Chapter Eight

Learning to Grow

The Solow Model

There is no greater tool for gaining insight into the world than math. Mathematical equations can be used to describe real-world events and, once identified, let us make predictions about future events. Economists use economic models to describe the real-world, and if the description is convincing, the model's variables can be adjusted to see how analogous real-world policy changes might impact the outcome. One such model is the Solow Model of Economic Growth.

Open Video Transcript

During World War II, Germany, Japan, and the other Axis powers suffered huge causalities and major loses in capital. They lost the war, and with it went their economies, which were decimated by bombings and invasion. Nevertheless, after the war, the growth rates in the countries that lost the war were significantly higher than those in the countries that won it. France, where the economic devastation was greatest among the allies, was the fastest growing, but the United States – which saw very little combat within its borders – stagnated for the 10 years following the war. But Austria, Germany, Italy, and Japan all had growth rates which were higher than the average for the world. These economies doubled in size after the war or more. So, what gives? --To find an answer to the mystery, we need an economic model. An economic model is a simplified description of reality, designed to yield hypotheses about economic behavior that can be tested. Often, these economic models are expressed as mathematical equations. For example, I might have a model to describe how many units of a product we will sell depending on the price we charge for it. Perhaps my model is that the quantity sold (Q) is equal to 100 minus the price charged (represented by P). We can plot this model on a graph, with each of the two variables represented by one of the axes of the graph. Here I have price on the vertical axis and quantity on the horizontal one. My model translates to this line on the graph. At every point on this line, the equation is true. So, if the price were $90 each, my model predicts that we would sell 10 units. You just plug in 90 for P, the price, and solve the equation for Q, the quantity. Likewise, if I plug in $10 for P, the model says we will sell 90 units. The value of a model is that it makes predictions about the world, and we can look at how things actually go and test whether or not the predictions were accurate. So, I might randomly change the price every day and see how many we sell and collect that data. I could then plot the price and quantity sold each day and see if it’s close to the predictions. Hmm, this data started off pretty good, but then veered away. I might need to modify my model so that the slope of the line isn’t so steep. We have seen quite a few economic models already, like the production possibilities curve, supply and demand, and the Malthusian model of the economy. Each of these models make predictions about the world which we can test. But now we need one for economic growth. --In the 1950s, the economist Robert Solow developed such a model. He wanted to explain the causes of economic growth based on certain key variables that we presumably have some control over. The goal of Solow’s model was to translate these variables into a prediction about GDP. Solow’s model made some surprising predictions, at least for their time, such as the one about the war-torn countries growing faster than those relatively untouched by war. For this work, Solow was awarded the Nobel Prize in Economics in 1987. --To begin, Solow needed to think about the economy’s production function. At a basic level, the economy transforms resources into goods and services. But that doesn’t just happen automatically. It takes people and tools to make that happen, or as we call them, capital and labor. These are the basic inputs for the economy. Using capital, which we represent with the letter K, and labor, which we represent with the letter L, we are able to transform resources into goods and services. The total amount of goods and services produced by this process is our Real GDP, aka the economy’s output, which we represent with the letter Y. This is a really simple description of the economy, but this is what models do, they simplify the world into something we can think clearly about. How do we make smartphone apps? Well, we use computer programmers and computers. Labor and capital. How do we make bread? Using mills and mixers, and the talents of a baker. Capital and labor. Pretty much everything in the economy boils down to those two broad categories. --Of course, to really be able to use our model and see what it says, we need to express it mathematically. Our real GDP (Y) will be determined by some mathematical function which takes capital (K) and labor (L) as the inputs. That’s what this italicized ‘f’ with parentheses means. It’s a placeholder for some yet undefined mathematical equation. So, what equation do we use? It can be anything. Maybe its Y is equal to K plus L. Or Y is equal to K times L. Or Y is equal to K raised to the ½ power times L raised to the ½ power. Heck, it can be as crazy as we want, like 5 times K-squared plus 0.8 times the quantity K minus L raised to the ½ power + L-cubed. Deciding between these is going to be a matter of observation. We want a production function which gives similar results as the real economy. But first, we will need to build on our theory of production.

Diminishing Marginal Returns

Economic models seek to describe the world as they are, not as we wish them to be. One important observation that needs to be incorporated into our model for production is diminishing marginal returns. It is just an unfortunate condition of reality that the first squeeze of the lemon will get us more juice than the second. The returns to our efforts diminish the more effort we put in - at least that is generally the case when it comes to production.

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The Law of Diminishing Marginal Returns states that adding additional factors of production will result in sequentially smaller increases in output. For example, let’s say that I have a business digging holes in the ground. To increase worker productivity, I probably want some shovels. These five shovels here are my capital, so in this case K would be 5. For the law of diminishing marginal returns to apply, we need to keep that quantity fixed. In other words, we aren’t going to buy any more shovels for the time being. To signify that the quantity of capital is fixed, we put a little bar over the letter K, indicating that what was once a variable that can change is now an unchanging constant. So now we need to hire some people. Since I have five shovels, I will hire five people. So, L is equal to five. For the sake of the example, let’s say that each of the five workers is able to dig one hole in 15 minutes when they are equipped with a shovel. That means our total output per hour will be 20 holes, since each digger will dig four holes per hour, and we have 5 workers. Alright, now let’s think about what happens when I hire an additional worker. With one more worker, we now have a total of six, but we still only have five shovels. That doesn’t mean this new worker won’t be able to help. They can probably borrow a shovel while one of the others is on break, and when all the shovels are taken, they would still be able to dig with their hands. Obviously, this 6th worker won’t be as productive as the original five because of our limited capital, but output still has to increase when we add them on, and perhaps they are able to fill enough gaps that they dig three additional holes per hour, bringing our total to 23. That is diminishing marginal returns. The return is how much our total output rises. In this case it rose by three. And when we add an additional factor of production, in this case one more worker, we added 3 to our total output, which is less than the 4 holes per hour the previous worker added. And we can keep going. If we hire a 7th worker, there are fewer gaps to fill, and more time spent digging with their hands. Perhaps they only add 2 holes per hour to our total, bringing us to 25. An 8th worker will see further diminishment of the marginal returns to hiring them, perhaps adding only 1 to our total. The law of diminishing marginal returns almost certainly applies to adding more labor when we hold capital fixed, and so it is something we want to bake into our choice of a production function to represent the entire economy. --Let’s put it on a graph to see what a function with diminishing returns looks like. Our graph has Y, or output, on the vertical axis and L, or labor, on the horizontal one. We are holding capital fixed. Each tick mark here represents one more worker added to our team. As we move along that horizontal axis, hiring workers, our total output rises but by less and less each time. It starts to sort of level off as we go, and if we connect the dots, we can see what a function with diminishing returns looks like. Whatever we choose as our production function for the economy, we want it to look like this when graphed with respect to output and labor. That is, when we keep our value for K constant, increasing the number we plug in for L needs to result in diminishing marginal returns. --Likewise, if we chart the production function with capital on the horizontal axis, which means we are holding labor constant, we want to see something similar. It doesn’t have to be exactly the same, but it still needs to show diminishing marginal returns. This video is all a bit mathy, but I want to mention that this is what economists do in real life. They build models to represent some sort of economic process, and building models means making certain assumptions that help simplify things. We are simplifying things here by saying that our aggregate economic production – all production combined – will exhibit diminishing marginal returns in both capital and labor. There are good reasons to think that this is a reasonable assumption to make, and one that isn’t violated in the real world, but if it is that would mean we need to rethink whatever conclusions we draw from this analysis. And that’s why economists are still busy after all these years.

Practice Questions

1. Which of the following functions show diminishing marginal returns in Y for increases in X?

(A) Y = X2 + 5

(B) Y = 10 - X

Reveal the Answer

We can test each one by plugging in 1 for X, and then 2, and then 3.

(A) Y = X2 + 5

When X = 1, Y = (1)2 + 5 = 6
When X = 2 Y = (2)2 + 5 = 9
When X = 3, Y = (3)2 + 5 = 14

When X increases from 1 to 2, Y went up from 6 to 9, a change of 3 units. But when X went from 2 to 3, Y jumped up by 5 units. The marginal returns are getting bigger, not smaller. So, (A) shows increasing marginal returns.

(B) Y = 10 - X

When X = 1, Y = 10 - 1 = 9
When X = 2 Y = 10 - 2 = 8
When X = 3, Y = 10 - 3 = 7

CAREFUL! Diminishing marginal returns does not mean Y is decreasing. In fact, by our usual meaning, increasing X will always increase Y. When X goes from 1 to 2, Y changes by -1. When X goes from 2 to 3, Y changes by -1 again. The marginal returns aren't changing. So, (B) shows constant marginal returns.

When X = 1, Y = 11/2 = 1
When X = 2 Y = 21/2 = 1.41
When X = 3, Y = 31/2 = 1.73

Note: X1/2 is the same as the square root of X. When X goes from 1 to 2, Y increases by 0.41 units. Then, when X goes from 2 to 3, Y increases by 0.32 units, slightly less than before. So, (C) shows diminishing marginal returns.

2. A production function shows diminishing marginal returns for labor. The last worker I hired increased output by 10 units per hour. If I hire one more worker, how much can I expect output to increase?

Reveal the Answer

Output will increase by less than 10 units per hour.

That's all we can really say. We can't say for sure if it will be 9 or 8 or 1. All we know is that it will be less than the last hire, so it will be less than 10.

Endogenous Growth

Our production function for the economy represents our ability to transform capital and labor into output. In that way, it represents our technology. But technology isn't something that comes to us from on high. It depends on our level of innovation and education. So, we need the very structure of our production function to change with innovation and education. In mathematical terms, we need those to become endogenous variables in our function.

Open Video Transcript

We are still on our way to building a production function which we can use to model real GDP and make predictions about economic growth. So far, we know that our two inputs are capital and labor, and that our production function should exhibit diminishing marginal returns when we hold one or the other constant. Since the Industrial Revolution, however, capital and labor have gotten more complicated. In Solow’s original model, they were treated as exogenous parameters, meaning they were not determined by anything, they are just chosen by the economy. Later, economists started to consider these variables to be endogenous, or variables which are determined by other factors. --You’ll remember that as the industrial revolution got underway, innovation led to improvements in the productivity of capital. Machines got better and better, increasing how much output a single machine was able to assist in making, which also made labor more productive. Innovation leads to higher productivity all around, and the degree to which labor and capital are able to produce output changes with it. --We want to incorporate innovation into our production function, but unlike labor and capital, new ideas don’t have diminishing marginal returns. As we have seen, the returns to new ideas can even be exponential! So, we want the variable for it to appear outside our mathematical function like this. The variable A represents ideas, specifically ideas which increase the productivity of labor and capital. These ideas are multiplicative with our production function, because innovative ideas are able to magnify total output. As we improve our stock of knowledge, innovate new and better machines and production processes, and find better ways of mixing our resources together, we increase our value for A. Since it effects all factors of production, sometimes this gets called ‘Total Factor Productivity’. --Another important change is that jobs in these modern economies require lots of skills, meaning that workers need more education in order to do the work that needs to be done. More education builds our stock of skills, and you can see that over the past one-and-a-half centuries, educational attainment has risen from virtually nothing to a couple decades of schooling on average. There is also good reason to believe that investments in education may result in increased worker skills and a greater productivity of labor. We want our model of the economy to recognize this. --So, the L in our production function needs a companion. It isn’t just blood, sweat, and tears that gets the work done. Not just raw human labor. But rather labor paired with our education. Together, education times the labor supply gives us what we call human capital, or the stock of worker ability. Notice there are two ways to increase human capital. We can increase the quantity of labor in the economy, or we can increase the education level of our workforce. We can work harder, or we can work smarter. But H is what we want now in our production function, representing the pair. These modifications endogenize our modern sources of growth, they incorporate innovation and education, rather than leaving them as something external to production. So, now that we have all of our pieces, what equation should we choose for our function? --There are a lot of different choices we can make, but the most common is called the Cobb-Douglas Production Function, named for the two economists who developed it. It’s quite simple. Output (Y) is equal to Total Factor Productivity (A), times Capital (K) raised to the power of alpha, times human capital (H) raised to the power of beta. Alpha and beta are just placeholders, and the numbers we choose to put in for them represent the share of output attributable to capital and labor respectively. To exhibit diminishing marginal returns, both alpha and beta needs to be less than one, and usually we pick numbers that add to one for convenience. For example, we might set alpha to 1/3 and beta to 2/3. This would mean that capital is doing about 1/3 of the work in the economy, and labor is doing 2/3 of the work. Since there are ways to estimate those shares, the Cobb-Douglas production function matches reality pretty well. Now that we have this fantastic little toy economy, which is able to take in ideas, capital, education, and labor and turn them into real GDP, we can go about making some predictions about economic growth. But, we will need a few assumptions about how capital gets created first.

The Steady State

Now that we have a functional form for production, we can build up our model with a theory of capital creation and depreciation. By setting these forces in action, we will be able to use our model to make predictions about how the economy will grow!

Open Video Transcript

Alright, we have settled on a production function for the economy. If we choose values less than one for alpha and beta, like this production function here, our function will exhibit diminishing marginal returns just like we wanted. We know that H represents human capital, which is our average level of education times our labor supply. We know how people get educations, and we know how more people are made. What we need now is how and why capital gets created. --Remember, capital is the tools we use for production, things like shovels and factories and roads and computers and computer programs and stuff like that. So, if we want more capital, we have to divert some of our resources towards making it. That means whatever resources we use to make capital can no longer be used to make goods and services for consumption. So, capital is created by foregone consumption. If we think about this from the perspective of income, there are two things we can do with our money. We can purchase goods and services for consumption now, or we can save our money for later. Where does that savings go? Well, as we will discuss in greater detail down the road, savings kept in banks is lent out to finance investment. And this makes sense. Whatever we are not spending on consumption now, we must be spending on the production of capital for consumption later. In the Solow model, all of our savings is turned into investment, and our income is generated through the production of output. That is, people will get paid when they sell the goods are services they produce, and so our income is determined by output, or Y. Thus, the amount of savings we will have will be the average savings rate for the economy, s, times output, y. And that savings will become our total investment, or the total amount of new capital that is created. In other words, we will, collectively as an economy, save a fraction of our income which will be used to finance the creation of new tools that we can use for future production. --It’s not all hunky-dory though. Capital wears down over time. Shovels break, factories need upkeep, roads get potholes, computers get dusty, and programs get buggy. This is what we call depreciation, and each year we will lose d times K worth of capital, where d is the depreciation rate. For example, we might lose 10% of our capital on average each year to this decay, which would mean our rate of depreciation is 10%. This gives us a theory of how the amount of capital we have each year will change. Since d is the fraction of capital we lose, 1 – d is the fraction of capital we retain. So, the capital stock this year will be equal to the quantity 1 – d times the capital stock last year. If we have 100 units of capital and we lose 10% to depreciation, then we will have 90 left, or 0.9 times 100. But wait! That isn’t necessarily all we will have, because we might have created some new capital through investment. So, our capital stock this year is actually what is left over after depreciation plus investment. And that sets us up for a special outcome. --The economy will reach a steady state when the new capital created exactly replaces the capital lost to depreciation. That is, when the amount of capital we lose to depreciate, d times K, is equal to Investment. Since investment is determined by our savings rate and income, we can rewrite the steady state as being where d times K is equal to s times Y. When this is the case, the economy won’t be changing from year-to-year, because the amount of capital we will have will be staying exactly the same. Everything we lose will be replaced exactly. So, how likely is this outcome? --Very likely, in fact. It will represent the equilibrium of our Solow model. Let’s graph it and see why. Since what we have here is a theory about capital creation, we will plot our production function on a graph with output, or real GDP, on the vertical axis, and the capital stock on the horizontal axis. Since these are the only two variables on the graph, it means we are holding all of the others, like labor, education, ideas, savings, and depreciation constant.When we hold the other variables constant, our production function exhibits diminishing marginal returns for capital, and so we would draw it like this. This is our production function. Plug in the amount of capital, and it will tell you what real GDP will be. As we discussed, we will save a fraction of this output which will be used for investment. That means the amount of investment will be determined by this line, which is just a fraction, s, of the production function. Lastly, we have the quantity of capital that will depreciate. That will be determined by d, the depreciation rate, times K. Since d is a constant, this will just be a straight line like this. With each of these functions plotted out on our graph, we can think about what will happen at different levels of capital. --Imagine that our capital stock was low. That means it is all the way over here on the left of our graph. The highest dot corresponds the total amount of output, or real GDP, that we are going to produce with this quantity of capital. You can see it is pretty far down. Not too good. But the second dot down represents the fraction of output that is being devoted to new capital. We are using something like half of our economy to produce new capital in this example. And the third dot down corresponds to the amount of capital that we are losing to depreciation. Notice that we are creating more new capital than is being lost to depreciation. This difference will be added to our capital, and so our amount of K is increasing, pushing us over to the right along the curves. At this new spot, real GDP is higher than before, and if we stick to our savings rate of 50% and devote half the economy to the creation of new capital, we will create more capital than is lost to depreciation, adding more to our total capital again. This growth in capital and real GDP will continue until we reach this point, where those two lower dots converge. At this point, we are creating the same amount of capital as we lose to depreciation. So, we don’t add anything and move to the right or lose anything and move to the left. We just stay right where we are. But say we get it a little wrong and create some new capital anyway. What then? Does it keep going up? Well, no. At this high amount of capital, the amount depreciated is higher than the amount produced. So, we end up losing this depreciated capital for next year, shrinking our capital stock and moving us to the left along the curves until we hit the steady state again. And here lies the magic of the Solow model. With some simple assumptions, we now have a working toy version of the economy that tells us what we should expect to happen to real GDP over time. If you remember the puzzle about countries after World War II, this model explains it. Germany, France, Italy, Japan, and Austria all lost a lot of capital to the war. And so, their economies saw a lot more capital creation than depreciation. But in the U.S., we had lost very little capital, and so we just remained close to our steady state, without much room for change. --This model is also an incredible tool for breaking down what factors lead to economic growth. For example, we can evaluate what happens when the savings rate increases. Do we get much economic growth? Well, if s increases, then it will shift the green curve, which represents our investment. More savings means more investment, and the whole thing will push upwards, representing a larger fraction of total output. This will move our steady state over to the right, and the economy will see growth in the capital stock and some corresponding increases in real GDP. However, because there are diminishing marginal returns to more capital, we don’t see a very big increase in real GDP. There isn’t much growth from higher savings. Back when Solow published his model, this was the big result. At the time, most people thought the Industrial Revolution and the growth that followed was the result of increases in capital due to more savings and investment. But Solow showed, convincingly, that such a change simply cannot explain the massive increase in economic growth over the past 250 years. --Ok, let’s reset and try another change. What happens when the depreciation rate decreases? A lower depreciation rate changes the slope of our red line, and it sort-of rotates like this. Once again, we have a new steady state. A new point at which dK = sY, which is over here to the right. Again, we see an increase in capital, but a pretty weak increase in real GDP. So, this isn’t a great source of economic growth either. --Perhaps we need a change to one of our other factors, like education, labor, or ideas. Each of those, unlike s and d, plug into our production function, which will shift both the production function and investment. With a higher production function, whatever fraction we have for s will mean more investment. If we save 50% of our income, and our income goes up, we are saving a larger quantity of money. So, both shift up like that. And now we have a big change. The new steady state pushes up the capital stock just like the other changes, but now we get a huge increase in real GDP to go with it. This is where economic growth comes from. --Human capital takes a lot of time and investment to acquire. So, it isn’t the best way to grow. Ideas and innovation are the best. But they are both a lot bigger contributors to growth than savings or lower depreciation. This meshes with our understanding of the Industrial Revolution now, when economic growth was born, and it gives us a nice recipe for keeping it going. We need new ideas and innovations. So, how do we get those?

Try it Yourself!

This Google Spreadsheet has been set up with the Solow model, and you can edit the values entered in for different variables and see how it impacts the economy!

The Solow Model

The Economics of Ideas

The conclusion of the Solow model is that capital accumulation alone isn't enough to produce economic growth. To see significant increases in real GDP, we need to come up with ideas and innovations which make better use of that capital and labor. So, how do we promote more ideas?

Open Video Transcript

As we have seen, the Solow model shows us that the biggest source of economic growth will be new ideas. Innovation which reforms old methods and creates new products that make people better off is how we improve our standard of living. So, how do we promote these ideas? --Ideas seemingly come to us randomly, when suddenly we think of a different way to do something. But acting on those ideas is another story. When we look at where the new innovations and major inventions come from, the distribution is not so random. They tend to come from countries with a liberalized economies and functional governments. In short, ideas need good institutions in order to succeed. Institutions which reward innovation. --Ideas are not like regular goods and services. If I eat an apple, that apple is no longer available for you to eat. But if I have an idea, that idea can spread from one person to another without the initiator losing anything. Ideas are what we call nonrival, meaning that when one person consumes something, it doesn’t stop another person from consuming something. Ideas can also be nonexclusive, meaning that we aren’t able to prevent someone else from using our idea. That makes it difficult to profit from ideas. If I come up with a new and cheaper way to make batteries which last much longer, but everyone else is just able to copy my idea, I won’t be able to make much money from it. And if coming up with that new and cheaper way required a lot of expensive research, and development, and testing, I might not be willing to undertake those expenses to come across such a discovery. --The Founders saw this as a problem too, and they wrote the solution right into the constitution. Patents grant the innovator a monopoly on their idea for a fixed period of time, usually about 20 years. The inventor is able to profit by being the exclusive producer of their new product. Patents create huge incentives for innovation, which is good, but they can also slow innovation down. Sometimes a new idea can be made better by someone else, but patents essentially put a wall around that idea and prevent it from spreading to others who might tinker with it. Patents can also be abused. Should Amazon be able to patent their one-click buying option? Is that really a brilliant innovation we wouldn’t otherwise get? So, patents are an imperfect solution to the problem of generating ideas. But some of their flaws can be remedied by other solutions. --Subsidies are a major way in which we can promote new innovation. A subsidy is a payment from the government, in this case for research and development. Most major research universities are, in part, subsidized by state governments. Gatorade was invented at the University of Florida. Google started at Stanford. Web browsers and plasma screens were created at the University of Illinois. The allergy medication Allegra was developed at Georgetown University. A lot of the money for that research also comes from government organizations like the National Science Foundation. Other government organizations like NASA and DARPA –where the internet was invented – do research directly. The federal government also gives subsidies, usually in the form of tax breaks, to companies engaged in a lot of research and development like Apple, Google, Tesla, Boeing, and Pfizer. This is one thing to keep in mind when you read headlines about big corporations “not paying any taxes”. Sometimes that is by design, and we all benefit from the innovations spurred on by those tax breaks. Many of these innovations still end up being patented though, which means they suffer from many of the same issues there. Nevertheless, subsidies drive more than 30% of research and development, and that is a big boon to economic growth. --Another incentive, which is as old as patents are, is prizes. Often, organizations will offer major prizes to anyone who can innovate a solution to some problem. The XPRIZE foundation offers millions of dollars to those who can solve some of humanity’s biggest challenges, including a $100 million prize for anyone who can figure out how to tackle climate change through carbon removal. The upside of prizes is that it incentivizes people to tackle problems where there isn’t a clear profit motive, or where we want to be able to build on the ideas going forward. The downside is that the innovations driven by prizes do not have to face a commercial test. A patent is only valuable if the innovation is something that benefits people so much, they are willing to pay for it. Prizes on the other hand can be given for anything, whether there is a clear use for it or not. For example, since the year 2000, there has been a $1 million prize for anyone who can solve one of 7 math problems. To date, only one of the problems has been solved, and the solver turned down the prize. Solving these problems will certainly advance mathematics, but whether they can used to benefit our material wellbeing remains to be seen. --But the underrated element in promoting innovation is simply people. More people. Albert Einstein was a one-in-a-billion genius and his development of the theory of relativity didn’t have clear commercial applications either. But the GPS on our phones wouldn’t work right if we didn’t have that breakthrough. The more people we have, the more likely we will get the one-in-a-billion genius and their discoveries. People are our greatest resource, and the more we have, and most especially the more we can lift out of poverty and into situations where they are able to innovate, the faster our civilization will be able to advance.

Quick Quiz

Now is the time to test yourself on these concepts! This quick quiz is a great way to see what has clicked and what needs more attention. It is also a good time to reach out to your instructor with questions.

Office Hours

Deeper Thoughts and Extra Practice

Dr. Goegan Explains...

Example Question
Country A produces GDP according to the following equation: GDP = 5*SQRT(K). The country has a savings rate of 20.4% and 9.7% of capital depreciates every year. What is this country’s steady state amount of GDP?

Round your final answer to two decimal places.

Correction: At the end of the video I say the answer is 52.84, but if you do the calculation yourself, you will find that the true answer is 52.58.

Video Solution

Professor Peate Explains...

Example Question
Country A produces GDP according to the following equation: GDP = 5*SQRT(K) and has a capital stock of 10,000. If the country devotes 25% of its GDP to making investment goods, how much is this country investing? Additionally, if 1% of all capital depreciates every year, is the country's GDP increasing, decreasing, or remaining constant - in steady state?

The Power of Ideas

Here is an interesting TED talk from the economist Alex Tabarrok (you saw him before in the video about the price system). He makes the case that promoting ideas is the single most important thing our policies should do.

Simple Solow Model

This is an interactive version of the Solow Model from Wolfram Demonstrations.

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