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Jul 18, 2013

The Illogic of the Keynes Multiplier - 1

John:  Here are my restaurant bills for the last 5 years.
Mike:  Another discovery?

John:  My restaurant spending is always close to 10% of my income. That is, my income is 10 times my restaurant spending. So now I know a fun way to increase my income.
Mike:  I have to hear this.

John:  I will increase my restaurant spending by $2,000, and I will enjoy every minute of it. Then, my income will go up by about $20,000. That is what the chart says.
Mike:  You're nuts.
John:  No, I'm Keynes.

Reading Notes:  If you are familiar with the Keynes Multiplier or are impatient, jump directly to the section An Unreal InterpretationA link there will return.. Then read Equations and CausationA link there will return.. Then read the rest, of course.

The Keynes Multiplier Is Important

The Keynes Multiplier is probably the most important equation in history, despite using only introductory algebra. It has changed the world. It supports the big spending (fiscal, borrow and spend) policies of the United States government and the European Union.

 Total Income = 5 * (Investment + Govt Spending) 

It supports the belief that increased government spending (stimulus) produces more wealth than the money spentSpending decreases your wealth. It is only what you get in return, if anything, which increases it., at least when there are people out of work or economic growth is slow. If government spending increases total wealth, then huge spending is always good.

The famous economist John Maynard KeynesSee a few criticisms of Keynes at The Political Dictionary (1883-1946) (M)See Wikipedia for his life story. proposed ending the Great Depression by having the government spend lots of money, by first taxing, borrowing, or printing it. Politicians heartily agree, spending to fix every problem in good times and bad. This doesn't seem to help much, so their plan is to keep spending until it does.

Something Is Wrong

Steven E. LandsburgMore about him  is a Professor of Economics at the University of Rochester. He has two excellent posts at his blog TheBigQuestions.com exploring an amazing defect in the logic used to explain the Keynes Multiplier.

•  The Landsburg Multiplier
  -   How to Make Everyone RichSee his post.

•  Comments on the Multiplier postSee his post.

Landsburg derives his multiplier by using the same logic that Keynesian economists use to derive the Keynes Multiplier. He analyzes the relation of national income to his own income.

The Keynes Multiplier is about spending. The Landsburg Multiplier is about income. Both derivations use the same procedures, math, and logic.

Landsburg shows:

 Total Income = 100 million * Landsburg's Income 

So, Landsburg wryly suggests that people send him all the extra dollars they have, this will add to his income, and $100 million of added production will appear in the economy for each dollar sent to him. This will make us all rich.

Landsburg says correctly that this is an absurd result. This derivation of the Landsburg Multiplier must be wrong, and so the derivation of the Keynes Multiplier must be wrong.

Where is the error? It is puzzling that the facts are true and the math is simple. Landsburg suggests that the derivations may be wrong because they do not allow for the changes in consumer spending which likely follow changes in government spending. Equations describing economic behavior are unlikely to remain valid after a change in economic policy.

Economics is supposed to guide policy. This general criticism is that any economic formula might not be true after changes in economic policy. The formula might not predict results because the participants change their actions in response to the policy that the formula recommends.

This applies to all formulas and derivations. They represent what you think is true at some time. Rarely does everything interact in the same way after a significant change. Whatever the formula, you must verify the results in reality.

That criticism is good and often overlooked. But, it can't be what is wrong with the Multiplier. We get impossible results from the logic alone.

The truth is that the Multiplier is a wildly unrealistic result derived from an illogical argument. That is bad enough in a high school paper. It is alarming for a false insight which is used to mismanage societies.

An irony. It is harder to unravel and identify the errors in a simple argument than in a complex one. Those errors contain subtle defects, changes in meaning, and illogical conclusions. Once a person believes a simple argument, his tendency is to reject criticism, because it is hard to stop believing something he thought was obvious.


The Keynes Multiplier is the statement that the total income of the United States (or any country) is about five timesThe actual claim varies according to a broad estimate of consumer spending. This is an omen that something is not right. the amount of money invested in businesses and spent by government.

 Total Income (GDP) = 5 * (Investment + Govt Spending) 

GDP is Gross Domestic Product, the total production of goods and services in the US in a year or other specified time interval. Income is the flip side of production. Each person's income is the value he receives from his production. Someone receives income for every bit of production.

Keynesian economists see this as a fundamental economic truth first discovered by Keynes. They continue to the Keynes Conclusion that the total income of the country increases by 5 times any increase in government spending, or some similar multiplier.

They could conclude the same thing about increases in Investment, but politicians and Keynesian economists don't understand investment.

That increased income is supposed to appear as increased production, namely increased real wealth.

The Keynes Multiplier presented here is derived from reasonable assumptions and is only complex enough to make it seem significant. The 7th grade mathematics is simple and correct. Yet:

  • The derivation presents a reversal of cause and effect.
  • The conclusion presents a fact about the equations as if it is a fact about reality.

These fatal errors hide in a few simple equations and a seemingly obvious conclusion. Like a magician pulling a rabbit from his hat, everything seems ordinary until the rabbit appears. You may think that I am exaggerating, but bear with me.

Landsburg reports that this derivation is widely taught. For example, it is presented in the popular textbook "Macroeconomics" by Robin Wells and Paul Krugman, and in a textbook by Paul Samuelson. Those noted economists apply an equation to economic policy without checking on reality, and they have made fundamental errors in logic.

How could Keynesian economists with advanced degrees and world acclaim present this idiotic derivation and conclusion about what they have studied in great detail? It is a puzzle.

Landsburg notes a subtlety. Defenders of the conclusion might agree that the Landsburg Multiplier is absurd, that we can't trust the reasoning which produced it, and so we can't trust that same reasoning to produce the Keynes Multiplier. But, the conclusion might be shown in some other way.

The conclusion confuses cause and effect. It is no wonder that it can be derived from a confused argument.

This Derivation of the Keynes Multiplier

We first allocate All Production into three categories depending on who buys it: Household Spending (Consumption), Business Spending (Investment), or Government Spending (Govt). The flip side of production is income; all production is someone's income, so we can write All Income (Y, GDP) for All Production.

K1:   All Income  =  Household + Business + Govt 

K2:   Household  =  0.8 * All Income 

K2 is the estimate that 80% of all production goes to households as Consumption spending. This estimate might be incorrect, but this doesn't change the logic and general conclusion. It is plausible.

This is somewhat different from Keynes' Propensity to ConsumeSee also Economics Exposed.com.. That is the fraction
  (consumption spending) / (disposable income)
for any level of disposable income.

K3:   Business + Govt  =  0.2 * All Income 

Business and government purchase the remaining 20% of production.

K4:   All Income  =  5 * (Business + Govt) 

Switch the sides in K3, then multiply each side by 5. The result is K4, the Keynes Multiplier.

K5:   Increased All Income  =  5 * Increased Govt 

This is the Keynes Conclusion. Supposedly, if we increase Govt SpendingPoliticians, report to your tax and spend stations!, All Income will go up by 5 times that increase, and the additional income (GDP) will heal the economy.

What is the mechanism that increases production by a factor of 5? Amazingly, that detail is not in the equations. A statistical observation is supposedly telling us that spending $1 creates $5 of new production.

If we do the above using the Household fraction (K2), we get a spending multiplier of 1.25 for any increase in Household Spending. What is it about Household Spending in reality that gives it a different multiplier than Government Spending or Business Spending? It seems that this puzzle has remained unexplained.

Equations and Causation

Equations are convenient for relating quantities. They can "model" known or proposed relationships between physical things in the real world. But, equations don't care about how they are interpreted or about cause and effect. That is the responsibility of the user. For example:

Economist Bill studies how people react to rain. He observes:

 Raining  =  Many people wear raincoats 

Bill reads this as "When it rains, many people wear raincoats", and the reverse is also true "When many people wear raincoats, it is raining". Bill has observed this for many years without exception. It is an established result in the physical world.

Bill knows that rain causes people to wear raincoats and that wearing raincoats does not causeThis has been extensively studied and verified. it to rain. The mechanisms of the physical world are not part of the equation.

Bill's inexperienced assistant John looks at the equation and says "Now I know how to make it rain. If I can get more people to wear raincoats, then it will very probably rain. If I change the right side of the equation, that will change the left side. That is what '=' means."

Bill explains to John. "If you hear that many people are wearing raincoats, then in all of my experience it is raining. If you hear that it is raining, then in all of my experience many people are wearing raincoats. The equation expresses what we observe in nature, but it doesn't say anything about one thing causing the other. You have to look at the physical world to see what is going on."

An equation can accurately describe the observed relationship between quantities in the physical world. Still, that says nothing about what will happen if you try to physically change the system that you hope is represented by the equation. It is the difference between observing a system of many parts and changing one of those parts. You can't know from your past observations what that system will do in its changed form. You can't know the real cause and effect unless you specifically determine that by examining the physical world.

John understands. He can't conclude from the equation that raincoats cause rain. But, clearly rain causes raincoats.

Maybe simulated rain also causes raincoats. He uses a water hose to sprinkle the public, just like rain. But, they don't put on raincoats. Instead, they punch him in the face.

What is it about real rain, but not simulated rain, which causes many people to wear raincoats?

Life is complicated. Even if you respect cause and effect, you cannot know how a change will affect reality unless you have studied the physical details and have shown by observation that the changes do what you predict.

This is especially true of economics, which is the study of amazing complexity, namely of people, their interactions, and motivations. You can record what people do. This may suggest that changing something will have a desired effect. In reality, you don't know unless you study the change and the effect.

The simulated rain example may seem contrived. John was punched for reasons completely apart from the quality of the sprinkled water. That is the point. There is immense complexity in simulating economic conditions. The participants will see the manipulation, include that information into their response, and may act in a different way than they did before.

Mathematics is only an aid to thinking. It represents insights and quantities in a way which can be easily written and combined. The results can suggest more things to think about and investigate.

Physics gives mathematics a reputation for predicting the future. The universe seems to follow mathematical results, but it is actually mathematics which has been constructed to model the universeSee also About.com - The role of mathematics in physics.

Physicists know that their mathematics is almost always a simplified model of reality. They are dismayed but not surprised to find that an equation makes bad predictions. They always compare their expectations to reality to correct inevitable mistakes. When in doubt, they distrust the math, not reality.

The inspiring particle physicist Richard Feynman (1918-1988) fought for open, transparent science. He wrote:Cargo Cult Science, from the 1974 Caltech commencement address.

When someone says, "Science teaches such and such", he is using the word incorrectly. Science doesn’t teach anything; experience teaches it. You should ask, "How does science show it?"

How did the scientists find out? How? What? Where? It should not be "science has shown". And, you have as much right as anyone else, upon hearing about the experiments and after hearing all the evidence, to judge whether a sensible conclusion has been arrived at.

Airplanes are designed with well understood equations representing tested physical principles. Still, every airplane design is thoroughly tested for flight worthiness. Mathematics is a sophisticated guide to building and testing, but assumptions and math may not predict reality.

Which airplane would you like to fly on? One has been designed by teams of engineers using super-computers, but never tested. The other has been built from rough estimates and is thoroughly tested.

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An Unreal Interpretation

We will look at the derivation again and keep track of cause and effect. To save space and thought, BusGovt means Business + Government Spending, and All is All Income

K1:  All  =  Household + BusGovt
K2:←   Household  =  0.8 * All
K3:BusGovt    =  0.2 * All
K4: → All  =  5 * BusGovt
K5: → Increased All  =  5 * Increased BusGovt

This derivation describes a relationship between numeric values. It doesn't care about cause and effect, and it may not represent reality even if the reader thinks that some or all of the relationships are realistic. Math models may do a good job of representing reality in some ways, and be wildly wrong in others.

Physicists know this about mathematics. It seems that Keynesian economists do not, or choose to ignore it. The following facts are so basic that it is embarrassing to explain them.

The red arrows above are a reminder of what we see as cause and effect. When I say "the equation tells us", remember that we are getting back in some form just what was put in.

( Click on the K1 .. K5 labels below to pop up the above equation being explained. )

K1    All  =  Household + BusGovt  is an accounting identity. The "=" tells us that these amounts always add up. The double arrow says that I expect both sides to be equal in reality, even if I meddle. If I increase my total spending by $100 to buy donuts, production of donuts also goes up by $100 (after some inventory fluctuations), and others get $100 more in income. K1 balances approximately even if I meddle with the physical world.

K2    Household  =  0.8 * All  says that Household Spending is 80% of All Income (All Spending). We might be thinking that it is approximately 80%, but we have specified that it is exactly 80%.

The red arrow points to the left. It reminds us that we think Household Spending is a result of total spending (total production). If we can increase total production through better government policy, we expect that Household Spending will get 80% of that production. We think Household spending is a part of total production.

Still, all of this would have to be checked against reality. Increasing total production might not increase the part going to households. For example, building tanks for war does not result in households receiving 80% of the tanks. If households buy 80 more donuts, they get just those donuts. There is no reason to believe that 100 donuts are produced so that households receive 80 and BusGovt receives 20.

K3    BusGovt    =  0.2 * All  is all spending other than Household. The proper interpretation is:

The red arrow for K3 points to the left. It reminds us that we start with All Income produced in unspecified ways. We know that production supports investment, taxes, and government borrowing. Somehow, that averages out to BusGovt getting 20% of production. Our knowledge of physical processes makes the cause and effect in K3 believable.

K3 is a summary of past observations. The economy is a natural system resulting from the complex interplay of people and politics. It has produced values approximately satisfying K3.

If we are interested in a particular value of BusGovt, then we can use the model to calculate the naturally occurring value of All Income that would produce that value of BusGovt.

If we meddle and change that natural system by redirecting spending, K3 is only a weak suggestion about how the economy will change. Only direct observation could tell us that.

K4    All  =  5 * BusGovt  is a restatement of K3, having the same meaning in all respects. The cause and effect arrow now points to the right. We still believe that All Income determines (causes) BusGovt.

But, the presentation of K4 suggests that we start with BusGovt to calculate All Income. The strong suggestion is to reverse cause and effect, to accept that somehow BusGovt causes All Income.

We must decide if that possibility has physical meaning. It is illogical to believe that the equation alone is enough to predict that BusGovt Spending causes 5 times its value in Income. It is laughable that this interpretation rests on no physical processes whatsoever.

K5    Increased All Income  =  5 * Increased BusGovt  is the conclusion repeated everywhere. It reverses cause and effect, and it assumes that we can predict the results of changing a natural system.

The cause and effect we specified starts with Increased All Income to produce a 20% increase in BusGovt, enough to balance the equation. That is truly boring.

It is much more exciting to accept a reversal of cause and effect merely because the usual computational flow of K4 suggests it. It is illogical to believe that the physical world cares about the order of the terms in K4 and the resulting K5.

Say that I increase BusGovt Spending by $1 billion. By what possible physical mechanism is the universe going to respond to this by increasing All Income just so K5 will balance? What has caused learned economists to interpret this model in this unphysical way?

Many cars have a dial odometer. A red pointer moves like the hand of a clock, pointing to the speed of the car. The odometer always shows the correct speed if you limit yourself to observing the odometer and verifying the speed. A Keynesian observes that the car is traveling 30 miles per hour and takes action. He grabs the red arrow and pulls it to the right, believing this will speed up the car. That is what the equations say to him.

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The Undead

The analysis above is enough to discredit this simple derivation of the Keynes Multiplier. Anyone using this derivation should be ashamed. But, as Landsburg points out, a bad derivation doesn't prove that the result is wrong, only that it remains unproven.

There is another more complicated derivation more closely associated with Keynes. I will pick that one apart in another post.

That complicated derivation must also be wrong. It has all of the characteristics of the derivation above, but with flashier mathematics. It is also an accounting derivation. It mentally tracks the supposed flow of money through the economy without providing any physical analysis. It does not consider how production is accomplished. Amazingly, it blames investment for lowered incomes. It says that we could all consume a multiple more if we invested less of our incomes.

That goes against the physical observation of how income is produced, and it ignores cause and effect. The problem of producing more cannot be solved by the act of spending. An individual can only spend the income that he earns from his production, or has saved from his past production, or has borrowed from the production or savings of others. The government can only spend what it first takes from current production and savings.

The idea that Spending causes Production is a reversal of cause and effect. We can only exchange what we produce (or promise that we will produce). Exchange with other people is called Spending.

All we need for a utopia is to reverse cause and effect. All we need for infinite energy is to reverse a few physical laws. All we need for a happy and productive society is to give all resources to the government for wise spending and multiplying.

- -
Keynes' Propensity to ConsumeGet a feel for it. I don't recommend reading it deeply.

This is a chapter from "The General Theory of Employment, Interest and Money" by Keynes. This type of dense writing impresses some people. To me, it is used to obscure any proposed theory and prevent any clear analysis. Any criticism can be met with "you didn't properly understand the detail in section 3.6".

Let's Counterfeit Our Way to WealthI wish it were true.
02/2009 - Easy Opinions

A fantasy supports the idea that our government increases the wealth of our society when it borrows and spends. Supposedly, government spending increases wealth by a conservative multiplier of 1.5.

There is no wealth multiplier from the flow of money. If there were, we would all be living in Aruba by now as a result of huge government borrowing and spending.

See posts about Keynes and his economics ...Search Label: Keynes at EasyOpinions


JB McMunn said...

I've often felt the same way about MMT. They mistake a descriptive equation for the thing itself and run off into the Wonderland of $5 trillion coins.

Andrew_M_Garland said...

To JB McMunn: Well said.

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