This paper is the mathematical proof of the statements I made in this earlier article. I am putting this out now as a means of getting comments. The paper is written in the same mathematical style I used to write in before I bailed from academia, and it is not light reading with punchy quotes.
Note: If the above embed from Scribd does not work, the following link should: http://www.scribd.com/doc/202841217/Solution-Properties-of-Stock-Flow-Consistent-Models
I welcome comments, particularly those that tell me that I have done something wrong. This solution seems too simple, but it appears to work. If this article is seen as interesting, I would be happy to receive advice from academics on how it could be published through a peer-reviewed process. (Since I have left academia, this would be done more for my personal entertainment than for career enhancement.) I have no idea what journal would be interested in a highly mathematical model that questions the mathematical coherence of DSGE modelling.
I will change the document in-place, and in fact already have done so. (Oops.)
- I dropped the text on the Household ``No Ponzi'' condition; I wrote that section from memory, and my memory was bad (I mixed it up with the governmental budget constraint).
For those who do not want to parse the mathematics, my interpretation is the following:
- Optimisation problems involving a unitary Household Sector (a Representative Household) are inherently trivial.
- When dividends are taken into account, a Representative Household has essentially no financial constraints (other than those created by taxes, which are omitted in this article). Since money just sloshes back and forth between the Household and the Business it owns, the Household is indifferent between dividends and wage income.
- The Household is aware of the real constraints (the trade-off between working and real output). These are the constraints that determine the solution. Financial constraints, such as those implied by the Euler equation, do not matter.
- The solution is slightly more interesting if inventories are allowed to accumulate from period-to-period. In this case, it is necessary to determine the optimal path of real inventories. Once that optimisation is done, the solution will not depart from this optimal solution.
The standard "solution" technique in the literature does not look like my solution at all. This is partially because the mainstream literature generally does not find the true solution, rather the analysis make an approximation (linearisation). Guesses are made about the solution properties, without being able to validate which constraints on the solution are active. Since the original nonlinear model framework is abandoned for the linearisation, modellers do not realise that their framework implies that the Business Sector will useless pile up money balances which it has no intention of ever utilising.
My reading between the lines of mainstream discussion of the topic is that financial constraints are thought to matter as they would matter for a household in the real-world economy. This is then transferred to the Representative Household. This runs afoul of the Fallacy of Composition. Since the problem is cast as an optimisation, the optimising agent is aware of all the constraints, and is presumably able to determine that financial constraints do not matter for a unified household sector.
The only way the author can see financial constraints as mattering is by arbitrarily imposing them on the solution, and demanding that the Representative Household choose a sub-optimal solution because that is the desired output by the modellers. If that is the case, those added constraints need to be properly specified mathematically.
(c) Brian Romanchuk 2014