Lecture # 11b Costs and Cost Minimization Lecturer: Martin Paredes

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  Definitions of Costs Long-Run Cost Minimization The constrained minimization problem Comparative statics  Input Demands Short Run Cost Minimization 2

  Explicit costs involve a direct monetary outlay.

Implicit costs do not involve a direct monetary outlay.

 Opportunity cost is the value of a resource in its next best alternative, which is foregone when another alternative is chosen.

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Example: Opportunity Costs    You are currently a student at TCD What is your opportunity cost?

The salary you could earn as a high-school graduate.

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  Accounting costs involve explicit costs that have been incurred in the past.

Economic costs are the sum of all decision relevant implicit and explicit costs, including opportunity costs.

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  Sunk (or unavoidable) costs involve all economic costs that have been already been incurred and cannot be recovered.

Nonsunk (or avoidable) costs are economic costs that are incurred only if a particular decision is made.

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Example: Sunk Costs  Suppose you want to build an hydroelectric dam to generate electricity  Suppose the cost is € 20M and takes 3 years.

   A hydroelectric dam has no alternative use. Should you build the dam?

Whatever the decision, € 20M is not a sunk cost: you can avoid it by deciding not to build the dam.

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Example: Sunk Costs  Suppose you decided to build the dam.

  Three years from now, the dam is operative.

However, market conditions have changed.   Should you operate the dam?

Whatever the decision, € 20M is now is a sunk cost: you already incur that cost, and cannot recover the investment.  Your decision should not take into account the (already sunk) cost of the built dam.

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Definition: The cost minimisation problem for a firm is the problem of finding a combination of inputs to minimise the cost of producing a given amount of output.

   Decision problem for a firm may depend on whether or not there are time constraints: Long run: No constraints Short run: Constraints on use of some inputs 9

 Assume a firm produces a good using only two inputs: K and L  Firm takes as given:    Price of K: Price of L: r w Technology: F(L,K)  Total spending on inputs: TC = rK + wL 10

Definition: The Isocost Line defines the set of combinations of labour and capital that yield the same total cost for the firm.

TC 0 = rK + wL …or… K = TC 0 – w L r r 11

TC 0 /r K

Example: Isocost Lines

TC 0 /w L

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TC 0 /r K Slope = -w/r

Example: Isocost Lines

TC 0 /w L

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TC 1 /r TC 0 /r K Slope = -w/r

Example: Isocost Lines

TC 0 /w TC 1 /w L

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TC 2 /r K TC 1 /r TC 0 /r Slope = -w/r

Example: Isocost Lines

TC 0 /w TC 1 /w TC 2 /w L

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TC 2 /r K TC 1 /r TC 0 /r

Example: Isocost Lines

Slope = -w/r

Direction of increase in total cost

TC 0 /w TC 1 /w TC 2 /w L

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Assumption:  Firms want to minimise cost for a particular level of output and given technology Firm’s Problem: Min TC = rK + wL subject to: Q 0 L,K = F(L,K) 17

 The cost minimisation is analogous to expenditure minimization for the consumer.

 In this case, the constraint is the satisfaction of the isoquant equation: Q 0 = F(L,K)  Two conditions for interior solution:  Tangency condition: MRTS L,K = MPL = w MPK r  Isoquant constraint: Q 0 = F(L,K) 18

K

Example: Cost Minimization

Isoquant Q = Q 0 L

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K

Example: Cost Minimization

TC 0 /r TC 0 /w Isoquant Q = Q 0 L

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TC 2 /r K TC 0 /r

Example: Cost Minimization

Isoquant Q = Q 0 TC 0 /w TC 2 /w L

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TC 2 /r K TC 1 /r TC 0 /r •

Example: Cost Minimization

Isoquant Q = Q 0 TC 0 /w TC 1 /w TC 2 /w L

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Example:   Suppose: Suppose: Q(L,K) = 50L 0.5

K 0.5

Q 0 = 1000 w = € 5 r = € 20  Which is the cost-minimising choice for the firm?

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Example (cont.):  Tangency condition  MRTS L,K = MP L MP K = (0.5)(50)L -0.5

K 0.5

(0.5)(50)L 0.5

K -0.5

= K L  w = 5 = 1 r 20 4  So L = 4K 24

Example (cont.):  Isoquant Constraint:  50L 0.5

K 0.5

=> = 1000 50(4K) 0.5

K 0.5

= 1000 => => K* = 10 L* = 40 25