## Monday, February 29, 2016

### A Cash-In-Advance Model Comparing Consumers and Suppliers

John Handley presents a simple Cash-In-Advance model which I think can be the genesis for presenting a clear contrast in the inflation effect between consumers and suppliers.

The heart of John's model is

(1)       K*Pt = Pt+1

where K is a proportioning factor relating price at measurement one (Pt) and price at measurement plus one (Pt+1).  K can be composed of any factor or a multitude of factors.

John sets K = Rt where Rt is the current "gross nominal interest rate" (his definition) expressed as one plus the current interest rate (example 1 + .05 = 1.05). With this assignment, John  shows us how a future price can be found based on an increase attributed to interest accumulation.

We can use equation 1 to  compare consumers and suppliers using factors that cause a price difference between the beginning and end of a period (which requires two measurements). Interest on money is a very simple example that readers can use and agree upon.

The Cash-In-Advance model is based on the concept that cash must be in hand before trade can occur. In the simple form, it does not need a definition for money; it is only required that money be a fact of commerce.

In the coming comparison, we assume that both the customer and supplier have all the cash they need to complete a purchase. The purchase price is only part of the available cash held by each party. We are looking for motivation to complete a purchase now and motivation to complete the purchase of the same item in the future.

Everyone is a consumer which makes it easy to tell the consumers story. Allowing that the consumer has cash, he has a choice of buying now or buying in the future. If he delays his purchase until past the second measuring point, he can increase his cash holding by the amount of interest received.

Here is a vital point for this analysis. The supplier has the ability to delay construction of a product. A delay would allow the supplier to increase his cash (which would otherwise be used to build a product) position in the identical fashion as the consumer.

Any interest received from money-in-hand will provide the identical incentive to both customers and suppliers, with both having equal incentive to delay commerce. For both consumer and supplier, we have

(2)     Pt+1 = Rt*Pt              (for both consumers and suppliers)

If interest provides both economic sectors equal incentive to delay commerce, what about inflation from any cause? Inflation (which here is simply a price increase found at the next price measurement) could be caused by multiple factors. We will assume that both interest (Rt) and a general term for inflation (Ft) will be the cause of an increase in price at the second measurement.

Before we write the equation(s), we must notice that inflation may affect the consumer unequally with the supplier. From the point of view of the consumer, inflation will act as a negative interest rate.  On-the-other-hand, the supplier is often the beneficiary of inflation (but not necessarily) so, in general, inflation will act the same as interest. We see that we must write two equations to express the effects of inflation, one for the consumer and one for the supplier. We write

(3)      Pt+1 = (Rt - Ft) * Pt                (for the consumer)

(4)      Pt+1 = (Rt + Ft) * Pt               (for the supplier)

We might be tempted to say "Eureka! The cause for the slow response to the QE programs!".  Slow down. Suppliers have much more than inflation to balance when making product building decisions.

We close with the observation that Cash-In-Advance models are capable of providing insight into the motivation of both customers and suppliers. It is easy to see that motivation for one sector may be disincentive for a second sector.

1. Roger,

You said: "John sets K = Rt where Rt is the current 'gross nominal interest rate' (his definition) expressed as one plus the current interest rate (example 1 + .05 = 1.05). With this assignment, John shows us how a future price can be found based on an increase attributed to interest accumulation."

The euler equation in the basic CIA model really relates the nominal interest rate to the real interest rate. Since, by definition, R_t = (1 + r_t)(P_t+1/P_t) -- where R_t is still the gross nominal interest rate, r_t is the real interest rate, and P_t is the price level, the only real content of the second equation in post is that the real interest rate is constant at r = 1/B - 1 (B corresponds to beta in my post). This happens to be the case in the model because of a few things that are too complicated to explain in a comment, but, suffice it to say that the model does not necessarily imply that inflation is related to interest payments, just that the inflation rate will jump to whatever level is necessary (since prices are perfectly flexible).

2. Thanks for looking at this post and commenting.

Your second equation was indeed my inspiration but I certainly did make a drastic twist to evolve into my analysis.

In my analysis, everything is nominal. Terms Ft and Pt+1 (which are locked in relative value) would either be estimates or measured/calculated depending upon whether the observer is looking ahead or looking back.

My goal was to find a math equation that described motivation . The result was that two equations were needed to describe the motivation of consumers and suppliers when the inflation term was present. The terms in the equations were the same for both cohorts, only the sign for the inflation term was different between cohorts.

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