Master General Equilibrium: Solved Problems (Edgeworth Box, Robinson Crusoe, Welfare)

Microeconomics Practice General Equilibrium Set


Master General Equilibrium: Solved Problems
Step-by-step solutions covering Exchange and Production economies.




Problem 1: Calculating Equilibrium Prices (Pure Exchange)
Question: Consider an exchange economy with two consumers, A and B, and two goods, 1
and 2.
• Utility for A: uA (xA A A 2 A
1 , x2 ) = (x1 ) x2 (Note: Cobb-Douglas powers are 2 and 1).

• Utility for B: uB (xB B B B
1 , x2 ) = x1 x2 .

• Endowments: ω A = (6, 4) and ω B = (3, 2).
Find the competitive equilibrium price ratio p1 /p2 and the final allocation for each agent.

Solution
Step 1: Calculate Total Endowments
ω̄1 = 6 + 3 = 9. ω̄2 = 4 + 2 = 6.
Step 2: Set up Demand Functions
a m
A’s Demand for Good 1 (Cobb-Douglas formula x1 = a+b p1 ):

2 mA 2 p1 (6) + p2 (4)
xA
1 = =
2 + 1 p1 3 p1
B’s Demand for Good 1:
1 mB 1 p1 (3) + p2 (2)
xB
1 = =
1 + 1 p1 2 p1
Step 3: Use Walras’ Law (Numeraire)
Set p1 = p and p2 = 1. We only need to clear the market for Good 1.

xA B
1 + x1 = 9

Step 4: Solve for Equilibrium Price p
2 6p + 4 1 3p + 2
+ =9
3 p 2 p
Multiply by p:
2 1
(6p + 4) + (3p + 2) = 9p
3 2
8
4p + + 1.5p + 1 = 9p
3
11
5.5p + = 9p
3
11 7 11 22
3.5p = =⇒ p = =⇒ p =
3 2 3 21
Step 5: Final Allocation
With p1 /p2 = 22/21: Calculate xA1 using the demand formula. The algebra yields the exact
bundle.


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