Heavy Industry

Continuing the discussion of debt growth, let’s look at leverage. Here we define leverage at time t to be the debt growth from the starting point to time t, divided by the integral of cash flows accumulated from the starting point to time t. This does not necessarily correspond to bank leverage, as government debt or other credit market instruments may be used to fund profits and support interest payments.

With the same notation, at time t, C(t) corresponds to aggregate cash flow profits at time t.

Assume for the moment that cash hoarding is zero — that all cash is used to acquire financial assets. This means that total profits, or the integral of the cash flow, is provided to banks for lending.

In this case, let

x(t) = ratio of accumulated debt to accumulated cash flows.
z(t) = logarithmic derivative of profits (= cash flow/profits)
r(t) = base-e annual interest rate of debt outstanding

Then our debt flow equation can be re-written to obtain:

In the case of hoarding, suppose only a fraction, k, of cash flows is made available to fund debt growth, and the remainder is hoarded. We assume that k is sufficently constant for the time period so that it drops out of the logarithmic derivates.

In this case, we have:

Now, note that if x is less than 1, positive interest rates and positive profit growth rates will drive the ratio higher. Barring a catastrophic collapse in r(t), x will naturally be pushed to 1.

At x = 1, we have

So x will continue to increase.

When x is greater than 1, then positive profit growth rates begin to slow the growth of leverage, and negative profit growth rates begin to increase the growth of leverage.

Cash flows tend to be very volatile whereas r(t) is stable, barring a sudden massive debt default. The reason for the stability is that r(t) does not measure the offered rate for new debt, but the overall realized rate on the entire debt stock. Therefore, r(t) moves on a time scale in proportion to the duration of debt outstanding and the growth of new debt relative to the existing stock.

z(t) will quickly oscillate with time, but assume that it oscillates around some average value. If the interest rate is set above this average value, then x growth will be unbounded. If the interest rate is set below this average value, then x will oscillate with r(t), however the peak points of oscillation can still be extremely high, dependent on the rate at which profits are collapsing in proportion to the prevailing interest rate.

The Need for Leverage

Ignoring hoarding, which mandates leverage in and of itself, assume that x is 1. In this case, we have

Now if, for some reason, there was an upper bound

then, realized nominal interest rates would need to be 0 to maintain x = 1, or would need to be negative, i.e. as a result of mass defaults. Positive interest rates would only resume after a substantial period of negative profits had driven x sufficiently below 1 to support positive realized interest rates. Note that throughout, we consider only nominal rates.

In other words, an environment in which leverage is bounded at 1 drives the value of financial assets to zero, and chokes of lending, and consequently profits. It is incompatible with money accumulation, and mandates only an exchange economy.

Escape from the Malthusian Trap

For most of the world’s history, incomes per person were stagnant. Technological innovation was stagnant. Beginning with the 19th century, and accelerating thereafter, incomes and productivity began to grow. There are many issues at play in this dynamic, but my contention is that the key driver is the transition to money accumulation, as allowed by the growth of the financial markets and their ability to supply leverage. Without leverage, innovation and profits are punished because there is insufficient purchasing power to support profits — there is a ceiling on the value of financial assets — that is, investment — and this value shrinks to zero as the ceiling is reached. Leverage breaks through this ceiling, and allows people to accumulate not cash flows, which are restricted, but debt, which can grow much larger than cash flows. This supports profits and rewards technological innovation and efficiencies. In fact, it requires these efficiencies as the output growth rate must exceed the interest rate. Therefore leverage both rewards and requires improvements in productivity and technology, and the drive for profits.

However, the price paid for this growth is that the accumulated profits (i.e. financial assets) may become illusory if the interest rate is too high relative to the underlying cash flows. But the interest rate r(t) is much more stable than the profit growth rate, z(t). This is because r is computed over the entire debt stock and debts have long duration. Cash flows, on the other hand, swing wildly from year to year.

Therefore, there are inevitable wide divergences between r(t) and z(t). The leverage evolution equation has a growth rate scaled by the difference between r(t) and z(t). This tends to destabilize x, causing it to explode as profits collapse.

When that happens — i.e. leverage is too high — the financial markets will devalue the existing debt stock, ratchet up the interest rate, and force a de-leveraging.

This is just a restatement that the risk of borrowing short and lending long, is that you are dependent on a constant stream of stable cash flows, which the economy is not able to provide. Therefore standard business cycle variations, if they occur in an environment of high leverage, can lead to a debt collapse, and a sharp downward adjustment of r(t), which is necessary to reduce x to sustainable levels. Such a de-leveraging must entail a loss of confidence, since the surplus nodes accept as payment financial claims which are revealed to be unsupportable by existing cash flows (i.e. by additional excess lending). This strikes at the heart of the accumulation economy. More on this later.

The goal of this post is to try to understand the role of money and debt in the economy. This is a complex story, but we will begin with some broad strokes.

Often discussions of money describe two “stages” in economic development. In the first stage, barter predominates. By the second stage, money is used as the medium of exchange. In between, there are various forms of specie competing for the role of money. However, present economic life is dominated by a third stage, namely money as the object of accumulation. To see the distinction, consider a simple example of an exchange economy.

The Exchange Economy

The farmer that brings his crops to market. The crops are exchanged for money. The farmer walks a few paces to the next stand and uses the proceeds to purchase consumption goods (a new copy of Adam Smith’s The Wealth of Nations) and capital goods (a new plow, or a deed to additional land). The farmer leaves the market carrying the new goods, but with no money profits. Profits are in the form of tangible goods, and wealth accumulation is primarily expressed as the accumulation of land.

Now, this of course is an ideal portrait. Before the 19th century, money accumulation did occur, primarily to fund military projects and a (proportionally small) trade in manufactures and luxury goods, but relative to the size of the economy, a negligible amount of profits were in the form of financial assets, as opposed to tangible assets. The wealthy of the world were so because of their command of land, and the ability to work the land.

In such an economy, the role of money can be ignored (what economists still trapped in the world of Adam Smith and Ricardo disparagingly term “nominal effects”). Money is simply a technical tool to facilitate transactions, and does not significantly alter the workings of the economy.

The Accumulation Economy

However, with the advent of the industrial era, and necessarily of banking, money itself grew in importance, becoming the primary object of accumulation. The presence of money accumulation introduces new instabilities in the system. In this model, transactions are motivated by profits. Businesses are required to spend less on wages and input goods than they receive from sales. Households are required to spend less on goods purchased from businesses than they receive from wages. The motivation is not to exchange goods, but to accumulate financial profits.

Imagine a system of nodes representing economic agents that are capable of incurring debt and earning a profit. The nodes represent households, businesses, and governments.

During a sample time, the nodes exchange cash with each other, spending and receiving money. The exchanges are done to facilitate the the market for goods, which is to say, the buying and selling goods and services, including labor.

At the end of the sample time, some nodes have accumulated a profit, and others a deficit. The key point is that we cannot all be cashflow positive. For one node to achieve a profit, there must be a corresponding deficit elsewhere in the network. The deficit spending, i.e. spending in excess of receipts, is achieved through borrowing. In practice, the deficit spending occurs through debts for purchase durable goods such as houses, cars, tuition, or government deficit spending. But, for the system to be maintained, ever greater amounts must be borrowed by succeeding generations in order to allow for the majority of nodes to be cash flow positive even as they are servicing the debt.

It is a requirement, that for Coca-Cola to achieve a profit — any profit — the public must be engaged in borrowing to buy a house or a car. Without deficit spending primarily by households and governments, there would be no possibility for businesses to be cash flow positive. There would be no opportunity for “thrifty” households to accumulate financial assets. Without an expansion in borrowing, all profits would be driven to zero, and the value of financial assets would be driven to zero as well. In such an environment, as soon as a single household attempted to spend less than they earn, that household would have its wages reduced, or costs increased, since there would be no balancing node willing to take on additional debt to fund the thrifty household’s profit.

So, in order for participants in the goods market to realize profits, those nodes which realize the profit must turn around and lend the proceeds to the deficit nodes, to fund the profit. So, there is a parallel market, the financial market, representing the buying and selling of claims on future cash flows from the goods market. Accumulated cash flow surpluses are channeled into the financial market, where the money is supplied to deficit nodes in exchange for interest payments.

The Flow Equation

Returning to our network of nodes, during the sample time, from a purely cash-flow analysis, we have the following equation:

D = total indebtedness
C = Aggregate positive cash flow in the goods market during the sample time.
I = interest payments on the existing debt stock made during the sample time.

Note that the sum of positive cash flows is defined as adding up all the surplus cash flows for those nodes which have positive cash flow — this amount is of course the same as the cash flow deficit for those nodes which are in the red. The integral of C(t) is a good approximation for total profits during a given period (it differs only when nodes switch between being cash flow positive and negative, so in theory the integral of C can be greater than profits during the same period. We will ignore this distinction.)

Also note that the interest rate, r(t), is not the offered rate, but the realized rate for the entire debt stock. Because of this, r(t) adjusts much more slowly than the market rate. It is the rigidity of r(t), together with the inability of prediction markets (i.e. the financial markets) to accurately predict cash flows which forces nominal effects to dominate debt deflation and inflation. Rather than “menu costs”, interest rate mismatch drives nominal effects.

At the next tick of the clock, in order to continue to produce output, additional profits need to be obtained, but now interest must be paid on the current debt stock.

So we have the debt flow equation, representing contributions from both markets. Assuming the unit of time is in years, we have:

where
C(t) = aggregate positive cash flow at time (t)
r(t) = (base-e) annual interest rate = logarithmic derivative of aggregate interest payments.
D(t) = total debt.

Evolution of the Debt to Output ratio

We are interested in ratios, specifically the debt to output ratio, in order to determine the sustainability of borrowing, and the potential mechanisms of reducing the debt to output ratio.

Setting:

a = ratio of debt to output
g = logarithmic derivative of output
r = continuous (base e) interest rate on existing debt stock
c = ratio of aggregate cash flow to economic output

we obtain the debt to output equation:

First Impressions

In order to obtain a bounded debt to output ratio without credit collapse, investors would need to anticipate cash flows and economic growth, and demand an appropriately matching interest rate. However, the mechanism for setting interest rates is determined by perceptions of risk and expected future cash flows. Specifically, low interest rates correspond to a low perception of risk and require high cash flows, whereas high interest rates correspond to an increased perception of risk and lower economic growth. Therefore individual profit maximization serves to drive up the ratio — no “irrationality” assumptions are necessary.

Second, the key to reducing the debt to output ratio is not to encourage households to “save” more, which would only increase the ratio, but rather to ensure that the difference between the realized interest rate and the output growth rate be negative. Fortunately, when the ratio is high, the term p(t) is negligible, and so rejoinders to be thrifty are ignorant but mostly harmless. The key is to either re-finance or default on a sufficient amount of debt to reduce the realized interest rate.

Third, it is clear from the equation that it is extremely difficult for the debt to output ratio to decrease when output is falling. Only when growth rates increase is there a real possibility of bringing the debt to output ratio down. Note that in the depression, the ratio almost doubled in the time period from 1930-33.

Fourth, government actions to purchase debt with printed money are an effective mechanism to “re-finance” debt, and are most effective when the purchased debt is of long enough duration that the interest rate remains low when growth resumes, and when the interest rate on the purchased assets is high relative to the interest rate of government debt.

Fifth, we see the impossibility of 100% reserve banking, because non-zero interest payments preclude the full funding of deficit nodes by surplus nodes. Banks (or other investement vehicles) must lend more to the deficit nodes than the surplus nodes can fund simply by accumulating positive cash flows in the goods economy. The financial market must expand above and beyond the profit needs of the goods markets.