Tuesday, 7 February 2012

An alternative sectoral balance equation.

This is a popular equation with Modern Monetary Theory sectoral balance enthusiasts:

(I – S) + (G – T) + (X – M) = 0

I=investment, S=savings, G=government spending, T=tax, X=exports, and M=imports. The equation is cited for example here, here, here and here.

And there is a picture of Warren Mosler displaying the equation here!

However, the equation and the reasoning leading to it are flawed. I’ll argue below that investment (I) should be omitted from the equation.

Here is Bill Mitchell’s reasoning (in grey italics), leading to the equation.

From the sources perspective we write:

GDP = C + I + G + (X – M)

which says that total national income (GDP) is the sum of total final consumption spending (C), total private investment (I), total government spending (G) and net exports (X – M).

However, there is a problem here: in the real world there is no clear distinction between consumption items (C) and investment items (I). To illustrate, is something designed to last three months an investment? How about one year . . . three years? The distinction between the two is arbitrary.

Bill continues:

From the uses perspective, national income (GDP) can be used for:

GDP = C + S + T

which says that GDP (income) ultimately comes back to households who consume (C), save (S) or pay taxes (T) with it once all the distributions are made

Hang on: why doesn’t investment (I) appear on the right hand side? If we define anything designed to last more than say five years as an investment (e.g. cars), then rather a large proportion of what the average household “consumes” has been omitted (cars in particular).

Bill continues:

Equating these two perspectives we get:

C + S + T = GDP = C + I + G + (X – M)

So after simplification (but obeying the equation) we get the sectoral balances view of the national accounts.

(I – S) + (G – T) + (X – M) = 0

That is the three balances have to sum to zero. The sectoral balances derived are:

• The private domestic balance (I – S) – positive if in deficit, negative if in surplus.
• The Budget Deficit (G – T) – negative if in surplus, positive if in deficit.
• The Current Account balance (X – M) – positive if in surplus, negative if in deficit.

I suggest the above should read:

Equating these two perspectives we get:

C + I + S + T = GDP = C + I + G + (X – M)

So after simplification (but obeying the equation) we get the sectoral balances view of the national accounts.

S + (G – T) + (X – M) = 0

Perhaps the above mistake occurred because some microeconomics was applied at the macroeconomic level. That is, if a household or firm makes an investment, it normally runs down its savings. However, that idea does not apply at the macroeconomic level. That is, all else equal (external balance and budget deficit in particular), one household or firm running down its savings must cause another household or firm accumulating savings.

P.S. 23rd Feb. More discussion of the above points here, and here.



  1. S + (G – T) + (X – M) = 0 is wrong
    It is not macroeconomic identity anymore. If foreign country invests in domestic country, let's say buys domestic investments already made to make It simple, then domestic private sector savings go down ceteris paribus. Yet your formula doesn't show that.

  2. (I – S) + (G – T) + (X – M) = 0
    let's say (G-T)=0 and (X-M)=0
    I is higher than S because of foreign investment in domestic economy.

    S + (G – T) + (X – M)=0 doesn't show that

  3. Sorry about posting my comments so many times, but to give you a crystal clear example:

    Your thinking is that X-M already reflects financial flows with the rest of the world. For the sake of argument let's assume that this country has very bad terms of trade with the rest of the world. Let's say this country doesn't have any oil ant It is paying $1000 a barrel. X-M shows financial flows. Now X-M=0 and G-T=0. By your accounting identity S + (G – T) + (X – M)=0 domestic private sector savings cannot run down. Yet Arab oil sheiks are taking over all domestic investments and domestic private sector savings are going down.

  4. Kristjan,

    Re your first comment, if any foreign entity invests in country Z, that means currency flows into country Z. If foreigners buy for example government bonds of country Z, then the deficit of that government declines (G-T) – or the government surplus expands. Or if they buy for example a house, then the private sector balance of country Z improves (S).

    Re your third comment you say “Your thinking is that X-M already reflects financial flows with the rest of the world.” I don’t see where the word “already” fits in. X-M is simply the definition of a country’s external position (exports less imports). That’s how Bill Mitchell defines it, and I’m happy with that definition.

    Next, you propose a country with “very bad terms of trade” and then say that the external position of this country is in balance: (X-M)=0. If that’s the case, then Arab Sheiks cannot be investing billions in the relevant country.

  5. I did not present my idea right Ralph.

    X-M is trade balance or financial flows balance associated with trade. X-M is not current account balance.

    Brazil in 1986.

    NX = + $ 8.3b
    CA = - $ 5.3b
    i NFA = -$ 13.6b

    In this example, Brazil had in 1986 a large current account deficit in spite of a trade surplus. In fact, Brazil was a heavy foreign debtor, having borrowed a lot in the 1970s and 1980s. By 1986 the total foreign debt of Brazil was above $100b and the net foreign interest payments on that debt (and profit repatriations of foreign firms owning assets in Brazil) equaled $13.6b.

    You agree that if in country Z private sector is making large interest payments to foreigners then in case of G-T=0 and X-M=0 private sector savings decrease. In case of I>S CA is in deficit.

  6. Kristjan,

    I can’t fully answer your point because I don’t know what you mean by NX (net exports?) or CA.

    However, I think you’ve spotted a flaw in both my equation and the standard equation I started with above (the one that included investment). This is that both equations mix up transfers of money (and other financial obligations) with transfers of goods and services.

    It strikes me as an obvious truism that movements of money etc between the tree sectors, private, public and external must sum to zero. But the problem with the two equations is that its money that is referred to in the case of the private and public sectors, but in the case of the external sector (X-M) - exports less imports – this refers to the movement of goods and services (on any normal definition of the words export and import).

    I.e. in the case of my equation, if the equation is going to refer just to money etc, then the word “export” needs to be replaced with something like “payments by foreign entities to domestic entities” (and the reverse for imports).

    Alternatively, the equation could refer to just goods and services, and the equation would also hold. But it needs to be made clear what the equation refers to: money or goods and services.

    Does that solve the problem?

  7. Thanks Ralph.
    let's see what Randall Wray is saying about It: "We can divide the economy into 3 sectors. Let’s keep this as simple as possible: there is a private sector that includes both households and firms. There is a government sector that includes both the federal government as well as all levels of state and local governments. And there is a foreign sector that includes imports and exports; (in the simplest model, we can summarize that as net exports—the difference between imports and exports—although to be entirely accurate, we use the current account balance as the measure of the impact of the foreign sector on the balance of income and spending)."

    So he is distiguishing the two (trade balance and current account balance).
    What do I mean by trade balance and CAB? (I think the MMTers are pretty much mainstreamers when it comes to accounting identities)

    This is Nouriel Roubini's explanation
    "To understand better why a country may be running a current account deficit or surplus, one should notice that the current account is the difference between what a country produces (GNP) and what the country spends (total consumption plus investment). In fact:

    CA = GNP - (C + G + I)
    Substituting this definition of savings in the expression for the current account, we get:

    CA = S - I"

    They are talking about GNP instead of GDP.

  8. I just don't understand why Kristjan bothers discussing your mess so much.

    First, this matter:

    GDP = C + I + S + T.

    Hell, yeah! That's just the cutest thing I've seen in ages (the previous prize winner was the Ukrainian "hryvnia-song").
    Let's try understanding the right equation in a more or less intuitive way.

    GDP = C + S + T.
    GDP equals (a lot of things, but especially) total income: everything spent is someone's income.
    Next we move to the components of GDP.
    T is share of income that belongs to government. Crossed. The further discussion is about the income after taxation only.

    There are two components of person's income. One is Consumption, the income spent on short-term goods. The other is Savings, the income NOT spent on short-term goods.
    Everything seems right, doesn't it?

    Then you add the Investments. Now the private income after taxation consists of three pieces of income: one spent on short-term goods, one not spent on short-term goods and one spent on capital goods. Does it still seem right to you?
    It seems quite a mess to me. Using your equation makes you count the capital good spending (I) twice.

    A Dark Age of macroeconomics, yeah?


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