Abstract
In the real world time passes and, if you think about it, you will agree with me that the only truly exogenous variable in economic models is time. Dynamic analysis takes time into consideration in an essential way. In this chapter we shall discuss dynamic optimization and, in the next three chapters, the modeling of economic behavior over time. In Chaps. 12 and 13, we studied static optimization to find the maximum or minimum point of a function with or without constraints on the variables involved. In this chapter we are interested in optimization over time.
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Notes
- 1.
This problem is from Recursive Methods in Economic Dynamics by Nancy Stokey and Robert Lucas (1989).
- 2.
It consists of two Greek words meaning the shortest time.
- 3.
The calculus of variations has its origin in the challenge that the Swiss mathematician John Bernoulli (1667–1748) issued to mathematicians in 1696. He asked them to solve the Brachistochrone problem. Prominent mathematicians of the day, Newton, Leibnitz, l'Hôpital, as well as John and his elder brother James Bernoulli (1654–1705) solved the problem. But it was James's solution and his call for the solution of the more general problem of the calculus of variations that led the way in the development of the subject. Leonhard Euler (1707–1783), one of the great mathematicians and the most prolific of them all, came up with the general solution embodied in the differential equation bearing his name. It was Joseph Lagrange (1736–1813) who devised the method of variation of a function, considering \(y(t) = y^* (t) + \varepsilon h(t)\), which Euler promptly adopted for finding the generalized solution of the problem of the calculus of variations; hence sometimes the equation is referred to as the Euler-Lagrange equation. It should also be noted that the genesis of the problem of the calculus of variations can be found in the work of Newton and even Galileo.
- 4.
For the French mathematician Adrien-Marie Legendre (1752–1833).
- 5.
The interested reader may want to consult more specialized texts, for example, Gelfand and Fomin (1963), Intriligator (1971), Bryson and Ho (1975), Kamien and Schwartz (1991), and Chiang (1992).
- 6.
Dynamic programming is the brainchild of the brilliant American mathematician Richard Bellman (1920–1984).
- 7.
A more general formulation recognizes that min or max of an objective function may be unattainable; therefore, they are replaced by inf and sup, respectively.
- 8.
Russian mathematician Lev Semenovich Pontryagin (1908–1988) and his colleagues proposed the method of maximum principle.
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Dadkhah, K. (2011). Dynamic Optimization. In: Foundations of Mathematical and Computational Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13748-8_14
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DOI: https://doi.org/10.1007/978-3-642-13748-8_14
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