## January 28, 2008

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

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.

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.

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)

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

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.

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.

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