Lecture

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
• Firms 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
• gt 50(4K)0.5K0.5 1000
• gt K 10
• gt L 40