# Velocity of Money

The most basic difference between the demand theory of money and exchange theory of money lies in the understanding of quantity equation

M . v = P . Y —– (1)

Here M is money supply, P is price and Y is real output; in addition, v is constant velocity of money. The demand theory understands that (1) reflects the needs of the economic individual for money, not only the meaning of exchange. Under the assumption of liquidity preference, the demand theory introduces nominal interest rate into demand function of money, thus exhibiting more economic pictures than traditional quantity theory does. Let us, however concentrate on the economic movement through linearization of exchange theory emphasizing exchange medium function of money.

Let us assume that the central bank provides a very small supply M of money, which implies that the value PY of products manufactured by the producer will be unable to be realized only through one transaction. The producer has to suspend the transaction until the purchasers possess money at hand again, which will elevate the transaction costs and even result in the bankruptcy of the producer. Then, will the producer do nothing and wait for the bankruptcy?

In reality, producers would rather adjust sales value through raising or lowering the price or amount of product to attempt the realization of a maximal sales value M than reserve the stock of products to subject the sale to the limit of velocity of money. In other words, producer would adjust price or real output to control the velocity of money, since the velocity of money can influence the realization of the product value.

Every time money changes hands, a transaction is completed; thus numerous turnovers of money for an individual during a given period of time constitute a macroeconomic exchange ∑ipiYi if the prices pi can be replaced by an average price P, then we can rewrite the value of exchange as ∑ipiYi = P . Y. In a real economy, the producer will manage to make P . Y close the money supply M as much as possible through adjusting the real output or its price.

For example, when a retailer comes to a strange community to sell her commodities, she always prefers to make a price through trial and error. If she finds that higher price can still promote the sales amount, then she will choose to continue raising the price until the sales amount less changes; on the other hand, if she confirms that lower price can create the more sales amount, then she will decrease the price of the commodity. Her strategy of pricing depends on price elasticity of demand for the commodity. However, the maximal value of the sales amount is determined by how much money the community can supply, thus the pricing of the retailer will make her sales close this maximal sale value, namely money for consumption of the community. This explains why the same commodity can always be sold at a higher price in the rich area.

Equation (1) is not an identical equation but an equilibrium state of exchange process in an economic system. Evidently, the difference M –  P . Y  between the supply of money and present sales value provides a vacancy for elevating sales value, in other words, the supply of money acts as the role of a carrying capacity for sales value. We assume that the vacancy is in direct proportion to velocity of increase of the sales value, and then derive a dynamical quantity equation

M(t) - P(t) . Y(t)  =  k . d[P(t) . Y(t)]/d(t) —– (2)

Here k is a positive constant and expresses a characteristic time with which the vacancy is filled. This is a speculated basic dynamical quantity equation of exchange by money. In reality, the money supply M(t) can usually be given; (2) is actually an evolution equation of sales value P(t)Y(t) , which can uniquely determine an evolving path of the price.

The role of money in (2) can be seen that money is only a medium of commodity exchange, just like the chopsticks for eating and the soap for washing. All needs for money are or will be order to carry out the commodity exchange. The behavior of holding money of the economic individuals implies a potential exchange in the future, whether for speculation or for the preservation of wealth, but it cannot directly determine the present price because every realistic price always comes from the commodity exchange, and no exchange and no price. In other words, what we are concerned with is not the reason of money generation, but form of money generation, namely we are concerned about money generation as a function of time rather than it as a function of income or interest rate. The potential needs for money which you can use various reasons to explain cannot contribute to price as long as the money does not participate in the exchange, thus the money supply not used to exchange will not occur in (2).

On the other hand, the change in money supply would result in a temporary vacancy of sales value, although sales value will also be achieved through exchanging with the new money supply at the next moment, since the price or sales volume may change. For example, a group of residents spend M(t) to buy houses of P(t)Y(t) through the loan at time t, evidently M(t) = P(t)Y(t). At time t+1, another group of residents spend M(t+1) to buy houses of P(t+1)Y(t+1) through the loan, and M(t+1) = P(t+1)Y(t+1). Thus, we can consider M(t+1) – M(t) as increase in money supply, and this increase can cause a temporary vacancy of sales value M(t+1) – P(t)Y(t). It is this vacancy that encourages sellers to try to maximize sales through adjusting the price by trial and error and also real estate developers to increase or decrease their housing production. Ultimately, new prices and production are produced and the exchange is completed at the level of M(t+1) = P(t+1)Y(t+1). In reality, the gap between M(t+1) and M(t) is often much smaller than the vacancy M(t+1) – P(t)Y(t), therefore we can approximately consider M(t+1) as M(t) if the money supply function M(t) is continuous and smooth.

However, it is necessary to emphasize that (2) is not a generation equation of demand function P(Y), which means (2) is a unique equation of determination of price (path), since, from the perspective of monetary exchange theory, the evolution of price depends only on money supply and production and arises from commodity exchange rather than relationship between supply and demand of products in the traditional economics where the meaning of the exchange is not obvious. In addition, velocity of money is not contained in this dynamical quantity equation, but its significance PY/M will be endogenously exhibited by the system.