Experiment with an Auction to create a Demand Function

Experiment with an Auction

to create a Demand Function

Berufsakademie Eisenach Staatliche Studienakademie

Thüringen University of Cooperative Education

Prof. Dr. Stephan Rometsch

Demand Function

0 2 4 6 8 10 12 140

2

4

6

8

10

12

quantity

price

EXPERIMENT

Auctions and the Theory of Demand

PART I • Second price closed seal bid auction

• English auction

• First price closed seal bid auction

• Dutch auction

PART II • Demand Function and implications

Second price closed seal bid auction

ASSUMPTIONS AND INTRODUCTION

A bidder knows not the bid of others

Person with the highest bid gets the item

But he/she pays the price of the second highest bid

Why is it sensefull to reveal the true willingness to pay?

Simple: Because there is no regret!

Every other decision doesn´t gain any advantage!

I want to know Your true willingness to pay!

Therefore, I want to know Your true evalution of the good to create Your true

demand function

The true willingness to pay for the item of person i is zi

Background ?

Example

A person i evaluates the item with € 25,- (zi=25)

zi = 25gj = 20 gi =p =

cs = 5

If person i bids gi=zi and no one bids more – then he/she gets the item and has to pay p=gj=20, the

price of the second highest bid (of person j) and i gets a consumer surplus of 5.

Well ?1.

Would person i better off if he does anything else?

The result is the same. He gets the item and has to pay

p=gj =20, the price of the second highest bid and gets a true

consumer surplus of 5. Therefore, gi=zi=25 was high enough.

zi = 25gj = 20 ĝi = 28p =

cs = 5

Well ?2.

Would person i better off if he reduces the bid?

The result is the same. He gets the item and has to pay

p=gj=20, the price of the second highest bid and gets a true

consumer surplus of 5. Therefore, gi=zi=25 was ok.

zi = 25gj = 20 ği = 23p =

cs = 5

gi = zi = 25 was high enough

But…

Be careful, if gi<gj, then person j gets the item…

Now person j gets the item and has to pay p=gk=18, the

price of the second highest bid (of person k) and person i

is very angry.

zi = 25gj = 20ği = 17

gk = 18p =

Person i is very angry, because he wants to have the item, because his true willingness to pay was zi=25.

ResultsIf person i has the highest evaluation

To higher the price over zi doesn´t gain any advantage.

To lower the price under zi doesn´t gain any advantage.

If you lower the price, be careful, there is an increasing probability that another person j with the

bid gj has the highest bid and gets the item.

And then person i is very angry, because he wants to have the item, because his

true willingness to pay was zi=25.

Results

Therefore, reveal your true willingness to pay with your true bid and you get the item

for the price p, which is not higher then zi and you realize some positive

consumer surplus.

Note: The probability that the price p equals your bid gi and your willingness to pay zi is

close to zero:

probability (p = gj = gi = zi ) = 0

Another person has a higher bid 1. zj = gj = 28

Would person i better off if he bids ĝi=30 and not gi=zi =25?

Person i gets the item and has to pay a price which is higher

than his true willingness to pay zi=25 < p=gj=28.

Then he gets the item…. --- But, why ?

zi = 25 gj = 28 ĝi = 30p =

cs = -3

Another person gets the item 2. zj = gj = 28

Would person i better off if he does not bid gi=zi=25?

Person i doesn´t get the item and person j has to pay

a higher price as before p=ĝi=27 < gj=28.

Reduced cs for j – but what is the advantage for person i?

zi = 25 gj = 28ĝi = 27p =

Another person gets the item 3. zj = gj = 28

Would person i better off if he bids ği<zi and not gi=zi=25?

Person i doesn´t get the item and person j has to pay

a lower price as before p=ği=21 < zi=25 < gj=28.

Persons i and j are friends? – Again: what is the

advantage for i?

zi = 25 gj = 28ği = 21p =

large cs for j

Another person gets the item

If your bid is higher than your true willingness to pay, there is an increasing probability

that you get the item for the price p, which is higher then zi and you have to

pay more than your true evaluation and you realize some negative consumer surplus.

If another guy gets the item, then he has to pay the price of the second highest bid.

Be careful and don´t try to influence the price of others!

Don´t try to influence the consumer surplus of others!

Second price auction is equal to an ”English auction”

The auctioneer starts with a reserve price and bidders successively offer higher prices -

bids. Each bid exceed the previous by some minimal bid increment.

When no participant is willing to increase the bid further, then the item is awarded to

the highest bidder.

Normally the auction stops below the willingness to pay of the highest bidder.

Therefore, he pays the second highest bid!

Result is the same1.

Would person i better off if he does anything else?

The result is the same. He gets the item and has to pay

p=gi=20, the price of the second highest bid and gets a true

consumer surplus of 5. Therefore, gi=zi=25 was high enough.

zi = 25gjgi p =

cs = 5

12 15

gi gigj

17 2018

First price auction is equal to an ”Dutch auction”

The auctioneer starts with a high price and lowers the price successively until the first bidder

rises his finger.

When no participant rises his finger he reduces the price more and more.

The auction stops until the first bidder shouts:

“ Mine! ”

Then he pays the price were the auction stops, when he rises his

finger.

This is the first price!

Another Result 1.

The auctioneer begins with highest price, say 100.

Person i gets the item and has to pay his bid!

p = 60 = zi

zip =

60 70 80 10090

„MINE !“

60

What´s the problem for person i ? 1.

The auctioneer begins with highest price, say 100.

Person i gets the item and has to pay his bid!

But: What is in between the gap to the next highest bid?

Person i wants to wait until the auctioneer achieves a border

zip =

60 70 80 1009040

gj

BID01

2,5356

7,59

10121415

# BIDS231221123121

first idea: average price = 7,08

second proposal: p = 10 and 3 customers

Create the data

BID01

2,5356

7,59

10121415

# BIDS231221123121

QUANTITY

431

QUANTITY2119161513111097431

QUANTITYQUANTITY

31

QUANTITY

1

Create the data

BID0

1

2,5

3

5

6

7,5

9

10

12

14

15

# BIDS2

3

1

2

2

1

1

2

3

1

2

1

QUANTITY21

19

16

15

13

11

10

9

7

4

3

1

REVENUE0

19

40

45

65

66

75

81

70

48

42

15

Create the data

BID0

1

2,5

3

5

6

7,5

9

10

12

14

15

# BIDS2

3

1

2

2

1

1

2

3

1

2

1

QUANTITY21

19

16

15

13

11

10

9

7

4

3

1

REVENUE0

19

40

45

65

66

75

81

70

48

42

15

Create the data

BID

0

1

2,5

3

5

6

7,5

p = 9

10

12

14

15

# BIDS

2

3

1

2

2

1

1

2

3

1

2

1

QUAN.

21

19

16

15

13

11

10

9

7

4

3

1

REV.

0

19

40

45

65

66

75

81

70

48

42

15

0 5 10 15 200

2

4

6

8

10

12

14

16

quantity

price

estimation:

p(q) = 15 - ¾ q

R = p*q = 9*9 = 81

y = -0,7863x + 15,536

R² = 0,9857

REV.

0

19

40

45

65

66

75

81

70

48

42

15

0 5 10 15 20

-10

0

10

20

30

40

50

60

70

80

90

f(x) = − 0.635199669982683 x² + 12.6080606553327 x + 8.4450309883674R² = 0.945545512412993

q

R

REVENUE

0 5 10 15 20 250

2

4

6

8

10

12

14

16

quantity

priceconsumers surplus

= zi – p

1 1 1

6

5 5

0 0

3

p=9

q=9

R = p * q = 81

= (15-9) + 2(14-9) + (12-9) + 3(10-9) + 2(9-9) = 22

profit = revenue - costs

BID

0

1

2,5

3

5

6

7,5

9

p = 10

12

14

15

costs per unit: c = 6

price ↑ p = 10

quantity ↓ q = 7

revenue ↓ r = 70

profit ↓ π = 28

q

21

19

16

15

13

11

10

9

7

4

3

1

r

0

19

40

45

65

66

75

81

70

48

42

15

c

126

114

96

90

78

66

60

54

42

24

18

6

- 126

- 95

- 56

- 45

- 13

0

15

27

28

24

24

9

0 5 10 15 20

-100

-50

0

50

100

150

q

costs

revenue

profit

profit = 70 - 42 = 28

70

28

42

28

q = 7

profit = revenue + subsidy

BID

0

1

2,5

3

p=5

6

7,5

9

10

12

14

15

q

21

19

16

15

13

11

10

9

7

4

3

1

r

0

19

40

45

65

66

75

81

70

48

42

15

s

- 147

- 133

- 112

- 105

- 91

- 77

- 70

- 63

- 49

- 28

- 21

- 7

147

152

152

150

156

143

145

144

119

76

63

22

Subsidy per unit s = 7

price ↓ p = 5

quantity ↑ q = 13

revenue ↓ r = 65

profit ↑ π = 156

But: price is not zero!

Additional costs per unit or subsidy

influences the marginal behaviour.

0 5 10 15 20

-200

-150

-100

-50

0

50

100

150

200

q

profit

subsidy

revenue

q=13

156

65

quantity increasing: q =13price reducing: p = 5

T-shirt as a present for every participant –

logical equivalent to the total price discriminating monopolist.

consumers surplus =

0 5 10 15 20 250

2

4

6

8

10

12

14

16

quantity

price

p = 0

zi – p = (15-0) + 2(14-0) + … + 2(0-0) = 138

= monopolistic profit

Compare Theory vs. Experiment I

1. Demand

EXP: q = 9

bqaqp q4315p

qbqaqqpqR qq4315R

²bqaq ²q43q15

02bqaR´ 0q2315R´

2baq 10q

2. Revenue

3. Marginal Revenue

! !

Compare Theory vs. Experiment II

EXP: p = 9

b2abapbqap

2a*p

q4315p

5,7*p

cqqC q6C

cqqbqaq

0cbq2a´

q6qq4315

06q2315´

4. Price

5. Costs

6. Profit maximization

! !

Compare Theory vs. Experiment III

EXP: p = 10

EXP: q = 7

EXP: π = 28

b2caq

2

cap

65,1

615q

cqqbqaq 666643156q

π

276q π

7. Quantity and Price

8. Profit

10,5064315p

Literatur

H. VARIAN, Intermediate Microeconomics - A modern Approach, Norton, Chapter 17: Auctions

A. ROTT, Die Nachfragefunktion: Ein ökonomisches Experiment für die Lehre, Dortmunder Diskussionsbeiträge zur Wirtschaftspolitik, Nr. 92, Juli 1999

T. Bergstrom, J. Miller, Experiments with Economic Principles, McGraw-Hill, 1998. (http://zia.hss.cmu.edu/miller/eep/eep.html)

Thank you for attention!

Vielen Dank für Ihre Aufmerksamkeit

Prof. Dr. Stephan M. Rometsch

Berufsakademie Eisenach Staatliche Studienakademie Thüringen