Cost Minimization - Great notes to help achieve a first class PDF

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Great notes to help achieve a first class...

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Cost Minimisation The Firm’s Problem • Consider a firm using two inputs to make one output • The production function is y = f(x₁, x₂) • Take the output level y ≥ 0 as given • Given the input price w₁ and w₂, the cost of an input bundle (x₁, x₂) is w₁x₁ + w₂x₂

• For given w₁, w₂ and y, the firm’s cost- minimisation problem is to solve:

• The levels x₁*(w₁, w₂, y) and x₂*(w₁, w₂, y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2

• The (smallest possible) total cost for producing y output units is therefore !

isocost • We are looking for the combinations that are going to give the level of cost, C:

- w₁x₁ + w₂x₂ = C • We can rearrange:

• The equation above gives us straight lines:

• Higher isocosts lines are associated with higher costs!

cost minimisation

EXAMPLE 1: • Long run cost function: - Cobb-Douglas Technology:

• We can use lagrangian to solve the problem:

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EXAMPLE 2: • Long run cost function: - Leontief Technology: - f(x₁, x₂) = min{ax₁, bx₂}

• Assume: - Q = min {2x₁, 3x₂} - w₁ = 10; w₂ = 50 • What is the least costly input bundle to produce a given level of output?

• Q = 2x₁ = 3x₂ • Set: x₁ = Q/2; x₂ = Q/3 • LTC = 10x₁ + 50x₂ = 10Q/2 + 50Q/3 = 65/3Q • LAC = LMC = 65/3

EXAMPLE 3: • Long run cost function:

- Perfect Substitutes Technology: - Q = 2x₁ + 3x₂ • w₁ = 10; w₂ = 50 • Minimize: C = w₁x₁ + w₂x₂ = 10x₁ + 50x₂ • The isoquants are straight lines • Corner Solution: do not apply tangency rule

• The firm finds x₂ too expensive and uses only x₁ • Q = 2x₁ + 3x₂; with x₂ = 0 • Q = 2x₁; x₁ = Q/2 • LTC = 10(Q/2) = 5Q; LAC = LMC = 5...