Abstract
The main purpose of this chapter is to build and explore a benchmark macro-model in a deterministic setting of financial markets. Although this model exhibits highly unrealistic features like a constant short-term interest rate and thus a flat term structure of interest rates, it has nevertheless some merits: first of all, the income and debt dynamics are linear so that we can calculate and analyze the time paths of these variables explicitly. This delivers clear-cut results with regard to economic growth and public indebtedness. Second, the stochastic macro-model to be developed later on converges to this deterministic version when risk disappears. We can check the correctness of the reduced-form dynamics of our stochastic model by simply looking at the deterministic model. Third, by comparing the outcomes of the stochastic version with the deterministic one, we can discern the qualitative effects brought about by the introduction of risk.
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Notes
Barro/Sala-i-Martin (1995, Chapter 1.3 and 4.1) call this production function ‘A-K technology’. They also show that this type of production function is the simplest one that generates endogenous growth.
We have omitted the time index and will continue to do so mainly for notational convenience. The reader should note that all variables in this chapter are described by capital letters and are functions of time.
From now on dots on variables denote derivatives with regard to time.
Government debt in form of short-term bonds can hence be thought to be rolled over continuously. We abstract from the issue of government default because every specification of a possible default of the government would be arbitrary and thus ad-hoc if we did not model government behavior endogenously.
Note from (2.3a) that any pair of stocks of the three assets could be thought to be control variables since by choosing two stocks the third stock is determined via (2.3a) given the stock of wealth.
The original reference is Pontryagin et al. (1962). Books including economic examples of how to use this Maximum Principle are Kamien/Schwartz (1991), Chiang (1992), Léonard/Van Long (1992) and Takayama (1993).
We do not consider the second-order condition here since it is extremely easy to see that it holds as long as γ is positive.
Of course, one could ask why we did not include capital accumulation as another dynamic constraint into the decision problem of the representative household since this would probably have led to a consumption/saving behavior of the private sector that does not include the danger of zero capital stocks. The reason is simple: it seems unreasonable that private households take into consideration the aggregate capital accumulation of the economy. If they did, the assumption of price taking behavior would become logically unsustainable.
Simultaneously, we can also see that it is sufficient in order to get always positive growth rates when τ ≤ αк-δ-p holds.
This means that „…a poor economy tends to grow faster than a rich one, so that the poor country tends to catch up with the rich one…“ (Barro/Sala-i-Martin (1995, p. 383)).
As Obstfeld/Rogoff (1996, Supplements to Chapter 2) argue for the case of infinite-horizon representative agent models, the government’s intertemporal solvency or no Ponzi-game constraint is always fulfilled when the transversality condition holds.
We are well aware of the fact that this interpretation has a kind of Malthusian-flavored character which many economists do not like and share.
For a general exposition of how to solve systems of first-order differential equations, see Boyce/DiPrima (1997, Chapter 7).
Compare Boyce/DiPrima (1997, Chapter 7.7).
We decided to put the first eigenvalue into the fundamental solution instead of the second. The other way round is, of course, also possible since we consider the case were both eigenvalues are identical.
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© 1999 Springer-Verlag Berlin Heidelberg
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Demmel, R. (1999). The basic deterministic macroeconomic model. In: Fiscal Policy, Public Debt and the Term Structure of Interest Rates. Lecture Notes in Economics and Mathematical Systems, vol 476. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58595-1_2
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DOI: https://doi.org/10.1007/978-3-642-58595-1_2
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