1

Consistent framing

Notes for chapter 8

2

Accounting and decisions

• In accounting decision problems are often modifiedFixed costs are not included– Often only incremental costs are considered– Costs are approximated

• Profit maximization is often assumed– Cost minimization

• Uncertainty is often not considered– Risk and risk premiums– Variance and covariance among projects

• Time preferences are also disregarded– No discounting

3

More accounting

• Product classification– Primary product – Secondary products– Scrap

• Cost allocations• Transfer pricing• ABC costing

4

Rational Behavior

• Intelligent, wise, and enlightened.• Economic setting pursue

– Self-interest – Wealth

• How do we describe rationality• How do we model rationality

5

Rational Behavior

• Consistency– Complete and transitive ranking

• Two statements equivalent. – Ranking complete and transitive. – Exists a function on A, ω(a),

• a`, a : A, ω(a`) ≥ ω(a :) ∈• Only when a` is ranked as good as a :. • (The set A is finite)

• Smoothness

6

A generic decision problem

max ( )subject to

aa A

We want to maximize a generic function subject to some constraints of feasibility.This could take the form as:

7

Decision with two variables

8

Irrelevance of Increasing Transformations

21

22 1

2 2 23 1

4 2 1

graph 1: ( ) 10 20

graph 2 : ( ) ( ) 20 10

graph 3: ( ) 1 [ ( )] 1 [10 20]

graph 4 : ( ) ln[ ( )] ln[ ( ) 20]

a a a

a a a a

a a a a

a a a

9

Irrelevance of Increasing Transformations

10

Irrelevance of Increasing Transformations

11

Irrelevance of Increasing Transformations

• Definition 19 Function T is an increasing transformation of function ω(a) if ω (a) > ω (â) if and only if T[ω(a)] > T[ω(â)] for every a and â in the domain of the original function

• The solution to a decision problem is unaffected by an increasing transformation of the objective function.

12

Shadow prices

max(40 42 ) (30 30 ) max10 12

400

2 500

x y x y x y

x y

x y

(300,100)

13

Local search - Shadow prices

0, 0max ( , ) 10 12

subject to 8

2 12

x yx y x y

x y

x y

Optimal choice is x = 4, y = 4

14

Component searches are possibleInteractions

0, 0max ( , ) 10 12

subject to 8

2 12

x yx y x y

x y

x y

First constraint

8

Second constraint

y 0.5(12-x)

y x

15

Component searches are possible

( ) min{8 ;.5(12 )}y g x x x

ˆ( , ( )) ( ) 10 12 min{8 ;.5(12 )}10 6(12 ) 72 4 , 0 410 12(8 ) 96 2 , 4 8

w x g x x x x xx x x if xx x x if x

This reduces to:

Then we get:

16

Component searches are possibleInteractions

17

Component searches are possible

1 21 2 1 20, 0

1 2

1

( ; ) min ( , ) 5 20

subject to15

z zC q P z z z z

q z zz

2

21

qz

z

1

2

1 10

1

1

ˆ( ; ) min ( ) 5 20

subject to 15z

qC q P z z

zz

18

Component searches are possible

,max ( , )

x X y Yx y

max{max ( , )}x X y Y

x y

ˆmax{max ( , )} max ( , ( )) max ( )x X y Y x X x X

x y x g x x

19

Component search

• When faced with an optimization problem of several variables we do component search when we solve the problem sequentially, by first optimizing with respect to one variable then the next etc.

20

( ) min{400 ;.5(500 )}

10 12 ( ) 10 12min{400 ;.5(500 )}

3000 4 0 300

4200 2 300 400

0 400

y g x x x

x g x x x x

x x

x x

x

21

22

23

24

Consistent Framing

• 3 principles:– Irrelevance of Increasing Transformation– Local searches are possible– Component searches are Possible

25

26

27

28

29

30

31

Application of framing principles and cost functions

32

New framing!

33

Yet another framing!

2 1 1 1( ) min{8 ;.5(12 )}q g q q q

1 1 1 1 1

1 1 1 1

1 1 1 1

ˆ( , ( )) ( ) 10 12 min{8 ;.5(12 )}10 6(12 ) 72 4 , 0 410 12(8 ) 96 2 , 4 8

w x g q q q q qq q q if qq q q if q

Using Component search this reduces to:

Then we get:

34

Cost function I

Product 1 Product 2

Direct laborDirect materialVariable overhead

203030

404060

Variable product cost 80 140

1 2 1 2 1 2 1 2 1 2( , ) 20( 2 ) 10(3 4 ) 15(2 4 ) 80 140

50000 1.5

C q q q q q q q q q q

OV DL

35

Cost function II

Direct laborDirect materialVariable overheadExternality

203030

6

20303012

Variable product cost 86 92

1 1 1

1 1 1

( ) (80 6) 72 0 4

( ) (80 12) 96 4 8

C q q q

C q q q

10 4q 14 8q

36

Cost function III

Frame Explicit choices

Implicit choices

Marginal cost of first

productI q1,q2,

z1,z2,z3

N/A N/A

II q1,q2 z1,z2,z3 80

III q1 q2,z1,z2,z3 86 or 92

37

Changed parameters

1 2 1 2 3

1 2

1 2

1 1 2

2 1 2

3 1 2

* * * * *1 2 1 2 3

1 2 3 4 5

1 2

max 90 152149 20 10 15

. .

8

2 12

2

3 4

2 4

Optimal solution: 8; 0; 8; 24; 16

Dual variables: 10, 0, 20, 10, 15.

Cost function: ( , ) 8

q q z z z

s t

q q

q q

z q q

z q q

z q q

q q z z z

C q q

1 20 140q q

38

Cost function –Changed parameters

Frame Explicit choices

Implicit choices

Marginal cost of first

productI q1,q2,

z1,z2,z3

N/A N/A

II q1,q2 z1,z2,z3 80

III q1 q2,z1,z2,z3 80

39

Frame I – Short Term

40

Frame II – Short Term

1 2 2 3

1 2 1 2 1 2

1 2

1 2

1 2

1 2

1 2

90 149 10 15

90 149 10(3 4 ) 15(2 4 )

(90 30 30) (149 40 60)

(90 60) (149 100)

max 30 49

.

8

2 8

q q z z

q q q q q q

q q

q q

or

q q

st

q q

q q

41

Frame II – Short Term

• Short run cost: C(q1,q2)=60q1 +100q2

• No labor cost• New use of the accounting library• What we mean by cost – depends!

Product 1 Product 2

Direct laborDirect materialVariable overhead

03030

04060

Variable product cost 60 100

42

Cost terminology

• Cost and benefit• F(z) = B(z) – C(z)

– Separation always possible– Separation hardly unique

• Relevant cost– Is simply the portion of the cost function that

varies with the options at hand– Depends upon framing

43

Our objective

• Are we maximizing profit?• Are we maximizing wealth?• Are we maximizing utility?

– Are these different?• What happened to uncertainty?• How do we cope with uncertainty?• Is risk aversion part of our story?

44

Back to accounting

• Cost function– Which products are included?– How is scrap accounted for?

• Cost allocation– Are externalities accounted for?

• Transfer pricing– Linear pricing – first order condition maintained?

• ABC costing– Approximation of cost function?

45

Conclusions

• Ease of analysis vs complete specification• Framing is this decision• Notion of cost follows frame• Cost allocation might be part of framing• Where did the problem set-up come from

– Out of the blue– A handy and clever representation of the problem at

hand

• Professional judgment