AAEC 3315Agricultural Price Theory

CHAPTER 6Cost Relationships

The Case of One Variable Inputin the Short-Run

Objectives

To gain understanding of: Cost Relationships

Fixed Costs, Variable Costs,& Total Costs Average and Marginal Costs

Cost Functions Relationships between product and cost

curves

Cost Relationships

A manager’s goal is to determine how much to produce to maximize profits.

We established earlier that Stage II is the rational stage of production, but realized that cost and revenue information are necessary to determine at which point in Stage II to produce.

Now, let’s introduce cost relationships into production.

Cost Definitions

Costs of Production or Economic Costs: The payments that a firm must make to attract inputs and keep them from being used to produce other products.

A firm’s cost functions show various relationships between its costs and output rate. Thus, the firm’s cost functions are determined by the firm’s production function and input prices.

Since the production function can pertain to the short run or the long run, it follows that the cost functions can also pertain to the short run or the long run.

Cost Functions in the Short Run

Fixed Costs: Costs which do not vary with the level of production - These costs are associated with the fixed factors of production. Incurred regardless whether any output is produced

Variable Costs: Costs that vary as the output level changes - These costs are associated with variable factors of production.

Short-Run Cost RelationshipsThe Case of One Variable Input

Costs Based on Total Output

Total Fixed Costs (TFC): costs of inputs that are fixed in the SR & do not change as the output level changes.

Total Variable Costs (TVC): costs of inputs that are variable in the Short Run, and change as output level changes, i.e., TVC = PXX

Total Costs (TC): TFC + TVC

Total Cost Curves(Assume TFC = $80 and Px = $25

0 0 $80 1 10 $80 2 25 $80 3 50 $80 4 70 $80 5 85 $80 6 95 $80 7 100 $80 8 101 $80 9 95 $80 10 85 $80

0

X Y TFC X Y TFC

TOTAL FIXED COSTS

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Output

Cos

t

TFC

Total Cost Curves(Assume TFC = $80 and Px = $25

0 0 $80 $0 1 10 $80 $25 2 25 $80 $50 3 50 $80 $75 4 70 $80 $100 5 85 $80 $125 6 95 $80 $150 7 100 $80 $175 8 101 $80 $200 9 95 $80 $225 10 85 $80 $250

0

XX Y TFC TVC = Y TFC TVC = PXX

TOTAL VARIABLE COSTS

0

50

100

150

200

250

300

0 20 40 60 80 100 120

Output

Co

sts

TFC

TVC

Total Cost Curves(Assume TFC = $80 and Px = $25

XX Y TFC TVC TC = TFC+TVC Y TFC TVC TC = TFC+TVC

0 0 $80 $0 80

1 10 $80 $25 105

2 25 $80 $50 130

3 50 $80 $75 155

4 70 $80 $100 180

5 85 $80 $125 205

6 95 $80 $150 230

7 100 $80 $175 255

8 101 $80 $200 280

9 95 $80 $225 305

10 85 $80 $250 330

0

TOTAL COSTS

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Output

Co

sts TVC

TFC

TC

Total Cost Curves

TOTAL COSTS

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Output

Co

sts

TFC

TVC

TC

Total Cost Curve Functions

TFC = 100 TVC = 6Q – 0.4Q2 + 0.02Q3

TC = TFC + TVC = 100 + 6Q – 0.4Q2 + 0.02Q3

Short-Run Cost Relationships

Costs Based on Per Unit of Output

Average Fixed Costs (AFC): Total fixed costs per unit of output, i.e., AFC = TFC / Q

Average Variable Costs (ATC): Total variable cost per unit of output, i.e., AVC = TVC/Q

Average Total Costs (ATC): Average total cost per unit of output, i.e., ATC = TC / Y = AFC + AVC

Marginal Cost (MC): The increase in cost necessary to increase output by one more unit, i.e.,MC = ∆TC/∆QMC = (∆TVC + ∆ TFC) / ∆QMC = ∆TVC / ∆Q MC = ∂TC/ ∂Q = ∂TVC/ ∂Q

Costs Based on Per-Unit Output

Average Fixed Costs (AFC): Average cost of fixed inputs per unit of output, i.e., AFC = TFC / Q

Average Fixed Costs

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0 20 40 60 80 100 120

Y TFC AFC

0 80

10 80 8.00

25 80 3.20

50 80 1.60

70 80 1.14

85 80 0.94

95 80 0.84

100 80 0.80

101 80 0.79

95 80 0.84

85 80 0.94

AFC

Costs Based on Per-Unit Output

Average Variable Costs (ATC): Total Variable cost per unit of output, i.e., AVC = TVC / Q

Y TVC AVC

0 0

10 25 2.50

25 50 2.00

50 75 1.50

70 100 1.43

85 125 1.47

95 150 1.58

100 175 1.75

101 200 1.98

95 225 2.37

85 250 2.94

Average Variable Cost

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120

AFC

AVC

Costs Based on Per-Unit Output

Average Total Costs (ATC): Average total cost per unit of output, i.e., ATC = TC / Q = AFC + AVC

Y TC ATC

0 80

10 105 10.50

25 130 5.20

50 155 3.10

70 180 2.57

85 205 2.41

95 230 2.42

100 255 2.55

101 280 2.77

95 305 3.21

85 330 3.88

Average Total Cost

0

2

4

6

8

10

12

0 20 40 60 80 100 120

AFC

AVC

ATC

Costs Based on Per-Unit OutputMarginal Cost (MC): The increase in cost necessary to increase output by one more unit, i.e.,MC = ∆TC/∆Q= ∆TVC / ∆Q = ∂TC/ ∂Q = ∂TVC/ ∂Q

Marginal Cost

-1

1

3

5

7

9

11

13

15

-10 10 30 50 70 90 110

Y TC MC

0 80

10 105 2.50

25 130 1.67

50 155 1.00

70 180 1.25

85 205 1.67

95 230 2.50

100 255 5.00

101 280 25.00

95 305 -4.17

85 330 -2.50

ATC

AFC

AVC

MC

Summary of Relationships Between Short-Run Cost Curves

AFC is a continuously decreasing function

AVC & ATC curves are U-shaped

The vertical distance between ATC & AVC at each output level is equal to AFC

MC crosses both AVC & ATC from below at their respective minimums

MC is not affected by fixed costs

Relationship Among Cost Curves

TOTAL COSTS

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Output

Co

sts

...

ATC

AVC

MC

TC

TVC

TFC

AFC

Cost

s/unit

Output

Inflection Point

Changes in Input Price

Price of Variable Input Increases The cost of producing each output level increases

VC & TC shift upward & left; TFC remains unchanged AVC, AC, & MC shift upward & left

Price of variable Input Decreases The cost of producing each output level decreases

TVC & TC shift downward & right; TFC remains unchanged AVC, ATC, & MC shift downward & right

Working With Cost Functions

Given the total cost functions:TC = 100 + 6Q – 0.4Q2 + 0.02Q3,

TFC = 100,

TVC = 6Q – 0.4Q2 + 0.02Q3,

Average and Marginal costs

functions can be derived.ATC = TC/Q = 100/Q + 6 – 0.4Q + 0.02Q2,

AFC = TFC/Q = 1000/Q,

AVC = TVC/Q = 6 – 0.4Q + 0.02Q2, and

MC = = ∂TC/ ∂Q = ∂TVC/ ∂Q = 6 – 0.8Q + 0.06Q2

With these given:

Can you calculate the level

of output at the minimum of

AVC, and MC?

Co

sts

Output

TOTAL COSTS

.

ATC

AVC

MC

TC

TVC

Cost

s/unit

Output

Inflection Point

AFC

TFC

Working With Cost Functions

Given the cost functions:

TC = 100 + 6Q – 0.4Q2 + 0.02Q3,

TFC = 100,

TVC = 6Q – 0.4Q2 + 0.02Q3,

ATC = TC/Q = 100/Q + 6 – 0.4Q + 0.02Q2,

AFC = TFC/Q = 100/Q,

AVC = TVC/Q = 6 – 0.4Q + 0.02Q2, and

MC = = ∂TC/ ∂Q = ∂TVC/ ∂Q = 6 – 0.8Q + 0.06Q2

Level of output at the minimum of AVC:

∂ AVC/ ∂Q = -0.4 + 0.04Q = 0 Q = 10

Co

sts

Output

TOTAL COSTS

.

ATC

AVC

MC

TC

TVC

Cost

s/unit

Output

Inflection Point

AFC

TFC

10

Working With Cost Functions

Given the cost functions:

TC = 100 + 6Q – 0.4Q2 + 0.02Q3,

TFC = 100,

TVC = 6Q – 0.4Q2 + 0.02Q3,

ATC = TC/Q = 100/Q + 6 – 0.4Q + 0.02Q2,

AFC = TFC/Q = 100/Q,

AVC = TVC/Q = 6 – 0.4Q + 0.02Q2, and

MC = = ∂TC/ ∂Q = ∂TVC/ ∂Q = 6 – 0.8Q + 0.06Q2

Level of output at the minimum of MC:

∂ MC/ ∂Q = -0.8 + 0.12Q = 0 Q = 6.67

Co

sts

Output

TOTAL COSTS

.

ATC

AVC

MC

TC

TVC

Cost

s/unit

Output

Inflection Point

AFC

TFC

106.67

Relationships among Product Curves and Cost Curves

The cost curves are derived directly from the production process.

TPP & TVC, APP & AVC and MPP & MC are mirror images of each other Therefore, the production function can be

transferred directly to the cost curves The three stages of a production function can

be transferred directly to the cost curves

Relationship Between TPP and TVC

TPP

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100.00

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0.00 2.00 4.00 6.00 8.00 10.00 12.00

TPP

TVC

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0.00 20.00 40.00 60.00 80.00 100.00 120.00

TVC

A A*

B

B*

The TVC is derived from the TPP: At “A” on TPP, 25 units of the output is being produced with 2 units of the input. The corresponding point “A*” on the TVC shows that the variable cost of producing 25 units of output is $50 (PX:$25 * 2 units of input =$50). Note similar linkage between point “B” on TPP and point “B*” on TVC.

Similar relationships can be derived between AVC & APP and between MPP & MC.

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