The relationship between velocity, surplus value and interest rate can be shown by a simple example:

A retailer takes delivery of an inventory of goods. He is able to turn over his inventory V times per year. To pay for the inventory he gets a loan for the delivered price of the goods which is the present value, or pv. The interest on the loan is i. The loan is for term T. T is the time at which he must repay the loan plus interest, which is the future value of the loan fv. T is equal to 1/V.

pv is the discounted future value of fv.

pv = (1-iT)*fv

The percentage the goods must be marked up to finance the loan is (fv-pv)/fv = 1-u where u = pv/fv. 1-u is the **surplus value **and is the creditor’s gain in savings from repayment of the loan.

Now, (fv-pv)/fv = iT and since T = 1/V and (fv-pv)/fv = 1-u the markup for finance cost is

1-u = i/V

From this we can deduce V = i/(1-u) which is an important expression for V, the velocity. **Velocity **is the ratio of the interest rate to the surplus value.

To make the example more concrete, assume the retailer is a clothier, the inventory is a line of seasonal clothes which he expects to sell out of in 90 days, or 1/4 year. T = 1/4 so V = 4. The interest rate i = 6% or 0.06. 1-u = 0.06/4 = 0.015 .

Thus the clothier has to mark up the delivered cost of the clothes 1.5% to cover financing.

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**Application to a macro model**

The **Quantity Theory of Money**, also known as the **Exchange Equation**, is given by the formula

pQ = mV

and is the fundamental statement of the equivalence of money to goods and services. It is a definitional tautology in that it is determinative of the quantity V, velocity. which can be calculated from the measurable quantities p, Q and m.

By taking the logarithms of both sides of the above equation and then differentiating with respect to time, it can be written in equivalent dynamic form

p’/p + Q’/Q = m’/m + V’/V wherein f’ = df/dt for all variables.

The quantity Q is the production rate of goods and services and Q’/Q is the real growth rate of the economy. Q can written as the product aL where **a **is the product rate per worker and **L ** is the size of the labor force.

Thus Q’/Q = a’/a +L’/L

L, in turn, is the product nE where n is the population and E is the employment percentage, so that L’/L = n’/n + E’/E

Q’/Q = a’/a + n’/n + E’/E

The terms a’/a and n’/n are slowly changing variables and can be taken as constants. a’/a is taken as .025 or 2.5%/year and n’/n as 1.23%/year . a’/a is the investment cost of productivity.

Q’/Q = .0373 + E’/E

At equilibrium, E’/E = 0 leaving

Q’/Q = .0373

The real interest rate **r** also includes the depreciation d’/d which is taken here as** **1%

so the equilibrium interest rate is

Q’/Q+d’/d = .0473 which is given by the blue curve in the figure above.

When i = V*(1-u) is determined by market forces, it converges to r, the real interest rate, once V, u, E are at equilibrium values. The figure above shows simulations starting from similar assumed initial values but with differing initial values for the interest rate.

The red simulation converges to higher values of both V and u and can be described as a high velocity, low debt equilibrium.

The black simulation converges to low V and u and can be described as a low velocity, high debt equilibrium.

Central bank interest rate policy, then, is determinative of what equilibrium the economy will arrive at. A low interest rate policy will push the economy to the right into high debt and low velocity while a high interest rate policy will push the economy to the left into higher velocity and lower debt.