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# Economic growth and labour

I created a page on economic growth and sustainability at

sofar containing link to a page on economic growth and labour at

Checks of the computations and other comments are appreciated.

One should maybe create another category for this.

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1.

A small comment concerning elasticity:

you wrote

the elasticity seems to be given by $\dot{L}/\dot{GDP}$

Wikipedia specifies

elasticity is the ratio of the percent change in one variable to the percent change in another variable.

the "x-elasticity of y", also called the "elasticity of y with respect to x", is:

$$E = \frac{\partial ln(y)}{\partial ln(x)}$$

According to the document (p4):

The employment elasticity is defined as the average percentage point change in employment for a given employed population group (total, female, male) associated with a 1 percentage point change in output over a selected period.

So I suppose it has to be $\frac{\dot{L} GDP}{L \dot{GDP}}$. I may easily be wrong, as I don't know much about economy. Maybe you implied this when you wrtoe the growth in but then it wasn't clear at least for me.

Btw, if you use Markdown+Itex and use double brackets for the wiki and single brackets- curly brackets for outside links (click Source to see), the links are automatically generated.

Comment Source:A small comment concerning elasticity: you wrote > the elasticity seems to be given by $\dot{L}/\dot{GDP}$ [Wikipedia](http://en.wikipedia.org/wiki/Elasticity_%28economics%29) specifies > elasticity is the ratio of the percent change in one variable to the percent change in another variable. > the "x-elasticity of y", also called the "elasticity of y with respect to x", is: > $$E = \frac{\partial ln(y)}{\partial ln(x)}$$ According to the [document](http://kilm.ilo.org/KILMnetBeta/pdf/kilm19EN-2009.pdf) (p4): > The employment elasticity is defined as the average percentage point change in employment for a given employed population group (total, female, male) associated with a 1 percentage point change in output over a selected period. So I suppose it has to be $\frac{\dot{L} GDP}{L \dot{GDP}}$. I may easily be wrong, as I don't know much about economy. Maybe you implied this when you wrtoe *the growth in* but then it wasn't clear at least for me. Btw, if you use Markdown+Itex and use double brackets for the wiki and single brackets- curly brackets for outside links (click Source to see), the links are automatically generated.
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2.
edited March 2011

I saw the explication in document p4 and it was the one what I was referring to as "non-formula" definition. The explication is quite ambiguous thats why I wrote the elasticity "seems" to be given. I mean what means "associated to"?! I guess they mean divide by, but who knows. I wrote actually an email to ILO but they wouldnt give me a formula, but back then I was quite sure that they must mean "divide by". The issue is especially annoying since they obviously use some formula. I think there should be a decent mathematical explanation to each entry of the KILM. The only thing which kept me from sending them further moanings was that this KILM is only a beta-version.

In some sense the above seems actually to boil down to the question what kind of growth economic growth and growth in employment is meant here. If you assume that $\dot{GDP}=(\delta GDP)/GDP$*(1) where \delta GDP is the (e.g. annual) change in GDP; do the same for L then this would -apart from the issue of differences vs differentials- go somewhat together with the Wikipedia definition.

I hoped that somebody could clarify this in this Forum. May be I should send ILO more moanings.

*(1) I used that definition of "growth" for the increase in wage as you can see in the text

Comment Source:Thanks Frederik for reading. I saw the explication in document p4 and it was the one what I was referring to as "non-formula" definition. The explication is quite ambiguous thats why I wrote the elasticity "seems" to be given. I mean what means "associated to"?! I guess they mean divide by, but who knows. I wrote actually an email to ILO but they wouldnt give me a formula, but back then I was quite sure that they must mean "divide by". The issue is especially annoying since they obviously use some formula. I think there should be a decent mathematical explanation to each entry of the KILM. The only thing which kept me from sending them further moanings was that this KILM is only a beta-version. In some sense the above seems actually to boil down to the question what kind of growth economic growth and growth in employment is meant here. If you assume that $\dot{GDP}=(\delta GDP)/GDP$*(1) where \delta GDP is the (e.g. annual) change in GDP; do the same for L then this would -apart from the issue of differences vs differentials- go somewhat together with the Wikipedia definition. I hoped that somebody could clarify this in this Forum. May be I should send ILO more moanings. *(1) I used that definition of "growth" for the increase in wage as you can see in the text
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3.

Let's forget for a moment that the economic system is there to support the people, rather than the other way around. Before the industrial revolution energy was usually the thing in chronic short supply, and wages were driven to the floor. Since then the thing in short supply has usually been skill-weighted labour, and energy prices have been driven to the floor: $10/bbl oil is a recent memory. We're not short of energy yet, but we do have the wrong infrastructure as we change energy input, and it is a big job to replace our oil-dependant transport infrastructure (and/or start producing diesel/petrol artificially from other energy). In summary: labour productivity is not always the most important issue for growth (positive or negative), and currently it is nearly irrelevant. Another thing to understand with respect to productivity is that it is often concentrated in a small number of workers but spreads. Consider barbers. The cost of a haircut has gone up in line with general wage levels. If it didn't there'd be no barbers: they'd all go off to more lucrative employment. Yet clearly a haircut hasn't changed. Yet people who look at "labour productivity" reckon Barber productivity must have gone up. I believe this "barber effect" is the source of most wage rises. Or look at the other side. Suppose we are trying to work out the EROEI (energy return on energy invested) of some energy source. The energy workers get paid, and that money is used for energy, which might reasonably be included in the Energy Invested. We can't afford EROEI to fall below 1 (indeed we need much more), and one way to improve the EROEI is to reduce wages to energy workers. However note that this can only be done in the context of all wages falling, otherwise the energy workers will leave. So in some way the need to reduce energy worker wages will have to happen in a way that transmits to all society. Sometimes you can see that something will have to happen even though you can't predict the mechanism, but understanding will be deeper if we understand the mechanism. Comment Source:Let's forget for a moment that the economic system is there to support the people, rather than the other way around. Before the industrial revolution energy was usually the thing in chronic short supply, and wages were driven to the floor. Since then the thing in short supply has usually been skill-weighted labour, and energy prices have been driven to the floor:$10/bbl oil is a recent memory. We're not short of energy yet, but we do have the wrong infrastructure as we change energy input, and it is a big job to replace our oil-dependant transport infrastructure (and/or start producing diesel/petrol artificially from other energy). In summary: labour productivity is not always the most important issue for growth (positive or negative), and currently it is nearly irrelevant. Another thing to understand with respect to productivity is that it is often concentrated in a small number of workers but spreads. Consider barbers. The cost of a haircut has gone up in line with general wage levels. If it didn't there'd be no barbers: they'd all go off to more lucrative employment. Yet clearly a haircut hasn't changed. Yet people who look at "labour productivity" reckon Barber productivity must have gone up. I believe this "barber effect" is the source of most wage rises. Or look at the other side. Suppose we are trying to work out the EROEI (energy return on energy invested) of some energy source. The energy workers get paid, and that money is used for energy, which might reasonably be included in the Energy Invested. We can't afford EROEI to fall below 1 (indeed we need much more), and one way to improve the EROEI is to reduce wages to energy workers. However note that this can only be done in the context of all wages falling, otherwise the energy workers will leave. So in some way the need to reduce energy worker wages will have to happen in a way that transmits to all society. Sometimes you can see that something will have to happen even though you can't predict the mechanism, but understanding will be deeper if we understand the mechanism.
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4.
edited March 2011

@Frederik I think you are right that one might misinterpret what I wrote, so I changed now $\dot L$ into $L_growth$ and $\dot GDP$ into $GDP_growth$

That makes

Let $\dot L$ denote the growth in employment, and $\dot GDP$ the growth of GDP,

into:

Let $L_growth$ denote the growth in employment, and $GDP_growth$ the growth of GDP

which looks a bit over-explicatory, on the other hand I think everything should be defined.

@rks Yes you are right productivity raises in some sectors more than in others. As I understood the productivity of KILM is averaged. But apart from that, I do think that with the onset of cut-n-go shops barbers cut more hair per hour than before on average. You wrote:

The energy workers get paid, and that money is used for energy, which might reasonably be included in the Energy Invested. We can't afford EROEI to fall below 1 (indeed we need much more), and one way to improve the EROEI is to reduce wages to energy workers.

That I don't understand.

Comment Source:@Frederik I think you are right that one might misinterpret what I wrote, so I changed now $\dot L$ into $L_growth$ and $\dot GDP$ into $GDP_growth$ That makes > Let $\dot L$ denote the growth in employment, and $\dot GDP$ the growth of GDP, into: > Let $L_growth$ denote the growth in employment, and $GDP_growth$ the growth of GDP which looks a bit over-explicatory, on the other hand I think everything should be defined. @rks Yes you are right productivity raises in some sectors more than in others. As I understood the productivity of KILM is averaged. But apart from that, I do think that with the onset of cut-n-go shops barbers cut more hair per hour than before on average. You wrote: > The energy workers get paid, and that money is used for energy, which might reasonably be included in the Energy Invested. We can't afford EROEI to fall below 1 (indeed we need much more), and one way to improve the EROEI is to reduce wages to energy workers. That I don't understand.
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5.

I think you are right that one might misinterprete what I wrote, so I changed now \dot L into Lgrowth and \dot GDP into GDPgrowth

Actually I meant that "the growth of GDP" was - at least for me - less clear then $\dot{GDP}$, because with $\dot{GDP}$ I assume $d GDP/dt$ while with "growth of GDP" one could perhaps implicitly refer to logarithmic growth, in which case the elasticity formula is satisfied, but the dot notation is wrong.

So -at least for me -

"Let Lgrowth denote the growth in employment, and GDPgrowth the growth of GDP"

is not much clearer, because I was wondering about the precise meaning of "growth of GDP", and the above line looks like a tautology to me. I would suppose it means GDP(year)-GDP(year-1) although this doesn't agree with the elasticity formula, I think. If it's log GDP(year) - log(GDP(year-1)) I would be happy ;-)

Comment Source:Nad said: > I think you are right that one might misinterprete what I wrote, so I changed now \dot L into Lgrowth and \dot GDP into GDPgrowth Actually I meant that "the growth of GDP" was - at least for me - less clear then $\dot{GDP}$, because with $\dot{GDP}$ I assume $d GDP/dt$ while with "growth of GDP" one could perhaps implicitly refer to *logarithmic* growth, in which case the elasticity formula is satisfied, but the dot notation is wrong. So -at least for me - > "Let Lgrowth denote the growth in employment, and GDPgrowth the growth of GDP" is not much clearer, because I was wondering about the precise meaning of "growth of GDP", and the above line looks like a tautology to me. I would suppose it means GDP(year)-GDP(year-1) although this doesn't agree with the elasticity formula, I think. If it's log GDP(year) - log(GDP(year-1)) I would be happy ;-)
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6.
edited March 2011

On page 2 they write:

An elasticity of 1 implies that every 1 percentage point of GDP growth is associated with a 1 percentage point increase in employment.

I might have a problem with the english language here, but this sounds to me as: if the elasticity is one then if GDPgrowth is 1 percent then emloymentincrease=employmentgrowth is also 1 percent, if GDPgrowth is 2 percent then employmentgrowth is also 2 percent etc. thats why I assumed that for the elasticity they just take the quotient. A little later they write:

Box 19b shows that globally, the world’s aggregate employment elasticity was between 0.32 and 0.37 during the four time periods between 1992 and 2008. This implies that for every 1 percentage point of additional GDP growth, total employment has grown between 0.32 and 0.37 percentage points during these periods."

That means if one accept that they take a quotient (?) then the quotient has to be Lgrowth/GDPgrowth and not the other way around.

I am just as you unsure what ILO means with $GDP_{growth}$, that was another reason why I wrote "seem". I used \dotG here just as an arbitrary symbol for "the growth of GDP as ever meant by ILO", thats why I wrote Let .... denote the growth of GDP. I should have taken a less ambigous symbol.

You say you would be happy with $GDP_{growth}=log(GDP(year) - log(GDP(year-1))$? I could also imagine it could be e.g. $GDP_{growth}= (GDP(year) - GDP(year-1))/GDP(year-1))$. I actually meanwhile wrote them again twice an email, begging them for a formula.

The important information was for me however that the GDP grows faster than employment thats why I dared to write further. But of course it is good to know "how" faster.

Comment Source:On page 2 they write: > An elasticity of 1 implies that every 1 percentage point of GDP growth is associated with a 1 percentage point increase in employment. I might have a problem with the english language here, but this sounds to me as: if the elasticity is one then if GDPgrowth is 1 percent then emloymentincrease=employmentgrowth is also 1 percent, if GDPgrowth is 2 percent then employmentgrowth is also 2 percent etc. thats why I assumed that for the elasticity they just take the quotient. A little later they write: > Box 19b shows that globally, the world’s aggregate employment elasticity was between 0.32 and 0.37 during the four time periods between 1992 and 2008. This implies that for every 1 percentage point of additional GDP growth, total employment has grown between 0.32 and 0.37 percentage points during these periods." That means if one accept that they take a quotient (?) then the quotient has to be Lgrowth/GDPgrowth and not the other way around. I am just as you unsure what ILO means with $GDP_{growth}$, that was another reason why I wrote "seem". I used \dotG here just as an arbitrary symbol for "the growth of GDP as ever meant by ILO", thats why I wrote Let .... denote the growth of GDP. I should have taken a less ambigous symbol. You say you would be happy with $GDP_{growth}=log(GDP(year) - log(GDP(year-1))$? I could also imagine it could be e.g. $GDP_{growth}= (GDP(year) - GDP(year-1))/GDP(year-1))$. I actually meanwhile wrote them again twice an email, begging them for a formula. The important information was for me however that the GDP grows faster than employment thats why I dared to write further. But of course it is good to know "how" faster.
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7.

On page 2 they write:

"An elasticity of 1 implies that every 1 percentage point of GDP growth is associated with a 1 percentage point increase in employment."

This concurs with what Wikipedia writes on the subject. In the limit, the ratio of percent changes becomes equal to the ratio of the differential of the logarithms.

You say you would be happy with GDPgrowth=log(GDP(year) - log(GDP(year-1))? I could also imagine it could be e.g. GDPgrowth= (GDP(year) - GDP(year-1))/GDP(year-1)).

Well, for my level of mathematical rigour (ahum) those two options are more or less the same if you move one of the round brackets in the first option such that it agrees with what I would be happy with.

Comment Source:Nad wrote: > On page 2 they write: > "An elasticity of 1 implies that every 1 percentage point of GDP growth is associated with a 1 percentage point increase in employment." This concurs with what [Wikipedia](http://en.wikipedia.org/wiki/Elasticity_(economics)#Mathematical_definition) writes on the subject. In the limit, the ratio of percent changes becomes equal to the ratio of the differential of the logarithms. Nad wrote: > You say you would be happy with GDPgrowth=log(GDP(year) - log(GDP(year-1))? I could also imagine it could be e.g. GDPgrowth= (GDP(year) - GDP(year-1))/GDP(year-1)). Well, for my level of mathematical rigour (ahum) those two options are more or less the same if you move one of the round brackets in the first option such that it agrees with what I would be happy with.
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8.
edited March 2011

Barbers are meant to be a thought experiment mostly: even if there were in fact no rise in productivity their wages would still have gone up in line with the general rise in wages. So productivity rises in one part of the economy [less than or equal to the set of non-barbers] spreads out.

Now let us imagine that world production is energy-constrained. [In fact it is only oil-constrained, but that is a complicated story since oil is substitutable, with difficulty, by other energy. Note that the substitution so far has nearly all been coal, with the rise in renewables unable to noticeably slow the rise in coal use, let alone reduce the total.] In an energy constrained world it is important to look at the productivity of energy. Let us first assume that the productivity of energy stays fixed. Now suppose there is a small rise in worker productivity in some section of the workforce. That means those workers get more "right to consume" than they had before [money is a "right to consume" token]. But by assumption that extra stuff they consume uses more energy, and there is only a fixed amount of energy with a fixed productivity. So the rest of the workforce has to get less consumption. So as far as the know-nothing economists sees it: all other workers have lower productivity, and total worker productivity doesn't change.

Now look at it the other way. Say there is a rise in the productivity of energy (i.e. a rise in energy efficiency). A little thought shows that total production goes up. However there is no need for this to appear as an increase in worker productivity if, as at present, there are lots of unemployed. Instead we can increase production at the same worker productivity by expanding the workforce.

In summary: we are used to a world where worker productivity increase requires more energy to be effective, and that extra energy has become available. This leaves us ill-equipped to think about economics in a world where that doesn't happen.

Comment Source:Barbers are meant to be a thought experiment mostly: even if there were in fact no rise in productivity their wages would still have gone up in line with the general rise in wages. So productivity rises in one part of the economy [less than or equal to the set of non-barbers] spreads out. Now let us imagine that world production is energy-constrained. [In fact it is only oil-constrained, but that is a complicated story since oil is substitutable, with difficulty, by other energy. Note that the substitution so far has nearly all been coal, with the rise in renewables unable to noticeably slow the rise in coal use, let alone reduce the total.] In an energy constrained world it is important to look at the productivity of energy. Let us first assume that the productivity of energy stays fixed. Now suppose there is a small rise in worker productivity in some section of the workforce. That means those workers get more "right to consume" than they had before [money is a "right to consume" token]. But by assumption that extra stuff they consume uses more energy, and there is only a fixed amount of energy with a fixed productivity. So the rest of the workforce has to get less consumption. So as far as the know-nothing economists sees it: all other workers have lower productivity, and total worker productivity doesn't change. Now look at it the other way. Say there is a rise in the productivity of energy (i.e. a rise in energy efficiency). A little thought shows that total production goes up. However there is no need for this to appear as an increase in worker productivity if, as at present, there are lots of unemployed. Instead we can increase production at the same worker productivity by expanding the workforce. In summary: we are used to a world where worker productivity increase requires more energy to be effective, and that extra energy has become available. This leaves us ill-equipped to think about economics in a world where that doesn't happen.
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9.
edited March 2011

@Frederik: Frederik wrote: "Well, for my level of mathematical rigour (ahum) those two options are more or less the same .." Depends on what you mean with more or less:

$Log(GDP(year) - log(GDP(year-1))= log(GDP(year)/GDP(year-1)) = log(((GDP(year)-GDP(year-1))/GDP(year-1)) +1) = \Sum_{n=0}^\infty Log^(n)(1)/n!(GDP(year)/GDP(year-1)-1)^n =$ (assuming that Log is Ln)$= 0 + (GDP(year)-GDP(year-1))/GDP(year-1) - 1/2 \cdot ((GDP(year)-GDP(year-1))/GDP(year-1))^2 +.....$

@rks: I think the rise in productivity is also to a great extend due to substituting labour by machines, which also needs extra energy.

Comment Source:@Frederik: Frederik wrote: "Well, for my level of mathematical rigour (ahum) those two options are more or less the same .." Depends on what you mean with more or less: $Log(GDP(year) - log(GDP(year-1))= log(GDP(year)/GDP(year-1)) = log(((GDP(year)-GDP(year-1))/GDP(year-1)) +1) = \Sum_{n=0}^\infty Log^(n)(1)/n!(GDP(year)/GDP(year-1)-1)^n =$ (assuming that Log is Ln)$= 0 + (GDP(year)-GDP(year-1))/GDP(year-1) - 1/2 \cdot ((GDP(year)-GDP(year-1))/GDP(year-1))^2 +.....$ @rks: I think the rise in productivity is also to a great extend due to substituting labour by machines, which also needs extra energy.
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10.
edited March 2011

I edited

putting it into more of the standard Azimuth style, and polishing the English.

I also couldn't resist polishing a bunch of Nad's posts here. Nad: if you pick "Markdown+Itex" when posting a comment, you can use TeX and Markdown to make your comments utterly beautiful. Links, quotes, and math formulas become easy.

As for the more important issue of elasticity, the elasticity of $y$ with respect to $x$ is defined as

$$\frac{d y / y}{d x / x} = \frac{d \ln y}{d \ln x}$$ It sounds like Frederik and I agree on this point now, and are just wondering whether people at the ILO are using the discrete approximation

$$\frac{\Delta y / y}{\Delta x / x}$$ or

$$\frac{\Delta \ln y}{\Delta \ln x}$$ Presumably in many real-world examples the difference between these two approximations is less than the uncertainty caused by noise in the data. Since nonmathematicians like division better than logarithms, I'd guess they use the first one. But that's just a guess.

Comment Source:I edited * [[Economic growth and labour]] putting it into more of the standard Azimuth style, and polishing the English. I also couldn't resist polishing a bunch of Nad's posts here. Nad: if you pick "Markdown+Itex" when posting a comment, you can use TeX and Markdown to make your comments utterly beautiful. Links, quotes, and math formulas become easy. As for the more important issue of elasticity, the elasticity of $y$ with respect to $x$ is defined as $$\frac{d y / y}{d x / x} = \frac{d \ln y}{d \ln x}$$ It sounds like Frederik and I agree on this point now, and are just wondering whether people at the ILO are using the discrete approximation $$\frac{\Delta y / y}{\Delta x / x}$$ or $$\frac{\Delta \ln y}{\Delta \ln x}$$ Presumably in many real-world examples the difference between these two approximations is less than the uncertainty caused by noise in the data. Since nonmathematicians like division better than logarithms, I'd guess they use the first one. But that's just a guess.
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11.
edited April 2011

Nice that you and Frederik agree!

But please note eventually that I had said similar things as you (please see above). That is I wrote that I don't know what ILO is using, but that I think that they probably use the first of the discrete approximations. I used Taylor approximation (for the case of the GDP, please see above) to display how the last two formulas differ.

Comment Source:Nice that you and Frederik agree! But please note eventually that I had said similar things as you (please see above). That is I wrote that I don't know what ILO is using, but that I think that they probably use the first of the discrete approximations. I used Taylor approximation (for the case of the GDP, please see above) to display how the last two formulas differ. Please note that there is also a discussion about this page [here](http://johncarlosbaez.wordpress.com/2011/03/18/energy-the-environment-and-what-mathematicians-can-do-part-1/#comment-4955).
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12.

I got meanwhile an answer from Steven Kapsos from the Economist, Employment Trends Unit at ILO about the calculations. He pointed out that there are details in an article by him at: http://www.ilo.org/public/libdoc/ilo/2005/105B09_255_engl.pdf

I didn't understand a couple of things in that article and wrote him an email about that. I will let you know more when I get his reply.

Comment Source:I got meanwhile an answer from Steven Kapsos from the Economist, Employment Trends Unit at ILO about the calculations. He pointed out that there are details in an article by him at: <a href="http://www.ilo.org/public/libdoc/ilo/2005/105B09_255_engl.pdf">http://www.ilo.org/public/libdoc/ilo/2005/105B09_255_engl.pdf</a> I didn't understand a couple of things in that article and wrote him an email about that. I will let you know more when I get his reply.
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13.

There is now a short sample calculation illustrating discrete exponential growth for a mathematically less inclined audience. I would be grateful to anybody who checks it.

Comment Source:There is now a short sample calculation illustrating discrete exponential growth for a mathematically less inclined audience. I would be grateful to anybody who checks it.