May 13, 2012

DIY Econ: Debtors and Inflation

Objective

In this post I want to clearly get over the myth that "inflation is always good for a debtor that has a fixed nominal payment obligation towards his debt as long as his nominal income increases or even stays put". Furthermore, I want to establish exactly when inflation works in favor of debtors and under what circumstances that is likely to materialize. Thereafter, we will switch to the deflation scenario to inspect whether it could possibly be good in some instance, linking to the one of the earliest posts on this blog.

KEY RESULTS

(This blogpost is a detailed, and extended, version of my comments on this topic on winterspeak.com)

There is a big misconception out there that the problems of debt overhang can be solved by creating inflation; that inflation helps debtors by “reducing the real burden of debt” (sic) and hurts only creditors. In this post, I first focus on the actual dynamic of inflation on a debtor in nominal terms. Then I translate it to inflation adjusted (aka “Real”) terms, in the process observing that “real burden of debt” is a lifeless economics expression that is not the same thing as pinch of debt on the debtor. The results are:

Inflation is only helpful to a debtor WHEN the inflation rate (i) and the increase in his nominal income (Ώ) satisfy the relationship: Ώ > i*(1-f0), where f0 is the proportion of his income that is servicing fixed debt payments. Else it is hurtful.

Equivalently, In inflations adjusted (“Real”) terms, Fractional or % Fall in Debtor's real income should be smaller than the fractional or % fall in his “Real Burden of Debt” times his Debt Service ratio before the inflation mounted. Else the debtor will hurt.

Deflation is helpful to the debtor WHEN the deflation rate (ð - expressed as a positive fraction) and the increase in his nominal income (Ώ) satisfy the relationship: Ώ > - ð *(1-f0), where f0 is the proportion of his income that is servicing fixed debt payments. Else it is hurtful. However, if deflation goes beyond a point, the firms might want to cut production or summarily exit the market, so the job itself may be lost.


The derivation of these and more follows.

Derivation

First, an example, then the algebraic derivation of the formula.

Example

A person has an income of $3,000 per month, and pays $1,200 as monthly payment for the debt he has taken. After making the payment, he is left each month with $1,800 to consume.

Going forward, in one year, an inflation of 15% occurs, while his income goes up by 7%.

His “real burden of debt” is supposed to have fallen approximately 15% (= inflation), while his real income has fallen approximately only 8% (inflation – wage increase). This is often taken as the proof that inflation always helps the debtor as long as he got some nominal wage increase. But this is misleading, including the term “real burden of debt” when absorbed as plain English. Let us see if really the debtor is better off:

At the end of the inflation year:

Nominal monthly Income = 3,000 * 107% = $3,210

Debt Service = (still) $1200

He is left to consume = 3210 – 1200 = $ 2010 Nominal to consume

Is he better off?

These nominal $2010 in inflation adjusted (real) terms = 2010/1.15 = $ 1,748

It is LOWER than his real consumption surplus that he had before (of $1,800)!

In order to come out indifferent, this person needed his income to rise by 15% of 1800 = $270.

But his actual wage increase was only 7% of 3000 = 210.

He is short by 270 – 210 = $ 60 nominal or 60/1.15 = $ 52 real.

Formula

Nominal Terms

In order for the debtor to be better off, his:

Real NEW (Income after debt service) > Real OLD (Income after debt service)

=> Real (New Nominal Income – Debt payment) > Real (Old Nominal Income – Debt Payment)

Now let:
Old Nominal Income = w0, Inflation rate = i, nominal wage increase = Ώ
(Debt payment/Original Income) ratio = f0 = Debt Payment/W0

So, we get:

[w0(1+ Ώ) – w0f0]/(1+i) > w0 – w0f0

Or (1+ Ώ) – f0 > (1-f0)(1+i)

Or Ώ > i*(1-f0)

The debtor is better off with the inflation ONLY IF the nominal wage increase he received is greater than the inflation rate deflated by the fraction of his income that he is left with after debt service. He does not gain for just ANY nominal wage increase, and definitely not when his nominal wage does not move. 

{So in the example above, the nominal wage increase needed to be > 15%*(1-1200/3000) = 9%.

But our guy managed to get only a 7% raise, so he loses to inflation = 2% of 3,000 = $60 nominal, even though his nominal income as well as nominal surplus after debt service has gone up!}

Real terms

Let's Go back to:

Real (New Nominal Income – Debt payment) > Real (Old Nominal Income – Debt Payment)

=> Real New Income – New Real Debt Payment > Old Income – Debt Payment

Rearranging:
Old Income – Real New Income < Debt Payment – New Real Debt Payment

=> Fall in Real Income should be less than the fall in “Real Burden of Debt” (i.e. Inflation adjusted Debt payment - NOT the same thing as real pinch of inflation)

The statement that was being evaluated at winterspeak was : “increase real wages relative to the real burden of debt” - Turns out to be true. But let’s work further:

Real New Income/Old Income – New Real Debt Payment/Old Income > 1- Debt Payment/Old Income

=> Real New Income/Old Income – f0/(1+i) > 1 – f0

Rearranging we get:

(Old Income – New Real Income)/Old Income < if0/(1+i)

=> Fractional Fall in Real Income < if0/(1+i) ……. (A)

Let’s compute:

Fractional Fall in Inflation Adjusted Debt Service = (Old Debt Payment – New Real Debt Payment)/Old Debt Payment

=> Fractional Fall in Inflation Adjusted Debt Service (ie.“Real Burden of Debt”)= [F0W0 – f0W0/(1+i)]/f0W0

=> Fractional Fall in “Real Burden of Debt” = i/(1+i) …. (B)

=> From A & B:

Fractional Fall in Real Income < f0* Fractional Fall in “Real Burden of Debt”

Is the condition for leaving the debtor better off. That is, when spoken in this way:

Fractional or % Fall in his real income should be smaller than the fractional or % fall in his “Real Burden of Debt” times his Debt Service ratio.

Why the Myth

There were times when people were able to get out of debt squeeze over a period of time when there was heavy inflation. But that does not mean causing top-down inflation will achieve that affect, as we just saw above. This author’s best conjecture is that when economy achieves traction owing to some real need, like having to go to war or create massive infrastructure or any other real need, and with a good fiscal balance from which to go into it if these are to be public funded (like the situation in early 2000s), then the employment and competition for workers can produce wage increases and inflation such that the relationship between these two satisfies the condition we have established here, i.e. Ώ > i*(1-f0).
However, there is not causation in the direction inflation -> job growth or falling real pinch of debt.

Deflation case

Is there a condition, if achieved, where deflation can be good for the economy?

Let’s go back to our original starting point:

Real NEW (Income after debt service) > Real OLD (Income after debt service)

=> Real (New Nominal Income – Debt payment) > Real (Old Nominal Income – Debt Payment)

=> Real (New Nominal Income – Debt payment) > Old Income – Debt Payment

Now let Old Nominal Income = W0,
Deflation rate = ð,
nominal wage change (let’s neither assume it is positive nor negative) = Ώ
Debt payment/Original Income ratio = f0 = Debt Payment/W0

So, we get:

[w0(1+ Ώ) – w0f0]/(1- ð) > w0 – w0f0

Assuming ð < 100% (of course!)

1+ Ώ – f0 > (1- ð)*(1-f0)

Or Ώ > - ð *(1-f0)

When general deflation = 10%, and the DTI = 30%, we need Ώ > -10%*(1 - 0.3),
i.e. Ώ > -7%

=> If the person can keep his job or business, he can take upto as much % income drop as the general deflation x fraction of current income that he is left with after the fixed nominal debt service.

=> In deflation, even when nominal income stays flat, there is gain for the debtor.
However, if deflation goes to price points where companies are simply not able to operate profitably, even with top executive compensations genuinely tight-linked to profit, or there is no demand uplift projected with lower prices – even the already existing demand can scarcely be preserved, you would get layoffs, with Ώ going to -100% if the guy is just not able to find another job.

Related Posts:

‘Deflationomics’ & It’s Utility in Today’s Economic Conditions
I argue that shrugging off a deflation in home prices with a mild deflation in consumer prices can get a new equilibrium where there is traction in the economic activity for upward movement.

The Price Drop Stimulant
When a Business is at equilibrium (that is, it can fire no more if it desires to hang on to its current sales level, and it is not a lost cause), THEN price cuts can only be delivered by cuts higher up the food chain, NOT by cutting any more workers. Businesses already have made the cuts in number of workers to meet the new normal in consumer demand after the home-equity bubble pop. In fact - Now - to swoop the incremental demand that can be shored through a price-cut strategy, more workers will have to be hired. But businesses will first try to extract doles from Government before trying price cuts and cuts in "performance-linked" Executive payouts.

7 comments:

  1. I printed out your post & just finished reading it (first read). A nice combination of symbol formulas (which I always have trouble with) and word formulas (which are easier) and examples (which make thing clear for me).

    Important, timely topic. (I also have been considering the inflation adjustment of debt lately, from a different angle.)

    Plus, where you summarize your analysis, these are real high spots. //No pun intended.

    Your comments at Winterspeak caught my eye. But the topic merited fuller treatment. It's good to see you thought so too.

    Great post.

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