The Simultaneous Choice of Investment and Financing

AlternativesA Calculus of Variations Approach

The Simultaneous Choice of The Simultaneous Choice of Investment and Financing Investment and Financing

AlternativesAlternativesA Calculus of Variations ApproachA Calculus of Variations Approach

Robert W GrubbströmRobert W GrubbströmDepartment of Production EconomicsDepartment of Production Economics

Linköping Institute of Technology, SwedenLinköping Institute of Technology, Sweden

rwg@ipe.liu.serwg@ipe.liu.se

Robert W GrubbströmRobert W GrubbströmDepartment of Production EconomicsDepartment of Production Economics

Linköping Institute of Technology, SwedenLinköping Institute of Technology, Sweden

rwg@ipe.liu.serwg@ipe.liu.se

- An Invited Lecture forThe A.M.A.S.E.S. XXIV Conveigna

Padenghe sul GardaSeptember 6-9, 2000

- An Invited Lecture for- An Invited Lecture forThe A.M.A.S.E.S. XXIV ConveignaThe A.M.A.S.E.S. XXIV Conveigna

Padenghe sul GardaPadenghe sul GardaSeptember 6-9, 2000September 6-9, 2000

Orlicky

To Lorenzo Peccati and Marco Li Calzi :

Many thanks for giving me and my wife Anne-Marie

the opportunity to come to Padenghe!Molte grazie!!

BackgroundBackgroundmotivesmotives

BackgroundBackgroundmotivesmotives

2 Reasons for thinking in Cash terms2 Reasons for thinking in Cash terms

The Cash Flow (and what you can do with it) is the ultimate consequence of all economic activities

The Cash Flow (and what you can do with it) is the ultimate consequence of all economic activities

The Cash Flow is the nearest you can get to finding a physical measure of economic activities

The Cash Flow is the nearest you can get to finding a physical measure of economic activities

“A principle for determining the correct capital costs of work-in-progress and inventory”, International Journal of Production Research, 18, 1980, 259-271

“A principle for determining the correct capital costs of work-in-progress and inventory”, International Journal of Production Research, 18, 1980, 259-271

Staircase functionStaircase function

D

SP

Staircase functionStaircase function

S P D

S P D

B D P

B D P

The Product Structure, the Gozinto The Product Structure, the Gozinto Graph and the Input MatrixGraph and the Input Matrix

0 0 0 0 0

0 0 0 0 0

4 1 0 0 0

2 0 1 0 0

0 3 2 2 0

H

A E B C D

A

E

B

C

D

The Product Structure, the Gozinto The Product Structure, the Gozinto Graph and the Input MatrixGraph and the Input Matrix

XC

Y

X

A E

D

B2

4

2

1

2

1

1

3

1

C

Y

A E

D

B2

4

2

1

2

1

1

3

1

A E

C

C

B

D D

D

D

Y

Y

Y

Y

XC

B

D

D

Y

Y

X

4*

2*

2*

2*

2*1*

3*

1*

1*

1* 1* 1*

1*

1*1* 1*

1*2*

2* Y

A E B C D

A

E

B

C

D

X

Y

0 0 0 0 0

0 0 0 0 0

4 1 0 0 0

2 0 1 0 0

0 3 2 2 0

0 0 1 0 0

0 0 0 0 1

H =

The Input Matrix and The Input Matrix and its Leontief Inverseits Leontief Inverse

The Input Matrix and The Input Matrix and its Leontief Inverseits Leontief Inverse

Leontief Inverse

1

1 0 0 0 0

0 1 0 0 0

( ) 4 1 1 0 0

6 1 1 1 0

2 0 7 4 2 1

I H

0 0 0 0 0

0 0 0 0 0

4 1 0 0 0

2 0 1 0 0

0 3 2 2 0

H

A E B C D

A

E

B

C

D

Input Matrix

Adding the Lead Time Matrix and Adding the Lead Time Matrix and Creating the Generalised Input MatrixCreating the Generalised Input Matrix

A E

C

C

B

D D

D

D

Y

Y

Y

Y

XC

B

D

D

Y

Y

X

4*

2*

2*

2*

2*1*

3*

1*

1*

1* 1* 1*

1*

1*1* 1*

1*2*

2* Y

Adding the Lead Time Matrix and Adding the Lead Time Matrix and Creating the Generalised Input MatrixCreating the Generalised Input Matrix

XC

Y

X

A E

D

B2

4

2

1

2

1

1

3

1

C

Y

A E

D

B2

4

2

1

2

1

1

3

1

A E B C D

A

E

B

C

D

X

Y

0 0 0 0 0

0 0 0 0 0

4 1 0 0 0

2 0 1 0 0

0 3 2 2 0

0 0 1 0 0

0 0 0 0 1

H =

s

s

s

s

s

e

e

e

e

e

6

8

1 0

1 2

1 4

0000

0000

0000

0000

0000

~

Lead Time Matrix

1 4 1 2

1 4 1 0

1 2 1 0 8

0 0 0 0 0

0 0 0 0 0

4 0 0 0

2 0 0 0

0 3 2 2 0

s s

s s

s s s

e e

e e

e e e

H

Generalised Input Matrix

Leontief Inverse and Generalised Leontief InverseLeontief Inverse and Generalised Leontief InverseLeontief Inverse and Generalised Leontief InverseLeontief Inverse and Generalised Leontief Inverse

Leontief Inverse

1

1 0 0 0 0

0 1 0 0 0

( ) 4 1 1 0 0

6 1 1 1 0

2 0 7 4 2 1

I H

1 4 1 2

1 4 2 4 2 2 10

2 4 2 2 3 2 1 2 2 2 3 0 1 0 1 8 8

1 0 0 0 0

0 1 0 0 0

4 1 0 0

2 4 1 0

8 4 8 3 2 2 2 2 2 1

s s

s s s s

s s s s s s s s s

e e

e e e e

e e e e e e e e e

1( ) I H

Generalised Leontief Inverse

~

Inventory and Backlog RelationshipsInventory and Backlog Relationships

A L L O C AT IO N S(W ork -in -P ro cess)

A S S E M B LY(in stan tan eou s)

AVA IL A B L EIN V E N T O R Y

R B~~~~ ~

Inventory and Backlog RelationshipsInventory and Backlog Relationships

P R O D U C -T IO N

P U R C H A S E S

P~

H P~

T O TA L IN V E N T O R Y S~

IN T E R N A LD E M A N D

H P~~

D E L IV E -R IE S

F~

E X T E R N A LD E M A N D

D~

ConclusionConclusionConclusionConclusion

•When studying complex multi-level, multistage production-inventory systems, there is an essential need to be aware of what the basic objective should be interpreted as.

•The discussion to follow has the purpose to justify the use of the Net Present Value as the sole objective.

•When studying complex multi-level, multistage production-inventory systems, there is an essential need to be aware of what the basic objective should be interpreted as.

•The discussion to follow has the purpose to justify the use of the Net Present Value as the sole objective.

ReferencesReferencesReferencesReferences

http://ipe.liu.se/rwg/mrp_publ.htm

For references please consult:

Grubbström, R. W., Tang, O., An Overview of Input-Output Analysis Applied to Production-Inventory Systems, Economic Systems Research,

Vol. 12, No 1, 2000, pp. 3-26

Recent survey:

Tang, O., Planning and Replanning within the Material Requirements Planning Environment – a Transform Approach, PROFIL 16,

Production-Economic Research in Linköping, Linköping 2000

Recent thesis:

CurrentCurrentproblemproblemCurrentCurrentproblemproblem

General ProblemGeneral ProblemGeneral ProblemGeneral ProblemThe problem considered is to choose from a finite set of inter-related investment and financing alternatives and also levels of consumption/work over time to maximise a utility functional.

Each investment and financing option is characterised by its cash flow over time. An inter-temporal budget requirement operates continuously. Application of the Calculus of Variations leads to consideration of the Euler-Lagrange equations combined with Kuhn-Tucker conditions.

It is shown that the solution (also when there are logical dependencies present) requires the maximisation of a Generalised Net Present Value measure in which the discount factor is formed from an integral of a Lagrangean multiplier function.

The problem considered is to choose from a finite set of inter-related investment and financing alternatives and also levels of consumption/work over time to maximise a utility functional.

Each investment and financing option is characterised by its cash flow over time. An inter-temporal budget requirement operates continuously. Application of the Calculus of Variations leads to consideration of the Euler-Lagrange equations combined with Kuhn-Tucker conditions.

It is shown that the solution (also when there are logical dependencies present) requires the maximisation of a Generalised Net Present Value measure in which the discount factor is formed from an integral of a Lagrangean multiplier function.

Calculus of VariationsCalculus of VariationsCalculus of VariationsCalculus of Variations Around 1755 Developed by Joseph-Louis Lagrange

(Giuseppe Lodovico Lagrangia), 1736-1813, (then only 19 years old).

Name suggested by Leonhard Euler, 1707-1783.

Around 1755 Developed by Joseph-Louis Lagrange (Giuseppe Lodovico Lagrangia), 1736-1813, (then only 19 years old).

Name suggested by Leonhard Euler, 1707-1783.

Original Problems

*Find the maximum area enclosed by a curve of given length.

*The Brachistochrone problem: To specify a path between two given points in space, such that a particle released at a given velocity will slide from the upper to the lower point under gravity in the minimum time.

Original Problems

*Find the maximum area enclosed by a curve of given length.

*The Brachistochrone problem: To specify a path between two given points in space, such that a particle released at a given velocity will slide from the upper to the lower point under gravity in the minimum time.

Joseph-Louis Lagrange 1736-1813 Leonhard Euler 1707-1783

A functional (an integral) is to be optimised. This is the objective function.

A functional (an integral) is to be optimised. This is the objective function.

A functional (an integral) is to be optimised. This is the objective function.

The problem is to find the path x(t). The integrand H is called the Hamiltonian.

The necessary local optimisation conditions read:

A functional (an integral) is to be optimised. This is the objective function.

The problem is to find the path x(t). The integrand H is called the Hamiltonian.

The necessary local optimisation conditions read:

( , , )b

a

H x x t dt

Basic MethodologyBasic MethodologyBasic MethodologyBasic Methodology A functional (an integral) is to be optimised. This is the objective function.

The problem is to find the path x(t). The integrand H is called the Hamiltonian.

The necessary local optimisation conditions read:

These are the Euler-Lagrange Equations. In the following these conditions will be extended slightly with the use of the ”modern” Kuhn-Tucker conditions in order to take care of non-negativity requirements.

A functional (an integral) is to be optimised. This is the objective function.

The problem is to find the path x(t). The integrand H is called the Hamiltonian.

The necessary local optimisation conditions read:

These are the Euler-Lagrange Equations. In the following these conditions will be extended slightly with the use of the ”modern” Kuhn-Tucker conditions in order to take care of non-negativity requirements.

( , , )b

a

H x x t dt

0H d H

x dt x

( , , )b

a

H x x t dt

0H d H

x dt x

A non-rigorous comparison with a Lagrangean discrete optimisation is the following. Let and optimise

by a suitable choice of the , where .

A non-rigorous comparison with a Lagrangean discrete optimisation is the following. Let and optimise

by a suitable choice of the , where .

( ), ( ) / , /n n nn

U H x t x t T t T ( )nx t 1( ) ( ) ( )n n nx t x t x t

nt n T

Discrete time comparisonDiscrete time comparisonDiscrete time comparisonDiscrete time comparison

A non-rigorous comparison with a Lagrangean discrete optimisation is the following. Let and optimise

by a suitable choice of the , where . Let be Lagrangean multipliers. The Lagrangean is written:

A non-rigorous comparison with a Lagrangean discrete optimisation is the following. Let and optimise

by a suitable choice of the , where . Let be Lagrangean multipliers. The Lagrangean is written:

( ), ( ) / , /n n nn

U H x t x t T t T ( )nx t 1( ) ( ) ( )n n nx t x t x t

nt n T

n

1

( ), ( ) / , /

( ) ( ( ) ( ))

n n nn

n n n nn

L H x t x t T t T

x t x t x t

Multiplier

Necessary Lagrangean conditions:

Necessary Lagrangean conditions:

1

( ), ( ) / ,10

( ) ( )n n n

n nn n

H x t x t T tL

x t T x t

( ), ( ) / ,10

( ( ) / ) ( ( ) / )n n n

nn n

H x t x t T tLT

x x t T T x x t T

Discrete time comparison IIDiscrete time comparison IIDiscrete time comparison IIDiscrete time comparison II

Necessary Lagrangean conditions:

Eliminating the leads to

Necessary Lagrangean conditions:

Eliminating the leads to

n

1

( ), ( ) / ,10

( ) ( )n n n

n nn n

H x t x t T tL

x t T x t

( ), ( ) / ,10

( ( ) / ) ( ( ) / )n n n

nn n

H x t x t T tLT

x x t T T x x t T

1 1 1

1

( ), ( ) / ,

( )

( ), ( ) / , ( ), ( ) / ,10

( ( ) / ) ( ( ) / )

n n n

n

n n n n n n

n n

H x t x t T t

x t

H x t x t T t H x t x t T t

T x t T x t T

Discrete time comparison IIIDiscrete time comparison IIIDiscrete time comparison IIIDiscrete time comparison III

Take the following limits and

while keeping .

Take the following limits and

while keeping .

n 0T

nn T t t

, 0,

1 1 1

1

( ), ( ) / ,

( )

( ), ( ) / , ( ), ( ) / ,1

( ( ) / ) ( ( ) / )

lim n n n

n T n T t n

n n n n n n

n n

H x t x t T t

x t

H x t x t T t H x t x t T t

T x t T x t T

, , , ,0

H x x t H x x td

x dt x

Then:

Utility functionalUtility functionalUtility functionalUtility functional

0

ˆ ˆ ˆ( ), ( ), ( , , )T

U u x t y t t dt u x y T 0

ˆ ˆ ˆ( ), ( ), ( , , )T

U u x t y t t dt u x y T

Types of Cash FlowsTypes of Cash FlowsTypes of Cash FlowsTypes of Cash Flows

•Money earned from work and money paid for consumption.

•Investment projects or loans with a fixed payment scheme. These cash flows can be accepted partially or completely.

• Variable loans or short-term investment alternatives which can be varied over time. An interest rate, possibly time varying, is attached to each such cash flow.

• Interest payments attached to each variable loan or short-term investment.

•Money earned from work and money paid for consumption.

•Investment projects or loans with a fixed payment scheme. These cash flows can be accepted partially or completely.

• Variable loans or short-term investment alternatives which can be varied over time. An interest rate, possibly time varying, is attached to each such cash flow.

• Interest payments attached to each variable loan or short-term investment.

The Budget BoxThe Budget BoxThe Budget Box

Andrew Vazsonyi

0

(loan or obtained repayment)ig

(work)y

(consumption)x

0

(investment return)if

0

(loan repayment or lending)ig

0

(cash outflow for investment)if

0

( ) 0

(interest received)

t

i ig d 0

( ) 0

(interest paid)

t

i ig d

Variable Loan + InterestVariable Loan + InterestVariable Loan + InterestVariable Loan + Interest

( )( ) ( )i

i i

dG tt G t

dt

( )( ) ( )i

i i

dG tt G t

dt

0

( ) ( ) ( ) ( )t

i i i if t g t t g d 0

( ) ( ) ( ) ( )t

i i i if t g t t g d

0 0( ) ( )

( )t t

i id d

i

de e G t

dt

0 0( ) ( )

( )t t

i id d

i

de e G t

dt

Budget ConstraintBudget ConstraintBudget ConstraintBudget Constraint

0

( ) ( ) ( ) 0,t

i ii

f x y d t T

0

( ) ( ) ( ) 0,t

i ii

f x y d t T

0

ˆ ˆ( ) ( ) ( ) 0T

i ii

f x y d x y

0

ˆ ˆ( ) ( ) ( ) 0T

i ii

f x y d x y

Other ConstraintsOther ConstraintsOther ConstraintsOther Constraints

1 0i 1 0i Decision variables for fixed payment schemes

Decision variables for fixed payment schemes

0

( ) ( ) 0T

i ig t dt G T 0

( ) ( ) 0T

i ig t dt G T Repayment constraints for loans

Repayment constraints for loans

0ji i ji

A b 0ji i ji

A b Possible logical constraints for investments etc.

Possible logical constraints for investments etc.

Lagrangean functionLagrangean functionLagrangean functionLagrangean function

0

ˆ ˆ ˆ( , , ) ( , , )T

t

L u x y t dt u x y T

0

ˆ ˆ ˆ( , , ) ( , , )T

t

L u x y t dt u x y T

0 0

( ) ( ) ( ) ( )T t

i iit

t f x y d dt

0 0

( ) ( ) ( ) ( )T t

i iit

t f x y d dt

0

ˆ ˆ( ) ( ) ( )T

i iit

f t x t y t dt x y

0

ˆ ˆ( ) ( ) ( )T

i iit

f t x t y t dt x y

(1 )i i j ji i ji j i

A b

(1 )i i j ji i ji j i

A b

Multiplier

Multiplier

Multiplier Multiplier

Lagrangean functionLagrangean functionLagrangean functionLagrangean function

Let ( ) ( ) and change order of integration:T

t

t d Let ( ) ( ) and change order of integration:T

t

t d

0

( , , ) ( ) ( ) ( ) ( )T

i ii

L u x y t t f t x t y t dt

0

( , , ) ( ) ( ) ( ) ( )T

i ii

L u x y t t f t x t y t dt

ˆ ˆ ˆ ˆ ˆ( , , ) ( )( ) (1 )i i j ji i ji j i

u x y T T y x A b

ˆ ˆ ˆ ˆ ˆ( , , ) ( )( ) (1 )i i j ji i ji j i

u x y T T y x A b

Hamiltonian

Euler-Lagrange Conditions IEuler-Lagrange Conditions IEuler-Lagrange Conditions IEuler-Lagrange Conditions I( , , )

( ) 0H u x y t

tx x

( , , )( ) 0

H u x y tt

x x

( , , )( ) 0

H u x y tt

y y

( , , )( ) 0

H u x y tt

y y

( , , )( ) 0

H u x y tx x t

x x

( , , )( ) 0

H u x y tx x t

x x

( , , )( ) 0

H u x y ty y t

y y

( , , )( ) 0

H u x y ty y t

y y

Euler-Lagrange Conditions IIEuler-Lagrange Conditions IIEuler-Lagrange Conditions IIEuler-Lagrange Conditions II

0, 0( ) ( ) ( )

0, 0i

iii i

GH d Ht t t

GG dt G

0, 0( ) ( ) ( )

0, 0i

iii i

GH d Ht t t

GG dt G

( ) ( ) ( ) 0i i ii i

H d HG G t t t

G dt G

( ) ( ) ( ) 0i i i

i i

H d HG G t t t

G dt G

( )( ) ( )i

i i i i

dG tf t G t

dt

( )( ) ( )i

i i i i

dG tf t G t

dt

Remember, for variable loans/short-term investments:Remember, for variable loans/short-term investments:

Then:Then:

Consequences IConsequences IConsequences IConsequences I

0( )

( ) (0)t

i dt e

0( )

( ) (0)t

i dt e

So, whenever , we have the important solution:So, whenever , we have the important solution:( ) 0iG t

Or, when :Or, when :consti

( ) (0) i tt e ( ) (0) i tt e

The normalised Lagrangean multiplier integral is therefore the discount factor!

The normalised Lagrangean multiplier integral is therefore the discount factor!

( ) / (0)t

Kuhn-Tucker ConditionsKuhn-Tucker ConditionsKuhn-Tucker ConditionsKuhn-Tucker Conditions

0

( ) ( ) 0T

i i j jiji

Lf t t dt A

0

( ) ( ) 0T

i i j jiji

Lf t t dt A

NPV 0i i i i j jiji

LA

NPV 0i i i i j jiji

LA

1 0ii

L

1 0ii

L

1 0i i i

i

L

1 0i i ii

L

, 0i j , 0i j

Net Present Value NPV

except for (0)

Consequences IIConsequences IIConsequences IIConsequences II

NPV 0i i NPV 0i i

NPV 0i i i NPV 0i i i

1 0i 1 0i

1 0i i 1 0i i

Disregarding (temporarily) logical constraints:Disregarding (temporarily) logical constraints:

lead to:lead to:

If then and then .If then and then .NPV 0i 0i 1i

If then, since , we have .If then, since , we have .NPV 0i 0i 0i

Consequences IIIConsequences IIIConsequences IIIConsequences IIIMarginal utilities and discount factor:Marginal utilities and discount factor:

For instance, when there is both consumption and work, then .Then, if marginal momentary utility decreases with x (becomes more negative with y) and time preferences (the t in u(x,y,t)) are disregarded, we must have a greater x for a greater t, and a lower y for a greater t.

For instance, when there is both consumption and work, then .Then, if marginal momentary utility decreases with x (becomes more negative with y) and time preferences (the t in u(x,y,t)) are disregarded, we must have a greater x for a greater t, and a lower y for a greater t.

consumption workMU ( ) MUt

( , , ) ( , , )( )

u x y t u x y tt

x y

( , , ) ( , , )( )

u x y t u x y tt

x y

with equalities whenever x > 0 or y > 0.with equalities whenever x > 0 or y > 0. Mathematical justification of Bertrand Russel’s statement:

“Forethought, which involves doing unpleasant things now, for the sake of pleasant things in the future, is one of the most essential marks of the development of man.”

Mathematical justification of Bertrand Russel’s statement:

“Forethought, which involves doing unpleasant things now, for the sake of pleasant things in the future, is one of the most essential marks of the development of man.”

Consequences IVConsequences IVConsequences IVConsequences IVIf there is both consumption and work, which keep at the same levels over time, x=const and y=const, then the time preference must follow the discount factor.

If there is both consumption and work, which keep at the same levels over time, x=const and y=const, then the time preference must follow the discount factor.

( , , ) ( , , ) ( , ) ( , )( ) ( ) ( )

u x y t u x y t u x y u x yt t t

x y x y

( , , ) ( , , ) ( , ) ( , )( ) ( ) ( )

u x y t u x y t u x y u x yt t t

x y x y

Assume, for instance, a separable case:Assume, for instance, a separable case:( , , ) ( , ) ( )u x y t u x y t

Then:Then:

which means that must be proportional to . which means that must be proportional to . ( )t ( )t

ExampleExampleExampleExample

10000

7945

28152240

-4103.06 14.26 17.30

20.0

1( )G t

2( )G t

( )x t

Utility function

Loan at 20 % p.a.

Short-term investment at 10% p.a.

Investment

initial outlay -10000, inflow +2000 p.a.

Utility function

Loan at 20 % p.a.

Short-term investment at 10% p.a.

Investment

initial outlay -10000, inflow +2000 p.a.

1( )G t

0.0011 xu e

2( )G t

1( )f t

ConclusionsConclusionsConclusionsConclusions

•The Net Present Value, appropriately defined, is a superior measure of the benefit of a cash flow.

•The Calculus of Variations appears to be tailor made for analysing investment and financing alternatives.

•The Net Present Value, appropriately defined, is a superior measure of the benefit of a cash flow.

•The Calculus of Variations appears to be tailor made for analysing investment and financing alternatives.

Main referencesMain referencesMain referencesMain references

Grubbström, R. W., Ashcroft, S. H., Application of the Calculus of Variations to Financing Alternatives, Omega, Vol. 19, No 4, 1991, pp. 305-316

Grubbström, R. W., Ashcroft, S. H., Application of the Calculus of Variations to Financing Alternatives, Omega, Vol. 19, No 4, 1991, pp. 305-316

Grubbström, R.W., Jiang, Y., Application of the Calculus of Variations to Economic Decisions: A survey of some economic problem areas, Modelling Simulation and Control, C, Vol. 28, No 2, 1991, pp. 33-44

Grubbström, R.W., Jiang, Y., Application of the Calculus of Variations to Economic Decisions: A survey of some economic problem areas, Modelling Simulation and Control, C, Vol. 28, No 2, 1991, pp. 33-44

There is more to life than making money…There is more to life than making money…

But, I do think it is But, I do think it is

important for a person to important for a person to

have something to do,have something to do,

when he’s not drinking when he’s not drinking

or making love ...or making love ...

Orlicky

Thank you for your kind attention!