Economic Growth II: Technology,
Empirics, and Policy
Chapter 9 of Macroeconomics, 8th
edition, by N. Gregory Mankiw
ECO62 Udayan Roy
Recap: Solow-Swan, Ch. 7
• L and K are used to produce a final good Y = F(K, L)
• k = K/L and y = Y/L = f(k) are per worker capital and
output
• The population is P, but a fraction u is not engaged
in the production of the final good. Therefore, L =
(1 – u)P.
• Both P and L grow at the rate n.
• A fraction s of Y is saved and added to capital
• A fraction δ of K depreciates (wears out)
Recap: Solow-Swan, Ch. 8
• In the long run, the economy reaches a steady
state, with constant k and y
sf (kt )  (  n)kt
kt 
n 1
Recap: Solow-Swan, Ch. 8
• In the long run, the economy reaches a steady
state, with constant k and y
• Like the per-worker variables, k and y, percapita capital and output are also constant in
the long run
• Total capital (K) and total output (Y) both
increase at the rate n, which is the rate of
growth of both the number of workers (L) and
the population (P)
Recap: Solow-Swan, Ch. 8
Variable
Symbol
Steady state behavior
Capital per worker
k
Constant
Income per worker
y = f(k)
Constant
sy
Constant
c = (1 – s)y
Constant
Labor
L
Grows at rate n
Capital
K
Grows at rate n
Income
Y = F(K, L)
Grows at rate n
Saving and investment
sY
Grows at rate n
Population
P
Grows at rate n
Capital per capita
(1 – u)k
Constant
Income per capita
(1 – u)y
Constant
Saving and investment per capita
(1 – u)sy
Constant
Consumption per capita
(1 – u)c
Constant
Saving and investment per worker
Consumption per worker
The sad lesson of Solow-Swan
• It is an undeniable fact that our standards of
living increase over time
• Yet, Solow-Swan cannot explain this! Why?
• Solow-Swan relies on capital accumulation as
the only means of progress
• Therefore, the model’s failure to show
economic progress indicates that we must
introduce some means of progress other than
capital accumulation
Technological Progress
• Maybe Solow-Swan fails to show economic
progress because there is no technological
progress in it
• We need to create a theory with technological
progress
• But how?
Technological Progress
• A simple way to introduce technological
progress into the Solow-Swan model is to
think of technological progress as increases in
our ability to multitask
Technological Progress
• Imagine that both population and the number
of workers are constant but that steady
increases in the workers’ ability to multitask
creates an economy that is equivalent to the
Solow-Swan economy with steadily increasing
population
Technological Progress
• In such an economy, total output would be
increasing—exactly as in the Solow-Swan
economy with steady population growth—but
without population growth
• That is, under multitasking technological
progress, per capita and per worker output would
be steadily increasing
• In this way, a simple re-interpretation of the
Solow-Swan economy gives us what we were
looking for—steadily increasing income per
worker
Efficiency of Labor
• Specifically, section 9−1 defines a new variable
– E is the efficiency of labor
– Specify some date in the past, say 1984, and
arbitrarily set E = 1 for 1984.
– Let’s say that technological progress has enabled
each worker of 2011 to do the work of 10 workers
of 1984.
– This implies that E = 10 in 2011.
Efficiency of Labor
• The old production function F(K,L) no longer
applies to both 1984 and 2011
• Suppose K = 4 in both 1984 and 2011
• Suppose L = 10 in 1984 and L = 1 in 2011
• The old production function F(K,L) will say that
output is larger in 1984
• But we know that output is the same in both
years because just one worker in 2011 can do the
work of 10 workers of 1984
• We need a new production function: F(K, E ✕ L)
Y = F(K, E ✕ L)
• In other words, although the number of
human workers is 10 in 1984 and 1 in 2011,
the effective number of workers is 10 in both
years,
• and that’s what matters in determining the
level of output
• The effective number of workers is E ✕ L
Efficiency of Labor
• Assumption: the efficiency of labor grows at
the constant and exogenous rate g
Production
• As the production of the final good no longer
depends only on the number of workers, but
instead depends on the effective number of
workers, …
• … we replace the production function
Y = F(K, L) by the new production function
Y = F(K, E ✕ L)
From “per worker” to “per effective
worker”
• Similarly, we will now redefine k, which used
to be capital per worker (K/L), as capital per
effective worker: k = K/(E ✕ L)
• Likewise, we will now redefine y, which used
to be output per worker (Y/L), as output per
effective worker: y = Y/(E ✕ L)
From “per worker” to “per effective
worker”
• As a result of the redefinition of k and y, we
still have y = f(k), except that the definitions of
y and k are now in “per effective worker” form
• sy = sf(k), is now saving (and investment) per
effective worker
• Only the growth rate of effective labor is
slightly different
From “per worker” to “per effective
worker”
• In Chapter 8, what mattered in production
was L, the number of workers, and the growth
rate of L was n
• Now, however, what matters in production is E
✕ L, the effective number of workers, and the
growth rate of E ✕ L = growth rate of E +
growth rate of L = g + n
From “per worker” to “per effective
worker”
• Recall from Chapter 8 that the break-even
investment per worker was (δ + n)k
• This will have to be replaced by the breakeven investment per effective worker
• We can do this by redefining k as capital per
effective worker (which we have already done)
and by replacing n by g + n
• Therefore, break-even investment per
effective worker is now (δ + n + g)k
Dynamics: algebra
Ch. 8 No technological change
Ch. 9 Technological Progress
sf (kt )  (  n)kt
kt  kt 1  kt 
n 1
sf (kt )  (  n  g )kt
kt  kt 1  kt 
n  g 1
Dynamics: graph
• As in Ch. 8, in the long run, k and y reach a
steady state at k = k* and y = y* = f(k*)
sf (kt )  (  n  g )kt
kt 
n  g 1
Describing the Steady State
• We just saw that k is constant in the steady
state
• That is, k = K/(E ✕ L) is constant
• Therefore, in terms of growth rates,
kg = Kg – (Eg + Lg) = Kg – (g + n) = 0
• Therefore, the economy’s total stock of
capital grows at the rate Kg = g + n
Describing the Steady State
• Capital per worker (K/L) grows at the rate Kg –
Lg = g + n – n = g
• Therefore, the per-worker capital stock,
which was constant in Chapter 8, grows at
the rate g
• As each worker’s ability to multitask increases
at the rate g, the capital used by a worker also
increases at that rate
Describing the Steady State
• y = f(k) is constant in the steady state
• That is, y = Y/(E ✕ L) is constant
• Therefore, in terms of growth rates,
yg = Yg – (Eg + Lg) = Yg – (g + n) = 0
• Therefore, the economy’s total output grows
at the rate Yg = g + n
– Recall that this is also the growth rate of the total
stock of capital, K.
Describing the Steady State
• Output per worker (Y/L) grows at the rate Yg –
Lg = g + n – n = g
• Therefore, the per-worker output, which was
constant in Chapter 8, grows at the rate g
– Recall that this is also the growth rate of perworker capital, K/L.
Progress, finally!
• We have just seen that if we
introduce technological
progress in the Solow-Swan
theory of long-run growth, then
in the economy’s steady state
– Per-worker output (Y/L) increases
at the rate g, which is the rate of
technological progress
• This is a major triumph for the
Solow-Swan theory
Solow-Swan Steady State
• Table 9.1 Steady-State Growth Rates in the
Solow Model With Technological Progress
Solow-Swan Steady State
• Remember from Chapter 8 that, when the
production function follows the Cobb-Douglas
form, the steady state value of k = k* was given
by the formula
• Now the formula changes to
 sA 

k  
  n g 
*
1
1
Technological Progress: where does it
come from????
• But a puzzle remains …
• So far, the rate of technological progress, g,
has been exogenous
• We need to ask, What does g depend on?
• We need to make g endogenous
Endogenous Technological Progress
• Remember that in Chapter 8 we had
distinguished between the population (P) and the
number of workers (L)
• We had defined the exogenous variable u as the
fraction of the population that does not produce
the final good
– Therefore, we had L = (1 – u)P or L/P = 1 – u
• In Ch. 8 we had interpreted u as the long-run
unemployment rate
• Now, we’ll reinterpret u as the fraction of the
population that does scientific research
Endogenous Technological Progress
• Once u is seen as the fraction of the
population that is engaged in scientific
research, it makes sense to assume that …
• Assumption: the rate of technological
progress increases if and only if u increases
• This assumption is represented by the
technology function g(u)
– Example: g(u) = g0 + guu
Endogenous Technological Progress
• We now have a theory that gives an answer to
the following question: Why is growth in living
standards slow in some cases and fast in
others?
• Growth in per-worker output is fast when u is
high.
• That is, our standards of living grow rapidly
when we invest more heavily in scientific
research
Productivity Slowdown
• There was a worldwide slowdown in economic
growth during 1972-1995. Why?
Growth Accounting
• Table 9.3 Accounting for Economic Growth in
the United States