economic models with fixed costs

8/18/2019 Economic models with fixed costs

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Economic Models withFixed Costs

John Howell and Greg Allenby

The Ohio State University

We wish to thank Sawtooth Technologies and an anonymous clientfor generously allowing us to share the data


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Our Problem

Agricultural Equipment Manufacturer

–Developed a new machine for detecting diseases

–Two components: a test reader and disposable

test strips–Three di! erent test strips to detect threedi! erent diseases

–Study is real client study, but attributes and

levels have been disguised

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Primary Objective

Estimate the volume of both readers and teststrips sold under di! erent pricing scenarios

–CBC not su"cient as it does not directly allow

for volume estimates–Volumes should depend on both the readerprice and the test strip prices

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Reader 1

Option 1 for $6,000

Feature Level 1

Service Level 1

Disease 1: $10/TestDisease 2: $7/TestDisease 3: $15/Test

Reader 2

Option 2 for $5,000

Feature Level 2

Service Level 2


Reader 3

Option 3 for $10,000

Feature Level 1

Service Level 4


Please select the option you would chose and indicate how many tests you would

purchase for system you selected.

I would not purchase

any of these readers

Disease 1 tests per year



Example Screen

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Prior Work

Tom Eagle presented a number of models atprevious conference

–Regression based approach

–Choice Modeling approaches•Volume = Share * MEV

– Joint Discrete/Continuous models

–Economic Models

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What’s New?

Fixed Cost: A one time cost that a consumermust incur in order to participate in a market

–Need not be monetary

Variable Cost: The cost associate with eachunit

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Examples

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Examples

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Examples

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Examples

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Examples

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Examples

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Fixed Cost

Make purchasing small quantities too costlyleading to no purchases with small quantities

–Expect to observe no data with small purchase

quantities

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Intuition

A consumer will buy a product as long as thebenefit outweighs the cost

With Fixed Costs

For small quantities the price per unit is high,but decreases as quantity increases

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0 2 4 6 8

0

1

2

3

4

Demand including fixed costs w/ varying preferences

Product 1

P r o d u c t 2

All products purchased

P2 & P3 Purchased

P1 & P3 Purchased

P1 & P2 Purchased

Only 1 Product Purchased


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Properties of Fixed Costs

Create local monopolies through switching

costs

Cause respondents to self-screen in to high

volume purchasers and no volumepurchasers

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Simple Economic Model

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. .


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12

. .

Product of InterestOutside Good


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12

. .

Baseline Preference for x

Fixed Cost for x Variable Cost for xExpenditure


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12

. .

Diminishing Marginal Benefit

Fixed cost only applies if x is positive


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Preference Parameter

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13

Part-worths


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One Product Example

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60 1 2 3 4 5

15

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Inside Good

O u t s i d e G o o d

!x

= .8

B u d g e t C o n s t r a i n t

Optimal Indi! erence Curves for various values of "x

!x = .4

Minimum Demand non-zero demand for inside good

A

B


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Reader 1

Option 1 for $6,000

Feature Level 1

Service Level 1


Reader 2

Option 2 for $5,000

Feature Level 2

Service Level 2


Reader 3

Option 3 for $10,000

Feature Level 1

Service Level 4


Please select the option you would chose and indicate how many tests you would

purchase for system you selected.

I would not purchase

any of these readers




Example Screen

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Bayesian Estimation

Bayesian modeling makes this easier

–Bayesian Data Augmentation

•Estimate # so $ and E easily estimated

–Gibbs Sampling with adaptive rejection samplingto e"ciently estimate #

Details available in paper

–Email me ([email protected]) for a copy

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mailto:[email protected]?subject=Fixed%20Costs%20Working%20Papermailto:[email protected]?subject=Fixed%20Costs%20Working%20Papermailto:[email protected]?subject=Fixed%20Costs%20Working%20Paper


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Sample Characteristics

3 categorical attributes plus 4 continuousprices

220 respondents from population of 3500

–Only 147 were usable for estimation12 choice tasks

Di"cult task

–Respondents asked to make product choice andthen provide volume estimates

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Raw Data

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0

2 0 0

4 0 0

6 0 0

8 0 0

Total Number of Tests per Year

N u m b e r o f O b s e r v a t i o n s

0 1600 3750 6000 8200 11000 14000 18000 21000


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Raw Data

18

0

2 0 0

4 0 0

6 0 0

8 0 0

Total Number of Tests per Year

N u m b e r o f O b s e r v

a t i o n s

0 1600 3750 6000 8200 11000 14000 18000 21000

0

1 0

2 0

3 0

4 0

5 0

0 60 80 100 125 143 160 190


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Raw Data

19

0

1 0

2 0

3

0

4 0

5 0

0 60 80 100 125 143 160 190

N o t e g

a p i n d

e m a n d


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Model Validation

Compared fit of two holdout tasks

–3 Products + None

–3 Volume estimates for chosen product

Total of 3 reader volume and 9 test volumeestimates

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Holdout 1-Reader Volume

21

0

25

50

Choice 1 Choice 2 Choice 3

Predicted Actual

*RMSE = 5.93%


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Holdout 2 - Reader Volume

22

0

25

50

Choice 1 Choice 2 Choice 3

Predicted Actual

*RMSE = 3.55%


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Comparison to CBC

CBC does not natively handle volumeestimation

Procedure

Calculate the max test volume for eachrespondent (MEV) for each test

Weight the share predictions by individual’saverage volume

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Volume = ShareReader*MEVDisease


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CBC Raw Predictions

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N o g a

p i n d e

m a n d

0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

0 23 50 75 100 125 150 175 200


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Comparison with CBC

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RMSE FCCA CBC

Holdout 1

Holdout 2

5.93% 10.31%

3.6% 5.0%


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Test Volume Comparison

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RMSE FCCA CBC

Holdout 1

Holdout 2

768.24 1375.66

642.89 687.27


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Example Simulation

Fixed costs screen out low volumepurchasers

–Low volume customer may be unprofitable

–Optimize allocation of sales, service, andsupport resources

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Example Simulation

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Base Case

Purchase Structure 1

$X,000 Reader

Feature Level 1

Service Level 3

Same Test Strip Prices

$0 Reader

Purchase Structure 1

$0 Reader

Feature Level 1

Service Level 3

Same Test Strip Prices

Screening Respondents


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Results

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Base Case $0 Reader %Change

Customers

Disease 1

Disease 2

Disease 3

Revenue

Revenue/Customer

17.21 71.3 314%

25,115 66,765 166%

10,309 29,566 187%

1,047 4,692 348%

- - 49%

- - -64%


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Summary

Pros:

– Theoretically Sound

– Accurate volume estimates

– Automatically enforces economic constraints

– Volumes sensitive to product configuration and allprices

– Allows wider range of possible simulations

Cons:

– Estimation more di"cult

– Simulation more di"cult

– No packaged software yet

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Thank You

economic models with fixed costs

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