Chapter 7: Costs and Cost Minimization

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Chapter 7 Costs and Cost Minimization

• Consumers purchase GOODS to maximize their
  utility.
• This consumption depends upon a consumers INCOME
  and the PRICE of the goods
• Firms purchase INPUTS to produce OUTPUT
• This output depends upon the firms FUNDS and the
  PRICE of the inputs

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Chapter 7 Costs and Cost Minimization

• In this chapter we will cover
• 7.1 Different Types of Cost
• 7.1.1Explicit and Implicit Costs
• 7.1.2 Opportunity Costs
• 7.1.3 Economic and Accounting Costs
• 7.2 Isocost Lines
• 7.3 Cost Minimization
• 7.4 Short-Run Cost Minimization

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7.1.1 Implicit and Explicit Costs
Explicit Costs Costs that involve an exchange
of money -ie Rent, Wages, Licence,
Materials Implicit Costs Costs that dont
involve an exchange of money -ie Wage that
could have been earned working elsewhere
profitability of a goat if used mowing lawns
instead of for meat
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7.1.2 Opportunity Costs

• Definition Value of the next best alternative
  total benefit of choosing the next best option
• IE Instead of opening his own Bait shop, which
  cost 5,000 per month to run (explicit cost),
  Buck could have worked for Worms R Us for 2,000
  per month (implicit cost).
• His opportunity cost is 2,000 (alternate wage)
  5,000 (the amount he WOULDNT have to pay each
  month) 7,000

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7.1.3 Economic and Accounting Costs

• Economists Accountants calculate costs
  differently
• Economists are interested in studying how firms
  make production pricing decisions. They include
  all costs.

Economic Costs Explicit Implicit Costs
Accounting Costs

• Accountants are responsible for keeping track of
  the money that flows into and out of firms. They
  focus on explicit costs.

Accounting Costs Explicit Costs
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Profit Economists vs Accountants
Economists View
Accountants View
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Defining Sunk Cost
Sunk Costs are costs that must be incurred no
matter what the decision. These costs are not
part of opportunity costs.
Example Giant Dancing Elasticity sign
It costs 5M to build and has no alternative
uses 5M is not a sunk cost for the decision of
whether or not to build the sign 5M is a sunk
cost for the decision of whether to operate or
shut down the sign
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Costs Example

• Last year, Hugo decided to open a box factory.
  Hugo built the factory for 200,000. Materials
  and wages required to make a box amount to 5
  cents per box.
• Before starting production, Hugo was offered a
  job at BoxMart that paid 4,000 a month.
• Classify Hugos costs (explicit, implicit,
  economic, accounting, and sunk)

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Costs Example

• Explicit Costs
• Factory (200K historic cost)
• Production (5 cents/box ongoing cost)
• Implicit Costs
• Forgone Wage (4,000/month)
• Accounting CostsExplicit Costs
• Economic Costs ExplicitImplicit Costs
• Sunk Costs Factory (200K)

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Cost Minimization

• One of the goals of a firm is to produce output
  at a minimum cost.
• This minimization goal can be carried out in two
  situations
• The long run (where all inputs are variable)
• The short run (where some inputs are not variable)

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The (Long Run) Cost Minimization Problem
Suppose that a firms owners wish to minimize
costs Let the desired output be Q0 Technology
Q f(L,K) Owners problem min TC rK wL
K,L
Subject to Q0 f(L,K)
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7.2 The Isocost Line
From the firms cost equation TC0
rK wL One can obtain the formula for the
ISOCOST LINE K TC0/r
(w/r)L The isocost line graphically depicts all
combinations of inputs (labour and capital) that
carry the same cost.
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K
Example Isocost Lines
Direction of increase in total cost
TC2/r
TC1/r
Slope -w/r
TC0/r
L
TC0/w TC1/w TC2/w
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7.3 Cost Minimization
Isocost curves are similar to indifference
curves, and the tangency condition of cost
minimization is also similar to the tangency
condition of consumers MRTSL,K -MPL/MPK -w/r
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K
Example Cost Minimization
TC2/r

Cost inefficient point for Q0
TC1/r
Cost minimization point for Q0
TC0/r

Isoquant Q Q0
L
TC0/w TC1/w TC2/w
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Cost Minimization Steps

• Tangency Condition
• - MPL/MPK w/r-gives relationship between L
  and K
• 2) Substitute into Production Function
• -solves for L and K
• 3) Calculate Total Cost

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Example Interior Solution
Q 50L1/2K1/2 MPL 25K1/2/L1/2 MPK
25L1/2/K1/2 w 5 r 20 Q0 1000
1) TangencyMPL/MPK w/r K/L 5/20orL4K
3) Total Cost L 4K L 40 TC0 rK wL TC0
20(10) 5(40) TC0 400
2) Substitution 1000 50L1/2K1/2 1000
50(4K)1/2K1/2 1000100K K 10
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K
Example Interior Solution
400/r
Cost minimization point

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Isoquant Q 1000
L
40
400/w
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Example Corner Solution
Q 10L 2K
MPL 10 MPK 2
w 5 r 2 Q0 200
a. MPL/w 10/5 gt MPK /r 2/2 Butthe bang
for the buck in labor is larger than the bang
for the buck in capital MPL/w 10/5 gt MPK/r
2/2 K 0 L 20
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Example Cost Minimization Corner Solution
K
Isoquant Q Q0

L
Cost-minimizing input combination
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Comparative Statistics

• The isocost line depends upon input prices and
  desired output
• Any change in input prices or output will shift
  the isocost line
• This shift will cause changes in the optimal
  choice of inputs

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Comparative Statics

• 1. A change in the relative price of inputs
  changes the slope of the isocost line.

• All else equal, an increase in w must decrease
  the cost minimizing quantity of labor and
  increase the cost minimizing quantity of capital
  with diminishing MRTSL,K.
• All else equal, an increase in r must decrease
  the cost minimizing quantity of capital and
  increase the cost minimizing quantity of labor.

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Example Change in Relative Prices of Inputs
K
Cost minimizing input combination w2, r1

Cost minimizing input combination, w1 r1
Isoquant Q Q0

L
0
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Example
Originally, MicroCorp faced input prices of 10
for both labor and capital. MicroCorp has a
contract with its parent company, Econosoft, to
produce 100 units a day through the production
function Q2(LK)1/2 MPL(K/L)1/2
MPK(L/K)1/2 If the price of labour increased
to 40, calculate the effect on capital and
labour.
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Example
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Example
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Example

• If the price of labour quadruples from 10 to
  40
• Labour will be cut in half, from 50 to 25
• Capital will double, from 50 to 100

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Comparative Statics

• An increase in Q0 moves the isoquant Northeast.
• The cost minimizing input combinations, as Q0
  varies, trace out the expansion path
• If the cost minimizing quantities of labor and
  capital rise as output rises, labor and capital
  are normal inputs
• If the cost minimizing quantity of an input
  decreases as the firm produces more output, the
  input is called an inferior input

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K
Example An Expansion Path
TC2/r
TC1/r
Expansion path, normal inputs

TC0/r

Isoquant Q Q0

L
TC0/w TC1/w TC2/w
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K
Example An Expansion Path
TC2/r
Expansion path, labour is inferior

TC1/r

L
TC1/w TC2/w
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Example
Originally, MicroCorp faced input prices of 10
for both labor and capital. MicroCorp has a
contract with its parent company, Econosoft, to
produce 100 units a day through the production
function Q2(LK)1/2 MPL(K/L)1/2
MPK(L/K)1/2 If Econosoft demanded 200 units,
how would labour and capital change?
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Example
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Example

• If the output required doubled from 100 to 200..
• Labour will double, from 50 to 100
• Capital will double, from 50 to 100
• (Constant Returns to Scale)

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Input Demand Functions

• The demand curve for INPUTS is a schedule of
  amount of input demanded at each given price
  level
• This demand curve is derived from each individual
  firm minimizing costs

Definition The cost minimizing quantities of
labor and capital for various levels of Q, w and
r are the input demand functions. L
L(Q,w,r) K K(Q,w,r)
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K
Example Labour Demand Function


When input prices (wage and rent, etc) change,
the firm maximizes using different combinations
of inputs.


Q Q0
W3/r
W1/r
W2/r
0
L
w
As the price of inputs goes up, the firm uses
LESS of that input, as seen in the input demand
curve



L(Q0,w,r)
L
L1 L2 L3
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K


A change in the quantity produced will shift the
isoquant curve.



Q Q0


Q Q1
0
L
w
This will result in a shift in the input demand
curve.





L(Q0,w,r)

L(Q1,w,r)
L
L1 L2 L3
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Calculating Input demand functions

• Use the tangency condition to find the
  relationship between inputs
• MPL/MPK w/rKf(L) or Lf(K)
• 2) Substitute above into production
  function and solve for other variable
• Qf(L,K), Kf(L) gtLf(Q)
• Qf(L,K), Lf(K) gtKf(Q)

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Example Input demand functions
Q 50L1/2K1/2 MPL/MPK w/r gt K/L w/r or
K(w/r)L This is the equation for the
expansion path
Q0 50L1/2(w/r)L1/2 gt L(Q,w,r)
(Q0/50)(r/w)1/2 K(Q,w,r) (Q0/50)(w/r)1/2
Labor and capital are both normal inputs
Labor is a decreasing function of w Labor is an
increasing function of r
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Price Elasticity of Demand (Inputs)

• Price elasticity of demand can be calculated for
  inputs similar to outputs

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Example
JonTech produces the not-so-popular
J-Pod. JonTech faces the following
situation Q5(KL)1/2100 MRTSK/L. w20 and
r20 Calculate the Elasticity of Demand for
Labour if wages drop to 5.
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Example
Initially MRTSK/Lw/r K20L/20 KL
Q5(KL)1/2 1005K 20KL
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Example
After Wage Change MRTSK/Lw/r K5L/20 4KL
Q5(KL)1/2 10010K 10K 40L
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Example
Price Elasticity of Labour Demand
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7.4 Short Run Cost Minimization

• Cost minimization occurs in the short run when
  one input (generally capital) is fixed (K).
• Total variable cost is the amount spent on the
  variable input(s) (ie wL)
• -this cost is nonsunk
• Total fixed cost is the amount spent on fixed
  inputs (ie rK)
• -if this cost cannot be avoided, it is sunk
• -if this cost can be avoided, it is nonsunk
• (ie rent factory to another firm)

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Short Run Cost Minimization

• Cost minimization in the short run is easy
• Min TCwLrK
• L
• s.t. the constraint Qf(L,K)
• Where K is fixed.

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Short Run Cost Minimization

• Example
• Minimize the cost to build 80 units if Q2(KL)1/2
  and K25.
• Q2(KL)1/2
• 802(25L)1/2
• 8010(L)1/2
• 8(L)1/2
• 64L
• Notice that price doesnt matter.

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K
Short Run Cost Minimization
TC2/r
TC1/r
Long-Run Cost Minimization

Short-Run Cost Minimization

K
L
TC1/w TC2/w
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Short Run Expansion Path

• Choosing 1 input in the short run doesnt depend
  on prices, but it does depend on quantity
  produced.
• The short run expansion path shows the increased
  demand for labour as quantity produced increases
  (next slide)
• The demand for inputs will therefore vary
  according to quantity produced. (The demand
  curve for inputs shifts when production changes)

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K
Example Short and Long Run Expansion Paths
TC2/r
Long Run Expansion Path
TC1/r

TC0/r



Short Run Expansion Path
K

L
TC0/w TC1/w TC2/w
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Short Run and Many Inputs

• If the Short-Run Minimization problem has 1 fixed
  input and 2 or more variable inputs, it is
  handled similarly to the long run situation

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Chapter 7 Key Concepts

• Costs can be explicit, implicit, opportunity,
  sunk, fixed and variable
• Accountants ignore implicit costs, but economists
  deal with them
• The Isocost line gives all combinations of inputs
  that have the same cost
• Costs are minimized when the Isocost line is
  tangent to the Isoquant
• When input costs or required output changes, the
  minimization point (and minimum cost) changes

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Chapter 7 Key Concepts

• Individual firm choice drives input demand
• As input prices change, input demanded changes
• There are price elasticities of inputs
• In the short run, at least one factor is fixed
• Short run expansion paths differ from long run
  expansion paths