The Simultaneous Choice of Investment and Financing Alternatives A Calculus of Variations Approach Robert W Grubbström Department of Production Economics Linköping Institute of Technology, Sweden - An Invited Lecture for The A.M.A.S.E.S. XXIV Conveigna Padenghe sul Garda September 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!!

Background motives

2 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 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,

Staircase function

The Product Structure, the Gozinto Graph and the Input Matrix

The Input Matrix and its Leontief Inverse Leontief Inverse Input Matrix

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

Leontief Inverse and Generalised Leontief Inverse Leontief Inverse Generalised Leontief Inverse ~

Inventory and Backlog Relationships

Conclusion 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.

References 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 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:

Current problem

General Problem 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. 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 Variations Around 1755Developed by Joseph-Louis Lagrange (Giuseppe Lodovico Lagrangia), , (then only 19 years old). Name suggested by Leonhard Euler, Around 1755Developed by Joseph-Louis Lagrange (Giuseppe Lodovico Lagrangia), , (then only 19 years old). Name suggested by Leonhard Euler, 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 Leonhard Euler

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: Basic 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.

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. Discrete 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: Multiplier

Necessary Lagrangean conditions: Discrete time comparison II Necessary Lagrangean conditions: Eliminating the leads to

Discrete time comparison III Take the following limits and while keeping. Then:

Utility functional

Types 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 Box Andrew Vazsonyi

Variable Loan + Interest

Budget Constraint

Other Constraints Decision variables for fixed payment schemes Repayment constraints for loans Possible logical constraints for investments etc.

Lagrangean function Multiplier

Lagrangean function Hamiltonian

Euler-Lagrange Conditions I

Euler-Lagrange Conditions II Remember, for variable loans/short-term investments: Then:

Consequences I So, whenever, we have the important solution: Or, when : The normalised Lagrangean multiplier integral is therefore the discount factor!

Kuhn-Tucker Conditions Net Present Value NPV except for

Consequences II Disregarding (temporarily) logical constraints: lead to: If then and then. If then, since, we have.

Consequences III 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. with equalities whenever x > 0 or y > 0. Mathematical justification of Bertrand Russels 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 Russels 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 IV 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. Assume, for instance, a separable case: Then: which means that must be proportional to.

Example Utility function Loan at 20 % p.a. Short-term investment at 10% p.a. Investment initial outlay , inflow p.a. Utility function Loan at 20 % p.a. Short-term investment at 10% p.a. Investment initial outlay , inflow p.a.

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

Orlicky Thank you for your kind attention!