# Proof of the Credit Accelerator (but not quite as we know it)

Take our formula for lending power as developed through previous posts

LP    Lending Power

EQ    Equity

R    Principal repayments

I    Interest

D    Dividends

R     reserve ratio

E    Excess Reserves

M    Money leant

C     Collateral

F    Short term funding costs

Now let us gather on one side of the equation all of those terms that equate to prior savings

Where S=Prior Saving

Now lets gather the net new investment terms

1.

Note we are dealing here with the maximum amount of the change in debt, only profitable loans will be made by a bank.

So the equation becomes:

Change in Lending = Change in Savings +Change in Investment

But because the investment creates new savings it becomes

1. D’=S+S’

This makes sense it means that a change in debt can come about through Crusoe like savings or through bank created money or some combination of the two.

However a change in debt is not a change in money as the Crusoe like savings is existing money in deposits. The change in money would be as follows:

1. M’= S’

Familiar keynsian grounds.

Now what about the approach to effective demand. This is a flow variable for transactions, and so money in balances not used for transactions will not count. It would be double counting to include factor payments where these are already accounted for in flow of funds. And so

Change in demand =GDP +change in money used in transactions.

The change in money used in transactions is given by (5) above not (4) so we have very nearly but not quite the Steve Keen Walras/Schumpeter/Minksy law for aggregate demand – the Credit Impuse – or as he refers to it the Credit accelerator. The difference being the elimination of change in debt due to savings. The approach though is correct as the formula includes only the net addition of bank created money, which was Schumpeter’s intention to show the bridging of the demand gap between output now and new expanded output following debt created growth later.

Note this formulation is in instantaneous time and shows the effect of the additional demand at the point the new money is deposited in an account and about to be spent. An instant later we see the impact of changes in output including the growth induced by the change in debt. At that later point when the money is transferred to an asset (likely in another bank) it is recorded as GDP and not change in debt.

Of course this assumes the money is all spent and none remains in the original bank account. If part remains unspent we have to include the C.A. Phillips K factor.

However unspent loans add to excess deposits and so increase lending power pro-rata and so with two provisos effect aggregate demand pro-rata. Those conditions being that banks lend to their lending power, which they wont throughout the credit cycle, secondly banks are already at their desired reserve ratio.

Having looked at the savings part of the equation lets look at the debts part (equation 3).

What it says is Change in debt equals amortization of loan minus cost of loan.

Now you might think they cancel so so what. They only cancel when capitalised at net present value. They will not cancel in instantaneous time.

What we have here is what used to be called Hawtry’s Wheel of Production, Hawtry in his own verbal explanation of the Credit Accelerator said the following:

at the beginning [of a period of credit induced production there] will be an excess of purchasing power and no goods to buy, and at the end an excess of goods and a shortage of purchasing power… But the economic activity of a civilized community is continuous… the real significance of the power of the banks to create or extinguish money is that it enables them to bring about the release or absorption of cash. If the net result of all the different causes at work is an absorption of cash, then there is deficiency of purchasing power; if the net result is a release of cash, then there is an excess of purchasing power. (Birmingham Debates 1933)

Hawtry however did not pursue his own logic. At the beginning of a period of credit induced production there will be a net increase in purchasing power from workers to be spent on the existing stock of goods as the new goods have not yet been produced. Banks have net released cash but have not yet started to receive it in terms of principal and interest payments. As production gears up if this makes a profit there will be a net paying down of loans as they are amortized, this will be a net reduction in purchasing power in terms of the purchasing power generated by the bank as the loan is paid down. However as the cost of the loan is paid down unless there is a high rate of depreciation there will by now be a net profit from the capital accumulation leading to greater deposits in reserves from capitalists bank accounts and potential from higher workers wages’ Also from the bankers point of view once the inflation adjusted IRR point of return is reached all payments to it of premium and interest are pure profit. The greater competitiveness of the loan induced production process enables more to be produced per unit of investment and so more to be consumed per unit of income, even if that income shifts from no longer competitive old processes to more competitive new ones.

So change of debt matters, from a producers view point, as although the increase in debt and purchasing power at the beginning of a loan is exactly cancelled out by a decrease at the end (assuming steady inflation and interest rates) over the period of the loan assuming productive investment not speculation on assets, purchasing power has increased by producing more of what is demanded per unit of money.

The next stage of this analysis will be to graph certain aspects of this equation and compare it to IS/LM, before we do so however we need to expound a compatible therefore of portfolio demands for money. I also hope to compare this approach to that of Kalecki, especially once we consider the returns on the four factors of production.