Technological Progress and Growth: A Mathematical Approach
There are several essential aspects due to technological progress:
- It leads to more significant quantities of output from given quantities of capital and labor. For example, a new tesla machine in the factory allows the efficiency of production, which will lead to a larger production of model 3.
- It leads to a better product or a new one that is more useful in daily activities (which leads to a more productive one). Like a lamp that Edison invented, people were able to work or study in the middle of the nights.
- Et cetera (I hope we can imagine these aspects in real life😊)
These technology aspects lead as a multiplier on human productivities, etc from the given products and services. Edison invented a lamp, made people able to work or create the new invention in the dark, leading to another progress and growth on the human being.
As we can see, the state of technology plays an essential role in increasing output for a given amount of capital and labor. We can think the state of technology as a variable that tells us how much output can be produced, based on production function (which later we extend this), as:
The equation stands as Y (output) depends on K (capital), N (labor), and A (The state of technology). An improvement from one of these variables will lead to an increase in output. Nevertheless, according to Blanchard (2017), it will be more convenient to use a more restrictive equation that the preceding, like this:
Why? This equation leads us to a better understanding of technology’s effect on the relation between output, capital, and labor. As we’ve seen the equation states that production (Y) depends on capital (K) and labor multiplied by the state of technology (A x N). The equation implies that we can think of technological progress in two equivalent ways (Blanchard, 2017):
· technological progress reduces the number of workers needed to produce a given amount of output. Such as 2A = 1/2N
· However, technology also increases the output that can produce with a given number of workers. For example, if A doubles (and the N constant), the output will greater than before. We can think of AN as the amount of effective labor, as long as the state of technology doubles (A), then the economy had twice as many workers.
To make it easier to mention, let’s call the last equation as the extended production function. Ohya, I just remembered that it’s reasonable to assume constant returns to scale. About the doubling (x), both amount of the state of technology (AN), and capital (K), will lead to a double amount on output (Y), like this:
It is also reasonable to assume decreasing returns to each of the two factors (capital and effective labor). Given effective labor, a capital increase is likely to increase output but at a decreasing rate, vice versa. (For more comprehensive on constant returns to scale, it better to look over on chapter 11 in Blanchard’s Macroeconomics.
From the equation, what conclusion that can we get? The state of technology influences the output by multiplying the worker’s effect on output and fabricating a more rapid growth.
By the way, we should think about output per worker and capital per worker because the steady-state of the economy was a state where output per worker and capital per worker were constant. However, it is more convenient to look at output per effective worker and capital per effective worker, as the same reason: in steady state, output per effective worker and capital per effective worker are constant.
How we justified the relation? We can take x = 1/AN in the preceding equation that gives:
Alternatively, we can define the function f so that (K/AN) = F(K/AN, 1):
In other words, output per effective worker (the left side) is a function of capital per effective worker (the function on the right side). The relation between output per effective worker and capital per effective worker is drawn at the graph below:
We can interpret from the graph that an increase in K/N led to an increase in Y/N, but at the decreasing rate. Also, an increase in K/AN lead to increases in Y/AN, but at a decreasing rate.
So, I think this is the end of my class note. I presume this articles only as a note, so yeah; all of you may found something similar from this article with the textbook. Nevermind. For all of you who spend time to read, Thank you very much.
Source:
Blanchard, Olivier. 2017. Macroeconomics. England: Pearson
