Ramsey model condition

Question

Why are transversality conditions and no-ponzi game conditions needed in the Ramsey model? And why do we assume that is a CIES utility function?

Answer 1

First of all, a sharp conceptual distinction must be made between transversality conditions and no-Ponzi game condition.

The transversality conditions are purely mathematical conditions, and they are optimality conditions, that is conditions that must be fulfilled when looking for an optimum of the maximization problem.

They are conditions that must be imposed in any kind problem of optimal control (with some particular mathematical features, some class of problems) irrespective of the fact it deals with economics, the flight of an airplane or the trajectory of a rocket.

In economics problems, it is usual to give them, in the second instance, some economic significance, a content interpretation in terms of economic theory. But, anyway, it remains only a condition of optimality, in the sense that it doesn't impose further constraints on the behavior of agents, and its rationale is mathematical.

In contrast, the no Ponzi game condition is an economic assumption, it is a constraint on the behavior of individuals, and it is not a mathematical condition needed for optimization. It is a consequence of the assumptions on the budget constraint of the consumer, of how the budget constraint is formulated. Its economic content can be illustrated as follows:

Expressed in this form the budget constraint states that the present value of the household's asset holdings cannot be negative in the limit.

Equation $(2.10)$ is known as the no-Ponzi game condition. A Ponzi game is a scheme in which someone issues debt and rolls it over forever. That is, the issuer always obtains the funds to pay off debt when it comes due by issuing new debt. Such a scheme allows the issuer to have a present value of lifetime consumption that exceeds the present value of her or his lifetime resources.$^1$

For a discussion of this distinction between transversality conditions and no Ponzi game see also the answer by Alec Papadopoulos to a similar question: No Ponzi game condition and transversality condition are the same?

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To have an idea of the nature of the transversality condition (the technicalities are very complicated) one must forget in the first instance the economic content of a model. The Pontryagin maximum principle, which is used in optimal control problems,$^2$ is a whole class of theorems, relative to problems with different mathematical assumptions, finite or infinite horizon problems, fixed end or open end, initial states and target states that can be points or sets, and so on.

In some classes of optimal problems (not all) transversality conditions are needed, and they arise from mathematical, in particular geometrical, considerations about optimality conditions.

And they are the same, irrespective of the field of application. $^3$

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$^1$ ^{Romer, Advanced Macroconomics, McGraw-Hill, 2012, p. 53.}

$^2$ ^{The other possible approach is calculus of variations but, when the calculus of variation is applicable, the results are the same, and Pontryagin Maximum principle is applicable under more general assumptions. See, for example, Takayama, Analytical Methods in Economics, Michigan University, 1993.}

$^3$ ^{To have an idea of the mathematical aspects of this complex subject and of trasversality conditions one can see for example Liberzon, Calculus of Variations and Optimal Control Theory. A Concise Introduction, University of Illinois at Urbana-Champaign. Consider that in the original work by Pontryagin, The mathematical Theory of Optimal Processes, the proof of the first theorem only is forty pages! The book of Takayama I quoted above is also a good reference, as to economic applications.}