Charles Jones: �US Economic Growth in a World of

Ideas� and other Jones Papers

January 22, 2014

U.S. GDP per capita, log scale

Old view: therefore the US is in some kind of Solow steady state(i.e. Balanced Growth Path: BGP ) with constant technologicalprogress.

Jones critique of this:

we know that rates of investment in human capital have risen overthis period. [see next 3 slides]

If technological progress were constant, this would lead to hightransitional growth

Also, investment in R&D has risen in this period, which might behard to reconcile with constant technological progress.

Therefore, it can't be a steady state.

Factors of Production in the United States

Average U.S. Educational Attainment, persons aged 25 and

over

Research Intensity in the G-5 countries

Jones' Model

Production function: Y = AσKαH1−αY

HY is human capital devoted to producing output ... in practice,will assume that Hy/H = Ly/Y , even though this is probably nottrue

Note: since A is never observed, I don't understand the point ofhaving σ in the model.

Jones' Model - continued

Capital accumulation:K = sKY = dK

Human capital per worker is determined by the usual Mincerformulation:

h = eψl where l (as script lowercase �L� in the paper) is years ofschooling

Technology Production Function

A = δHλAA

φ

where HA = hLA is the total human capital devoted to R&D

This is the Jones technology production function from hisearlier papers. Two key components [next two slides]

The Parameter φ

A = δHλAA

φ

e�ect of existing knowledge on growth of new knowledge.

The standard model assumed that φ =1. That said that newinventions are proportional in size to the existing stock ofknowledge.

Jones says that this is an arbitrary assumption. If anything, hesays, φ = 0 is the more natural case. This would say thatinventions are of the same size.

You could even argue for φ < 0. This would say that the moreknowledge there is, the smaller every new invention is � forexample, if all of the good ideas had been taken. This is calledthe �shing out e�ect.

On the other hand, we might expect that φ > 0 because of the�better tools� e�ect, which is to say that when technology isbetter, each worker has better tools for inventing newtechnologies.

The Parameter λ

A = δHλAA

φ

l captures the idea that having more people working oninvention does not necessarily lead to a proportional increasein invention. The standard case is that l = 1.

But if there is crowding out (i.e. twice as many scientists doesnot lead to twice as many inventions) then l < 1.

Implications

A = δHλAA

φ

If φ < 1, then if HA is constant, growth rate of A willasymptote to zero.

In this case, having constant growth of A will require constantgrowth of HA. Call this rate of growth n

AA= δHλ

AAφ−1 di�ertiate w.r.t. time...

d( AA)

dt= λδHλ−1

A Aφ−1HA+δ(φ−1)HλAA

φ−2A = λn AA+(φ−1)

(AA

)2

Along the Balanced Growth Path, technology growth is constant,so the above is zero, so

AA= λ

1−φn

Measuring HA

Jones allows for technology to be advanced by R&D in all of thecutting edge countries. So, while the other variables above were allsupposed to have country subscripts (but I was too lazy to putthem in), the HA was not (nor was the A). Speci�cally:

HA =∑M

i=1hθi LA,i

where θ > 0 (I don't really understand this part. I would just makeθ = 1).

labor market adding up

N(1− sh) = LY + LA

where sh is the fraction of the adult population that is in school

de�ne

sR = LA/N

Rewriting the Production Function

Original Production Function: Y = AσKαH1−αY

de�ne y = Y /L and k = K/L

so we can rewrite production function as

y =(KY

) α1−α LY

LhA

σ1−α

This way of re-writing is similar to Hall and Jones developmentaccounting. The logic is that along a BGP where sR and share constant, the �rst three terms on the RHS will also beconstant.

Slight simplifying cheat: Jones de�nes output per worker asy = Y /LY instead of putting L in the numerator. This doesn'tmatter much because share of labor force doing R&D is verysmall.

Balanced Growth Path

Rewritten production function: y =(KY

) α1−α LY

LhA

σ1−α

Along the BGP, everything on the RHS except A is constant, so

y = σ1−α A = γn

where γ = λ1−φ

σ1−α

but we know that we are not on a balanced growth pathbecause h is in fact rising.

Growth of h

human capital production function (Mincer): h = eψl (l isyears of schooling)

take logs and di�erentiate w.r.t. time: h = ψ dldt

ψ is the (Mincerian) return to education. He uses 7%.dldt

averages .090 years per year

so h will be assumed to be .090× .07 = .0063

Average Annual Growth Rates, 1950 - 1993

Growth of A calculated as a residual (top half of table)

Note: this is really σ/(1− α)A and not A that is being calculated,but we will deal with that in a minute.

Derivation of γ

BGP equations: y = σ1−α A = γn where γ = λ

1−φσ

1−αJones makes normalization that σ = 1− α , which makes mewonder why we have been carrying around σ all this time....

so now we have: y = A = γn where γ = λ1−φ

that holds when R&D labor force has been growing at aconstant rate for a long time

that is not true � but for lack of anything better, say it weretrue. Then we could derive γ by dividing the growth of A by n

[Note: on the BGP, y = A, even though that is not true in ourdata because h is growing. But we asssume A is on a BGP, sowe use its growth rate.]

This yields γ = .0146/.0483 = .30

Jones has a fancier econometric section

Transitional vs. Non-Transitional Growth

Rewritten production function: y =(KY

) α1−α LY

LhA

σ1−α

Take logs and di�erentiate w.r.t. time:

y =(

α1−α

)(k − y) + h +

ˆ(LyL

)+(

σ1−α A− γn

)+ γn

Note we have subtracted and added the term γn on the right side.This is the only term that is non-zero on the BGP. The second tolast term is technology growth in excess of what will obtain on theBGP.

See Table [next slide]: output growth decomposed into transitionaland non-transitional parts.

Accounting for U.S. Growth, 1950 - 1993

What are We to Make of Growth Being Constant?

remember our intial motivating fact

Jones has shown that we are not on a BGP. Both h and�excess idea growth� are important (and K/Y growth before1950, maybe)

Jones introduces �Constant Growth Path� as an alternative toBGP

Constant Growth Paths

BGP arises when rates of accumulation (saving rate, level ofeducation, fraction of labor force doing R&D) are constant(and population is growing).

CGPs arise when all these accumulation rates are themsevlesgrowing at a constant rate (except for years of education,which has to have a constant time derivative).

Of course, that can't go on forever like a BGP can, but itcould last a while

Observed output could be a CGP

My view: yes, it could be, but it could also be a result of anyof a zillion other patterns of varying accumulation rates(ACGP - Arbitrary Consant Growth Path).

Deeper question: why would either CGP or ACGP happen?

Jones Conclusions

Jones says: constantcy of growth of output (key motivatingfact) is an illusion

US has experienced �grand traverses� in accumulation rates:

investment rate rose in 19th centuryschooling rose in 20th; R&D share still rising (temporarily)

Once all this traversing is over, growth will slow down.

Brown University critique: Hello? Why take population growthas �xed and permanent while allowing all these otheraccumulation rates to change?