The Closed Economy National Income Identity vs. Aggregate Demand

Assume a closed economy, no government spending and no taxes, and no depreciation. National income accounting states unambiguously:

C + I ≡ Y ≡ C + S

Where C, I, S are ex post. This is an identity. Sometimes it is simplified by putting “=” in place of “≡”, but this simplification often leads to confusion, because of this statement,

C + I = AD = Y

Which is a definition of aggregate demand as the sum of ex ante or “planned” consumption and planned investment. In equilibrium, AD = Y, that is production ramps up or down to match aggregate demand. That is why such a model is sometimes called “demand determined”.

When planned consumption plus planned investment equal ex post consumption and ex post investment, then AD = Y and planned saving equals planned investment.

When is there a “multiplier”? When planned consumption takes the linear form C = a + bY, investment is exogenous I*, then substituting in:

C + I = a + bY + I* = AD = Y

a + I* = Y – bY


a + I* = (1-b)Y


Y0 = (1/(1-b))(a + I*)

That is the level of equilibrium income is a multiple (since 1/(1-b) > 1 for 0 < b < 1) of the level spending that takes place irrespective of the level of income. a and I* qualify, and that is why they are called autonomous spending.

To find the multiplier, take the total differential of last equation:

ΔY = (1/(1-b))(Δa + ΔI*)

So the multiplier for a change in autonomous investment is found by holding the change in autonomous consumption at zero. Then:

ΔY = (1/(1-b))(ΔI*)


ΔY/ΔI* = (1/(1-b))

What are the key assumptions (in my mind)?
1. Consumption behavior is such that the marginal propensity to consume is less than unity, b < 1.
2. Output is demand is determined.

Bottom Lines:
1. Don’t reason from identities.
2. Understand what are ex ante and ex post quantities before trying to reason out causality.

36 thoughts on “The Closed Economy National Income Identity vs. Aggregate Demand

  1. pgl

    I used to call this model Ronald Reagan’s dream world. No government – no trade with the rest of the world. OK, it is the most basic model but at least you laid it out clearly. And of course you do not pretend to model fiscal policy in a model with no government.

    I might point to others who try and get all confused but we are trying to be nice here!

  2. Paul Mathis

    Speaking of Identities

    When is the federal debt not debt? A new bridge or airport is a capital asset and when financed with debt is zeroed out on a balance sheet. That is how state and local governments “balance” their budgets. But the federal budget is “unified” so all expenditures are financed with taxes or debt.

    Likewise, when the federal government “prints” dollars that does not represent debt, but when it prints a bond, that is debt even though the only difference is that the bond pays interest in dollars which are printed.

    Obviously, the ongoing debate about the federal debt is simply claptrap.

    1. pgl

      A good point. Private entities distinguish between current budgeting v. capital budgeting and so should government entities. Of course in the massive amount of data provided by the BEA – one can find what is government investment v. government consumption if one really wants to.

    2. 2slugbaits

      Here’s another weird one. Social Security Trust Bonds that represent excess FICA taxes beyond FICA expenditures are counted as part of the public debt, but in macroeconomics those same bonds count as national saving. So it’s possible for the national debt to rise as a result of increased national saving. It’s all perfectly reasonable, but it can be a bit counterintuitive for some folks.

  3. Moses Herzog

    I actually had thought that Menzie had certainly built up a large enough callus over the years that he would be the last person to make this kind of change or “rule”. In fact these kind of rules make the situation more dangerous. You really want to follow in the footsteps of these types?? I’m not arguing you host the blog and you have that “right”. I’m NOT arguing this is a First Amendment issue, as it’s clearly not. But, do you really wanna follow in the footsteps of Tim Cook?? He’s real good at filtering people:

    Zuckerberg (if you believe Russian influence actually determined the result of the 2016 elections) Zuckerberg should be your pubic enemy #1. And you want to follow in his steps?? Zuckerberg is just like Jimmy Swaggart—he’s really really incredibly and demonstrably “sorry” and “wants forgiveness” after he has pocketed and deposited the money in the bank he got from Russian propaganda. Zuckerberg is “oh so sorry”.

    So stopping a few “Joe Six-Packs” from openly discussing race and “hammering out” some things is going to help American society?? According to Larry “Sergeant Schultz” Kudlow he “never discussed” the “dirty topic” of race with Peter Brimelow. How’s that working out for him??

    1. Menzie Chinn Post author

      Moses Herzog: James Hamilton and I consulted on the proliferation of pure name-calling posts and posts purely calling out ethnic/racial origins. I believe that such a limited intervention will have no impact on the discussion of substantive economic matters.

      Please re-read the comment policy. It does not prohibit discussion of race. It prohibits discussion of a commenter’s racial or ethnic characteristics as a basis for analysis.

  4. 2slugbaits

    There were several times in his analysis in which PeakTrader went off the rails; e.g., incorrectly using an identity, talking about tax cuts without including a government sector, not recognizing the concept of national saving versus private saving, etc. But things really got bad when he started commenting on the balanced budget multiplier (BBM). He actually got the sign wrong! In a nutshell, the BBM says that increasing government expenditures by $100B and increasing taxes by $100B has no net effect on the deficit, but increases GDP by a multiplier of 1.00. But notice that in his discussion PeakTrader got the signs reversed, so he ended up claiming that a $100B decrease in spending coupled with a $100B decrease in taxes also increases GDP. He forgot that the BBM works in reverse, so decreasing government spending and decreasing taxes results in a negative multiplier of -1.00. To see this, let’s take an ultra-simple example:

    Let “k” = the multiplier

    “c” = the marginal propensity to consume such that 0 < c < 1, which we will set = 0.8

    "G" = government expenditures = $100B

    "T" = lump sum tax = $100B

    "Y" = GDP

    "d" refers to the change, or delta

    The expenditure multiplier = dY = dG / (1-c) = dY / dG = 1 / (1-c) = k, so

    k = $100B * (1 / (1 – 0.8)) = $100B * 5 = $500B

    The multiplier for a flat tax reduction of $100 = dY / dT = -c / (1 – c), so

    k = $100B * (-0.8 / (1 – 0.8)) = $100B * (0.8 / 0.2) = $100B * 4 = $400

    But if you raise taxes by $100B, which you need to do in order to make it a BBM, then you have to multiply -$100B rather than +$100B. So you end up with a net stimulus to aggregate demand of $500B – $400B = $100B, which tells you that the BBM is exactly 1.00. You changed spending and taxes by $100B and ended up stimulating demand by $100B.

    Clear enough? The problem is that PeakTrader decreased government spending (i.e., -$500B rather than +$500B) and decreased taxes. So even though there is no net effect on the deficit, it ends up reducing GDP by $100B. In other words: $400B – $500B = -$100B. PeakTrader got the signs wrong.

    FYI: You can combine the expenditure and tax multipliers into a single formula, but you still end up with a multiplier of 1.00, but I’ll spare you the algebra unless there’s a hue-and-cry for it.

    1. pgl

      All true but remember Peaky gets paid by Kudlow and company so what do you expect from him?

      Back in the 1980’s a lot of economic advisers told us that reducing G would raise aggregate demand to which the great James Tobin noted that in their large econometric models they got the sign wrong!

  5. joseph

    Confusion related to identities isn’t limited to laymen. Recall that Nobel Laureate Eugene Fama and his son-in-law John Cochrane seized upon the identity that savings equals investment to argue that fiscal stimulus by government spending crowds out private spending and that government debt decreases private investment by the same amount. This sort of stupid economics did great harm to the discourse during the Great Recession.

    It goes to show that being clever in one small area of economics, finance, does not necessarily translate to expertise in the broader field of macroeconomics.

    1. pgl

      I did not know John Cochrane was related to Fama. OK they generally get finance right but their macroeconomics is indeed weak.

  6. PeakTrader

    2slugbaits, obviously, you didn’t understand anything I wrote. And, you brought up the balanced budget multiplier, which of course I acknowledged that government spending has a higher multiplier, because the marginal propensity to consume is less than one, e.g. with a MPC of 0.8, $100 billion in government spending financed by a $100 billion tax hike will increase GDP $500 billion minus $400 billion or $100 billion. Of course, spending 100% gives you a higher multiplier than spending 80%. There are other mechanisms at play. For example, higher taxes can cause a disincentive to work, like we see in the E.U.. So, we get fewer taxpayers and less taxes from less work. A decrease in disposable income lowers consumption and saving. You want to achieve full employment not create disincentives to work through higher taxes or government paying people not to work. And, low interest rates induce spending and borrowing. You need to save to pay for the induced consumption or borrowing, and higher interest rates increase saving and limit spending beyond autonomous consumption.

    1. Menzie Chinn Post author

      PeakTrader: The statement:

      ou need to save to pay for the induced consumption or borrowing, and higher interest rates increase saving and limit spending beyond autonomous consumption.

      makes absolutely sense to me.

      1. PeakTrader

        Menzie Chinn, isn’t it true low interest rates induce consumption and reduce saving, and higher interest rates increase saving with less consumption beyond autonomous consumption?

        1. Menzie Chinn Post author

          PeakTrader: We tend to think so, and the intertemporal model of the household budget constraint with the coefficient of relative risk aversion assumed to be low, would indicate a big impact. Almost all empirical evidence I am familiar with indicates that this empirical model is woefully inadequate, where the coefficient of relative risk aversion has to be assumed to be so high as to make the intertemporal channel nearly irrelevant.

          Also, in basic models, with income and substitution effects of a particular configuration, if the household sector is a net creditor, you will not get the posited impact. See for instance discussion in Barro (Modern Business Cyclte Theory, Harvard, 1989).

          My (simpler) point is that in a world where Say’s Law does not hold, i.e., there are assets, interest rates are not the only channel of adjustment to equilibrium. In fact the multiplier is kind of giving you an alternative channel.

        2. pgl

          Peaky – you strike me as one who never learned the difference between a movement along a curve versus a shift of a curve. Hey – an idea. Find your local community college and take Econ 101. I’m sure the instructor will cover this during the first couple of weeks.

      2. pgl

        I’m sure there is some identity that makes sense of this odd statement but we may need Einstein to interpret this for us!

    2. pgl

      “2slugbaits, obviously, you didn’t understand anything I wrote.”

      Actually he did. What you did just here was to abandon the Keynesian multiplier to switch gears to some supply-side silliness. As usual, it is Peaky who has no clue what Peaky wrote.

      Please stop and it has become really embarrassing.

    3. 2slugbaits

      PeakTrader Your memory is failing you. Recall that you were the one who brought up the balanced budget multiplier, all I did was give it a name. Let’s have the court reporter read back what you said: you can reduce some taxes by $100 billion and reduce some government spending by $100 billion, basically shrink government, to increase GDP by much more than $100 billion. Reducing taxes by $100B and reducing government spending by an equal and offsetting amount is the very definition of the balanced budget multiplier. The only problem is that you’ve got the direction of the signs all wrong. Normally you want to INCREASE taxes by an amount that offsets the INCREASE in government spending. But the math is otherwise the same. It’s the balanced budget multiplier.

      I acknowledged that government spending has a higher multiplier, because the marginal propensity to consume is less than one, e.g. with a MPC of 0.8, $100 billion in government spending financed by a $100 billion tax hike will increase GDP $500 billion minus $400 billion or $100 billion.

      Then why did you say REDUCE taxes by $100B and REDUCE spending by $100B?

      1. PeakTrader

        2slugbaits, I stated reduce some taxes and some spending in support of everything else I said.

        1. pgl

          Oh Lord! You really do not get this. If G and T are reduced by the same amount, then aggregate demand FALLS. 2slug and I have both tried to walk you through the basics. This is what the basic balanced budget multiplier says.

          But you still are lost? Even a 3 year old would figure this out by now but not you!

    4. 2slugbaits

      higher taxes can cause a disincentive to work

      Well, that’s half true…the substitution effect half of the story. The other half is that lowering taxes can cause a disincentive to work…that’s the income effect half of the story. In any event, I’m not aware of any empirical evidence that supports the kind of strong supply side effect you have in mind. I might be wrong, but the labor supply elasticities I’ve seen are pretty small. In any event, most people don’t have the luxury of fine tuning their labor effort to optimize the labor/leisure trade-off. You don’t get to tell your boss that this week you’d like to work 35 hours, but because of a new tax cut you’d like to start working 45 hours a week.

      You need to save to pay for the induced consumption or borrowing, and higher interest rates increase saving and limit spending beyond autonomous consumption.

      I’ll note the time of your post (8:43 PM PDT) and generously attribute it to being past your bedtime. It’s just incoherent to me.

      1. PeakTrader

        2slugbaits, I guess, you never heard of anyone unwilling to work overtime to avoid high marginal taxes, working part-time to keep most or all government benefits, one couple not working at all to keep government benefits, etc.. Transfer payments can be large. Americans have flexibility in how many hours they want to work or not work at all.

        1. 2slugbaits

          PeakTrader If you’re looking to reduce marginal tax rates below some threshold such that tax cuts pay for themselves, then you’re living in the wrong century. You could have made a good case for that back in the 1960s when the top rate was 90% plus a surtax. But guess what? This isn’t the 1960s. At current marginal tax rates the reason people don’t want to work overtime has more to do with the income effect than the substitution effect. If the income effect dominates, then cutting taxes only makes people less willing to work the marginal hour. And it’s because of that reluctance that firms have to pay an overtime premium.

          one couple not working at all to keep government benefits

          In some cases that’s true, but keep in mind that those are exactly the cases in which the effective marginal tax rate is very high…on the order of 60%. So if you’re advocating policies that would allow low income folks to keep more of their transfer payments, then I’m all onboard. Sign me up. But somehow I don’t think that’s the demographic you have in mind. Correct me if I’m wrong, but my guess is that you’re really advocating tax cuts for the upper income groups.

          You’re hanging your hat on very elastic labor supply curves and relatively strong substitution effects. Just for yucks, tell us your estimate of the labor elasticity of supply.

        1. 2slugbaits

          And I’ve stated several times that there are two effects, not one. You only want to look at the substitution effect. You also need to look at the income effect. Which effect dominates is an empirical question.

        2. pgl

          Menzie tried to walk you through income effects v. substitution effects. And you get this backwards too?

          Lord – no wonder the Univ. of Colorado did not admit you to their Ph.D. program!

  7. PeakTrader

    2slugbaits, real wages and the minimum wage for most of the workforce have been depressed. The top 20% are doing well, but the rest are more or less struggling. A middle class tax cut and a higher minimum wage would likely attract idle labor into the workforce and increase hours worked (e.g. part-time to full-time). Of course, for those working two or more jobs to make ends meet, the income effect will likely exceed the substitution effect and will reduce their hours worked. However, their labor-leisure balance will improve from an extreme position.

    Also, if you want to help the poor and needy, then you want to maximize employment to generate the tax revenue to pay for government services, including a stronger safety net.

    1. 2slugbaits

      PeakTrader I could agree with much of that. But there’s a simpler way to help the poor and the needy; just raise taxes on the top decile, and especially the top 1 percent. No one is talking about going back to the LBJ tax rates, but the rich seem to have done pretty well under the top Clinton tax rate of 39.6 percent. Then use the extra revenue to beef up the Dept of Labor Wage & Hour Division, strengthen Obamacare, drug rehab, expanded post-secondary education, relocation assistance, etc. Those are things that would actually help the poor.

      1. PeakTrader

        2slugbaits, the effective tax rate was much lower under LBJ. And, under Clinton, actual output exceeded potential output because of demographics at the height of the Information Revolution, raising GDP growth, including faster real wage growth for low income workers, and the economy able to absorb higher taxes in the economic boom.

        The key is to raise GDP growth to increase tax revenue, and control entitlements, which is crowding out other government spending and economic growth. The top 20% may be able to absorb a slightly higher effective tax rate, but the net positive effect may be small.

        1. pgl

          “The key is to raise GDP growth to increase tax revenue, and control entitlements, which is crowding out other government spending and economic growth.”

          Cut and pasted from some right wing garbage and devoid of ANY economic reasoning. Look – if you think Paul Ryan is an economist, then you are dumber than we give you credit for.

          1. PeakTrader

            So, you believe people in the top rate actually paid 91%.

            Or, just like to change the subject to argue with yourself.

          2. baffling

            peak, i think you are confusing marginal and effective tax rates. or you made a poorly worded statement. i stand by my statement that you are wrong, as worded by you. feel free to clarify your statements intent.

    2. baffling

      “A middle class tax cut and a higher minimum wage would likely attract idle labor into the workforce and increase hours worked (e.g. part-time to full-time).”
      as the current president loves to say, the economy is the best ever. we have record low unemployment at this time. so why would you provide tax cuts? not against a minimum wage, but would that not result in loss of jobs (at least this is usually what i hear from the echo chamber whenever an increase in minimum wage is mentioned)? more tax cuts probably result in inflation at this point.

  8. Julian Silk

    Dear Menzie,

    It is rather difficult to argue with this if you accept that macroeconomics as a field has any validity at all. My only comment would be that this is a simplification, and it seems to me to assume a completely horizontal aggregate supply curve. It’s absolutely right for that, under the assumptions. If aggregate supply isn’t flat, or if there are changes in expectations (which would conflict with output being completely demand determined), then the multiplier may be somewhat less.

    By the way, this shows up in applied work with input-output analysis, or economic impact analysis.


  9. tom

    Wow, this is my first macro lecture which I’ll deliver on Friday:( Always useful to get the accounting right from the get go.

    I prefer to say “planned” vs “actual” since “ex ante” and “ex post” may be confusing/pretentious to some ears.

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