# What does it mean for a technology to have a $|\rho|<1$?

Nowadays, when trying to model (log)technology $a_t$, I've seen many times the following: $a_t = \rho a_{t-1}+\epsilon$, where epsilon is given a certain probability distribution. What does it mean for $|\rho|<1$? I know that in this way, the time series is stationary. My question is related towards more the interpretation. With such a $rho$ does this mean that our technological discoveries are being lost in time? We're forgetting our past? or That the discoveries give a surge of increase in 'productivity', but then the 'productivity' decreases...

Any help would be appreciated.

New technology makes past technology obsolete. How many people know these days how to light a fire by rubbing wood together? How many people know the nuts and bolts of tending to the engine of a train powered by coal? Heck, how many coal-trains are still operational?

So yes, we do forget our past discoveries and technologies, as they are replaced by new ones. But the technology as inserted in the production function, operates as a production shifter, a "mark-up", it does not represent the composition of technology, but rather its final effect on the production inputs.

Modeling the technological production shifter as a stationary auto-regressive scheme (without drift), we essentially assume that, random shocks aside, we are set today to produce less efficiently than yesterday.

Does this accord with observed reality? Do we depend on positive random discoveries to produce more efficiently? My impression is that, on the contrary, humans very purposefully try to make their production processes more efficient.

Now, if you were to add a drift (a constant term) in the log-technology process,

$$a_t = \delta + \rho a_{t-1}+\epsilon_t$$

you would capture, the rising trend in efficiency we have observed historically, and also implicitly, the fact that we discard old technologies. This also has the benefit that it envisions rising efficiency while also, an eventual level-off (which seems wise to me).

• +1 Alecos, I'll wait a bit for some other answers, but yours seems very promising. ;) thanks Oct 17, 2017 at 16:44
• By the way, then I wonder if this doesn't go against empirical evidence, and if so, has anyone tried to improve upon this? For example, Galí in his book «Monetary Policy, Inflation, and the Business Cycle» constantly uses this specification for tech shocks... in fact for any shock. Oct 17, 2017 at 17:15

when $|\rho|<1$, it means that that specific shock loses its influence over time.

for example, upon invention of the car engine there was a productivity shock to the transportation of goods and services.

As innovation and integration of newer transportation technology occurs (i.e. development of a newer and better engine,development of different transportation options) the original engine's contribution to an economy's level of technology loses its influence over time.

In short it tells us that as time moves on the integration of a specific technology has a diminishing returns to an economy's level of technological development.

• can the absolute value of $\rho$ be negative? This must be a typo. Oct 17, 2017 at 16:36