Intermediate Macro Exercise: How to Construct a Maximally Contractionary Budget Deficit Reduction

Consider an economy called Käseland, with gross output equal to approximately $475 billion, and unemployment rate of 7.5%, so considerable underemployment of factors of production exists; consistent with this interpretation, the general nonfarm wage rate has been relatively constant, growing at only 1.2% on a 12 month basis through 2010, and the price level has risen by about 1.5% from the second half of 2009 to second half of 2010.


Suppose there is a budget deficit, that you wish to close. How do you maximize the negative impact on output?

First, consider the definition of a government budget balance. Assume interest costs away, since they are determined exogenously (or at least predetermined, for the current period):


BuS ≡ T-TR-G = (TA0 + t1Y) – (Tr0) – (GO0)


Where BuS is the budget balance, TA0 is lump sum taxes, t1 is the marginal tax rate, Y is output, or income, Tr0 is lump sum transfers, and GO0 is exogenous government spending on goods and services.


Effecting a positive change in the budget balance can be accomplished in three ways. To see this, consider the total differential of the budget balance (assuming the marginal tax rate is held constant).


Δ BuS = Δ TA + t1 Δ Y – Δ TR – Δ GO


One can (i) raise taxes; (ii) decrease transfers; or (iii) decrease government spending. For simplicity, consider cutting government spending (on civil servants, for instance), say by $1 billion. In a demand determined model of output (i.e, Y = Z, where Z is aggregate demand), the resulting output reduction is:


Δ Y = γ Δ GO where γ ≡ [1-c1(1-t1) + m1]-1


where m1 is the marginal propensity to import from other economies, and should be relatively large in a small open economy like Kaseland. “Gamma” (γ) is the Keynesian multiplier. (Solving this simple model is laid out in detail in this handout). Recalling that the change in government spending on goods and service is less than zero, viz., Δ GO < 0, output falls by greater than a billion dollars.


One sees that an alternative means of closing the budget deficit is by raising taxes; for simplicity consider an increase in lump sum taxes.


Δ Y = -γ c1 Δ TA


Notice that because the parameter, c1 (the marginal propensity to consume out of disposable income, 0 ≤ c1 ≤ 1) premultiplies the change in lump sum taxes, then the corresponding reduction in output resulting from the increase in taxes is smaller than that for a cut in government spending on goods and services.


This has an additional ramification for the budget balance. Note that because tax receipts are endogenous, then the larger negative impact on output and hence income arising from government spending reduction manifests itself in a correspondingly larger decrease in tax receipts. To see this, repeat the expression for the impact on the budget balance:


Δ BuS = Δ TA + t1 Δ Y – Δ TR – Δ GO


For a decrease in government spending of $1 billion, holding all other fiscal measures constant:


Δ BuS = t1 γ Δ GO – Δ GO


For an increase in lump sum taxes of $1 billion, holding all other fiscal measures constant:


Δ BuS = Δ TA – t1 γ c1 Δ TA


Notice that the improvement of the budget balance is greater for an increase in lump sum taxes of $1 billion than for a $1 billion decrease in government spending.


Postscript, 4:53 Pacific: One can increase the contractionary effect of the budget balancing if one cuts taxes and increases the size of cuts to government spending to compensate, unless the elasticity of supply is sufficiently high. See a policy aimed at accomplishing this here Forbes (2/15/11).


Addendum 1: Typically, in examining the impact of tax policy changes, one analyzes changes in the marginal tax rate, or the parameter t1 (and not lump sum taxes) in this model. In order to accomplish this, one would need to examine the following expression for the budget balance, which uses the product rule for differentiation.


Δ BuS = (Δ t1) Y + t1 Δ Y


Holding all other fiscal policies constant. In addition, the product rule for differentiation has to be invoked for the impact on output:


Δ Y = – c1 γ 20]


Where Λ 0 is the sum of autonomous spending (e.g., investment unrelated to the level of income).


Without solving out for the analytical answer, what is true is that for a given improvement in the budget balance, a tax rate increase will have a smaller negative impact on output than a given cut in government spending on goods and services.


Addendum 2: The foregoing model has assumed perfectly elastic supply. Suppose the economy is at capacity, i.e., there are no underutilized factors of production. Then one would need to drop Y=Z, and substitute in AS=Z, where AS = fn(K, L). Then, under the assumption of sufficiently high elasticities of labor with respect to wages and capital with respect to the rental cost of capital (and assuming away accelerator effects), one can overturn the demand-determined model-based results obtained above. [1]


Addendum 3: The last IMF WEO assessed contractionary expansions. In that cross country assessment, spending (both goods and services and transfers) reductions effected smaller reductions in output than tax increases. The authors concluded that this outcome was due to the monetary authorities tending to offset spending cuts more than tax increases. In the model above, I have assumed Kaseland has no independent monetary authority (i.e., is a small open economy, with fixed exchange rate).


Addendum 4: If the marginal propensity to consume for government workers is higher than the marginal propensity to consume of the households that would be taxed at higher rates due to a tax increase, then the bottom line result would be strengthened.



Statistics on employment, wages, in Wisconsin here; on CPI growth in the Midwest here (search Class A cities). Statistics on Wisconsin GSP (gross state product) here.


Implied impact on government payrolls of current proposals: Institute for Wisconsin’s Future, page 3.


21 thoughts on “Intermediate Macro Exercise: How to Construct a Maximally Contractionary Budget Deficit Reduction

  1. Jeff

    Menzie, I’m sure I’m not pointing out anything new to you, but I it should be noted that the above Keynesian analysis implicitly assumes that (a) one dollar of government spending is equivalent to one dollar of private spending and (b) any reduction in government spending will not be met with any off-setting increase in private investment.

    And of course (a) and (b) are central to the main criticisms against increasing the size of government, i.e. that the returns to government investment are not as large as the private sector and that and increase in government spending will tend to crowd-out private spending

    I’m not saying that (a) and (b) are correct, but in general I’m skeptical of any analysis that “shows” one side to right by assuming the alternative to be wrong.

  2. Menzie Chinn

    Jeff: You are absolutely correct. Aggregate demand, Z is equal to the simple sum of autonomous spending components, as is typically the case in intermediate and introductory textbooks (and, as far as I know, Ph.D. level textbooks as well). One could put different weights on different components, but I assume what you want to do is welfare analysis. In that case, spending is not the only thing to be weighted, but also leisure, as well as dis-amenities, and negative/positive externalities.

    Furthermore, your “crowding-in” hypothesis works in a world where the investment function is I = b0 – b2i, and decreased state borrowing can affect the interest rate faced by firms. But we are at the zero interest boundary for the short term interest rate, and it is unclear to me that a reduced level of state borrowing will impact in a material fashion the borrowing cost of a typical firm in Wisconsin. Finally, note if I = b0 + b1Y – b2i, as seems plausible, and is incorporated in the Ec302-001 textbook, then your conclusion does not necessarily follow even if we are above the zero interest bound.

  3. CoRev

    Ah, if it were just so simple, but you forgot those important multipliers, (P) Politics and (pw) political will

  4. bturnbull

    According to this post, if government of Kaseland takes $100 from a private citizen (by increasing marginal rate) and spends it all, the economy of Kaseland shall benefit.

    What if a private citizen takes $100 back from the government and spends it, should it also benefit the economy of Kaseland?

    Of course, not. It cannot happen both ways, or your theory could not be used to justify taking money from private citizens? So, for your theory to work, there must be a set of implicit/explicit assumptions and/or set of past metrics that show that government spends money in a better way.

    What is the better way: maximizing short term GDP?

    But are all the underlying assumptions and conclusions of such theory necessarily true:

    1. Should maximizing short term GDP be the ultimate goal in itself? What if maximizing short term GDP happens at the expense of long term capital investments and therefore in reduction of future growth in GDP?
    2. Would government spend the money faster than a private citizen? (If a private citizen spends money as fast as government, how can GDP be maximized?)
    3. Can US government be justified to take money from US citizens and businesses, because Washington can spend our money faster?
  5. Get Rid of the Fed

    It seems to me you have left out the medium of exchange supply (currency plus demand deposits) and its composition.

  6. aaron

    “Z is equal to the simple sum of autonomous spending components”
    So it doesn’t matter what we do, just that it be big enough.
    That seems like a hella bad assumption.

  7. aaron

    “Would government spend the money faster than a private citizen? (If a private citizen spends money as fast as government, how can GDP be maximized?)”
    Yes, they would likely consume faster. But they are more likely to consume than invest.
    Short-term GDP us maximizes, but long-term GDP is decreased.

  8. Menzie Chinn

    bturnbull: I didn’t say anything about welfare. This was an exposition on output and income. You are free to do your own weighting in your calculation, depending on your social welfare function. If, for instance, you weighted corporate investment double that of consumption by households, you might conclude that it would be better to cut government spending. Even then, you might not wish to cut government spending, as the impact depends upon the nature of the investment function, and the magnitude of the parameters (not only in the investment function, but all the equations in the model).

  9. Ricardo

    Assuming a fantasy Keynesian world such Sudoku is spot on.
    Why not make the wisest man in the kingdom absolute dictator. He can determine production and distribution of products and he can by decree achieve total employment. He can dictate interest rates and decree unlimited money supply with no inflation, a chicken in every pot and two cars in every garage.
    Since this is the wisest man he will of course make all the right decisions determining the economy. The economy will grow, the people will be prosperous, the birds will sing, the sun will shine, and we will all live happily every after.
    If you are going to make fantasy assumptions you might as well go all the way.
    Oh, but then there is no Sudoku fun. Never mind!

  10. tj

    The smaller multiplier effect on a tax increase relative to the multiplier on a cut in government spending drives the result.
    No big surprise.
    The national debate, and the more interesting issue, is the optimal size of government.
    The only way a tax increase makes sense, is if you believe the current size of government is too small.
    In other words, a tax increase can eliminate government borrowing by raising tax revenue to the point at which it equals the current level of government spending. In contrast, the budget can be balanced by reducing spending to the a lower level that requires no borrowing.
    Thus, a tax increase preserves current spending levels. A spending cut reduces current spending levels.
    The model Menzie outlines above is not useful at capturing potential long run benefits to growth that result if you claim the current size of government is above the optimal level, e.g. fiscal conservatives, tea party, etc. (Some claim it is below the optimal level, e.g. Progessives, Utopians, etc.)

  11. markg

    How about Kaseland spend its own government created fiat currency based on the needs of the public (infrastructure, education, defense, social needs, etc). And then tax low enough to keep unemployment low but high enough to keep aggregate demand from exceeding the supply of goods and services leading to inflation. The resulting govt deficit is because the private sector decided not to spend all of its income.

  12. kharris

    Behind some of the criticism here, there is a small child with fingers stuffed firmly in ears, refusing to hear. This is, of course, a model. Models are used to learn, and to illustrate. Because the real world is tough to comprehend all at once, we simplify, in order to see if we can comprehend it a piece at a time. Done right, we would then build up a filler picture as we comprehend the parts.
    Leaving out political will, for instance, takes away the opportunity to simply claim that something is impossible because politics won’t stand for it. One useful feature of a model is to show why political capital might be worth shelling out in service of a particular policy.
    It is a more profound misunderstanding of how models work – or any objective effort to understand how the work operates – to suggest just imposing a dictator. Policy is made by those in power. People in power get their ideas from somewhere. Sometimes, they dredge up their own prejudices. Sometimes, they are influenced by howling partisans. Sometimes, though, they look at the analysis of objective people who have thought about problems, in the hope of getting as close to a good answer as possible.
    We don’t need to shy away from learning the right answer. We don’t need dictators to impose the right answer. But we do need to give people line Menzie the room to look for the right answer, and have the broad-mindedness to accept that not every right answer is going to fit our own preconceptions.

  13. John Schwaner

    “unemployment rate of 7.5%, so considerable underemployment of factors of production exists”
    One must answer the question as to WHY underemployment of factors of production exists.” and then solve that problem.
    Example, I have a friend who is sitting on a mothballed manufacturing facility — at no level of demand will he put it back into production.

  14. David Fiene

    How would the results change if you skip ahead a few lectures to the forward looking theory of consumption? Consumers who have been expecting a tax hike to cover the deficit under Doyle now see how serious Walker is about not raising taxes now or in the future. This could increase c.

  15. Frank

    I think people here forget the difference between normative (what ought to be) and positive economic statements (what is)…
    Menzie is making a positive statement based on simple calculus. The detractors are making normative statements. The equations above simply describe the simple linear first order effects of policy changes in either government spending or taxes and how they impact output in the short-term.
    I don’t think any “real” economic analyst or academic economist could look at this result and go “oh..hey! Lets raise government spending to infinity, increase tax receipts, and we’ll be out of this in no time.”
    We can make normative statements, talk about how we think things are and out to be, add in new functions which assume there is some level of G or T where the rate of change in I or/and C starts to decrease and offset the effect of the increase/decrease in G or T to the equation which bias it one way or another to analyze policy effects based on different scenarios, then add some type of welfare function which shows that at some level of G or T we decrease total welfare and on and on down the proverbial rabbit hole, but if you’re going to make normative statements and call out the Professor, at least do the calculus and show your results.
    Perhaps it’s intellectual insecurity more than anything else which forces people to post negative comments here without any real counter-analysis.

  16. Ben Claassen

    Your article as relates to unemployment, standard of living and taxation. An economy can be viewed a chamber with money suspended in an air draft. Exchangers of goods and services are fairly taking money in and out of the chamber with some “cut” going to taxes.
    Government spending is not a single commodity–
    1. infrastructure spending should be raising productivity (more money in the chamber),
    2. defense and security spending do not add to GDP but avoid large downside risk while putting the money into the hands of US spenders,
    3. overseas spening removes money from the system and adds to unemployment,
    4. payments not to work or grow a crop or to live for free, all reduce GDP and add to unemployment and lower overall standard of living.
    5. taxes effectively used for payments for insurance (avoidance of downside risk) funds for big malpractice/liability cases inflate costs, lower standard of living and take money out of the chamber.
    Deficit reduction needs to look at how the tax money is flowing out, seek to add to jobs, raise productivity , thus generate more exports (less money leaving the chamber). How about go back to the budget from 2000? Then finds cuts to litigation costs, defensive spending, to get balanced as soon as more people are working?

  17. Menzie Chinn

    David Fiene: Ah, a true believer in Ricardian equivalence. Empirical estimates I have seen indicate that it’s not a dollar for dollar effect; more like 40 cents on the dollar. If you have seen estimates to the contrary, I would welcome seeing them.

    Drop the infinite-lived representative agent, perhaps use an overlapping generations model, or allow for distortionary taxes, or presence of some Campbell-Mankiw type hand-to-mouth consumers, and the implications could change.

  18. David Fiene

    I wouldn’t call myself a true believer, I just remember your slides from last semester in 302. Does the 40 cents on the dollar affect the model above? If so, how?
    I think I remember you mentioning infinite-lived representative agent, I’ll check out the other three sometime tonight.

  19. Anonymous

    INteresting. Looks like the situation of a small EUR country that wants to cheat by not conytributing to generating the demand necessary to lift the laggards. A strategy only available to a country without banks etc heavily invested in PIGS debt. And, in that case, admirably patriotic.

  20. 2slugbaits

    Once again the usual suspects don’t seem to understand the difference between the AD and AS curves. Menzie is talking about a situation in which there is an output gap; i.e., we have a magneto problem. He is talking about which fiscal policies expand (or in this case contract) GDP the most. Menzie is talking about pushing out (or in this case pulling in) the AD curve. Arguments about private sector productivity versus public sector producivity are completely irrelevant here. Menzie is not talking about pushing out the AS curve. In fact, Addendum 2 makes this point pretty clearly, but it seems that many posters here just didn’t get it. One can argue whether or not it’s actually true that private sector growth is more productive than public sector growth, but it’s not necessary to make that argument because it doesn’t have anything to do with what Menzie is talking about. Kaseland is not at full employment and there is a significant output gap. In fact, with interest rates at zero it’s pretty easy to show that pushing out the AS curve with higher productivity would actually reduce output (see Tobin and Krugman).

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