Tfp2020 (c uong)

Total Factor Productivity CASED/DEPOCEN Why Does Productivity Matter?Cuong Le Van
Total Factor Productivity
CASED/DEPOCEN
Why Does Productivity Matter?
Cuong Le Van
December 2020

Presentation Plan 2
I. Total Factor Productivity (TFP)
TFP and economic growth
TFP and competitiveness
II. Labour Productivity

I. Total Factor Productivity (TFP) 3
Solow’s result in 1956 (he got the Nobel award for this result)
Using the data of the US economy of the 50 beginning years
of the 20th century Solow ran a regression
∆ ln(Yt) = B + α∆ ln(Kt) + β∆ ln(Nt) (1)
where Yt, Kt, Nt are respectively the GDP, physical capital
and number of workers.
He found that the term B explained more than 87% the GDP
growth of the US economy for these 50 years
This regression is derived from a production function
(Cobb-Douglas function)
Yt = AtKα
t Nβ
t (2)
Obviously, B = ∆ ln(At) in the regression
B is called Solow residue

I. Total Factor Productivity (TFP) 4
TFP is the portion of growth in output not explained by
growth in traditionally measured inputs of labour and capital
used in production
TFP is measured as the ratio of aggregate output (e.g., GDP)
to aggregate inputs (here, this is the quantity kαNβ when
β = 1 − α)
The rate of TFP growth is calculated by subtracting average
growth rates of labor and capital inputs from the growth rate
of output

TFP and economic growth 5
Harrod Model
Consider an inﬁnite horizon closed economy starting with an initial
capital stock k0 > 0:
ct + St = Yt
It = St
kt+1 = kt(1 − δ) + It
St = sYt
Yt = Atkt, where:
ct, St, It are consumptions, saving, investment at date
t, t = 0, 1, . . . , +∞
s ∈ (0, 1) is the exogenous saving rate
kt is the physical capital at date t, δ ∈ [0, 1] is the capital
depreciation rate
Yt is the output

Harrod Model
The production function in this model (Yt = Atkt) is interpreted
either, as a special case of the general form of Cobb-Douglas
function with β = 0
or, the labour Nt has an exogenous rate of growth
Nt = N0(1 + n)t. In this case the TFP becomes
AtNβ
0 (1 + n)βt
or , if β = 1 − α, the function can be written as:
Yt
Nt
= At
kt
Nt
α
i.e. we consider the output per capita as function of capital
per capita

Harrod Model
We then obtain
∆Yt
Yt
=
At+1
At
(1 − δ) + sAt+1
where ∆Yt = Yt+1 − Yt.
Suppose At → A > 0 when t → +∞.
Then ∆Yt
Yt
→ 1 + sA − δ
If sA − δ > 0 then Yt → +∞
If sA − δ < 0 then Yt → 0

Solow Model
The model is quite similar to the Harrod Model, excepted the
production function.
ct + St = Yt
It = St
kt+1 = kt(1 − δ) + It
St = sYt
Yt = Atkα
t L1−α
t , α ∈ (0, 1)
At = a(1 + γ)t
Lt = L0(1 + n)t
Here γ > −1 is the rate of growth of the TFP At, n > −1 is the
rate of growth of the labour force. Both of them are exogenous.

Solow Model
One can show that
∆Yt
Yt
− 1 → g where g satisﬁes
1 + g = (1 + n)(1 + γ)
1
1−α
The longterm rate of growth g of the output depends strongly on
the rate of growth of the TFP At.

The previous models have two limits: the rate of saving is
exogenous and also is the rate of growth of the output. With the
Ramsey model, we can endogeneize the rate of saving but we do
not resolve the question of the exogeneity of the rate of growth of
the output. This question will be resolved with endogenous growth
models.
Ramsey Model
We assume there exists a representative consumer who lives for an
inﬁnite number of periods. She/he maximizes her/his
intertemporal utility +∞
t=0 βtu(ct), β ∈ (0, 1) under the constraints
∀t = 0, . . . , ∞, ct + It ≤ Ft(kt)
kt+1 = kt(1 − δ) + It
k0 > 0, is given
Ft is the production function, concave, strictly increasing,
Ft(0) = 0.

Remark 1 The representative consumer represents the consumers
of all the generations .
Explicitly, ∞
t=0 βtu(ct) = u(c0) + βu(c1) + β2u(c2) + . . ..
If β = 0 we do not care about the future generations. The higher
β the higher importance we attach to the future generations.
Remark 2 As in the Harrod Model, in the Ramsey Model,
implicitly,
either we consider the number of workers is exogenous and
has an exogenous rate of growth
or we consider in fact output per capita and capital per capita

Ramsey Model
If (c∗
t , k∗
t+1, I∗
t )t≥0 is the list of the optimal solutions, the the
optimal rates of saving are s∗
t =
I∗
t
Ft (k∗
t ).
Example 1 Suppose u(c) = ln(c), Ft(k) = Atk.
Let At = At + 1 − δ. Then the optimal output Y ∗
t satisﬁes
Y ∗
t = βt
(A0A1 . . . At)Y0
with Y0 = A0k0. The optimal rate of saving is
s∗
t =
βAt + (1 − δ)(β − 1)
At
≤ β < 1
If At ≤ At+1 then s∗
t ≤ s∗
t+1.

Ramsey Model
We also get
Y ∗
t+1
Y ∗
t
= β(At + 1 − δ)
Now suppose At → A > 0 as t → +∞. In this case
Y ∗
t+1
Y ∗
t
→ β(A + 1 − δ) and s∗
t → s = βA+(1−δ)(β−1)
A
If β(A + 1 − δ) > 1 ⇔ sA − δ > 0 then
Y ∗
t+1
Y ∗
t
→ +∞
If β(A + 1 − δ) < 1 ⇔ sA − δ < 0 then
Y ∗
t+1
Y ∗
t
→ 0
We get the same results as in the Harrod model.

Ramsey Model
Example 2.
u(c) = ln(c), Ft(k) = Akα, α ∈ (0, 1).
In this case the optimal output is
y∗
t+1 = A
1−αt+2
1−α (αβ)
α−αt+2
1−α kαt+2
0
When t goes to inﬁnity, the output y∗
t+1 converges to a steady
state
ys
= A
1
1−α (αβ)
α
1−α
There is no growth. It is due to the fact the production function is
of strictly decreasing returns to scale. However, observe when A
increases, the steady state becomes higher.

How to obtain economic growth or TFP is not a black box
So far the TFP At seems to be a blackbox in a production function
of the type
yt = Atkα
t Nβ
t
where kt, Nt are the number of machines and the number of
workers. We do not take into account the quality f the machines,
nor the skill of the workers. The production function should be
written as
yt = amt(Kt)α
(Nt)β
where mt is the quality of the management, the macroeconomic
environment (stability, law rule), Kt is the eﬀective capital stock,
Nt is the eﬀective labor. Let ζt denote the technology embedded
in the machines, θt denote the working time, ht the human capital
(education, training, health) of the workers. We then have
K = ζ k , N = θ h N

The production function now is
yt = Atkα
t Nβ
t
where the TFP At = amtζα
t (θtht)β
If we assume θt depends positively on wages or bonus (incentive
mechanism) then
yt = Atkα
t Nβ
t (3)
t (θ(wt)ht)β
(4)
The TFP is not anymore a black box. If we invest in the quality of
management, in technology, in training, education, health and if
the salaries of the workers are suﬃciently incentive we will have
high TFP. Using endogenous growth models (Romer, Lucas...) we
can prove there may be economic growth even with strictly
decreasing returns to scale production function.

Call Gt = amtζα
t (θtht)β . Consider an optimal growth model
max
∞
t=0
βt
u(ct), β ∈ (0, 1)
for t ≥ 1 ct + St+1 = GtF(kt)
where ct, St+1 are consumption, saving.

The saving St+1 is shared in investment in physical capital kt+1
and in investment Tt+1 in TFP, i.e. kt+1 + Tt+1 = St+1. Hence
Gt+1 = G(Tt+1). We rewrite the model as follows
max
∞
t=0
βt
u(ct)
for t ≥ 1 ct + St+1 = H(St) ≡ max{G(Tt)F(kt) : Tt + kt = St}
where kt+1 + Tt+1 = St+1.

Assume F(k) = kα, α ∈ (0, 1), G(T) = (λT + 1)ξ, ξ ∈ (0, 1) and
λ ≥ 1. The parameter ξ measures the quality of the TFP
investment technology. The higher ξ the more eﬃcient the TPF
investment. The parameter λ measures the utilization of Tt. For
instance λ is small because of diversion of Tt. We prove
If St ≤ α
ξλ then Tt = 0. It is not optimal to invest in TFP,
when St is small. In this case St = kt.
If St > α
ξλ then Tt > 0 (If St is high enough then it is
worthwhile to invest in TFP.) In this case
H(λ, St) = a
(λSt + 1)α+ξ
λα
where a is a constant depending on (α, ξ. The function H is
increasing in λ when S > α
ξλ

Result: If there exists τ such that Sτ > α
ξλ and if α + ξ ≥ 1 then
the optimal sequence {S∗
t }t, and hence the optimal sequence of
outputs {y∗
t = H(λ, S∗
t )} converge to inﬁnity.
The rate of growth (
y∗
t+1
y∗
t
− 1) is now endogenous. It is obtained by
an optimal share between investing in physical capital and
investing in HC, Technology, Management Quality, incentive
mechanisms. For that reason, we call these types of models
Endogenous Growth Models.

The results above deserve some comments.
The country will wait until some date τ, when the optimal
output generates enough saving Sτ > α
ξλ , before investing in
TFP.
If the diversion of the Tt is high (i.e. λ is close to zero, the
country may never invest in TFP and will not have growth).
If λ < 1 (the diversion exists), the date τ becomes larger. The
country has to wait longer before starting to invest in TFP.
We have H(λ, St) < H(1, St): the output with diversion of
saving is smaller than the one without diversion.

TFP and Competitiveness 22
Competition between physical capital and financial asset
Consider a two-period economy with one consumer, one producer.
In period 0, the consumer has a revenue R0 and consumes c0 and
saves s0. She wants to invest k1 in capital stock, ξ0 in financial
asset. We suppose the numeraire is the consumption good. Let q0
denote of the price of asset in period 0, and q1 is the price of
financial asset in period 1. The consumer wants to maximize the
revenue R1 in period 1. We have R1 = Akα
1 + q1ξ0, α ∈ (0, 1). She
solves the problem
max
k1≥0,ξ0∈R
{Akα
1 + q1ξ0 : k1 + q0ξ0 = s0}
The optimal value ξ∗
0 solves the equation
Aα(s0 − q0ξ∗
0)α−1
=
q1
q0

It is easy to see that ξ∗
0 is a decreasing function of the TFP A
(crowding out eﬀect). In particular, when A is very small, the
optimal value k∗
1 is also very small. Observe if the consumer
anticipates a high value of q1 (speculation), k∗
1 is small,
equivalently, q0ξ∗
0 is close to s0.

Competitition between two countries
Assume we have two countries a, b.
Country a: production function Aakα, saving in period 0: s0.
Country b: production function Abkα, saving in period 0: s0. Each
country maximizes its revenue in period 1: Aa(ka
1 )α + q1ξa
0,
Ab(kb
1 )α + q1ξb
0 .
The two countries exchange consumption good and ﬁnancial
assets. The optimal values are k∗a
1 , ξ∗a
0 , k∗b
1 , ξ∗b
0
and c∗a
1 = Aa(k∗a
1 )α + q1ξ∗a
0 , c∗b
1 = Ab(k∗b
1 )α + q1ξ∗b
0

Competitition between two countries
Equilibrium
1. c∗a
1 + c∗b
1 = Aa(k∗a
1 )α + Ab(k∗b
1 )α
2. ξ∗a
0 + ξ∗b
0 = 0
We then obtain the equilibrium price q∗
1 from these equilibrium
relations.
If the TFP Aa of country a is smaller than the one of country b,
Ab, then at equilibrium
ξ∗a
0 > 0 (Country a buys ﬁnancial asset), ξ∗b
0 = −ξ∗a
0 < 0
(Country b sells ﬁnancial asset)
k∗a
1 < k∗b
1
c∗a
1 < c∗b
1 , i.e. the consumption of country a is lower than the
consumption of country b.

II Labour Productivity 26
Deﬁne Labour Productivity:
πL
=
y
N
Where y is the output and N is the number of workers. Go back to
relations 3, 4 with β = 1 − α. We obtain
yt = Atkα
t N1−α
t (5)
t (θ(wt)ht)1−α
kα
t N1−α
t (6)
From these relations we obtain
πL
t =
yt
Nt
= amtζα
t (θ(wt)ht)1−α kt
Nt
α
(7)

We cite Baumol:
Baumol, W. J., Blackman, S. A. B., Edward N.. Wolﬀ. (1989).
Productivity and American leadership: the long view. MIT press.
"Key long-run drivers Historically, labor productivity growth has
been driven by innovation, better education, and investment in
physical capital. Innovation and investment by private sector
require a growth-friendly environment, with supportive institutions
and policies, including policies that promote macroeconomic
stability and the rule of law ".
Here, we add: incentive mechanisms (w) for the short term.

A wrong statement: Labour productivity is crucial for economic
growth or Increasing labor productivity will increase economic
growth
Consider relations 5 and 6. We then obtain
yα
t = Atkα
t
Nt
yt
1−α
= Atkα
t πL
t
−(1−α)
Hence
yt+1
yt
α
=
At+1
At
kt+1
kt
α πL
t+1
πL
t
−(1−α)
If At+1 = At (no investment in TFP by human capital,
organization, incentive mechanisms, technology), kt+1 = kt (no
investment in physical capital) then
πL
t+1 > πL
t ⇒ yt+1 < yt
One can increase labour productivity by laying oﬀ workers !!

Which kind of workers will be laid off? Obvious answer : inefficient
ones. How to define efficiency of a worker?
Consider a firm using one capital stock K, with quality ζ and two
types of workers N1, N2 with human capitals h1, h2. The
production function is
y = (ζk)α
(h1N1)β1
(h2N2)β2

We say that worker 2 is not eﬃcient if the labour productivity is
higher with just one worker 2.
The extreme case is β2 = 0.
Labour productivity with worker 2: πL = (ζk)α(h1N1)β1
N1+N2
Labour productivity without worker 2: πL = (ζk)α(h1N1)β1
N1
Obviously πL > πL: lay oﬀ worker 2.
More generally
Labour productivity with worker 2: πL = (ζk)α(h1N1)β1 (h2N2)β2
N1+N2
Labour productivity with one worker 2:
πL = (ζk)α(h1N1)β1 (h2×1)β2
N1+1
One can check that πL > πL ⇔ β2ln(N2) < ln(N1+N2
1+N1
).
We can do the same exercise with worker 1 too.

Let β1, β2 be given. We define the functions ν1, ν2 as follows
Given N1, the efficient number of workers 2 is N∗
2 = ν2(N1)
‘ Given N2, the efficient number of workers 1 is N∗
1 = ν1(N2)
Equilibrium N∗
2 = ν2(N∗
1 ), N∗
1 = ν1(N∗
2 )

Thank You

How to calculate ζ, h in the aggregate production function?
Consider the production function
y = A(ζ1k1)α1
. . . (ζpkp)αp
(h1N1)β1
. . . (hmNm)βm
Let
k = k1 + . . . + kp, α = α1 + . . . + αp, β = β1 + . . . + βm
Deﬁne the shares of machines of type j = 1, . . . , p by µj =
kj
k .
Then
ζα
= (ζ1µ1)α1
. . . (ζpkp)αp
⇔ ln(ζ) =
α1
α
ln(ζ1µ1) + . . . +
αp
α
ln(ζpµp)
Observe α1
α + . . . +
αp
α = 1

We do the same to obtain h as
hβ
= (h1N1)β1
. . . (hmNm)βm

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