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

Lecture # 11b Costs and Cost Minimization Lecturer: Martin Paredes. Outline. Definitions of Costs Long-Run Cost Minimization The constrained minimization problem Comparative statics Input Demands Short Run Cost Minimization. Definitions. Explicit costs involve a direct monetary outlay.

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

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

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

  3. Definitions • 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.

  4. 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.

  5. Definitions • 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.

  6. Definitions • 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.

  7. 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.

  8. 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.

  9. Cost Minimisation Problem 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

  10. Long-Run Cost Minimisation Problem • Assume a firm produces a good using only two inputs: K and L • Firm takes as given: • Price of K: r • Price of L: w • Technology: F(L,K) • Total spending on inputs: TC = rK + wL

  11. The Isocost Line Definition: The Isocost Line defines the set of combinations of labour and capital that yield the same total cost for the firm. TC0 = rK + wL …or… K = TC0 – w L r r

  12. K Example: Isocost Lines TC0/r L TC0/w

  13. K Example: Isocost Lines Slope = -w/r TC0/r L TC0/w

  14. K Example: Isocost Lines TC1/r Slope = -w/r TC0/r L TC0/w TC1/w

  15. K Example: Isocost Lines TC2/r TC1/r Slope = -w/r TC0/r L TC0/w TC1/w TC2/w

  16. K Example: Isocost Lines TC2/r TC1/r Slope = -w/r Direction of increase in total cost TC0/r L TC0/w TC1/w TC2/w

  17. Long-Run Cost Minimisation Problem Assumption: • Firms want to minimise cost for a particular level of output and given technology • Firm’s Problem: • Min TC = rK + wL subject to: Q0 = F(L,K) L,K

  18. Long-Run Cost Minimisation Problem • The cost minimisation is analogous to expenditure minimization for the consumer. • In this case, the constraint is the satisfaction of the isoquant equation: Q0 = F(L,K) • Two conditions for interior solution: • Tangency condition: MRTSL,K = MPL = w MPK r • Isoquant constraint: Q0 = F(L,K)

  19. K Example: Cost Minimization Isoquant Q = Q0 L

  20. K Example: Cost Minimization TC0/r Isoquant Q = Q0 L TC0/w

  21. K Example: Cost Minimization TC2/r TC0/r Isoquant Q = Q0 L TC0/w TC2/w

  22. K Example: Cost Minimization TC2/r TC1/r TC0/r • Isoquant Q = Q0 L TC0/w TC1/w TC2/w

  23. Interior Solution Example: • Suppose : Q(L,K) = 50L0.5K0.5 • Suppose : Q0 = 1000 w = € 5 r = € 20 • Which is the cost-minimising choice for the firm?

  24. Example (cont.): • Tangency condition • MRTSL,K = MPL = (0.5)(50)L-0.5K0.5 = K MPK (0.5)(50)L0.5K-0.5 L • w = 5 = 1 r 20 4 • So L = 4K

  25. Example (cont.): • Isoquant Constraint: • 50L0.5K0.5 = 1000 => 50(4K)0.5K0.5 = 1000 => K* = 10 => L* = 40

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