Simple ideas on non linear DSGE

We want to keep it simple.  We assume households, firms and government are adapted, that is they transact goods at the proper size and frequency such that the two period model is met, they are adiabatic.  How do we find the number of goods that can be moved for some given, short period?

Assume perfect linearity and measure real growth rate level and variance of growth rate over some stable period.  The variance is the power deviation from linearity. It is a measure of the value of e in the limit and the value of e in actual use, where e is Eulers number. That variance should be small, equal to the variation in prices. 

The signal in this case is e, everyone is trying to keep up. The noise is the variance from trend. So the signal to noise ratio is high, the variation in the checkout counter is not that bad.  How much value can be transacted in aggregate? 

We get C/(2*B) from the Shannon condition, value rate over potential transaction rate if everyone knew e. Of the total bandwidth, 2*B, 2*B-C is the clock rate of the economy, the speed at which the check out counter can work the register.

Now I used value instead of good because when the economy is adapted, packing occurs, and value it the -iLog(i), not i alone.

Now we know then the economy is two period adapted, then there is an index that can be uniquely assigned to each transaction.  That index is:

2^(C/2*B)-1.  C/B is large, and let's just take it up to the next integer. Then we get a two bit counter which assigns a two bit number to each good. The checkout person takes the good package and runs it through a decoder selecting the proper index.

Now we go back to our linear DSGE model and go a ahead and solve it, getting a short series sum like e^(at) + e^(bt) + e^(ct), for example, the sum computing the real GDP growth over time. But t counts from 1 to 2*B. The a,b andc are ordered and diagonal. The number to clock ticks they get is proportional to their covariance (or deviation, see note below), and the bits in their index  proportion to the coefficient,a,b,c. So from there we get the relative transaction value and probability of occurence.

Non linear effects occur with the number of agents, N in each group.   Or a slow down in transaction rate, real GDP growth variance changes. Quantization requires one agent per transaction. Check for the probability that some agents will crowd the checkout counter.

Note of caution!

The two period model implies the agents can plan two periods ahead, an handle that determine whether you divide spectrum by the covariation of the sectors or the square root of covariance.  So you have to keep that straight. that is where the checkout count samples a trice the rate of purchases.