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January 28, 2008

Comments

Emre Mangir

The most significant factor for determining changes in output/worker growth was an increase in technology. A three percent increase in technological innovation, g, resulted a nearly four-fold increase in output/worker when compared to projected pre-shock levels. Interestingly enough, an increase in technology resulted in a lowered capital-output ratio, which makes sense because workers are producing more goods and services, while capital is being replaced at a slower rate. For every 1% increase in technological efficiency, a 2.5% increase in the savings rate is required to maintain the capital-output ratio constant at 2.5. This is a somewhat redundant fact, but something that did not seem obvious to me. If the rate of technological innvation goes to zero, the output/worker stagnates, with very little growth. The savings rate is required to keep long-term growth. As s tends to zero, both the capital-output ratio as well as the output/worker goes to zero which makes sense, since the capital depreciates over time, leaving workers with no tools to use to produce goods and services. Thus, according to this model, a high rate of savings-investment, a low level of capital depreciation, a stable labor force, and high rate of innovation will result in the highest possible quality of life. Finally, in the shorter term, the model is most useful (read accurate) for relatively small changes in the capital-output ratio.

Emre Mangir

The most significant factor for determining changes in output/worker growth was an increase in technology. A three percent increase in technological innovation, g, resulted a nearly four-fold increase in output/worker when compared to projected pre-shock levels. Interestingly enough, an increase in technology resulted in a lowered capital-output ratio, which makes sense because workers are producing more goods and services, while capital is being replaced at a slower rate. For every 1% increase in technological efficiency, a 2.5% increase in the savings rate is required to maintain the capital-output ratio constant at 2.5. This is a somewhat redundant fact, but something that did not seem obvious to me. If the rate of technological innvation goes to zero, the output/worker stagnates, with very little growth. The savings rate is required to keep long-term growth. As s tends to zero, both the capital-output ratio as well as the output/worker goes to zero which makes sense, since the capital depreciates over time, leaving workers with no tools to use to produce goods and services. Thus, according to this model, a high rate of savings-investment, a low level of capital depreciation, a stable labor force, and high rate of innovation will result in the highest possible quality of life. Finally, in the shorter term, the model is most useful (read accurate) for relatively small changes in the capital-output ratio.

Michael Leung

The capital-output ratio drops when s decreases, delta increases, n increases, or g increases, which makes sense from the algebra. In the first three cases, this translates into reduced output per worker over time, but in the last case there is a significant increase in output, despite the drop in the capital-output ratio. This demonstrates that the direct effect of increased labor efficiency on E, which increases growth, outweighs its indirect effect on the capital-output ratio, which presumably would decrease growth alone, by a surprising margin.

On another note, while the model explains how and why growth occurs (through changes in s, delta, n, g), it doesn't explain how and why these variables themselves change, which seems to be an important question to ask.

Valerie Cheung

As alpha decreases, the shape of the capital-output ratio curve tends to be more concave; while it flattens out when alpha increases. As s increases, the curve becomes more concave. When delta, n, and g increase, the capital-output ratio becomes convex. Also, the output per worker and the output per worker curve flatten out when s decreases. The balanced-growth path output per worker falls below the curve of output per worker when delta and n increase to 5%. The output per worker and balanced-growth path output per worker becomes very convex with g at 5% as compared to 0%. The above comparisons are all done altering one variable at a time, holding other variables constant. It becomes difficult to analyze the effects once we have more than one variable changing at the same time.

Yanik Jayaram

I played with both initial values and shock values, but I didn't understand what a shock value was, or how the model was taking it into account. I also didn't understand the balanced growth path in terms of how it was calculated or what exactly it was representing. So what follows is my results from adjusting the initial conditions and my understanding of those results:

Increasing 'g' leads to:
1) greater output per worker- makes sense because technology improvements should allow each individual to produce more
2) lower capital/output ratio- makes sense because you now can produce more output for a given amount of capital

Increasing 'delta' leads to:
1) greater output per worker- this didn't make sense to me, because if your capital is depreciating at a quicker rate, then a person should be able to produce less
2) greater capital-output ratio - I couldn't decide if this made sense or not, because although the output amount is decreasing due to the wearing-out of the capital, isn't it also true that there is less capital since it is breaking down?

Increasing 'n' leads to:
1) Greater output per worker- this didn't make sense to me because I would think that if a population was growing rapidly, this would overcrowd the work place (less capital per person), and therefore, output per person should go down
2)Lower Capital-Output ratio: This makes sense because there is now more output (due to more people working) but the same amount of capital, resulting in a lower ratio

Increasing 's' leads to:
1)Lower output per worker- this didn't make sense to me. If 's' increases, this means investment in capital increases, and therefore should result in more efficient production. Yet output per worker appears to have gone down...strange
2) Greater Capital-Output ratio: This might make sense, because we've invested in the creation of more capital. However, I would think that the creation of that new capital would also lead to greater output, which would make the final outcome of the ratio dependent on the extent of the increase in output.

An increase in 'a'
1)Greater output per worker- this makes sense because, as you decrease the amount of diminishing returns for a given level of capital, this makes your population more productive, resulting in more output per worker
2) Lower capital-output ratio - this makes sense, because for a given level of capital, you have less diminishing returns, and therefore, greater output, which reduces the size of the ratio.

So I would say I understood about half of it. Still uncertain about shock values, balanced growth, if we are supposed to ignore everything before time=0, and some of the data interpretations.

Cindy Wu

The lecture cleared up some confusion that I had from just reading the book. The definition of a balanced-growth path is still a little hazy for me, but it’s starting to clear up. From my investigations, this is what I found:

1) As savings increased both the output per worker and the capital output ratio increased. This means that more money is invested into capital, leading to more efficient output per worker.
2) As the depreciation rate increased both the output per worker and the capital output ratio decreased. This means that the capital used in making goods are not as efficient each successive year, causing output per worker to decrease, despite the same amount of effort used. It also means that capital is losing value, which decreases the second value.
3) As the annual rate of growth increased both the output per worker and the capital output ratio decreased. This is because more workers are crowding onto the job market, resulting in more people sharing an overall amount of capital. With more output because of more workers, the same amount of capital is split into much smaller proportions, which decreases the capital output ratio.
4) As technological progress increased, output per worker skyrocketed, but the capital output ratio dropped. This occurred because more technology means each worker can produce much more efficiently using the same given amount of capital as before. To produce the same amount as before, a smaller amount of capital is needed to be used.
5) As the capital share increased, output per worker increased, but the capital output ratio decreased. This happened because when the diminishing returns become less, the workers are actually less unproductive the more they make. Because they are making more goods, spread over the capital, it becomes less, which decreases the capital output ratio.

Individually manipulating the data makes perfect sense because of all the algebra and intuitive reasoning, but once multiple variables are changed, the graphs seem to change almost unexpectedly. Some variables are much more dominant than others and it’s difficult to predict their actual interaction in the real world.

Anela Chan

A simple bit of derivatives (or an even simpler look at the algebra) would tell you that an increase in savings increases output per worker but creates more diminishing returns, hence the concavity. Indeed, savings--increasingly not domestic--are typically invested in capital goods, which is reflected in the increased balanced-growth capital-output ratio. As for barriers to growth, an increase in the depreciation rate obviously decreases the balanced-growth capital-output ratio, as the capital goods are breaking down, and decreases Y/L, though Y/L is still free to grow convexly. An increase in the rate of labor force growth also leads to decreased K-Y ratio and Y/L, as now there are too many people and not enough capital to go around. However, this rate of efficiency growth g sitting in the denominator of the balanced-growth K-Y fraction actually has positive effects. Though the balanced-growth capital-output ratio initally drops, its negative effects are overtaken by the positive effects on the growing Efficiency of labor, which ultimately contributes the most to economic growth despite the parameters. Increased efficiency, spurred by innovation and technological progress, puts an economy on a delightfully convex path to riches, and so the variables E and g become the most important factors in growth. Though many economies are far from rich, at least the Solow growth model shows that with a little education, savings, and demographic transition, any economy can dig itself out of its initial conditions and put itself on a journey towards growth. So simple, huh.

Elizabeth Creed

For Web Assignment 2, I considered the effect of a population shock on an economy presently on its balanced growth path when faced with a change in “n” from 2% to 4% per year (a rate perhaps relevant if the poor and densely populated regions in the Andes and/or Himalayans experienced a strain on their water resources due to climate change). Holding “s,” “delta,” “g” and “E” constant at 17.5%, 4%, 1% and $45,000, the one-time, permanent shock to the economy (a) diminishes the overall potential of the economy, (b) knocks it off its balanced growth path and (c) changes the capital-output ratio and output per worker of both its balanced growth path and projected levels.

In terms of the capital-output ratio, the balanced growth path K/Y immediately drops from its pre-shock level of 2.5 to a post-shock level of 1.944. Projected K/Y in the economy diminishes slowly until it reaches its steady state level over a period of more than 60 years: at year 1 (following the population growth shock), projected K/Y has diminished from 2.5 to 2.495, at year 10, it’s projected to be at 2.296, it is expected to be less than 2 near year 50 and, at year 60, the amount of investment per output is approaching its steady state at 1.98. Changes in the balanced growth path of K/Y and projected K/Y signify that less capital per productive output is available in an economy that experiences a one time, permanent increase in the rate of population growth.

In terms of output-per-worker, the balanced growth path of Y/L drops significantly at year zero from $111,386 to $87,500, requiring 24 years to recover to its pre-shock level of Y/L. Projected Y/L is not expected to drop in any significant way, but it grows at a rate less than pre-shock conditions. The difference between post-shock projected Y/L and pre-shock balance growth of Y/L grows over time both in dollar and proportional amount. At year 10, for example, projected Y/L ($114,127) is about $10,000 and 9% less than it would have been if it had stayed on its balance growth path ($124,270) whereas in year 50, projected Y/L ($148,094) is about $37,000 and 25% less than it would have been if it had stayed on its balanced growth path ($185,021). Changes in the balanced growth path of Y/L communicate the diminished potential of the economy following a one time, permanent increase in the population growth rate “n.” Changes in the projected Y/L show that the level of per capita wealth is not expected to diminish, but it is expected to grow at a decreased rate.

Continuing to hold “s,” “delta” and “E” constant at 17.5%, 4% and $45,000, the model shows that the diminishing effect of wealth potential resulting from a population growth increase from 2% to 4% can be offset by slight improvements in the rate of technology improvement, “g.” For example, an increase in “g” as small as .05% from 1 to 1.05% diminishes the balance growth path of Y/L but maintains projected Y/L levels so close to pre-shock levels that little difference in per capita wealth is expected. At pre-shock levels of “s,” “delta” and “E” and new, increased levels of “n” and “g,” projected Y/L follows a path close to pre-shock Y/L for two decades, after which projected Y/L levels rise above pre-shock conditions. This effect of improvements in technology demonstrates the long run importance of implementing resource-efficient technologies in a world of increasingly constrained resources.

Michael Beckman

In experimenting with the Solow Growth Model, I wanted to try to model some potential events that could happen to the US economy. My first experiment was to see the effect of immigration on the US economy. The balanced growth output ratio went from 3.5 to about 2.7 when labor force growth grew 2%. It took an increase in savings of almost 7.5% to bring the capital output ratio back on the long term path to 3.5. However, assuming the immigrants are well-educated, I increased the technology growth rate 1%. While this increased balanced growth path output per worker, it also decreased the capital-output ratio. I assume this is true because not enough investment is going toward implementing the new technology. I concluded that immigration is positive if the immigrants are educated, and if investment increases so that they can utilize capital at the same rate as others.

My next scenario would be a decline in the birth rate due to restrictions on immigration and declining birth rates. If s=17.5, delta =.04, n=.02, and g=1, then balanced growth path output per worker is about $145,000. But if say, due to an epidemic that causes n to decrease by -.03, then balanced growth path output per worker is about $254,00. I wasn't surprised to see that the capital output ratio increased, but I was surprised to see that balanced growth path output increased. I suppose that this causes s/n+g+d to become larger, and therefore K/Y must become larger. When there is more capital stock per worker, workers can be more productive.

My last experiment was to see what would happen if a decrease in investment, say due to the high credit costs of the sub-prime mortgage mess, occurred. Holding the original variables constant, I created an investment shock of -10%, meaning post-shock saving became 7.5%. Both output per worker and capital output ratio decreased dramatically. This explains why we are now entering a recession: business are not investing or re-investing enough in the capital stock. The government realizes this, which is why Ben Bernanke is still cutting rates,

Tim Wang

Parameter Initial Values Shocks Post-Shock Value
s 10.00% 0.00% 10.00%
delta 4.00% 100.00% 104.00%
n 0.00% 0.00% 0.00%
g 2.00% 0.00% 2.00%
a 0.1 0.1

In the event of a huge disaster where the country's entire capital stock has depreciated completely, the capital-output ratio obviously decreases rapidly but it was interesting to me that the output per worker still increases over time, albeit at a slower rate.

Even in the impossible case that depreciation is greater than 100% meaning all capital goods lose in value more than what was originally invested, Y/L still increases.

Lastly, as alpha approaches 0, capital is less important and so the shock has less impact on the economy. This implies that natural disasters will affect the economies of capital-intensive, more developed countries more than poorer, low-alpha countries.

Nick Broten

I set up a hypothetical shock situation similar to what one might find
in South Africa or Uganda following the outbreak of HIV/AIDS.  The
size of the labor force drops significantly, in this case by 6%, the
saving-investment rate falls (to around 3%) as citizens lose
confidence in their economic institutions, and the degradation of
capital quickens as maintenance and general care becomes harder to
come by. I'll make the degradation of capital fall by more than the population to represent some spiraling negative impact on infrastructure.  I will be bold, however, and say that the rate of technological innovation increases as foreign support agencies--NGOs,
development organizations, etc.--swarm to the country in question,
introducing new technologies related to health, communications, and
business management.  Using the Solow model, this small increase in E, or the efficiency of labor, carries the day, actually increasing the growth rate in the long run. This is the takeaway from Solow--technology matters. And it provides some hope for the future of the impoverished world.

Xueyao Liu

I played around with shocks on labor growth rate (n) and the model reflects a sad reality of developing countries – an overwhelming increase in labor growth rate actually decreases output per worker. This explains why even in the booming economy of China, the living standard of the vast majority is rising at a much slower rate than the growth rate of national economy. Too much new labor force is entering the market. If capital is fixed, the labor market is very inefficient as supply far exceeds demand. Therefore, labor will be hired at relatively low wages and also utilized at relatively low productivity because there is not enough capital for all the labor force to work on. Employers can afford to waste labor in a country like China since people are willing to settle for lower wages as long as they can find a job. Hence, maximizing labor efficiency is not of primary concern because labor is relatively cheap any ways. Luckily for China, new capital is planted as the economy expands rapidly, but it takes a much bigger shock in value of s to balance out the negative results caused by an increase in n, as the model suggest. The crazy growth of Chinese economy is very worrisome because it does not seem very sustainable. I just hope the labor growth rate will slow down before the rate of increase in capital wears out so that everyone could have a job and lead a life they deserve.

Diana Lee

I attempted to answer the question: what are the long term economic effects of brain drain as predicted by the Solow model? Namely, what happens when the best and brightest workers leave developing countries? And how are these negative effects counteracted by the fact that these workers often send money back to their countries to help support their families?

In thinking about the model, I assumed that brain drain would affect the savings rate, the labor force growth rate, and the growth of technology. I looked at these separately before examining their simultaneous effects. First, since workers are leaving the country, we would expect the labor force to either decrease or grow more slowly. This, as we discussed in class, causes output per worker to increase, somewhat counter-intuitively. Next, brain drain depletes the country of its capacity to develop new technology since the most highly trained workers have left. By setting the technological growth rate to zero, we see that output per worker flattens out very rapidly and reaches a fairly stable, low rate at the end of 60 years. Finally, I modeled the additional income generated by overseas workers as an increase in the savings rate. As expected, this change alone results in an increased rate of output per worker growth.

In combining the shocks, I set the technological growth rate to zero and reduced the labor force growth rate by 25%. I then incrementally increased the savings rate to see what would happen. With no increase in savings rate, we find that output per worker grows more slowly than the preshock model, and the curve becomes concave as opposed to convex. With a 20% increase in the savings rate, the model shows an interesting result. For the first 20 years, the economy grows slightly faster than it would have under the initial conditions, but since the new curve is concave, it crosses the pre-shock path and falls significantly below the pre-shock levels in the long run. With a 30% increase in savings rate, we see a similar result, except the short run boost in growth lasts for longer (approximately 30 years). Ultimately, it looks like the Solow model predicts that brain drain could possibly result in faster short run growth but will ultimately result in long-run stagnation because of the lack of technological growth.

Daniel Yeghiazarian

I decided to look at a fast growing developing country such as China. I increased the savings-investment rate to a shock of 10 percent since China is investing in so much capital such as machinery and new buildings. Also, many companies have been outsourcing in China and so many jobs have been created increasing the labor force. Finally, all this investment usually results in an increase in technological progress. This results in the amount of money a worker produces doubles every 20 years. This is why China’s stock market is always booming right now and many people are buying stocks from China and profiting from there rapid economic growth. The capital-output ratio is decreasing slowly after the shocks apply since they produce less but are worth more money and again displays the rapid positive growth of the Chinese stock market and why everyone is investing so much money to reap the benefits of this stock market. I feel though that the graphs do not display the skepticism people have of how long this economic growth can be sustained and whether the numbers China says they are making in profits and production are true.

Daniel Yeghiazarian

I decided to look at a fast growing developing country such as China. I increased the savings-investment rate to a shock of 10 percent since China is investing in so much capital such as machinery and new buildings. Also, many companies have been outsourcing in China and so many jobs have been created increasing the labor force. Finally, all this investment usually results in an increase in technological progress. This results in the amount of money a worker produces doubles every 20 years. This is why China’s stock market is always booming right now and many people are buying stocks from China and profiting from there rapid economic growth. The capital-output ratio is decreasing slowly after the shocks apply since they produce less but are worth more money and again displays the rapid positive growth of the Chinese stock market and why everyone is investing so much money to reap the benefits of this stock market. I feel though that the graphs do not display the skepticism people have of how long this economic growth can be sustained and whether the numbers China says they are making in profits and production are true.

Nan Lu

Let me start with a list of inputs and outputs. s, delta, n, and g are %age changes in the shock. Y/L and K/Y are the "ending values" of the two graphs.
s delta n g Y/L (thousands) K/Y
1 0 0 0 153 2.62
1 1 0 0 134 2.329
1 0 1 0 134 2.329
1 0 0 1 242 2.329
2 0 0 0 161 2.751
findings:
1. changes of the the three denominators (delta, n, g) have same effects on K/L. This is very reasonable given their identical "roles" in the denominator. However, things are a little different in Y/L. Changes in technology has a much bigger impact than the other two factors.
2. if we use the first row as control and compare the rest with it, we will find only growth in technology and savings rate will increase Y/L. growth in delta is obviously bad. growth in n, although increasing Y, affects L more directly. Having more babies is not the way to make the economy stronger.
3. advancement in technology does not increase increase the capital-output ratio. Only savings rate does. I am a little surprised, because intuitively we would think that technology will make your investment more efficient.

Gerardo Zaragoza

I decided to use the initial values given, and then I proceeded to play around with the shock values. Here is what I found interesting.
1) With the savings-investment rate set at 5% , increasing the capital share from .1 to .9 in intervals of .1 affected the output per worker and capital-output ratio curves, widening the initial distance between them and their respective balanced-growth paths.
2) Leaving the variables n and g at 0%, and increasing only delta and s showed that it took a greater and a significantly higher savings rate to return to the pre-shock balanced growth path of output per worker.
3) Again, leaving the variables n and g at 0%, and delta at 3% showed that increasing technological progress initially decreases the balance-growth path output per worker, but over the 60 year span, increasing technological progress increases output per worker substantially.
4) Doing as in three, but setting n, and g to 3% one at time while leaving the other and delta at 0% and increasing technological progress, showed similar results as in #3.

There is no doubt that technological progress is important in increasing output per worker.

Jonathan Ong

Using this model, I wanted to see the results of a catastrophic shock to the economy and compare them to a real life shock. The shock I wanted to use was a -12% shock in labor force, representing a plague of some sort. For example, the bubonic plague from 1347 to 1351 caused the European population to drop from 75 to 50 million, which is about a 10 million annual loss in population. In addiction, I left the initial values default, but changed the shock values. S = 0 since I don’t know how the savings-investment rate would be affected by the plague, delta = 2 since with less people, less work overall would be done and the tools and land would depreciate more, n = -12, making a post-shock value of -10%, and g = -1, as I believe that during these years people are more worried about surviving than technological progress.

The results were interesting. According to wikipedia, the black death exacerbated an ongoing recession. However, labor did gain some wages and freedom due to a lack of labor. In the Solow Growth model, post-shock, the output per worker grew faster than pre-shock values, but balanced growth path output per worker was negative. This doesn’t make sense to me, because back in these days one hour on the farm only made a certain amount of food, no more no less. Maybe my knowledge of farming is wrong though. But how can a less people do the same or more work with no technology? Similarly, the capital-output ratio increases exponentially on the graph, but the balanced-growth path capital-output ratio is negative. However, this seems conflicting as the capital-output ratio is not headed towards equilibrium in both cases. This may be because a negative balanced-growth path capital-output ratio is hard to understand. During these bubonic years, capital didn’t fall, but population and depreciation did, causing output to decrease, thus causing K > 0 and Y < 0, making K/Y < 0. In the chart, the balanced growth path capital-output ratio post-shock was -5.833333. This balanced-growth path capital-output ratio doesn’t really make sense to me. What if n + g + delta = 0? Then would you have an infinite balanced growth path capital-output ratio?

Thus the equilibrium of this model doesn’t really make sense to me, but it may just be my lack of understanding of this model. The model does not predict the outcome of a catastrophic event. Instead of a recession, the effects of the plague seem to be the opposite with growing economic efficiency and furthermore with a contradictory negative balanced-growth path output per worker. The model has me asking questions of its basis, of how it works, what it means, what the heck I’m doing. Maybe I should have made a shock in variable S, but I am unsure if that would be the case in such a catastrophe. Perhaps this model is too simple for this kind of modeling and needs more parameters for me to calculate this catastrophe accurately.

Zhihui Zhang

I investigated the effects of an increase in the levels of technological innovation. I.e. what would happen if the next great discovery was made and were to spawn an influx of new ideas. It is interesting that relatively small increases in g can result in quite drastic changes in our growth model. But it would seem that historically this makes sense. During the renaissance era, there was suddenly a new interest in technological invocation and as we can observe, the result lead to an exponential increase in the growth of society. I also looked at what would happen if we experienced a large scale disaster and s was significantly reduced and found that both output per worker and the capital output ratio decreased significantly. This seems like an interesting exercise and I'll probably play with this more over the weekend.

Ronald Park

I decided to use the data from South Korea's economic history as inputs into the spreadsheet. The data was imperfect though; the savings and labor force growth rates are average rates but from overlapping and different time periods (savings rate: 2% rate from the mid-1960s and a 10% rate from 1970-1972; labor force growth: 2.8% from 1955 to 1966 and 1.7% from 1966-1985). I left the depreciation rate and technology rates as they were since there wasn't really data on those parameters. What the Solow model spit out was that at time zero output per worker was $80,000 which then rose to $399,898 27 years after the shock - an increase by a factor of about 5. In reality, per capita GNP went from $87 in 1962 to $4,830 in 1989 - an increase by a factor of about 55.5! Granted the data doesn't all fit. The various rates aren't from the same time periods, the savings rate no doubt changed after 1972, and the "shock" should have been from 1970 to 1989 - not from 1962. Nonetheless the different results between the model and reality are too striking to be ignored. The Solow model is either lacking drastically in accounting for some feature of reality or the depreciation and technology rates must be very different from the "default" settings.

Dena Fehrenbacher

I first attempted to find the highest levels of output-per-worker an economy could reach if their growth of g did not increase (thus indicating that the derivative of E is constant).

Keeping initial conditions, even with the obscene and highly unrealistic shock-increase of 80% in the investment rate, (while the balanced growth path of the capital-output ratio increased to 14), the balanced growth path for output per worker only increased to $800,000 after 60 years.

However, when the savings rate increases by only a 10% shock, and the growth of the efficiency of labor rate g grew 10% as well, the output per worker balanced growth path sky rocketed. After 60 years, it had reached $27,000,000, even though the capital-output ratio balanced growth path had dropped to 1.6.

Thus, I was able to fully comprehend the overwhelming importance of the growth of the efficiency of labor (g), relative to the growth in s. (Even when s shock=2% and g shock=10%, the increase in the capital-output ratio was still enormous.)

Leslie Edwards

I began playing around with the Solow Growth model and became fascinated with the tremendous effect a change in technology can induce on the output per worker. A 1% shock value multiplied the worker output by more than 1.5 times the initial value. If our society were in for another industrial and technological revolution, the worker-output ratio would skyrocket. Although slightly unrealistic, a 10% shock to g would make workers produce $25,000,000 per worker. Though this may be a large number, this would be bad for social welfare because the more technology invented, the less need for human capital and unemployment rates may skyrocket.

I also checked how the retiring baby boomer generation, accounting for 30% of the US population, would affect the worker output and capital output ratios. With less people entering the workforce, the output per worker increased (probably due to more available jobs than workers, causing more work per worker). By slowing down the labor force by 5%, worker output increased by $100,000.

Dmitri Krupnov

As predicted from corresponding equations, as rates n and delta increase (keeping else constant), the slopes of both Y/L and K/Y decrease. As g increases, Y/L curves up due to increased efficiency. However, K/Y decreases because output increases, while capital investment stays constant.

This raises a question: if Y/L increases, does it really matter what happens to K/Y?

Then, as s increases, the growth-paths of both Y/L and K/Y go to a higher level. This is because higher savings result in greater capital investment and a higher K. However, an interesting thing happens when you increase g and either n or delta at the same time. Y/L drops initially after the shock (understandably, because of more workers), but with time the balance-growth path intersects and rises above the original path. It appears that the increase in efficiency outweighs the rise in population.

So, ideally (for an economy to maximize growth), the rate of innovation g should be great, with a decent savings rate g to compensate for capital depreciation.

Joe Cummins

I hoped to use the model to determine the optimum savings level for a given society. The hope was to then modify certain parameters and attempt to determine the changes in the optimum savings level for different economic conditions, and look for trends. However, when I increased the savings rate, first mildly, and then to seemingly impossible (and then to actually impossible) levels, the growth rate and overall production per worker continued to rise.
This seems to be a problem. As the savings rate increases, the percentage of money being spent on consumption goes down. Even supposing that, at certain savings levels, this drop in percentage of spending on consumption could be more than made up for in the long run by a greater increase in overall output per worker, it still is clear that at some point (say s approaches 100%), the overall output per worker would have to fall, because no one would be producing if no one was buying. The logic implies an optimal savings level, where consumption and savings are jointly, maximally contributing to economic growth. However, at least in this particular representation of the Solow model, it appears that higher savings leads to higher output, regardless of the savings level

Stuart Jaffe

I decided to concentrate on "s," the savings-investment rate. It is of particular interest since the savings rate in the United States has decreased from 11% in the 1960's to about 1% today. The population of the United States saves far less than our European or Japanese counterparts.
Initial Values Shock Values Post Shock
s: 11% -10% 1%

Every other parameter was left unchanged. I found that the capital/labor ratio and the capital per worker dropped significantly, which isn't a surprise. With the decreased amount of capital, it is also no shock that output per worker decreased as well. However, the efficiency of labor (E) continued to rise after the savings rate shock. This came, well, as a shock to me. I would expect a decrease in the efficiency of labor without all that capital, especially since the labor force is still growing at 2% per year. This could mean that there is massive unemployment and the workers who still have jobs are working very effectively.
I suppose having a sudden shock of -10% on the savings rate creates more drastic results than a steady decline over 40 years; however, capital consumption in the United States has increased. Which could lead one to the conclusion that we have financed our increase in capital consumption with borrowing, and more borrowing. Cheap credit has been a substitute for a healthy savings rate and we're starting to feel the fallout of that cheap credit. Maybe a consumption tax is in order to reverse this process, but that's a little beyond the scope of the model.

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