Example Does the shape of the utility function matter? A group of towns intends to develop a resort....

Example

Does the shape of the utility function matter?

A group of towns intends to develop a resort. Two possible sites have been identified. The council is currently conducting studies on the merits of the two sites. The final decision will be made next year. Land values around the two possible sites are increasing as investors speculate that property values will rise sharply in the vicinity of the new resort.

20.1

Friday 21 April 2023 07:17 AM

Example

A private real estate developer would like to build houses near the new resort. They have identified parcels of land near the two sites. Current prices (£,000) and estimated future values of developable land are summarised below.

20.2

Parcel of land near location

A BCurrent purchase price £1,200 £700Present value if future cash flows if resort is built at this location

£1,500 £1,100

Present value of future sales price of parcel of land if resort is not built at this location

£950 £350

Example

The developers can purchase land now either at site A only, at site B only, or at both. If land is purchased and the resort is not built at that location, the land will sold at a loss. If the land is not purchased now, it will not be available at a later date.

It is assessed that there is a probability of 0.4 that the new resort is built at location A (0.6 at location B).

20.3

Financial Decision Tree

20.4

0.4

Site A s elected Payoffs

£300

Buy A £1,500 ##

-£1,200 -£30 0.6

Site B s elected

-£250

£950 ##

0.4

Site A s elected

-£350

Buy B £350 ##

-£700 £100 0.6

Site B s elected

£400

2 £1,100 ##

£100

0.4

Site A s elected

-£50

Buy A and B £1,850 ##

-£1,900 £70 0.6

Site B s elected

£150

£2,050 ##

Buy nothing

£0

£0 £0

Generating cash flows from left to right

Probabilistic Decision Tree

20.5

0.4


£300

Buy A £1,500 ##

-£1,200 -£30 0.6

Site B s elected

-£250

£950 ##

0.4

Site A s elected

-£350

Buy B £350 ##

-£700 £100 0.6

Site B s elected

£400

2 £1,100 ##

£100

0.4

Site A s elected

-£50

Buy A and B £1,850 ##

-£1,900 £70 0.6

Site B s elected

£150

£2,050 ##

Buy nothing

£0

£0 £0

Generating cash flows from right to left

Combined Decision Tree

20.6

0.4


£300

Buy A £1,500 ##

-£1,200 -£30 0.6

Site B s elected

-£250

£950 ##

0.4

Site A s elected

-£350

Buy B £350 ##

-£700 £100 0.6

Site B s elected

£400

2 £1,100 ##

£100

0.4

Site A s elected

-£50

Buy A and B £1,850 ##

-£1,900 £70 0.6

Site B s elected

£150

£2,050 ##

Buy nothing

£0

£0 £0

So the selected strategy is option 2, to buy B.

Assessment of the Managers Utility

20.7

But does the “shape” of the utility function matter?


20.8

Enter the value that makes you indifferent to the two values below.

0.7

W ith certain ty

£350 £3,500

£2,240

0.3

-£700

You would accept £350, rather than take the gamble.

Feed forward to the two proposed models.

An Exponential Utility Function

20.9

U x

0.00 -£700

0.7 £350

1 £3,500

A 700.00

B 4200.00

C 0.26

-70.00

-28.00

14.00

56.00

98.00

140.00

182.00

224.00

266.00

308.00

A - RT - Ris k Tolerane Level

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-1000 -500 0 500 1000 1500 2000 2500 3000 3500 4000

U

x

C/xBeAU

Using the values from the previous slide.

The probabilities are fed into the tree.

Decision Tree Employing An Exponential Utility Function

20.10

0.4 A 700.00

Site A s elected Payoffs Utilities B 4200.00

£300 0.69 C 0.26

Buy A £1,500 0.69 Payoffs Utilities

-700 0.00

-£1,200 0.61 0.6 -500 0.46

Site B selected -300 0.55

-£250 0.56 -100 0.61

£950 0.56 100 0.65

300 0.69

0.4 500 0.72

Site A s elected 700 0.75

-£350 0.53 900 0.78

Buy B £350 0.53 1100 0.80

1300 0.83

-£700 0.64 0.6 1500 0.85

Site B selected 1700 0.87

£400 0.71 1900 0.88

3 £1,100 0.71 2100 0.90

0.65 2300 0.92

0.4 2500 0.93


-£50 0.62 2900 0.96

Buy A and B £1,850 0.62 3100 0.97

3300 0.99

-£1,900 0.65 0.6 3500 1.00

Site B selected 3700 1.01

£150 0.66 3900 1.02

£2,050 0.66 4100 1.03

4300 1.05

4500 1.06

Buy nothing

£0 0.63

£0 0.63

So the selected strategy is option 3, to buy A and B.

Cccccc

ccc

A Power Utility Function

20.11

U x

0 -£700

0.7 £350

1 £3,500

A 1.01

B -0.46

C 887.11

^r

886.72 0.0003 Loop

887.11 0.0000

887.50 -0.0003

C - RT - Ris k Tolerane Level

Regula Falsi For RT

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-1000 -500 0 500 1000 1500 2000 2500 3000 3500 4000

U

x

C

B

xAU

Using the values from before.

The probabilities are fed into the tree.

Decision Tree Employing A Power Utility Function

20.12

0.4 A 1.01

Site A s elected Payoffs Ut ilities B -0.46

£300 0.68 C 887.11

Buy A £1,500 0.68 Payoffs Utilities

-1000 -0.41

-£1,200 0.51 0.6 -800 -0.12

Site B s elected -600 0.11

-£250 0.40 -400 0.29

£950 0.40 -200 0.43

0 0.55

0.4 200 0.64


-£350 0.33 600 0.78

Buy B £350 0.33 800 0.82

1000 0.86

-£700 0.56 0.6 1200 0.89

Site B s elected 1400 0.91

£400 0.72 1600 0.93

3 £1,100 0.72 1800 0.95

0.58 2000 0.96

0.4 2200 0.97


-£50 0.52 2600 0.98

Buy A and B £1,850 0.52 2800 0.99

3000 0.99

-£1,900 0.58 0.6 3200 1.00

Site B s elected 3400 1.00

£150 0.62 3600 1.00

£2,050 0.62 3800 1.00

4000 1.00

4200 1.00

Buy nothing 4400 1.01

£0 0.55 4600 1.01

£0 0.55

Cccccc

ccc

Experimentally Observed Utility Function – Low Stimuli

20.13

Using the previous values and the low stimuli Booij and van de Kuillen 2009 model.Where low stimuli reflect the financial levels of the decision the participants had to make.

Decision Tree Employing Regression For Experimentally Observed Utility Function

– Low Stimuli

20.14

0.4Site A selected Payoffs Utilities

£300 0.03Buy A £1,500 0.03

-£1,200 -0.1674 0.6Site B selected

-£250 -0.30£950 -0.30

0.4Site A selected

-£350 -0.45Buy B £350 -0.45

-£700 -0.1436 0.6Site B selected

£400 0.064 £1,100 0.06

0.000.4

Site A selected-£50 0.00

Buy A and B £1,850 0.00

-£1,900 -0.0098 0.6Site B selected

£150 -0.02£2,050 -0.02

Buy nothing£0 0.00

£0 0.0000

Uses linear regression

Low-Stimuli For Data From Booij And Van De Kuilen 2009 Table 5.2

So the selected strategy (just) is option 4, to buy nothing.

Decision Tree Employing Piece Wise Interpolation For Experimentally

Observed Utility Function – Low Stimuli

20.15


£300 0.05Buy A £1,500 0.05

-£1,200 -0.1274 0.6Site B selected

-£250 -0.25£950 -0.25

0.4Site A selected

-£350 -0.42Buy B £350 -0.42


£400 0.074 £1,100 0.07

0.000.4

Site A selected-£50 -0.04

Buy A and B £1,850 -0.04

-£1,900 -0.0005 0.6Site B selected

£150 0.03£2,050 0.03

Buy nothing£0 0.00

£0 0.0000

Uses piece wise linear interpolation

Low-Stimuli For Data From Booij And Van De Kuilen 2009 Table 5.2

So the selected strategy (just) is option 4, to buy nothing.

Experimentally Observed Utility Function – High Stimuli

20.16

U x0 -£700 x6

0.7 £350 x51 £3,500 x4

x3x2

high-stimuli x1

x U x0x6 £3,500 1.00 originx5 £2,860 0.83 y0x4 £2,264 0.67 y1x3 £1,726 0.50 y2x2 £1,248 0.33 y3x1 £800 0.17 y4x0 £0 0.00 y5

origin y6y0 £0 0.00y1 -£204 -0.17y2 -£295 -0.33y3 -£393 -0.50y4 -£494 -0.67y5 -£597 -0.83y6 -£700 -1.00

U intercept gradient R2positive -0.0261 0.000297 99.52%negative 0.0708 0.001489 98.48%

Select high or low stimuli

Data from Booij and van de Kuilen 2009 Table 5.2

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

-1000 0 1000 2000 3000 4000U

x

Data fromBooij and vande Kuilen 2009Table 5.2Fit

Using the previous values and the high stimuli Booij and van de Kuillen 2009 model.Where high stimuli reflect the financial levels of the decision the participants had to make.

Decision Tree Employing Regression For Experimentally Observed Utility Function

– High Stimuli

20.17


£300 0.06Buy A £1,500 0.06

-£1,200 -0.1557 0.6Site B selected

-£250 -0.30£950 -0.30

0.4Site A selected

-£350 -0.45Buy B £350 -0.45


£400 0.093 £1,100 0.09

0.010.4

Site A selected-£50 0.00

Buy A and B £1,850 0.00

-£1,900 0.0096 0.6Site B selected

£150 0.02£2,050 0.02

Buy nothing£0 0.00

£0 0.0000

Uses linear regression

High-Stimuli For Data From Booij And Van De Kuilen 2009 Table 5.2

So the selected strategy (just) is option 3, to buy A and B.

Decision Tree Employing Piece Wise Interpolation For Experimentally

Observed Utility Function – High Stimuli

20.18


£300 0.06Buy A £1,500 0.06

-£1,200 -0.1257 0.6Site B selected

-£250 -0.25£950 -0.25

0.4Site A selected

-£350 -0.43Buy B £350 -0.43


£400 0.083 £1,100 0.08

0.000.4

Site A selected-£50 -0.04

Buy A and B £1,850 -0.04

-£1,900 0.0024 0.6Site B selected

£150 0.03£2,050 0.03

Buy nothing£0 0.00

£0 0.0000

Uses piece wise linear interpolation

High-Stimuli For Data From Booij And Van De Kuilen 2009 Table 5.2

So the selected strategy (just) is option 3, to buy A and B.

Experimentally Observed Utility Function – High Stimuli versus Low

Stimuli

20.19

But they hardly differ!


20.20

But does the “shape” of the utility function matter?

Yes!!

Example Does the shape of the utility function matter? A group of towns intends to develop a resort....

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