y_A \), and \( y_B \) that satisfy the feasibility conditions and give B their initial utility of 1600.

B's utility constraint:

\[
x_B \cdot y_B = 1600 \quad \text{(2)}
\]

From (2), solve for \( y_B \) in terms of \( x_B \):

\[
y_B = \frac{1600}{x_B}
\]

Substitute this into the feasibility condition for \( y \):

\[
y_A + \frac{1600}{x_B} = 90 \quad \Rightarrow \quad y_A = 90 - \frac{1600}{x_B} \quad \text{(3)}
\]

Substitute \( y_A \) from (3) into the MRS condition (1):

\[
(90 - \frac{1600}{x_B}) \cdot x_B = y_B \cdot x_A \quad \Rightarrow \quad \text{relation involving }x_B
\]




Consider an exchange economy with consumers A and B. A has an endowment of 10 units of x and 50 units of y, while B has 40 units of x and 40 units of y. A has utility function u_A(x,y) = x_A * y_A and B has a utility function u_B(x,y) = x_B * y_B. Find the Pareto efficient allocation that gives B the same utility as he gets at his endowment.

Discover how to find the Pareto efficient allocation of resources between two consumers in an exchange economy. This solution explores utility maximization, the Marginal Rate of Substitution (MRS), and resource feasibility conditions. Learn how to ensure one consumer's utility remains constant while maximizing efficiency. Suitable for economics undergraduates.

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