Web Assignment 2

• Due before lecture on Feb 4
• Download the Solow Growth Models Scenario Spreadsheet
• Read through it
• Conduct an experiment--enter your own values into the grey "initial conditions," "shocks," and "capital share," boxes, and examine the tables and charts that result.
• Write a paragraph and post it as a comment on this web page describing what you did and what it told you--or what went wrong and why you were unable to accomplish the assignment.

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.

[Konrad Knusel]

I have primarily played with the parameter n (annual rate of growth of the labor force).

The most simple results I found were:
1) following a positive shock to n
a) a decrease in output per worker growth
- the marginal productivity of workers
begins to decrease after a population
boom because there is not a sufficient
increase in capital/technology to
maximize the effectiveness of the
labor of the workers
b) a capital-output decrease - production
is more labor intensive (technology,
savings, depreciation all held
constant) so output increases more
dramatically than capital
2) following a negative shock to n
a) an increase in output per worker
growth
b) an increase in the capital output
ratio

What was interesting to me was comparing the difference between a shock and an equivalent change in the initial value.
Example) comparing n = 2.00%, shock = -3.00% to n = -1.00%, shock = 0.00%
Some Results:
1) no difference in output per worker
growth between -1.00% and
2.00% when shock = 0.00% in both
cases - output per worker growth is
the same for a country with a growing
population and an identical (in every
other way) country with a decreasing
population (the first country will
have a faster growing GDP because it
has more workers)
2) there is a difference in the capital
-output ratio - a decrease in the
initial value results in a new steady
ratio which exceeds the one which is
eventually approached with an equal
negative shock to the prior initial
value

[Seng Tat Chua]

A positive 'shock' to the rate of labor-augmenting "technological process" causes a greater increase in growth of output per worker than the savings-investment rate does. On the other hand, output per worker decreases when the rate of growth of labor force and the annual depreciation rate increase, corroborating what we know from the Solow Growth model equation. So having a huge population may not lead to a bigger economy. Interesting to note, considering how aging countries are frantically trying to arrest their declining birth rates.

Also, increasing rate of technological progress actually causes a decline in capital-output ratio, which puzzles me, for I would intuitively expect the reverse to occur.

[Kevin Hong]

I considered what would happen to the balanced growth path if there were a shock in technological progress. Holding s, delta, and n constant, I increased g, the technological growth rate, to 4% for its year-zero shock. This resulted in an upward jump in both capital per worker an output per worker. The output per worker growth graph shows that as you increase g, the output per worker increases faster exponentially and does not deviate from the balanced-growth path output per worker line. This makes intuitive sense, since one would expect output per worker to increase as these workers become more efficient and productive. As g increases, the average worker has more capital (K/L increases) to amplify his or her productivity, since technological improvements would introduce more value-adding machinery, equipment, and so forth. As for the capital-output ratio, however, it begins to decrease from year zero. The rate at which it is decreasing slows down as time goes on. The capital-output ratio graph shows that the capital-output ratio eventually converges to the balanced-growth path line. This means that the effect of the shock eventually settles approximately 55 years after.

What surprised me was an increase in g to 2% moves the balanced-growth path for capital-output up to about 2.72, but an increase in g to 3% moves the balanced-growth path for capital-output down to 2.45 on the graph. I believe this is the case, because after a certain amount of technological progress, less amount of capital is needed to produce the same amount of output.

[Jonathan Ellithorpe]

Attempting to model economic disaster, I set up the model with the following parameters:

savings rate: 7% to -0.03%
depreciation rate: 4% to 4% (unchanged)
labor force growth rate: 2% to -1% (on the decline)
tech growth: 2% to -0.07% (regression in technology)

I found that the equilibrium capital output ratio went from about 0.85 to 0, while the transient behavior was to halve this distance every 40 years. This economic resilience (40 year kappa half life) I found surprising considering that the parameter shocks practically cause economic growth to come to a rigid halt.

On the other hand, I found that the output per worker declines faster that it was growing before year zero. This makes sense and is not too surprising, given that capital is still depreciating at 4% while technology is actually experiencing a regression. This effect is of course softened a bit by a decreasing labor force, but not by much (given that it's decreasing by 1% and alpha = 0.5)

[Amanda Bradshaw]

In applying the model, I concentrated primarily on shifting the economy's saving-investment, rate and efficency of labor.

Maintaining initial conditions, I incrementally increased the savings rate, s, by positive shocks of 1%. This unsurprisingly increased the capital-output ratio and output per worker, although at a less than proportional rate. Doubling the savings rate also doubles the balanced-growth capital-output ratio, since the economy's investment effort and investment requirements are in balance.

Again, keeping other variables constant, I decided to toy with the growth rate of the efficency of labor, g. Unremarkably, continuous shocks in the rate of labor-augmenting "technological" progress reduces the balanced-growth equilibrium capital-output ratio, likened to an increase in the labor force.

Summarily, I found this spreadsheet to make calculations with the solow growth model more readily available, which more importantly facilitates economic and policy interpretations.

[Jerry Wang]

The Capital-output ratio changes due to changes in different variables. As S decreases the ratio drops and the curve becomes more convex. This holds true for variables delta, n, and g. Because it is a Ceteris Paribus assumption, only one variable is changed at a single time. This change means that the output per worker is reduced over time. However, that does not hold for G, which is annual rate of labor-augmenting “technological” progress. An increase in G has the most significant impact on the change in outpour per worker ratio growth. For a decrease in capital-output ratio, there is a dramatic increase in output. This makes sense for there are less workers working, but there output increases due to more efficiency. Also there is less capital used for more output. A three percent increase in G increased the output per worker by nearly four times. As depreciation, delta, increases, the capital per output and output per worker increases. This didn’t make sense for both statements couldn’t be true at the same time. As n, number of workers, increased, output per worker increased and lower capital per output. The lower capital per output is lower is true because there is now more output with the same amount of capital. The increase in output per worker is true to an extent for at a certain point, we will be experiencing diminishing returns.

[Deviyani Gurung]

So first I checked to see how efficiency of labor would effect the both the output per worker and the balanced growth output per worker. In my example, I used n= 1%, g= 2%, delta = 3%, alpha = 0.5, s= 18% and labor efficiency = $ 10,000. From this first trial I learned that the balanced output became $ 30,000. Then when I raised the labor efficiency to $ 20,000 the balanced growth output per worker rose to $ 60,000. I realized that because balanced growth output per worker is a constant multiple of the efficiency of labor, its growth rate is the same as g, so as efficiency of labor doubles, the output per worker also doubles, by the same percent as g. In the second trial I kept increasing the rate of technology to see how it would effect the output per worker and the balanced growth output per worker and what I learned was that when you increase g, the output per worker and balanced growth output per worker not only increase faster but they converge. Also in the last trial I slowly decreased the savings rate to see the results in output per worker and capital output ratio. The savings rate is required to keep long-term growth. As s goes to zero, both the capital-output ratio and the output/worker goes to zero which makes sense, because the capital depreciates over time, resulting in the labor force having nothing left to produce anything. So for the Solow growth model the best standard of living occurs when the s is high, g is high, efficiency of labor if high, a steady level of n and a low delta.

[Yusuf Amir-Ebrahimi]

In order to test out the Solow growth model, I wanted to simulate shocks that could possibly occur and see the model is anything similar to what we have seen in the US economy. First I simulated a technology boom, like what occurred in the 90s. When the technology growth rate jumped from 1% per year to 4% per year, the output per worker over a twenty year period jumped from $80,000 to $150,000 per year while the capital to output ratio dropped about half a point. Then, looking at a drop from 3% to 2% in technological progress, the output per worker growth falls significantly from its usual trajectory. Overall, it appears that a technology boom has a very positive effect on worker output growth, but a shock makes a very significant difference in the long run. Another probably shock would be an increase in capital investments. This would likely happen when the interest rate drops like it has in recent years. Unlike technological advancements, increasing capital investments only has a moderate impact, raising the output per worker growth by only about $10,000 in 20 years and the capital to output ratio rises by less than .2 points. Based on the two parameters that I experimented with, the Solow growth model shows that it would be more worthwhile to invest in new technology that increases worker productivity than simply invest in new capital.

[Benjamin Turk]

For this assignment I wanted to examine the difference between changing the growth rate of the labor force vs changing the rate of technological growth. Before I even did the experiment I expected a technological increase to do much more for output per worker than an influx of more workers. After plugging in some numbers this was obviously true. This makes sense because what is the use of having 1,000 workers to build a car if no one has the technology to weld. All in all it seems output is affected much more profoundly by technology.

[David Liu]

Something of interest that I noted was the effect of a large negative shock to the labor force growth. As the size of the labor force decreases, the output per worker grows exponentially, not an incredibly realistic prediction. The capital output ration also increases exponentially. A substantial increase in growth of the labor force the output per worker for a short time but output slowly returns to the original growth rate. The economy is initially not able to handle the sudden influx of people but eventually balance is restored and we return to equilibrium growth.

Having just watched I am Legend, I tried to simulate Will Smith’s post apocalyptic world with solow’s growth model. Assuming that all old technology is still available, but no new technology is being developed, and that the savings rate is drastically lowered, due to needing all available money for consumption. In addition, the growth of the labor force is zero, as mortality rates are much higher in the new, dangerous world. The end results of the solow model tell me that output per worker decreases dramatically, eventually reaching zero, as the depreciation destroys capital and no new technology is developed. However, if more people die off, and the rate of death is greater than the rate of depreciation, the output per worker increases, which makes intuitive sense.

The last thing I worked with is changing the savings rate. Any amount of increase in savings increases the output per worker and capital-output ratio, even to the point where consumers save 100% and spend no money, so there is no money flowing in the economy. In a situation where the savings rate is zero, the eventual output per worker goes to zero, unless the growth of technology is very high, which also makes intuitive sense.

[Eugene Kur]

I put the initial values of s=10%, delta=4%, n=2%, g=1%. I put a shock on the savings rate (s) of +10% leading to a post-shock value of 20%. I then changed the capital share parameter (alpha) from 0.5 to 0.8 in increments of 0.1 and watched what happened to the graphs. This shifts the output per worker along the pre-shock balanced growth path down and also makes the output per worker take longer to reach the new balanced growth path. The capital output ratio also takes longer to recover from the shock at higher values of capital share. Thus, the more dependent the output of an economy is on the amount of capital, the longer it takes the economy to regain the balanced growth path after an increase in savings shock (although the new balanced growth path tends to be higher in terms of output per worker for higher capital share parameters).

[Kevin Ahn]

From the four different parameters: the savings rate (s), the annual depreciation rate (delta), the annual rate of growth of the labor force (n), and the rate of labor-augmenting progress (g), I analyzed independently the effects of the positive and negative shocks of each parameter. In the analysis of each parameter, I did not change the fixed initial values and I used 5% as the number for a positive shock and -5% as the number for a negative shock. When s was affected by a positive shock, the output per worker increased and the capital-output ratio converged to a number greater than that in the absence of a positive shock. When s was affected by a negative shock, the output per worker decreased and the capital-output ratio converged to a number less than that in the absence of a negative shock. In the case of an increase in delta, the output per worker decreased and the capital-output ratio converged to a smaller value. Likewise, a decrease in delta would produce the opposite effect. An increase in n decreases the output per worker as well as the capital-output ratio and a decrease in n increases the output per worker as well as the capital-output ratio. An increase in g increases the output per worker and the capital-output ratio; and a decrease in g decreases the output per worker and the capital-output ratio. After analyzing the behavior of each parameter, I looked closely into the relation between n and g and found that that the negative effects of the annual rate of growth of the labor force on productivity can be easily fixed by the rate of technological progress, and that technology has a greater effect than the growth of the labor force in this model.

[Lisa Sweeney]

I decided to look at the effects of negative shocks to the various parameters. As could be expected, a decrease in the savings rate s by 5% resulted in a dramatic drop in the capital-output ratio as well as a decrease in the rate of output per worker growth. Since there is less capital available in the economy, the decrease in output per worker growth is logical, though it is interesting that there is growth nonetheless. A decrease of 3% in the annual depreciation rate δ led to increases in output per worker growth and the capital-output ratio. This would mean that capital lasted longer and wouldn’t need to be replaced as often, freeing up resources that would have otherwise been used on maintenance instead of progress. Decreasing the rate of labor force growth n by 1% led to increases in both output per worker growth and, more dramatically, the capital-output ratio. This makes a lot of sense, and serves to illustrate the economic incentives behind population control and greater access to birth control. Finally, I decreased g, the rate of technological growth, by 0.5% and found that while output per worker growth slowed considerably, the capital-output ratio maintained an upward trend, albeit at a reduced pace. This is understandable because while g decreased, I didn’t reduce it to below 0, which I felt would be unrealistic, as the only situations I can think of in which technology was lost involve the death of a civilization, and I prefer to think that ours will hold out long enough for that not to be relevant to my own economic studies.

[Chanwoo Myung]

The model was interesting. I altered different parameters to see the effect that it had on the capital-output ratio curve. As expected, as ‘g’ increased, the output per worker rose sharply as it allows each worker can produce more. However, the capital output ratio dropped, as workers will produce more efficiently with same amount of capital. When ‘s’, or savings increased, both of the values increased, which seemed obvious, because there is more capital invested. The change in delta also resulted in lower output per worker as well as lower capital-output ratio because the capital invested does not efficiently result in more output therefore making the capital to be less valuable. Finally, the increase in the labor force growth resulted in lowering both values, since the capital must be shared with more people, which makes them less efficient. Overall, the graph was very easy to use and definitely helped me to understand this model better.

[Max Zheng]

I decided to see what effect the credit crunch would have on our economy if it becomes more severe. If we reduce "s" by 7.5% to a final value of 10% while leaving the other variables unchanged, a significant chasm opens up between the pre-shock growth path and the new balanced-growth path. Output per worker as well as the capital output ratio both dropped immediately, as expected. By extending the chart, I saw that it took about 100 years to return to the balanced growth path. And while the new balanced growth path continued to be concave upwards, the gap between the pre-shock and post-shock paths continues to widen. The value for the balanced-growth path capital-output ratio dropped from 2.5 to 1.43. But, it took at least 120 to 140 years for the output per worker ratio to drop to the new value.

[Dingjiao Xu]

I didn’t use the spreadsheet to model any particular economic changes but rather, to understand the implications of the Solow Growth model with regards to the factors affecting permanent growth rates, with a focus on the savings rate and “technological progress”.
I first adjusted s to have a shock of +22.50% to a post shock value of 40% (common in East Asian countries such as Singapore where economic growth has been attributed to high saving rates and prudent macroeconomic policies). There is a short period of high rapid growth which then stabilizes to the same growth (output-per-worker line running close to parallel with the continuation of the pre-shock balanced-growth path), showing that s has only a short-run effect on growth rate.
Next, I increased g from 1% to a post-shock value of 6%. There is a significant increase in growth, and we can see that the increase in growth rate is permanent (blue line is significantly steeper than then the green dashed line).
I also adjusted the delta and n values with shocks of +9% and +5% respectively. They produced predictable results: both brought about lower output-per-worker levels with no change to the long-run growth rate (close to parallel lines again).
Given that g is the only parameter that affects the permanent growth rate, we are able to gain a better understanding of governmental policies that encourage technological and scientific innovations, and the underlying macroeconomic principles.
However, there's something from the graph that I don't understand. There was a slight dip in the balanced-growth path output per worker for approximately a decade. Is this due to the adjustments and retraining that workers have to go through to adapt to the new technology?

[Andrew Nhieu]

One of the parameters which I decided to change was technology. Though it is obvious that a change in technology will increase the growth model, it is surprising by how much it does. The other parameters will slightly affect the growth model when changed, but technology has a noticeable one. Visibly, the other parameters, when changed by one percent, cause the graph just to shift up slightly. A one percent change in technology however, causes the graph’s slope to change, as if increasing exponentially. This makes sense, for one discovery can revolutionize many fields, causing a kind of snowball effect that increases efficiency over many fields. I also investigated what would happen if there was a huge increase in the labor force. If the growth in the labor increases too rapidly, then there will a decline in output per worker. There are only so many people the workplace can accommodate, after a certain point; extra people just get in the way. In the US, we will face the opposite phenomenon, since the baby boomers are retiring soon and there are not enough new workers to replace them.

[Ken Amornnopawong]

I'm first experimenting a rough comparison between two economies, one being similar to a dragon Asian economy and another being close to the U.S. economy.
s=50%, 10%
delta=10%, 7%
n=4%, 3%
g=8%, 15%
alpha=.4, .3
The economy with a much lower savings rate, lower depreciation rate, lower labor force growth rate, but much higher innovation has more than doubled the efficiency of labor. It has much lower K/L but the same Y/L. This means technological advancement contributes a major improvement to efficiency of labor. Substatially lower savings rate could dampen capital per worker ratio. Better innovation helps offset the drop in savings rate. Hence, the two economies yeild the same output per worker ratio.
Now, let's compare the two economies like the U.S. and Japan with following parameters for Japanese economy.
s=25%
delta=5%
n=1%
g=10%
alpha=.6
The Japanese economy has the lowest efficiency of labor because the economy depends more heavily on capital as it has higher share of capital. It has quadruppled capital per worker ratio comparing to the U.S. due to higher savings rate, lower depreciation rate, and much lower labor force growth. Comparing to the U.S., laggard in technological advancement results in the same output per worker ratio despite enjoying higher capital to labor ratio.

[Chen Xu]

For this exercise in comparative statics, I decided to examine the effect of increasing the rate of labor force growth as well as technological process and depreciation rate on capital-output ratio and output per worker growth. While we do this, we examine the effect that the initial efficiency of labor has on these numbers. The motivation behind this exercise is to note the effect of a constant influx of workers (either highly skilled or unskilled) into the economy. An example would be the illegal immigration of unskilled workers from Mexico or the immigration of highly skilled tech workers from the East.
To make the changes more apparent, we hold constant the savings-investment rate. First we assume that the rate of labor force growth in the economy is 2.00%, but that suddenly, that a natural disaster or change of government causes an increase in immigration. We do not change the rate of technological progress or depreciation rate of capital because we assume that the vast majority of these workers are unskilled laborers. We find immediately that the balanced path Capital-Output ratio drops by .5 but that the rate at which the economy tends to this does not change. This makes sense since there is there is more output and no change in capital or rate of technological growth. Before the influx of people, we see that the Output per worker was around $80,000 and rose steadily towards the $200,000 after 60 years. After the shock, however, the we see that the initial balanced-path output per worker decreased from $112,000 to around $98,000 and that it increased at a lesser rate, ending up after 60 years at $178,000. These results make sense, since the influx of workers was not accompanied by an increase in the efficiency of capital. This meant that each worker would have to work with less capital and thus would produce less on average. In contrast, we now assume that there is an increase in immigration from a country will a largely skilled worker population, including scientists and inventors. (Similar to what happened in the US in the period surrounding World War II.) Now we assume not only a 1% growth in rate of worker population, but also an increase in technological growth by 1% and a decrease in depreciation rate by 1%. Not surprisingly, we see that the rate of change of output per worker grows tremendously, with balanced path output per worker after 60 years at $321,541. It took 20 years less to achieve the control level after 60 years of $200,000. Also not surprisingly, the capital-output ratio was identical with that of the first simulation, because it seems that the increase in technological growth and the decrease in depreciation rate cancelled each other out. (An increase in technological growth made the capital more efficient i.e. more output, but the decrease in depreciation made the capital wear out less quickly i.e. more capital. Finally, we vary the initial efficiency of labor, and found that it just changed the magnitude of the events, and did not substantially change our understanding of what was going on.

[Seung Eun(Rina) Park]

Through out many trials and errors, I've chosen to change the annual rate of growth of the labor force(n) initial values. U.S senate has shown that from 1950s to 1970s, the rate has increased. (i.e. 1950s-1.1%, 1960s 1.7%, 1970s 2.7%) seeign how the change in the annual rate of growth of the labor force effected other values is my efficient goal of the experiment.
Graphically, increased in growth rate has made the output per worker's slope more bigger, closer to balanced-growth path output per worker line.
nummeratically, effected efficiency of labor (E), Capital-OUtput Ratio, balanced-growth path capital-output ratio, capital per worker but did not effect output per worker, balanced groth path output per worker, and output per worker along the continuatin of the pre-shock balanced growth path. Efficiency of labor E increased while n grow, capital output ratio decreased, balance-growth path capital output ratio also decreased and capital per worker decreased. Which was understandable because the more labor force the prodictivty increase so the capital output increases and the efficency for labor has grown and due to large labor, capital per output should decrease by limited amount of capital produced and so the capital per worker will also decrease.

[Pablo Barrientos]

I considered the case of a country where suddenly capital’s depreciation rate, ‘delta,’ decreased due to an increase in technological progress, g, that made capital’s productivity last longer. I picked values for g and ‘delta’ such that the negative impact of an increase in g was more than counterbalanced by the decrease in ‘delta’ and as a result increasing the capital-output ratio instead of decreasing it. As expected, the balanced-growth path output per worker increased along with its slope and thus increased the growth rate of the standard of living.

I further assumed that because of the decrease in ‘delta,’ investments would decrease since capital does not have to be replaced that often. However, I picked such value for s, the savings-investment rate, so that capital-output ratio would still increase rather than decline due to the negative impact of the decrease in s. Although the balanced-growth path output per worker decreased by the latter assumption on s, the overall impact of the shock I described would positively affect the economy on the long run.

I decided to impose these conditions because I wanted to visualize the potential positive impact that technological progress could have on a country’s economy and thus, since technological progress is closely related to a country’s level of education, show the importance of education in the world economy. Whether the technological advances took place on the country of interest, imports and exports could alleviate this issue and consequently arrive to the same conclusion.

[Stephen Gu]

I decided to test the effects that an increase in technology would have on the model. I adjusted g to have a 5% shock for a final post shock value of 6%. While this is a pretty significant increase in technical progress, it has an even greater impact on the output per worker growth. In the first 20 years, output per worker more than doubled the output per worker from the pre shock path. By the end of the 60th year, the output per worker growth is not even comparable. Not surprisingly, the capital-output ratio dropped significantly because the technology became better. This makes sense because it is obvious that increasing the technological progress of a society is the most efficient way of improving the economy. Increasing the technological progress is also easier to implement than increasing the other factors because it doesn't require changing the behavior of individuals and businesses.

[Emerson Green]

I decided to focus strictly on a single case. Specifically, I was curious to see what the model predicted in the case of countries with sufficiently low fertility and immigration rates such that population is actually declining. This seems to be occuring in certain eastern european countries, such as Russia. I set an initial population growth of n, the annual growth rate of the labor force, 2 and a shock value that brought n down to -1% after year zero. This action yielded a dramatic increase in output per worker. Also, the capital-output ratio climbed sharply. Could this be a factor in the recent economic growth of Russia? Perhaps, although alot of the growth is definitely explainable by the increase in energy prices. Still, I wonder if the dramatic increase in productivity is enough to offset the fears that a large increase in the number of retirees with pensions coupled with a smaller labor force will hinder the economy. Sadly, this appears to be outside the scope of the spreadsheet.

[Namgui]

Here are what I found out:

When s increases,
-output per worker increases
-capital-output increases
this makes sense because increase in investment will result in more efficient and better production.

When delta increases,
-output per worker decreases
-capital-output decreases
when depreciation rate is high, capital used in process is depreciating at faster rate, so workers can produce less. Also, capital itself is losing its value at faster rate.

When n increases,
-output per worker decreases
-capital-output decreases
When population is growing at quick rate, the working places will be crowded and workers will be less efficient. Now we have more output from more workers but same capital, resulting in decreases in capital-output.

When g increases,
-output per worker increases a lot
-capital-output decreases
better technology makes workers to produce more efficiently. given a certain amount of capital, workers can produce more. so, capital-output is decreased.

It was a very interactive and interesting chance to investigate Solow growth model with this excel sheet. However, sometimes, it is somewhat hard to catch the difference after I made some changes to variables. To see this, I intentionally changed some variables too much, so this findings might be wrong. Also, if we can see many instances on only one chart at the same time, it would be easier to see the pattern.

[Kelland Chan]

One thing I wanted to see was what the effect a policy like China’s one-child policy should have on the growth of the economy. I kept all the rates constant by setting all the shock values except n to zero. I decreased the annual rate of growth of the labor force, and as expected, the capital-output ratio increases, since the number of workers in the economy declines over the years while the amount of capital grows at the same rate.

Another thing I wanted to test was difference between a capital-oriented economy and a labor-oriented economy. To do this, I played around with the alpha parameter. When alpha is low and therefore more labor-oriented, an increase in population growth produces a relatively smaller change in output per worker than when alpha is high. When the savings-investment rate does not change, there will be less capital per worker. If an economy relies heavily on capital, a higher labor force growth rate means there will be even less capital per worker than an economy that relies mostly on labor. When I increased the savings-investment rate in a capital-oriented economy, the increase in output per worker was much bigger than that of the labor-oriented economy as expected. The same result occurred when I increased the technological rate. The Solow Growth Model was worked well in displaying the characteristics of different economies.

[Coleman Maher]

I tried to model the economic situation of countries that were rapidly aging. Countries such as Italy and Japan are seeing older people live longer and less people being born. This means that there are less workers and more pensioners to support.

To approximate this 'demographic shock', I lowered the savings-investment rate from 15 to 10% and the growth of the labor force from 0 to -3%. To emphasize the shift from a worker to a pensioner economy, I also lowered technological growth from 1% to 0.75%.

The results showed slow growth in the efficiency of labor, a capital-output ratio that went from 3 to 4 in 50 years, and a larger Balanced-Growth Path Output per Worker compared to the pre-shock path.

It is hard to see the negative effects of this demographic slowing in this model because as the number of workers and capital investment declines, the capital-output ratio increases and each worker produces more. Even though the size and health of the economy has probably gone down quite a bit, the capital output ratio and output per worker has grown.

It is hard to make a guess as to whether or not economies with rapidly aging and declining labor forces with slow growth in technology and productivity will suffer. This model seems to imply that any positive capital-investment and growth in technology will simply increase productivity faster than the labor force declines.

[Anne Wu]

For simplicity's sake, I first increased each of the variables one by one,
keeping the others unchanged, in order to observe the effects of each
variable. To be consistent, I increased each by 2%. An increase in s, the
savings-investment rate, produced a slightly higher rate for the
increasing output per worker. The balanced-growth path output per worker
jumps higher than the output per worker in the first year after the shock,
but it appears to converge to output per worker eventually. The
capital-output ratio increases as well.

I found that a 2% increase in delta, the annual depreciation rate, had
the exact same results as a 2% increase in n, the annual rate of growth of
the labor force. The output per worker decreases slightly in the first
year after the shock, then increases at a slower rate that what would have
been. The balanced-growth path output per worker jumps lower than the
output per worker before apparently converging to output per worker again.
The capital-output ratio decreases and the balanced-growth path
capital-output ratio becomes lower.

An increase in g, the annual rate of technological progress, caused the
rate of increase of output per worker to skyrocket. The capital-output
ratio decreased gradually before converging to the balanced-growth path
capital-output ratio, which dropped.

Next, I decided to find out how changes in one variable can be cancelled
out by changes in another variable. I found that for every percent
increase in s, a 0.4% increase in any of the other three variables would
keep the capital-output ratio constant. A 0.4% increase also kept the
output per worker increasing as it would have before the changes, except
with g, which caused output per worker to increase at a higher rate.

Expectedly, delta and n produced the same results again. For every percent
increase in delta or n, a 2.5% increase in s cancelled out the effect. I
suppose this makes sense because the capital-output ratio K/Y is 2.5. A
percent increase in delta was cancelled out by a percent decreasein n, and
vice versa. A 1% decrease in g kept both capital-output ratio and output
per worker constant.

If there were a 1% increase in g, capital-output ratio could be kept
constant by a 2.5% increase in s, a 1% decrease in delta, or a 1% decrease
in n. However, in all cases the output per worker ended up increasing at a
higher rate than before. This would mean that capital also increased at a higher rate, so that the capital-output ratio would be constant.

[Anshul Shah]

When I first made a 2% shock to the growth rate of labor efficiency g (initial value 5%), with all other shocks =0% (s=1&, delta=4%, n=2%), i was a little confused because the capital-output ratio dropped when I assumed that it would rise. However I think the capital output ratio is dropping because the denominator Y (potential output) increases due to the increase in the rate of labor efficiency, g.

Another interesting thing I noticed was with the following values:
s initial=4, shock =-4
all other initial values positive and shocks zero.
With a -4% shock to the initial 4% savings-investment rate, the 2 graphs show output per worker heading towards zero, and the capital-output ratio also going down to zero over time.
What is interesting, which I don't quite understand, is that if I set initial delta to 0%, and even if I enter a negative shock for delta, the capital-output ratio still goes to zero eventually. I don't understand why it would do this since K/Y = s/delta. If delta becomes infinitely small, then the according to the equation, the capital-output ratio should increase.

[Yikam Law]

I decided to explore the effects of saving-investment rate by setting “s” from initial value of 17.50% to higher values (with other parameters left unchanged). After several attempts, I noticed that the increase in saving rate has a rising effect on both the output per worker and capital-output ratio. This result is not surprising because investments raise new capital, which in turn leads to a higher output per worker. In the short-run, the increase in saving-investment rate augments the growth rate of Y and K. However, it has no effect on permanent growth rate of Y and Y/L.

[Ben Bednarz]

Well I tried a couple of things. The first thing I wanted to see is if what if there was a technological boom. Like a real big one.. kind of simple I know. Anyway, putting g (technological advance in labor-augmenting) at 1% pre-boom and 3% post-boom there as large increase in productivity per-worker that by 40 years meant that workers were outputting 3 times as much in goods compared to if there hadnt been any boom. Pretty impressive, but expected.

More interesting was the capital per worker/output per worker ratio declined from the "optimal" 2.5 to 1.9. Perhaps this represents the fact that workers can effectively make things faster and dont need as much capital anymore. I'm not really sure.

I then decided to make the rate at which capital wears out 15% instead of 4% (after the incident). Now why I did this, well, just because I curious, I'm not really sure I can think of a situation in real life where that would happen. Maybe the material we are using runs out and we are forced to use weaker material. Anyways here's what happened:

Output per worker dropped. It dropped by a pretty hefty margin, and by 20 years it was cut in half. However it continued to grow at a new slower rate. So its still possibly for the economy to keep growing even with a whoping 15% capital depreciation rate. The capital/output per worker ratio went way down unsuprisingly. Workers are now a better way of making money through raw labor then relying on capital that wears out really fast. In fact the ratio went from 2.5 to 1. And it got to about 1.3 in about 40 years. Pretty impressive.

[Ryan Helbert]

I decided to see just how much the efficiency of labor constant, g, affects economic growth. I knew that an improvement in the efficiency of labor greatly increases output/worker in spite of the fact that it also lowers the steady-state balanced-growth capital/output ratio. But what I found surprised me.

For my initial values I used:
s = 15%, delta = 4%, n = 2%, g = 1%, E = $37,000, alpha = 0.5

My shock values were:
s = -10%, delta = 0%, n = 0%, g = 3%

By making s fall by a significant amount (10%), and g rise by a comparatively small amount (3%), I was attempting to compare the relative strength of g versus s in how they affect the output/worker ratio. Initially, output/worker fell below the pre-shock balanced-growth path, and continued to remain below the pre-shock levels until about 33 years after the shock. But amazingly, beyond this point the output/worker ratio quickly surpassed the pre-shock path and by year 60 had reached $226,792 (compared to $144,038 for the pre-shock path). This illustrates the profound effect that even a small increase in g can have on the long-term economic growth of a country. In spite of the fact that the savings rate for capital stock was significantly reduced, the increase in labor technology mitigated these affects over the long-term, and indeed vastly improved the country's output/worker ratio from what it would have achieved under the pre-shock path.

[Elliot Gold]

Using the spreadsheet it is easy to confirm the relationship between the parameters s, g, n, delta and the steady state capital output ratio. As s goes up through a shock in the savings-investment rate, the capital output ratio rises and as n, delta or g go up, it has a negative impact on the capital output ratio.

Keeping s, delta and g constant and varying the level of shock n to see what happens when suddenly the labour force changes size. It is interesting to see that even if there were a relatively small sudden reduction of 5% in the number of workers in the economy, this has a massive effect on output per worker. Furthermore with this small 5% reduction output per worker tends to the new balanced growth path output per worker extremely slowly and likewise with the capital output ratio. Even after time = 60 years the capital-output ratio is still well away from reaching its balanced growth path. It we reduce the negative shock on n to 2%, the capital output ratio tends towards the asymptote of balanced growth path capital-output ratio much, much more quickly. Indeed after 60years the capital-output ratio has already nearly reached the balanced growth path value. However if we add to this shock an equal 2% negative shock in g, output per worker growth actually becomes negative and the rate at which the capital-output ratio tends to the balanced growth path is reduced more than proportionately.

[Konniam Chan]

My focus of the experiment is to simulate economies of emerging markets. One feature of emerging markets is the high rate of growth of technology and organization. I decided to provide a shock of 1% to g, doubling the value from 1% to 2%. Although g decreases the value of the equilibrium capital-output ratio, the effect on output per worker is extremely positive. In the new economy, Y/L is nearly twice that of the old economy by the end of 60 years. This experiment demonstrated that although the factor in front of the efficiency of labor is reduced, the increase in the exponent in the growth of the efficiency of labor more than offset it. As has been reiterated in the text and lecture, the growth in the efficiency of labor is primarily responsible for driving long term economic growth.

[Victor Ho]

One thing that particularly interested me was the massive effect which technology has on output per worker. If technology growth were to experience a 1% shock, output per worker steeply declines. Though it is true that 1% represents double the initial value, a period of strong technological innovation in which that may be possible does not seem at all unlikely. A stronger technological growth rate probably also requires higher dependence on machines, and we see that the savings investment rate required to offset this negative effect is 2.5 fold - for every percentage shock of g, a 2.5% shock in s in the same direction is required to steady output per worker. In terms of delta and n, an equivallent percentage shcok is needed in the opposite direction. With this observation, the benefits of technological innovation seem quite clear; this is because the initial values for delta and n are generally much higher than g, so a much smaller shift (in terms of percentage) will suffice in balancing out the negative effects of technological growth (delta begins at 4%, so a 1% shock is only 25%, and n begins at 2% so a 1% shock represents 50%, either of which will suffice in offsetting a 100% shock in technology growth).

Finally, it is interesting to note that the input of technology growth adds a degree of complexity to this formula that the other inputs do not. If you set all shocks to 0, you can see that the equation is working to find equillibrium, and there are small erratic movements as it attempts to converege on 2.5. Set delta or n to 0%, and you see that this does not change. However, if g is set to 0, the economy easily converges and we see a perfectly flat line. Additionally, it is interesting to note that the calculation is impossible with a 0% savings rate, which does make sense because if all the equipment were to break down output per worker should be 0.

[Sanjay Nimbark Sugarek]

The effect of a considerable drop in the savings rate is a drop in the initial value of the capital-output ratio, but also a faster rate of growth therein. The output/worker growth smoothes out when this drop occurs, but otherwise stays in relatively the same position on the graph.

An increase in the depreciation rate delta forces the capital-output ratio to drop precipitously, which is to be expected, ceteris paribus. The output/worker growth’s initial value actually increases, but the subsequent rate of growth is slower than before.

An increase in the population rate forces a similar drop in the capital-output ratio, probably because of a lack of capital availability to support the burgeoning population. While this increase raises the initial value of the output/worker growth, such growth increases very slowly over time compared with that of the previous values.

Surprisingly, an increase in the tech. growth rate causes the capital-output ratio to drop. The effect on the output/worker growth is more expected: a considerable increase in this rate of growth, both initially and over time.

An increase in alpha flattens out the curves of both graphs, but leaves the relative positioning of the graphs about the same. The growth rates, however, are now obviously slower as a result of the flatter curves (as opposed to very convex ones with quick growth rates).

A good increase in efficiency of labor, E, does not affect the capital-output ratio, because it does not directly concern capital. The shock does, however, increase both the initial value and the growth rate of the worker/output ratio.

[Hovhannes Harutyunyan]

Working with this simulation program, I found many of the processes and economic activities described both in the lecture and in the book to be true. For example, by increasing the savings rate, both output per worker and the capital output ratio increased because more savings demonstrates the attempt by firms to invest into the expansion of their productive methods. On the other hand, there is a negative relationship between the depreciation and capital output ration because depreciation implies that the stock of capital deteriorating in quality and becoming obsolete which makes the economy poorer. As n, the growth rate of the labor force rate, increased capital output ratio also decreased because more workers with the fixed amount of capital would lead to crowded work rooms. As I increased g, the growth rate for the efficiency of labor, capital output ratio increased which would be expected. According to the textbook, this type of change in the economy is the only positive shock that can cause a permanent growth in the capital output ratio. In conclusion, I found the simulation to be depictive of what would be expected in the economy according to the Solow Growth Model introduced in lecture. In particular, I feel that the program would be quite helpful in testing, and thus evaluating the effectiveness, of certain hypotheses regarding the effects of various changes in the economy on its overall long-run growth potential.

[Alex Hornof]

The interplay between the saving-investment rate, s, and the annual rate of labor-augmentation, g seemed to produce some of the spreadsheets most interesting and counterintuitive results. If the s value is lowered to zero then the Capitol-Output Ratio exponentially decays towards the origin, which is to be expected since depreciation should cause the slop of K to be negative. In this situation, the steady state of both the Output per Worker and the Capital-Output Ratio is at zero, and yet if the labor-augmentation is increased sufficiently (to between 8-9%), the graph shows the Output per Worker increasing exponentially, rather than suffering the expected drop off. Mathematically, this result makes sense, as the exponential growth caused by the improved technology are able to offset the exponential decay caused by depreciation. Logically it seems that the unchecked degradation of the infrastructure used to produce output should result in a total collapses, because in the real world there is a point at which capital stock (such as manufacturing plants) that have not been maintained cease to work at all (they break), and the decay at this point immediately to zero.

[Samuel Leiboff]

I am unsure of what I gained from using this tool. While it was nice to see the effect that the different constants have upon long term growth, some aspects of the tool left me puzzled. Why, for instance, if there are no shocks to any of the constants, does the capital-output ratio create a stepwise function that moves away from the predicted slow-growth value? Is that just an imperfection of the tool or actual predictions?

Still there were some things that were interesting to see. It's nice to imagine that each worker will eventually produce hundreds of millions of dollars of output. All we have to do is develop technologies 20 times as fast! Easy.

[Tianqi Zhu]

I used a savings rate of 10%.
As the savings rate changed, the output per worker growth did not change significantly between 10% and 20%. However, the capital output ratio changed from an upward sloping curve to a downward sloping curve, which is really strange.
I then tried changing the annual depreciation which drastically changed the slope of the capital output ratio but didn't change the output per worker growth. So essentially, these first two parameters deal more with the capital that is invested rather than the actual worker efficiency.
Strangely enough when n, the labor growth increased, the efficiency per worker decreased. I guess by having more workers, the efficiency of the workers overall decreases because there are more workers to do the job, so everyone does less on the job.