Master Producer Theory: Solved Problems (Cost Minimization & Profit Maximization)

Microeconomics Practice Producer Theory


Master Producer Theory: Solved Problems
Step-by-step solutions based on Varian’s approach.




Problem 1: Cost Minimization (Cobb-Douglas)
1/2 1/2
Question: A firm has a production function given by f (x1 , x2 ) = x1 x2 . The prices of factors
are w1 = 4 and w2 = 1. The firm wants to produce y = 10 units. Find the cost-minimizing
combination of inputs (x∗1 , x∗2 ) and the minimum cost.

Solution
Step 1: Calculate Marginal Products (MP)
1 −1/2 1/2 1 1/2 −1/2
M P 1 = x1 x2 , M P2 = x1 x2
2 2
Step 2: Find the Technical Rate of Substitution (TRS)

1 −1/2 1/2
M P1 x x2 x2
|T RS| = = 2 11/2 −1/2 =
M P2 1 x1
2 x1 x2
Step 3: Apply the Tangency Condition (T RS = w1 /w2 )
x2 4
= =⇒ x2 = 4x1
x1 1
This is the expansion path.
Step 4: Use the Production Constraint
We need to produce y = 10.
1/2 1/2
x1 x2 = 10
Substitute x2 = 4x1 :
1/2
x1 (4x1 )1/2 = 10
1/2 1/2
x1 · 2 · x1 = 10
2x1 = 10 =⇒ x∗1 = 5
Step 5: Solve for x2

x∗2 = 4(5) = 20
Step 6: Calculate Total Cost

C = w1 x1 + w2 x2 = 4(5) + 1(20) = 20 + 20 = 40

Final Answer: Inputs: (5, 20). Minimum Cost: $40.




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