# Explaining the Size of the Economy

## 1 The Questions

We learned how to measure the size of the economy using the concept of real GDP. Now, we want a theory -or, a model - that can explain how an economy arrives at its measured size.  In particular, we are interested in the following questions.

• What is a country's economic standard of living?
• Some countries are rich, and some are very poor.  Why is that so?
• Some emerging economies, like China, seem to grow much faster than developed economies like the United States and Germany. Does that mean that eventually emerging economies will surpass the U.S. as the most productive economy?
• What are the growth prospects for the U.S. and China?
• Are there sets of policies that governments can institute that can speed up economic growth and enhance a country's standard of living?
• Many developed economies are aging as the "baby boom" generation hits retirement age. What are the implications for economic growth and standards of living?

## 2 The Approach

To answer these questions, we need a theory.  Our theory is based on the Solow Growth Model, developed by the economist Robert Solow in the 1950's. There are two components to the theory:

1. A theory of production based on the concept of a production function.
2. A theory of how capital (i.e. machines, equipment, factories, software, etc.) is accumulated.

Before continuing, we first need to be specific about what we mean by "standard of living." A country's income is how much it produces, i.e. its output or real GDP. However, this doesn't give a good sense of a country's overall standard of living as you could have a large economy with most of the population living in subsistence poverty and one very wealthy person. The income of this hypothetical country is, therefore, very close to the income of this one wealthy individual. But, if we looked at the country's average income, then the country would look very poor.  For that reason, we define standard of living as output or income per capita.

Definition. A country's economic standard of living can be measured by output per capita, or equivalently, income per capita, and real GDP per capita.

The graph below plots real GDP per capita for a variety of countries.  Use the pull-down menu to select the country you are interested in seeing.  In each case, you will see that countries' standards of living are improving over a several decade period of time.  However, the rate of growth can vary, with periods of rapid improvement in standards of living and slower periods of growth for stretches of time.  The graph for China is especially interesting.  It looks like real GDP for countries like the U.S., U.K., and Germany from the Industrial Revolution onward; except, for the case of China it is over a 50 year period rather than a 200 year period!

## 3 The Production Function

When an economy is functioning smoothly, how much it can produce depends on the nature of the production process, i.e. how inputs to production are transformed into output that can be used as consumption by households (foreign and  domestic) and governments, or by firms as an investment.  Later on in the course, we study the business cycle (i.e. economic recessions and expansions) when the economy does not operate smoothly.  When the economy is functioning smoothly we call that the long-run and the business cycle the short-run.  This is because we expect the temporary ups and downs of the business cycle to wash out over a sufficiently long stretch of time.

Definition. A production function is a mathematical representation of the production process.  That is, it describes how inputs are combined with production technology to produce output or real GDP.

Inputs are factors in the production process.  Hopefully, your production function included things like labor, capital, land, human capital, and oil, and left out things that are better described as consumption goods - that is, things that are not obviously useful for producing other goods - like meat, roller coasters, houses, and surfboards.

As you can see from this simple interactive exercise, there are various types of inputs to the production process.  These can, in general, be divided into two groups:

1. Inputs that can be accumulated: such as labor, capital, and human capital.
2. Inputs that are a part of a nation's endowment: such as land and natural resources (e.g. oil, precious metals, minerals, etc.)

Although we will include both types of inputs in our production function, our theory focuses on the first type, those inputs that a country can accumulate over time.  Now, of course, labor can be increased, or decreased, depending on the birth/death rate and immigration/emigration policies.  But, these tend to be slow to change: a higher birth rate won't show up in the labor force for 16-18 years, at least! So, our theory will really be about how economies acquire physical capital (e.g. factories, machines, software) and human capital (e.g. education, vocational training).

The simple interactive exercise above let you select a few inputs for the production function.  Here's the general representation that we use in the class: $$Y=AF\left( K,L,H,N\right)$$ where

• $$Y=$$ output, income, or real GDP. Remember these are all equivalent ways of saying the same thing (Hint: recall the circular flow diagram).
• $$A=$$ the state of technology.  This is a key variable that describes how technology evolves to improve the production process. We write $$A$$ outside the production function $$F$$ because we think of technology as a general purpose input that makes all of the inputs more productive. So an increase in technological progress is as if all of a country's inputs have increased.
• $$K=$$ the physical capital stock.
• $$L=$$ the labor force.
• $$H=$$ the human capital stock.
• $$N-$$ natural resources, such as land, minerals, oil.

Our primary interest is a theory to explain how economies can increase their average incomes, $$Y/L$$, which is also called productivity as it gives the average output per worker.  To come up with a meaningful theory we need to make two assumptions on the characteristics of the production function.  Specifically,

1. Constant returns to scale: "Double the inputs, double the output."
2. Diminishing marginal returns: an increase in a single input increases output, but at a diminishing rate.

These two features of the production function, and how they lead to the graph of the production function are discussed in the following short (7 min) video.

The following interactive graphic gives you the opportunity to see the production function in action.  There are two important forces that influence the graph of the production function (in $$Y/L$$ space, as seen in the video):

• Capital's share in production.  There are a variety of factors in the production function.  How a change in a single input translates into additional output depends on how important that input is in the production process.  Since we are focusing on capital per worker as an input to productivity (output per worker), we can alter capital's importance in production and see how it affects the shape of the production function.  A simple way to measure capital's importance in production, is it's share in production, i.e. the amount of output directly attributable to capital.  You can slide the button to increase or decrease capital's share.
• Technology.  In the production function, an increase in the state of technology makes all inputs more productive.  You can slide the button to increase or decrease the state of technology.

### Self-check Quiz

In the graph above, the production function has a slope, starting from the origin, that becomes progressively flatter. What property of the production function accounts for this?     Hint  |  Answer
 productivity constant returns to scale increasing returns to scale diminishing marginal returns

Hint: watch the video.

### Self-check Quiz

In the graph above, what happens to the slope of the production function if there is an increase in capital's share of production?     Hint  |  Answer
 becomes flatter becomes steeper becomes negative no change

Hint: also try graphing it on a piece of paper.

## 4 Solow Growth Model

Having developed a theory of production, based on the production function, we are now ready to explain how an economy determines its productivity level in the long-run.  The production function tells us given an amount of capital per worker, human capital per worker, and natural resources per worker, the level of  productivity (output per worker) in an economy.  But, we need a theory that tells us how an economy accumulates the inputs.

• capital per worker: discussed in this section on the Solow growth model.
• human capital per worker: discussed in the next section.
• natural resources per worker: this is fixed by the physical characteristics of the particular country.

So, the Solow Growth Model is about how an economy accumulates capital per worker.  The specifics are discussed below but, first, a reminder that the capital stock comes from Investment. Recall from the GDP accounting that $$Y=C+I+G+NX$$ and $$I$$ are investment goods, i.e. production of new plants, equipment, software, factories, etc.  Thus, an economy accumulates new capital through investment.

What, then, determines the level of investment.  We can use the GDP accounting identity to see that the following identity holds:

Investment=Savings

Take the GDP accounting identity, and manipulate it in the following way:

$$\begin{eqnarray*} Y&=&C+I+G+NX \\ \Rightarrow Y-C-G-NX&=& I\\ \Rightarrow S&=&I \end{eqnarray*}$$

Why is $$Y-C-G-NX$$ equal to savings?  Well, a country's savings is whatever output left over that is not consumed.  Output is consumed by households ($$C$$ ), governments ($$G$$ ), and by people in other countries via net exports ($$NX$$ ).  Therefore, $$Y-C-G-NX=S$$, i.e. savings.  The identity tells us that savings equals investment.  This makes sense.  Whatever output that isn't a consumption good, must be a capital good.

Why is this identity useful?  It turns the capital accumulation question from figuring out investment to figuring out savings.  If we spend just a minute thinking about it, it's pretty easy to come up with a theory of savings.  This, as well as other factors affecting capital accumulation are discussed in the following video.

Use the interactive graphic below, to see how changes in the savings rate, depreciation rate, capital's share in production, and technology affect the long-run equilibrium for the economy (the red dot).

### Self-check Quiz

In the graph above, what happens to the long-run equilibrium productivity of the economy if the savings rate increases?     Hint  |  Answer
 stays the same, no effect increases decreases does not exist

Hint: also try graphing it on a piece of paper.

### Self-check Quiz

In the graph above, what happens to the long-run equilibrium productivity of the economy if the depreciation rate increases?     Hint  |  Answer
 stays the same, no effect increases decreases does not exist

Hint: also try graphing it on a piece of paper.

### Self-check Quiz

In the graph above, what happens to the long-run equilibrium productivity of the economy if the state of technology increases?     Hint  |  Answer
 stays the same, no effect increases decreases does not exist

Hint: also try graphing it on a piece of paper.

## 5 Applications

Now that we understand how an economy grows: savings leads to investment; investment leads to capital accumulation; capital accumulation leads to higher average incomes or productivity; higher average incomes leads to more savings... And, we also understand how there is a long run equilibrium where the capital stock per worker is constant, and this occurs where savings equals depreciation, i.e. all investment is just to replace capital that has broken down.  So we are now in a position to answer some of the questions posed at the top of this page.

### Income Convergence

At the top of this page, we asked why it is that some emerging economies, like China, seem to grow much faster than developed economies like the United States and Germany.  A closely related question is, why do developed economies like the United States and Germany seem to have similar income levels?  To answer this question, we will see that the Solow model predicts income convergence for similar economies.

Imagine the U.S. and Germany following the end of World War II.  These are similar countries in terms of technologies and production processes, and have the same household savings rates.  So, what is different between them in 1945?  Well, a very large fraction of Germany's capital stock was destroyed during World War II.  But now, 70 years later, the two economies are virtually identical in terms of average incomes and growth.  The Solow model can explain this phenomenon of income convergence, and makes predictions about China's future income growth prospects.

In the above interactive graphic, the growth path for the U.S. and Germany post-WWII is illustrated. (In the graphic, the red and blue dots automatically move up the production function.  Depending on your internet connection, they may not move smoothly.  You can use the slide bar to manually move the graphic through time.)  They share a common production function, technology, saving rate and depreciation rate.  The green dot, then, is the long-run equilibrium productivity and capital stock per worker for both the U.S. and Germany.  Starting in 1945, in this example, Germany (red dot) has zero capital stock and the U.S. (blue dot) has a capital stock per worker of 0.25.  Thus, at 1945 the U.S. has a higher productivity than Germany.  Then the growth process, as described above and in the video, takes place with both countries investing in their capital stock.  They both continue to grow until they reach the long-run equilibrium.  This is convergence in incomes.  Even though the U.S. started at a higher productivity, they both end up at the same level of development.  This is a big insight of the Solow model, and can explain why most developed countries have very similar average incomes.

The interactive graphic also can explain why some countries, while developing, grow faster than the U.S.  Notice in the graph that Germany starts way behind the U.S. but grows fast initially and seems to be catching up to the U.S.  If you were following these growth directories at about the halfway point to the long-run equilibrium, you might forecast that Germany will eventually overtake the U.S. -- much like many people think about China today.

What accounts for Germany's faster initial growth rate, and its failure to overtake the U.S.?  This comes from the diminishing returns property of the production function.  Germany starting in 1945 with a very low capital stock receives large jumps in productivity from accumulating capital.  The U.S. does not see as large jumps in the productivity because in 1945 they are on the "flatter" part of the production function.  But, Germany can't maintain such a fast growth rate as it too eventually hits the flat part of the production function.

### Policy Implications

A goal of economic policy can be to increase a country's average income, i.e. economic standard of living.  In the long-run, this is accomplished by policies that lead to higher productivity.  The following video discusses policies that lead to:

1. higher savings rates: for example, taxation policies that encourage savings like tax deferred savings accounts (e.g. IRA's), sheltering investment and interest income (e.g. Roth IRA's, lower capital gains tax rates), etc.
2. higher technological progress: for example, explicit government expenditure on research and development, tax credits for businesses that conduct research and development, or patents that give incentives for businesses to invent new technologies.
3. human capital accumulation: public provision of education, tuition grants and scholarships, subsidizing the financing of education, re-training programs.