Discrete Time Or Continuous Time?

There is always some debate by people on how continuous time is better modelling. A James Tobin quote from his Nobel Lecture comes to mind.

Macroeconomic Modeling Strategy: Continuous or Discrete Time

The issues just discussed are related to the modeling of time. The equations introduced above count time in discrete periods of equal finite length. Within any period, each variable assumes one and only one value. In particular, clearing of asset markets determines one set of asset prices per period. From one period to the next asset stocks jump by finite amounts. Therefore the demands and supplies for these jumps affect asset prices and other variables within the period, the more so the greater the length of the period. They will also, of course, influence the solutions in subsequent periods.

The same modeling strategy can be used with continuous time. The specific saving functions, as well as the total saving function, then tell the rate at which savers want to be increasing their stocks of particular assets and of total wealth. They will reflect both the continuous execution of long run saving and portfolio plans and the speeds of adjustment of stocks to deviations from these plans that arise because of surprises, news, and altered circumstances or preferences.

Either representation of time in economic dynamics is an unrealistic abstraction. We know by common observation that some variables, notably prices in organized markets, move virtually continuously. Others remain fixed for periods of varying length. Some decisions by economic agents are reconsidered daily or hourly, while others are reviewed at intervals of a year or longer except when extraordinary events compel revisions. It would be desirable in principle to allow for differences among variables in frequencies of change and even to make those frequencies endogenous. But at present models of such realism seem beyond the power of our analytic tools. Moreover, many statistical data are available only for arbitrary finite periods.

Representation of economies as systems of simultaneous equations always strains credibility. But it takes extraordinary suspension of disbelief to imagine that the economy solves and re-solves such systems every microsecond. Even with modern computers the task of the Walrasian Auctioneer, and of the market participants who provide demand and supply schedules, would be impossible. Economic interdependence is the feature of economic life and we as professional economists seek to understand and explain. Simultaneous equations systems are a convenient representation of interdependence, but it is more persuasive to think of the economic processes that solve them as taking time than as working instantaneously.

In any event, a model of short-run determination of macroeconomic activity must be regarded as referring to a slice of time, whether thick or paper thin, and as embedded in a dynamic process in which flows alter stocks, which in turn condition subsequent flows.

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