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Technological Progress and
the Production Function
Let’s denote the state of technology by A and rewrite the
production function as:
Y F K N A ( , , )(+ + +)
A convenient form is
Y F K AN ( , )
Output depends on both capital and labor (K and N), and on the
state of technology (A).
AN = effective labor.
The relation between output per effective worker and capital per effective worker is (given the property of constant returns to scale):
Y Kf
AN AN
, 1Y K
FAN AN
Or:
Interactions between Output
and Capital Accumulation
◦ The relation between output and capital accumulation is.
I S sY
I
ANs
Y
AN
In per effective labor terms:
I
ANsf
K
AN
Or:
Interactions between Output and Capital
Capital per effective worker
and output per effective worker
converge to constant values in
the steady state.
The Dynamics of Capital
per Effective Worker and
Output per Effective
Worker
Figure 12 - 2
We can now give a graphical description of the dynamics of capital per
effective worker and output per effective worker:
If actual investment exceeds the investment level required to
maintain the existing level of capital per effective worker, K/AN
increases.
Starting from (K/AN)0, the economy moves to the right, with the
level of capital per effective worker increasing over time.
In the long run, capital per effective worker reaches a constant level,
and so does output per effective worker. At that point, total output
(Y) is growing at the same rate as effective labor (AN).
In steady state,
◦total output (Y) grows at the same rate as effective labor (gA+gN);
◦Effective labor grows at a rate (gA+gN);
◦Therefore, output per effective labor (Y/AN) grows at 0 rate in
steady state equals.
◦And output per worker (Y/N) grows at rate of technological
progress (gA)
The steady state growth rate of output is still independent of the
saving rate.
Because output, capital, and effective labor all grow at the same rate,
(gA+gN), the steady state of the economy is also called a state of
balanced growth.
Fast growth may come from two sources:
◦ i) A higher rate of technological progress. If gA is higher, balanced
output growth (gY=gA+gN) will also be higher. In this case, the
rate of output growth equals the rate of technological progress.
◦ ii) Adjustment of capital per effective worker, K/AN, to a higher
level. In this case, the growth rate of output exceeds the rate of
technological progress.
Capital Accumulation versus Technological Progress
in Rich Countries since 1950
Table 12-2 Average Annual Rates of Growth of Output per worker and
Technological Progress in Four Rich Countries since 1950
Rate of Growth of Output per Worker (%)
1950 to 2004
Rate of Technological
Progress (%) 1950 to 2004
France 3.2 3.1
Japan 4.2 3.8
United Kingdom 2.4 2.6
United States 1.8 2.0
Average 2.9 2.9
Table 12-2 illustrates two main facts:
First, growth since 1950 has been a result of rapid
technological progress.
Second, convergence of output per worker across
countries has come from higher technological progress
(rather than from faster capital accumulation) in the
countries that started behind (e.g., Japan).
Consider the production function:
𝑌 = 𝐹 𝐾,𝑁, 𝐴 = 𝐹 𝐾, 𝐴𝑁
Assume constant returns to scale: 𝑌 = 𝐾𝛼(𝐴𝑁)1−𝛼
Logarithm transformation:
𝑙𝑛𝑌 = 𝛼𝑙𝑛𝐾 + 1 − 𝛼 𝑙𝑛𝐴 + 1 − 𝛼 𝑙𝑛𝑁
Calculate the change: 𝑑𝑌
𝑌= 𝛼
𝑑𝐾
𝐾+ 1 − 𝛼
𝑑𝐴
𝐴+ (1 − 𝛼)
𝑑𝑁
𝑁
Solve for the change in technology:
1 − 𝛼𝑑𝐴
𝐴=𝑑𝑌
𝑌− 𝛼
𝑑𝐾
𝐾− (1 − 𝛼)
𝑑𝑁
𝑁
So, technological progress is a residual:
1 − 𝛼 𝑔𝐴 = 𝑔𝑌 − 𝛼𝑔𝐾 − 1 − 𝛼 𝑔𝑁
Growth decomposition – technological progress as a “residual”
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