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?