Solow Model and the Steady State

… a core concept in Macroeconomic Policy

Concept description

The Solow model implies that the growth contribution from physical capital will be zero when physical capital has grown to a point where depreciation is equal to investment.

Alex Tabarrok (reference below and link to right) introduces his lesson video with:

“Remember our simplified Solow model? One end of it is input, and on the other end, we get output. What do we do with that output? Either we can consume it, or we can save it. This saved output can then be re-invested as physical capital, which grows the total capital stock of the economy. There’s a problem with that, though: physical capital rusts. Think about it. Yes, new roads can be nice and smooth, but then they get rough, as more cars travel over them. Before you know it, there are potholes that make your car jiggle each time you pass. Another example: remember the farmer from our last video? Well, unless he’s got some amazing maintenance powers, in the end, his tractors will break down.

The depreciation of capital can be graphed as follows (all graphs from MRU video references below):

Depreciation of physical capital at 2% per year

If the investment from savings is assumed to be 30% of output, the investment can be graphed as follows

Investment in physical capital at 30% of output

Putting the depreciation and investment lines on the same graph produces:

When depreciation (orange) and investment (purple) lines meet

When the steady-state level of physical capital is reached, all the investment is going to depreciation and none to increasing output so output also reaches its steady state, as illustrated in the graph below:

Steady-state levels of capital and output 

Tabarrok explains how the Solow model shows that an increase in savings and investment (to, say 40% of output) will temporarily move out of steady state to a higher level of output, but that as capital is added a new steady state will be achieved where depreciation is equal to the rate of investment and growth stops. This is illustrated in the graph below.

Increased savings and investment leads to a higher steady-state level of output

Tabarrok ends the lesson with a reminder that there is more to growth than physical capital and points to the next factor – human capital.

“A further examination of the steady state can help explain the growth tracks of Germany and Japan at the close of World War II. In the beginning, their first few units of capital were extremely productive, creating massive output, and therefore, equally high amounts available to be saved and re-invested. As time passed, the growing capital stock created less and less output, as per the logic of diminishing returns. Now, if economic growth really were just a function of capital, then the losers of World War II ought to have stopped growing once their capital levels returned to steady state. But no, although their growth did slow, it didn’t stop.

“Why is this the case? Remember, capital isn’t the only variable that affects growth. Recall that there are still other variables to tinker with. And in the next video, we’ll show two of those variables: education (e) and labor (L).”

MRU practice questions

See http://www.mruniversity.com/node/330826, accessed 23 April 2016.

1. For the next two questions, consider the following: Country A has K=10,000 and produces GDP according to the following equation: GDP=5√K. If the country devotes 25% of its GDP to making investment goods, how much is the country investing?
   1. 12.5
   2. 25 
   3. 125 
   4. 1,250 
   5. Can not be determined with the given information.
2. If 1% of all machines become worthless every year (they depreciate, in other words) in Country A, GDP is:
   1. Increasing. 
   2. Decreasing. 
   3. In steady state. 
   4. Can not be determined with the given information.
3. Suppose instead that 2% of all machines become worthless every year. In Country A, the capital stock (K) will be:
   1. increasing. 
   2. decreasing.
   3. in steady state. 
   4. Can not be determined with the given information.
4. According to the Solow model, when a country is in steady state, 
   1. Depreciation > investment.
   2. Depreciation < investment. 
   3. Depreciation = investment.
   4. Depreciation < output. 
   5. Depreciation > output. 
   6. Depreciation = output.

Source

Alex Tabarrok, Solow Model and the Steady State, at http://www.mruniversity.com/courses/principles-economics-macroeconomics/solow-model-and-steady-state, accessed 23 April 2016.

Atlas topic and subject

Growth, Capital Accumulation, and the Financial System (core topic) in Macroeconomic Policy.

Page created by: Ian Clark, last modified on 23 April 2016.

Image: Alex Tabarrok, Solow Model and the Steady State, at http://www.mruniversity.com/courses/principles-economics-macroeconomics/solow-model-and-steady-state, accessed 23 April 2016.