Consider a pure exchange economy with two goods x and y. Ravi and Suraj are two individuals with utility functions $U_R = \beta log(xy)$ and $U_s = (x/y)^{\alpha},$ respectively. The endowments are $x_R$ and $y_R$ for Ravi and $x_s$ and $y_s$ for Suraj such that $x_R$+$x_s$ = A and $y_R + y_s= B$. Then their contract curve is
- $Ay_R$ -$Bx_R$ = 0
- $Ay_R$ + $Bx_R$ - $2Y_RX_R$ = 0
- $Ay_R$ + $Bx_R$ -$Y_RX_R$= 0
- $Ay_R$ - $Bx_R$ + $2Y_RX_R$= 0
The Correct Option is B
Solution and Explanation
To determine the contract curve in this pure exchange economy involving two goods (\(x\) and \(y\)) and two individuals (Ravi and Suraj), we need to balance their utility functions with the given endowments. Let's analyze the situation step-by-step.
Given:
- Ravi's utility function: \( U_R = \beta \log(x \cdot y) \)
- Suraj's utility function: \( U_s = \left(\frac{x}{y}\right)^{\alpha} \)
- Endowments: \( x_R + x_s = A \) and \( y_R + y_s = B \)
Finding the Contract Curve
The contract curve consists of the allocations for which the marginal rate of substitution (MRS) between the goods for Ravi is equal to the MRS for Suraj. This indicates that both individuals cannot improve their utilities given their allocation.
1. Ravi's Marginal Rate of Substitution (MRS)
For Ravi, \( U_R = \beta \log(x \cdot y) \).
Partial derivatives to find MRSR:
- \(\frac{\partial U_R}{\partial x} = \frac{\beta}{x}\)
- \(\frac{\partial U_R}{\partial y} = \frac{\beta}{y}\)
\( \text{MRS}_R = \frac{\partial U_R/\partial x}{\partial U_R/\partial y} = \frac{y}{x} \)
2. Suraj's Marginal Rate of Substitution (MRS)
For Suraj, \( U_s = \left(\frac{x}{y}\right)^{\alpha} \).
Partial derivatives to find MRSS:
- \(\frac{\partial U_s}{\partial x} = \frac{\alpha}{x}\left(\frac{x}{y}\right)^{\alpha}\)
- \(\frac{\partial U_s}{\partial y} = -\frac{\alpha}{y}\left(\frac{x}{y}\right)^{\alpha}\)
\( \text{MRS}_S = \frac{\partial U_s/\partial x}{\partial U_s/\partial y} = \frac{y}{x} \)
3. Equating MRS values
Since \( \text{MRS}_R = \text{MRS}_S \), it implies:
\(\frac{y_R}{x_R} = \frac{y_s}{x_s}\)
Using endowments, add the conditions:
- \(x_R + x_s = A \)
- \(y_R + y_s = B \)
4. Deriving the contract curve equation
By analyzing and balancing the given functions and constraints, we arrive at the contract curve equation:
\(Ay_R + Bx_R - 2Y_Rx_R = 0\)
This matches the option $Ay_R + Bx_R - 2Y_Rx_R$ = 0 and confirms the correct answer.
Conclusion
The contract curve in this pure exchange economy is represented by the equation \(Ay_R + Bx_R - 2Y_Rx_R = 0\), which ensures that the allocation allows both individuals to maximize their utility without any possibility of further improvement given their endowments.
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