Microeconomics Practice Consumer Theory
Master Consumer Choice: Solved Problems
Detailed step-by-step solutions based on Varian’s approach.
Problem 1: Cobb-Douglas Optimization (The Classic)
Question: A consumer has preferences represented by the utility function u(x1 , x2 ) = x0.5 0.5
1 x2 .
The price of good 1 is p1 = 10, the price of good 2 is p2 = 5, and income is m = 200. Find the
optimal consumption bundle (x∗1 , x∗2 ).
Solution
Step 1: Calculate the Marginal Utilities (MU)
∂u
M U1 = = 0.5x−0.5
1 x0.5
2
∂x1
∂u −0.5
M U2 = = 0.5x0.5
1 x2
∂x2
Step 2: Find the Marginal Rate of Substitution (MRS)
M U1 0.5x−0.5
1 x0.5
2 x2
|M RS| = = 0.5 −0.5 =
M U2 0.5x1 x2 x1
Step 3: Apply the Tangency Condition (M RS = p1 /p2 )
At the optimal point, the slope of the indifference curve equals the slope of the budget line:
x2 10 x2
= =⇒ = 2 =⇒ x2 = 2x1
x1 5 x1
Step 4: Substitute into the Budget Constraint
The budget constraint is p1 x1 + p2 x2 = m.
10x1 + 5x2 = 200
Substitute x2 = 2x1 :
10x1 + 5(2x1 ) = 200
10x1 + 10x1 = 200
20x1 = 200 =⇒ x∗1 = 10
Step 5: Solve for x2
x∗2 = 2(10) = 20
Final Answer: The optimal bundle is (10, 20).
1
, Microeconomics Practice Consumer Theory
Problem 2: Perfect Complements (Fixed Proportions)
Question: A consumer always consumes 2 units of sugar (x2 ) for every 1 unit of coffee (x1 ).
Prices are p1 = 2, p2 = 1, and income is m = 40. Find the demand.
Solution
Step 1: Identify the Utility Function and Optimal Path
Since goods are perfect complements, the utility function is min{x1 , 21 x2 }. The consumer is
only efficient at the vertex of the L-shaped curves, where:
x2 = 2x1
(Because for every 1 coffee, she buys 2 sugars).
Step 2: Set up the Budget Constraint
p1 x1 + p2 x2 = m =⇒ 2x1 + 1x2 = 40
Step 3: Substitute the Proportion into the Budget
Substitute x2 = 2x1 into the budget equation:
2x1 + 1(2x1 ) = 40
4x1 = 40
x∗1 = 10
Step 4: Solve for x2
x∗2 = 2(10) = 20
Final Answer: The consumer demands 10 units of coffee and 20 units of sugar.
Problem 3: Perfect Substitutes (Corner Solution)
Question: A consumer views Red Pencils (x1 ) and Blue Pencils (x2 ) as perfect substitutes,
with utility u(x1 , x2 ) = x1 + x2 . If p1 = 2, p2 = 3, and m = 30, what is the optimal choice?
Solution
Step 1: Compare Marginal Utility per Dollar
M RS = −1 (The consumer trades 1 for 1)
The market price ratio is p1 /p2 = 2/3. Since 1 > 2/3, the consumer gets more utility per dollar
spending on the cheaper good (relative to the utility gained). Basically: Good 1 costs $2. Good
2 costs $3. They provide the same utility.
Step 2: Determine the Corner Solution
Since p1 < p2 , the consumer will spend all income on Good 1.
m 30
x∗1 = = = 15
p1 2
x∗2 = 0
2
Master Consumer Choice: Solved Problems
Detailed step-by-step solutions based on Varian’s approach.
Problem 1: Cobb-Douglas Optimization (The Classic)
Question: A consumer has preferences represented by the utility function u(x1 , x2 ) = x0.5 0.5
1 x2 .
The price of good 1 is p1 = 10, the price of good 2 is p2 = 5, and income is m = 200. Find the
optimal consumption bundle (x∗1 , x∗2 ).
Solution
Step 1: Calculate the Marginal Utilities (MU)
∂u
M U1 = = 0.5x−0.5
1 x0.5
2
∂x1
∂u −0.5
M U2 = = 0.5x0.5
1 x2
∂x2
Step 2: Find the Marginal Rate of Substitution (MRS)
M U1 0.5x−0.5
1 x0.5
2 x2
|M RS| = = 0.5 −0.5 =
M U2 0.5x1 x2 x1
Step 3: Apply the Tangency Condition (M RS = p1 /p2 )
At the optimal point, the slope of the indifference curve equals the slope of the budget line:
x2 10 x2
= =⇒ = 2 =⇒ x2 = 2x1
x1 5 x1
Step 4: Substitute into the Budget Constraint
The budget constraint is p1 x1 + p2 x2 = m.
10x1 + 5x2 = 200
Substitute x2 = 2x1 :
10x1 + 5(2x1 ) = 200
10x1 + 10x1 = 200
20x1 = 200 =⇒ x∗1 = 10
Step 5: Solve for x2
x∗2 = 2(10) = 20
Final Answer: The optimal bundle is (10, 20).
1
, Microeconomics Practice Consumer Theory
Problem 2: Perfect Complements (Fixed Proportions)
Question: A consumer always consumes 2 units of sugar (x2 ) for every 1 unit of coffee (x1 ).
Prices are p1 = 2, p2 = 1, and income is m = 40. Find the demand.
Solution
Step 1: Identify the Utility Function and Optimal Path
Since goods are perfect complements, the utility function is min{x1 , 21 x2 }. The consumer is
only efficient at the vertex of the L-shaped curves, where:
x2 = 2x1
(Because for every 1 coffee, she buys 2 sugars).
Step 2: Set up the Budget Constraint
p1 x1 + p2 x2 = m =⇒ 2x1 + 1x2 = 40
Step 3: Substitute the Proportion into the Budget
Substitute x2 = 2x1 into the budget equation:
2x1 + 1(2x1 ) = 40
4x1 = 40
x∗1 = 10
Step 4: Solve for x2
x∗2 = 2(10) = 20
Final Answer: The consumer demands 10 units of coffee and 20 units of sugar.
Problem 3: Perfect Substitutes (Corner Solution)
Question: A consumer views Red Pencils (x1 ) and Blue Pencils (x2 ) as perfect substitutes,
with utility u(x1 , x2 ) = x1 + x2 . If p1 = 2, p2 = 3, and m = 30, what is the optimal choice?
Solution
Step 1: Compare Marginal Utility per Dollar
M RS = −1 (The consumer trades 1 for 1)
The market price ratio is p1 /p2 = 2/3. Since 1 > 2/3, the consumer gets more utility per dollar
spending on the cheaper good (relative to the utility gained). Basically: Good 1 costs $2. Good
2 costs $3. They provide the same utility.
Step 2: Determine the Corner Solution
Since p1 < p2 , the consumer will spend all income on Good 1.
m 30
x∗1 = = = 15
p1 2
x∗2 = 0
2
