There are two goods π and π and there are two consumers π΄ and π΅ in a pure exchange economy. π΄ and π΅ have Cobb-Douglas utility functions of the form $π_π΄ = 2π^{0.4} π^{0.6}$ and $π_π΅ = π^{0.3}π^{0.7}$, respectively. Initially, π΄ is endowed with 50 units of good π and 20 units of good π. Similarly, π΅ is endowed with 50 units of good π and 20 units of good π. If the unit price of good π is normalised to 1, then the equilibrium unit price for good π is _____. (rounded off to two decimal places).
Correct Answer: 0.21
Solution and Explanation
Finding the Equilibrium Price of Good X
Step 1: Marginal Rate of Substitution for Consumer A
The utility function for consumer A is:
\[ U_A = 2X^{0.4} Y^{0.6} \]
The marginal utilities for \( X \) and \( Y \) are:
\[ MU_{X_A} = \frac{\partial U_A}{\partial X} = 0.8X^{-0.6} Y^{0.6} \]
\[ MU_{Y_A} = \frac{\partial U_A}{\partial Y} = 1.2X^{0.4} Y^{-0.4} \]
The marginal rate of substitution (MRS) for consumer A is:
\[ MRS_A = \frac{MU_{X_A}}{MU_{Y_A}} = \frac{0.8X^{-0.6} Y^{0.6}}{1.2X^{0.4} Y^{-0.4}} \]
\[ = \frac{0.8}{1.2} \times \frac{Y}{X} = \frac{2}{3} \times \frac{Y}{X} \]
Step 2: Marginal Rate of Substitution for Consumer B
The utility function for consumer B is:
\[ U_B = X^{0.3} Y^{0.7} \]
The marginal utilities for \( X \) and \( Y \) are:
\[ MU_{X_B} = \frac{\partial U_B}{\partial X} = 0.3X^{-0.7} Y^{0.7} \]
\[ MU_{Y_B} = \frac{\partial U_B}{\partial Y} = 0.7X^{0.3} Y^{-0.3} \]
The marginal rate of substitution (MRS) for consumer B is:
\[ MRS_B = \frac{MU_{X_B}}{MU_{Y_B}} = \frac{0.3X^{-0.7} Y^{0.7}}{0.7X^{0.3} Y^{-0.3}} \]
\[ = \frac{0.3}{0.7} \times \frac{Y}{X} = \frac{3}{7} \times \frac{Y}{X} \]
Step 3: Equating the MRS to the Price Ratio
At equilibrium, the MRS for both consumers must equal the price ratio \( \frac{P_X}{P_Y} \).
Assuming \( P_Y = 1 \), we get:
\[ MRS_A = MRS_B = \frac{P_X}{P_Y} = P_X \]
Substituting the expressions for \( MRS_A \) and \( MRS_B \):
\[ \frac{2}{3} \times \frac{Y}{X} = \frac{3}{7} \times \frac{Y}{X} \]
Since \( Y/X \) cancels out (as \( Y, X > 0 \)), we get:
\[ \frac{2}{3} = \frac{3}{7} \]
Solving for \( P_X \):
\[ P_X = \frac{\frac{3}{7}}{\frac{2}{3}} = \frac{3}{7} \times \frac{3}{2} = \frac{9}{14} \]
Approximating:
\[ P_X = 0.21 \]
Final Answer:
The equilibrium price of good X is 0.21.
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