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I am new to using double-entry bookkeeping for personal finances.

GnuCash uses a variant of the accounting equation that has five variables, and which seems to be known colloquially as an "expanded" or "extended" accounting equation:

assets - liabilities = equity + (income - expenses)

This can be re-arranged as:

equity = assets - liabilities - income + expenses

Rearranging it in this way shows that, all other things being unchanged, an increase in income results in a decrease in equity. Conversely, an increase in expenses results in a increase in equity.

To me, this is totally counter-intuitive. I would have expected (if all else remains unchanged): increased income to increase equity; and increased expenses to decrease equity.

My question is: have I misunderstood the equation or the meaning of the terms; and if so, what is the correct interpretation; or if not, then why is this the correct interpretation?

  • You could also say expenses = equity - assets + liabilities + income. But that says that (if all else remains unchanged) increased income causes increased expenses! But increased expenses also cause increased liabilities! Doesn't that mean that a business in debt will never get out of debt?!? (answer: no, it doesn't mean that) – user253751 Apr 4 '17 at 1:36
  • This fits our topic at the Accounting site (currently in proposal at Area 51). Follow it! area51.stackexchange.com/proposals/113560/… – Jacob Sep 20 '17 at 3:44
  • @Jacob: FYI, it's very risky to place questions on Area51 sites, because they can be wiped out after being active for years. – Dan Dascalescu Sep 6 '18 at 4:04
  • See also this excellent answer which notes the difference between the "Equity" account in GnuCash and a different concept also called "Equity" (or "Overall Equity" or "Net Assets"). – Lily Chung Jul 4 at 19:31
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If your income stream goes up, it would usually increase both your "income" term and your "assets" term since that money sits in your bank account as an asset. (Even more likely a combination of assets and expenses go up if you have cost associated with the increase in income.) In this case, they balance in the equation and your equity doesn't change.

The question as you posed it is true mathematically, but the "paradox" happens because you're not taking into account where the money form the increased income falls in other terms of the equation.

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    Thanks :) In other words, while my point is indeed correct if all else remains unchanged, that is not what happens in practice. In practice, the equity remains unchanged and the assets increase by the same amount as the income. – sampablokuper Jul 3 '16 at 17:03
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The accounting equation, in short, is:

ASSETS = LIABILITIES + EQUITY

This can be further broken down into:

ASSETS = LIABILITIES + PAID-IN CAPITAL + RETAINED EARNINGS
(Equity is made up of shareholders' equity and retained earnings)

Which can be further broken down into:

ASSETS = LIABILITIES + PAID-IN CAPITAL + INCOME - EXPENSES
(Retained earnings is increased by income and decreased by expenses)

The GnuCash equation is right, though I would substitute the word equity in that equation with a more-specific paid-in capital.

Equity is (simply put) made up of 2 parts: shareholders' equity and retained earnings. Shareholders' equity is the amount invested by shareholders. Retained earnings is the amount earned by the business on behalf of the shareholders. Retained earnings is directly affected by your net income (which is income minus expenses).

An increase in income will result in an increase in retained earnings. This must be balanced somewhere. Usually an increase in an asset. It may also be balanced by a decrease in equity. Likewise, increase in expenses will result in a decrease in retained earnings, which must also be balanced.

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Equity is the term to make things balance. In a simple transaction, you get $100 paid to you. Income goes up by $100 and the asset of whatever bank account or petty cash drawer you put it into also goes up by $100. Equity is unchanged.

If for some reason you had to take some income into your books, but no asset increased, no debt decreased, and you had no way to take an offsetting expense into your books, then this would lower your equity. How else to explain having "earned" $100 but having nothing to show for it?

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    I notice a couple of people have downvoted this answer without explanation. That isn't very constructive. If you are one of them, or are thinking of downvoting it, please could you comment to explain why? Thanks :) – sampablokuper Jul 19 '16 at 19:27

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