Technological Progress and

Growth (Ch 12)

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”

Suggested practice problems

Read the Appendix at the end of chapter

12

Solve problems 6, 8