- 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.

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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.

Posted by: Coleman Maher | February 04, 2008 at 01:28 AM

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.

Posted by: Anne Wu | February 04, 2008 at 01:36 AM

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.

Posted by: Anshul Shah | February 04, 2008 at 01:55 AM

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.

Posted by: Yikam Law | February 04, 2008 at 02:02 AM

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.

Posted by: Ben Bednarz | February 04, 2008 at 02:09 AM

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.

Posted by: Ryan Helbert | February 04, 2008 at 02:39 AM

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.

Posted by: Elliot Gold | February 04, 2008 at 03:08 AM

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.

Posted by: Konniam Chan | February 04, 2008 at 06:44 AM

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.

Posted by: Victor Ho | February 04, 2008 at 07:55 AM

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.

Posted by: Sanjay Nimbark Sugarek | February 04, 2008 at 08:32 AM

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.

Posted by: Hovhannes Harutyunyan | February 04, 2008 at 08:40 AM

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.

Posted by: Alex Hornof | February 04, 2008 at 08:50 AM

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.

Posted by: Samuel Leiboff | February 04, 2008 at 09:07 AM

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.

Posted by: Tianqi Zhu | February 04, 2008 at 10:15 PM