chap003-balacing costs and benefits

Chapter 3

Balancing Costs and Benefits

McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.

Main Topics

Maximizing benefits less costsThinking on the marginSunk costs and decision-making

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Maximizing Net Benefit

Net benefit: total benefit minus total costTotal cost must include opportunity costOpportunity cost: the cost associated

with foregoing the opportunity to employ a resource in its best alternative use

Right decision is the choice with the greatest difference between total benefit and total cost

3-3

Car Repair Example:Benefit Schedule

Mechanic’s time is available in one-hour increments

Maximum repair time is 6 hours

The more time the car is repaired, the more it is worth

Table 3.1: Benefits of Repairing Your Car

Repair Time

(Hours)

Total Benefit

($)

0 0

1 615

2 1150

3 1600

4 1975

5 2270

6 2485

3-4

Car Repair Example:Cost Schedule

Table 3.2: Costs of Repairing Your Car

Repair Time

(Hours)

Cost of Mechanic and Parts

($)

Lost Wages from Pizza Delivery Job

($)

Total Cost

($)

0 0 0 0

1 140 10 150

2 355 25 380

3 645 45 690

4 1005 75 1080

5 1440 110 1550

6 1950 150 2100

3-5

Car Repair Example:Maximizing Net Benefit

How should you decide how many hours is the “right” number to have your car repaired?

Recall that every hour in the shop will bring both benefits and costs

Choose the number of hours where benefits exceed costs by the greatest amount

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Table 3.3: Total Benefit and Total Cost of Repairing Your Car

Repair Time (Hours)

Total Benefit ($)

Total Cost ($)

Net Benefit ($)

0 0 0

1 615 150

2 1150 380

3 1600 690

4 1975 1080

5 2270 1550

6 2485 2100

Car Repair Example:The Right Decision

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Repair Time (Hours)

Total Benefit ($)

Total Cost ($)

Net Benefit ($)

0 0 0 0

1 615 150 465

2 1150 380 770

3 1600 690 910

4 1975 1080 895

5 2270 1550 720

6 2485 2100 385


3-8


Repair Time (Hours)

Total Benefit ($)

Total Cost ($)

Net Benefit ($)

0 0 0 0

1 615 150 465

2 1150 380 770

3 1600 690 910

4 1975 1080 895

5 2270 1550 720

6 2485 2100 385

Best Choice


3-9

465

Car Repair Example:Graphical Approach

(Figure 3.1)

Data from Table 3.3 are shown in this graph

Costs are in red; benefits are in blue

The best choice is where benefits > costs and the distance between them is maximized

This is at 3 hours, net benefit = $910

Total Benefit,Total Cost($)

Repair Hours

1 2 3 4 5 6

400

800

1200

1600

2000

2400

710

910

Best Choice

3-10

Maximizing Net Benefit:Finely Divisible Actions

Many decisions involve actions that are more finely divisible

E.g. mechanic’s time available by the minute

In these cases can use benefit and cost curves rather than points or a schedule to make the best decision

Underlying principle is the same: maximize net benefit

3-11

Car Repair Example:Finely Divisible Benefit

Horizontal axis measures hours of mechanic’s time

Vertical axis measures in dollars the total increase in your car’s value

B(H)=654H-40H2

Total Benefit ($)

Hours (H)

(a): Total Benefit

0 1 2 3 4 5 6

614

1602

2270

B

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Car Repair Example:Finely Divisible Cost

Vertical axis measures total cost in dollars

Includes opportunity cost

C(H)=110H+40H2

0 1 2 3 4 5 6

150

690

1550

Hours (H)

(b): Total Cost

Total Cost ($)

C

3-13

Car Repair Example:Finely Divisible Net Benefit

Best choice is 3.4 hours of repair, maximizes net benefit

Net benefit with finely divisible choices is greater than in previous example; more flexibility allows you to do better

836.40

1761.20

Total BenefitTotal Cost($)

(c): Total Benefit versus Total Cost

0 1 2 3 4 5 6

Hours (H)

3.4

B

C924.80

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Net Benefit Curve(Figure 3.3)

Can also graph the net benefit curve

Vertical axis shows B-C, net benefit

Best choice is the number of hours that corresponds to the highest point on the curve, 3.4 hours

0 1 2 3 4 5 6

924.80

Hours (H) 3.4

Net Benefit($)

B – C

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Thinking on the Margin

Thinking like an economistAnother approach to maximizing net

benefitsCapture the way that benefits and costs

change as the level of activity changes just a little bitFor any action choice X, the marginal units

are the last X units, where X is the smallest amount you can add or subtract

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Marginal Cost

The marginal cost of an action at an activity level of X units is equal to the extra cost incurred due to the marginal units, divided by the number of marginal units

X

XXCXC

X

CMC

)()(

3-17

Car Repair Example:Marginal Cost

Marginal cost measures the additional cost incurred from the marginal units (H) of repair time

If C(H) is the total cost of H hour of repair work, the extra cost of the last H hours is C = C(H) – C(H-H)

To find marginal cost, divide this extra cost by the number of extra hours of repair time, H

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Car Repair Example:Marginal Cost

So the marginal cost of an additional hour of repair time is:

Using the data from Table 3.2, if H= 3, we see:

H

HHCHC

H

CMC

)()(

3103806901

)13()3(

CCMC

3-19

Car Repair Example:Marginal Cost Schedule

Table 3.5: Total Cost and Marginal Cost of Repairing Your Car

Repair Time

(Hours)

Total Cost ($)

Marginal Cost (MC) ($/hour)

0 0 -

1 150 150

2 380 230

3 690 310

4 1080 390

5 1550 470

6 2100 550

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Marginal Benefit

The marginal benefit of an action at an activity level of X units is equal to the extra benefit produced due to the marginal units, divided by the number of marginal units

X

XXBXB

X

BMC

)()(

3-21

Car Repair Example:Marginal Benefit

Marginal benefit measures the additional benefit gained from the marginal units (H) of repair time

This parallels the definition and formula for marginal cost

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Car Repair Example:Marginal Benefit

The marginal benefit of an additional hour of repair time is:

Using the data from Table 3.1, if H= 3, we see:

H

HHBHB

H

BMC

)()(

450115016001

)13()3(

BBMC

3-23

Car Repair Example:Marginal Benefit Schedule

Table 3.6: Total Benefit and Marginal Benefit of Repairing Your Car

Repair Time (Hours)

Total Benefit ($)

Marginal Benefit (MB) ($/hour)

0 0 -

1 615 615

2 1150 535

3 1600 450

4 1975 375

5 2270 295

6 2485 215

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Marginal Analysis and Best Choice

Comparing marginal benefits and marginal costs can show whether an increase or decrease in a level of an activity raises or lowers the net benefit

Increase level if MB of doing so is greater than MC; if MC of last increase was greater than MB, decrease level

At the best choice, a small change in activity level can’t increase the net benefit

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Marginal Analysis and Best Choice

Table 3.7: Marginal Benefit and Marginal Cost of Repairing Your Car

Repair Time

(Hours)

Marginal Benefit (MB)

($/hour)

Marginal Cost (MC) ($/hour)

0 - -

1 615 > 150

2 535 > 230

3 450 > 310

4 375 < 390

5 295 < 470

6 215 < 550

Best Choice

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Marginal Analysis withFinely Divisible Actions

Can conduct the same analysis if choices are finely divisible by using marginal benefit and marginal cost curves

Derive marginal benefit and marginal cost from total benefit and total cost curves

Marginal benefit at H hours of repair time is equal to the slope of the line drawn tangent to the total benefit function at point

Usually called simply the “slope of the total benefit curve” at point D

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Marginal Benefit

Using Calculus

This is the slope of the Total Benefit Curve B(H).

dHHdB

HMB

Car Repair Example:Finely Divisible Marginal Benefit

Let H' = the smallest possible change in hours of car repair

Adding the last H‘ of repairs increases total benefit from point F to point D in Figure 3.4 (on the next slide), this equal to:

Recall that marginal benefit is B' /H'

)()( HHBHBB

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Slope = MB

'HH

)'HH(B

'H

''HH H

''B

''H

'H/'B

)''HH(B

)H(B

'B

Slope = MB =

Slope= MB = ''H/''B

Hours (H)

Total Benefit ($)

Relationship between Total Benefit and Marginal Benefit (Figure 3.4)

F

E

D

3-30

Relationship between Total Benefit and Marginal Benefit

Tangents to the total benefit function at three different numbers of hours (H = 1, H = 3, H = 5)

Slope of each tangent equals the marginal benefit at each number of hours

Figure (b) shows the MB curve: note how the MB varies with the number of hours

Marginal benefit curve is described by the function MB(H)= 654-80H

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Relationship between Total Benefit and Marginal Benefit (Figure 3.5)

Total Benefit ($)

Hours (H)

0 1 2 3 4 5 6

614

1602

2270

B

(a): Total Benefit

Slope = MB = 574

Slope = MB = 414

Slope = MB = 254

0 1 2 3 4 5 6

254

414

574

Marginal Benefit ($/hour)

MB

(b): Marginal Benefit

654

Hours (H)

3-32

Relationship between Total Cost and Marginal Cost

Parallels relationship between total benefit curve and marginal benefit

When actions are finely divisible, the marginal cost when choosing action X is equal to the slope of the total cost curve at X

Using Calculus

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dHHdC

HMC

Relationship between Total Cost and Marginal Cost

Tangents to the total cost curve at three different numbers of hours (H = 1, H = 3, H = 5)

Slope of each tangent equals the marginal cost at each number of hours

Figure (b) shows the MC curve: note how the MC varies with the number of hours

Marginal cost curve is described by the function MC(H)= 110+80H

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Relationship between Total Cost and Marginal Cost (Figure 3.6)

110

190

350

510

150

690

1550

Total Cost ($)

(a): Total Cost (b): Marginal Cost

1 2 3 4 5 6

Hours (H)

0

C

Slope = MC = 190

Slope = MC = 350

Slope = MC = 510

0 1 2 3 4 5 6

Marginal Cost ($/hour)

Hours (H)

MC

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Using Calculus to find Best Choice

Net Benefit = Total Benefit – Total CostMaximize Net Benefit (NB(H)).Using Calculus;

HMCHMBdHHdC

dHHdB

dHHdC

dHHdB

dHHdNB

0

Using Calculus to find Best Choice

So, at the maximum; MB(H) = MC(H). MB(H)= 654-80HMC(H)= 110+80HSolving, we obtain H* = 3.4.

Marginal Benefit Equals Marginal Cost at a Best Choice

At the best choice of 3.4 hours, the No Marginal Improvement Principle holds so MB = MC

At any number of hours below 3.4, MB > MC, so a small increase in repair time will improve the net benefit

At any number of hours above 3.4, MC > MB, so that a small decrease in repair time will improve net benefit

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Marginal Benefit Equals Marginal Cost at a Best Choice (Figure 3.7)

Marginal Benefit, Marginal Cost ($/hour)

Hours (H)

MC

MB

3.4

0 1 2 3 4 5 6

110

382

654

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Slopes of Total Benefit and Total Cost Curves at the Best Choice

MC = MB at the best choice of 3.4 hours of repair

Therefore, the slopes of the total benefit and total cost curves must be equal at this point

Tangents to the total benefit and total cost curves show this relationship

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Slope of Total Benefit and Total Cost Curves (Figure 3.8)

0 1 2 3 4 5 6

Hours (H)

Total BenefitTotal Cost($) B

C924.80

3.4

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Sunk Costs and Decision Making

A sunk cost is a cost that the decision maker has already incurred, or

A cost that is unavoidable regardless of what the decision maker does.

Sunk costs affect the total cost of a decision

Sunk costs do not affect marginal costsSo sunk costs do not affect the best

choice

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Car Repair Example: Best Choice with a Sunk Cost

Figure 3.9 shows a cost-benefit comparison for two possible cost functions with sunk fixed costs: $500 and $1100.

In both cases, the best choice is H = 3.4: the level of sunk costs has no effect on the best choice

Notice that the slopes of the two total cost curves, and thus the marginal costs, are the same

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Best Choice with a Sunk Cost (Figure 3.9)

500

Hours (H) 3.4

0 1 2 3 4 5 6

C

C´

B

-175.20

424.80

Total Benefit,Total Cost($)

1100

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chap003-balacing costs and benefits

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