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Economic Growth and Investment, Technological Progress: Solow Growth Model

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Economic Growth and Investment, Technological Progress: Solow Growth Model

This article will discuss the principles and important conclusions of the Solow Growth Model, and discuss the relationship between economic growth, investment and technological progress.

What is the essence of economic growth? The Solow Growth Model, proposed by MIT scholar Robert Solow in the 1950s, will provide an idea for this question. For his outstanding contributions to economic growth, Solow was awarded the 1987 Nobel Prize in Economics. Let’s take a brief look at this model.

Cobb-Douglas social production function

According to neoclassical economics, society conducts large-scale production by inputting various factors, which can be roughly divided into two categories: capital (K) and labor (N), and technical level (A) is also a determinant of output. which is

Y=A * K^α * N^(1-α) (0

The above is the Cobb-Douglas social production function at the technological scale A, where Y is the output level, A is the technological level, K is the capital level, N is the labor level, α is the weight of capital, and 1-α is the weight of labor.

The above formula says that the level of social output is determined by three parts. The first is the technical level, the higher the technical level, the higher the output; the second is the capital level, the higher the capital level, the higher the output; the last is the labor, the higher the labor level, the higher the output level.

It is obvious that the marginal productivity of capital and labor decreases (0

To further study productivity (output per capita), this social production function simplifies to

Y= K^α * (AN)^(1-α)

AN is the product of skill level A and labor N, that is, effective labor. That is to say, the effective labor level was also backward in the age of technological backwardness, but in today’s technologically advanced era, the effective labor level is higher, the social production efficiency is high, and therefore the output level is also high.

The above formula can be transformed into

Y/(AN) is the output per effective labor. We hope that the higher the level, the better, because it represents the high output efficiency under a certain technical level A and labor level N. K/(AN) is the effective capital per labor, and we also hope that the higher the level, the better, because it represents the high efficiency of capital utilization under a certain technical level A and labor level N.

The above formula means that the output per effective labor Y/(AN) is an increasing function of the capital K/(AN) per effective labor. If we want to increase output per effective labor, we must increase capital per effective labor. How to increase effective capital per labor? Our first thought was to increase investment. How to increase investment?

Saving equals investment

In the long run, higher investment requires higher savings. The long-term here refers to a period of more than 50 years. In the process of studying economic growth, Solow considered a longer-term time frame. All the long-term in this paper refer to the time frame in which Solow conducted his research.

The IS relationship refers to the equivalence relationship between saving and investment.

Output Y = Consumption C + Investment I + Government Expenditure G + Net Export NX

can be transformed into

(Output Y – Tax T – Consumption C) + (Tax T – Government Expenditure G) = Investment I + Net Export NX

The first term on the left (output Y-tax T-consumption C) is private saving, the second (tax T-government spending G) is public saving, and the first investment I on the right refers to domestic investment, The second net export, NX, can be understood in a sense as investment abroad.

The result is that the sum of private and public saving should equal the sum of domestic and foreign investment.

Solow Growth Model

Solow’s research shows that effective capital per labor increases as follows:

On the left is the change in effective capital per labor, if positive, it means that the effective capital per labor in year t+1 has increased compared to year t.

The first item on the right is investment, s is the savings rate, and the product of the savings rate and the output per effective labor Y/(AN) refers to the part of output that is converted into investment.

The second item on the right is the depreciation of several effective labor capital K/(AN), so it is a negative sign. δ is the depreciation rate of capital, and the level of capital will naturally decline due to depreciation. GA is the growth rate of technology, and GN is the growth rate of labor. It can be understood that the growth of technology and labor will lead to a decrease in the effective capital per labor K/(AN), because AN is in the denominator, and an increase in the denominator will lead to a decrease in the fraction.

To sum up, the effective capital per labor will increase due to investment on the one hand, and decrease due to capital depreciation, technological progress, and population growth on the other hand. The combined effect of these two aspects will cause the level of effective capital per labor to fluctuate over time.

Solow believes that in the long run, capital per effective labor will stabilize. This is an ideal result, that is, capital per effective labor and output per effective labor adjust and stabilize in the long run.

When the effective capital K/(AN) does not change or changes very little, the right-hand side of the above equation is equal to 0, so

At this time, the subscript t can be removed, and then combined with Y/(AN )= [K/(AN)]^α, the result is

This is a simplified version of the Solow Growth Model. What does it say?

Static Analysis and Dynamic Analysis

First of all, under the condition that other conditions remain unchanged, increasing the savings rate is conducive to improving the long-term equilibrium level of capital KAN per effective labor and output YAN per effective labor.

However, the savings rate cannot be increased indefinitely. In the composition of output, the household sector typically spends part of its after-tax income on consumption and another part on saving.

Therefore, this conclusion can be further expressed as, in some countries with low savings rate, an appropriate increase in the savings rate will help to increase the effective capital per labor. In other words, long-term adherence to investment is conducive to increasing the capital stock and thus promoting economic development.

Secondly, output per effective labor Y/(AN) and capital per effective labor K/(AN) tend to the long-term equilibrium level, which means that the output growth rate, capital growth rate, labor growth rate, and technological progress rate remain stable relationship, that is

GY=GK=GA+GN (simple mathematical inference, proof omitted)

From another perspective, economists are more concerned about the growth rate of per capita output Y/N (on the premise that the ratio of the labor force to the total population is roughly unchanged, the per capita output Y/N can be used as the growth rate of per capita output indicator), i.e.

GY – GN = GA

That is, the growth rate of output per capita should be roughly equal to the rate of technological progress in the long run. This highlights the importance of technological development.

Finally, when we further consider the components of output growth rate, Solow proposes an important concept – Solow Residual.

Solow Residual = GY – α * GK – (1-α) * GN

Solow residual, also known as total factor productivity, refers to the contribution of technological progress to the output growth rate in addition to the contribution of capital and labor to the output growth rate.

To sum up, the Solow Growth Model emphasizes the positive effect of saving (investment) on the per capita output level of effective labor, and the importance of technological progress on the per capita output growth rate, and gives a measure of the impact of technological progress on output. An indicator of the contribution level of growth rate – Solow Residual.

The author finally points out that in Y/(AN )= [s/(δ+GA+GN)]^[α/(1-α)]In the formula, each factor on the right side is changing, and these changes lead to changes in the long-term equilibrium level. We must understand the so-called equilibrium of economists from a dynamic point of view.

Reference: Olivier Blanchard “Macroeconomics fifth edition”

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