Ultimately, long-run economic growth is the most important aspect of how the economy performs. Material standards of living and levels of economic productivity in the United States today are about four times what they are today, in say, Mexico because of favorable initial conditions and successful growth-promoting economic policies over the past two centuries. Material standards of living and levels of economic productivity in the United States today are at least five times what they were at the end of the nineteenth century and more than ten times what they were at the founding of the republic. --Brad DeLong

(This lesson draws heavily on Brad DeLong's Macroeconomics textbook, particularly Chapter 5.)

Economic growth is defined as the change in output per capita. To measure output, we take the total value of the goods and services produced by an economy in a year, called Gross Domestic Product (GDP). Then we divide by population to get the average standard of living. Alternatively, we can divide output by the size of the working population in order to measure labor productivity. We use either of these measures of output per capita to compare economic performance across time or across countries.

Here is a table (taken from DeLong) showing estimates of the growth in world population and average output per person from ancient times to the present.

A few remarks about the table:

1. Measuring the value of output historically is very tricky. There were goods that were very important at some points in time (such as covered wagons) which did not exist in earlier times and which are obsolete today.

2. Through 1800, average output per worker was less than $1 a day in today's terms. Nearly everyone lived in what we would call a state of poverty. The middle class is a very recent phenomenon.

3. Most of the growth in the average standard of living has taken place in the past 100 to 150 years. Other evidence that corroborates this GDP-based perpective includes the following:

   • Height and Longevity.

     According to Ward Nicholson, around the time of Christ, the average height of males was 171.9 centimeters and the average lifespan was 41.9 years. In 1400-1800, these figures were 172.2 centimeters and 33.9 years, respectively. Even by 1900, longevity had not reached 50 years. Today, the figures are 174.2 centimeters and 71.0 years, respectively.

   • Food.

     According to DeLong (p. 440-441), around the time of Christ, at least 90 percent of the populace had to be employed in agriculture in order to produce enough food. By 1800, this proportion still was over 50 percent. In the United States today, the proportion stands at around 2 percent.

   • Destructive power.

     According to Niall Ferguson (The Cash Nexus, p. 34), "between the seventeenth and twentieth century, the capacity of war to kill rose by roughly a factor of 800."

   • Mechanical power.

     According to data cited in this paper by DeLong, the conversion from steam engines to electric motors helped to increase the total mechanical power in the United States by a factor of 40 in the seventy years from 1869 to 1939.

   • Computational power.

     In the article cited above, DeLong notes that the exponential increase in the instructions performed per second by a computer chip, along with the growth in the number of computers, implies a million-fold increase in total computational power in the last forty years.

     Projecting this trend into the future, Ray Kurzweil makes this astonishing claim (The Age of Spiritual Machines, p.3):

     our most advanced computers are still simpler than the human brain--currently about a million times simpler (give or take two orders of magnitude depending on the assumptions used). But...Computers...are now doubling in speed every twelve months. This trend will continue, with computers achieving the memory capacity and computing speed of the human brain around the year 2020.

These quantitative indicators of cumulative exponential growth are significant. However, it may be easier to grasp the dramatic nature of growth by comparing the life of a particular rich man one hundred years ago to our lives today.

High atop Anacostia Park, a rundown, working "poor" section of Washington, DC, sits the mansion of Frederick Douglass, the great nineteenth century orator and agitator for the rights of women and African Americans. Douglass, although born a slave, became a wealthy newspaper publisher. He came to Washington late in his life, as a U.S. Marshall in 1879. His 21-room mansion was on a 15-acre site and employed three servants. A reasonable guess is that he was in the top one or two percent of the wealth distribution at that time.

The Douglass mansion has been preserved today as a museum in its condition as of 1895, when he died. Below is a partial list of the appliances that can be found there, compared with their modern equivalents.

Today's residents of Anacostia Park, although many would be considered poor by today's statistical measures, have all of the modern conveniences on the right hand side of the table. In addition, they can drive to work, while Frederick Douglass had to walk five miles to his job in the Capitol building. They have radios, televisions, and many other goods that the wealthy Douglass never possessed.

Future Growth

Since 1500, economic growth has accelerated. The main elements of this acceleration have been:

1. The "demographic transition," toward fewer children, so that population growth places less pressure on the food supply.

2. Modern science and technology, which has vastly improved our ability to grow food, use machinery in place of human labor, and harness information to use resources efficiently.

3. Modern democratic states, which encourage individual freedom and promote economic growth.

All of these elements promise to contribute to economic growth in the future. The "demographic transition" has begun in the underdeveloped countries of the world. Science continues to open new frontiers in biotechnology and nanotechnology (manipulating matter at the molecular level), while innovation continues in computing and communications. Most economists believe that we can achieve growth of at least 1.5 percent per year.

To extrapolate the effect of economic growth, we can compute what the average income will be in fifty years based on a given rate of growth. For example, if average income in the U.S. today is $30,000 per year, and income grows at 1.5 percent per year, what will income be in fifty years? To answer this, we multiply ($30,000)(1.015)⁵⁰, which gives $63,157. Average income will more than double if real growth is just 1.5 percent per year. This is an example of the power of compounding, or exponential growth.

What this sort of economic growth means is that in fifty years the average person will have an income that today would be considered upper middle class. Although some people will have less income than others, absolute poverty is something that can be eliminated by economic growth and policies that assist those who are mentally and physically disabled.

A difference in growth rates that might seem small--say, 2 percent vs. 1 percent--is one that economists would deem to be very important, because of the cumulative effect over many years. We would argue that such a difference is large enough to affect the outlook for many major social concerns, including:

• our ability to continue to reduce poverty
• our ability to deal with an aging population, with its impact on Social Security
• the quality of the environment

Ten years from now, the outlook for these issues will be brighter if economic growth averages over 2 percent than if economic growth averages less than 1 percent. If we have more growth, then the poor will enjoy a higher standard of living, social security will be solvent, and our ability to maintain clean air and water will be greater.

More Growth Arithmetic

Suppose that we look at average income at two points in time, and we want to compute the average rate of economic growth over that span. We take the ratio of the two levels of income, and then compute the nth root, where n is the number of years. Then we subtract one to get the growth rate. For example, the table above says that from 1500 to 1800 average per capita GDP grew from $175 to $250. The average rate of growth was
($250/$175)^(1/300) - 1.0 = .00119, or 0.119 percent.

Review

1. How is economic growth measured? Explain why growth is important.
2. DeLong uses the term "Malthusian growth" to describe the period in which total population and total output grew at the same rate, leading to very little change in output per capita. During what time periods was growth Malthusian? During what periods has output per capita increased significantly?
3. If one country has per capita income of $15,000 and its economic growth rate is 5 percent per year, what will its per capit income be in 10 years? About how many years will it take to catch up to a country where the per capita income is $30,000 per year and the economic growth rate is 1 percent per year?
4. If we have a per capita income of $30,000 today and we want it to be $35,000 in 10 years, what growth rate do we need?

We saw that as recently as 1800, average world GDP per capita was only $250 per year, according to DeLong's estimates. Even today, there are poor countries in Africa where GDP per capita averages less than $500 per year. On the other hand, the countries that belong to the Organization for Economic Cooperation and Development (OECD), consisting of the U.S., Canada, Japan, and many nations of Western Europe, have average GDP per capita today of over $20,000.

According to Angus Maddison's data, real per capita GDP for various countries and regions in 1998 was

DeLong makes the following remarks about GDP in different groups.

1. In 1950, the gap between the U.S. and other OECD economies was wider than it is today. In 1950, Japan's GDP per capita was only 20 percent of the U.S. level, and only Britain's was over 50 percent of the U.S. level. Now, most OECD countries are over 70 percent of the U.S. level.

2. Communism had an enormous adverse impact on relative GDP. In 1997, Communist North Korea's GDP per capita was only $700, but South Korea's was $13,600. Russia's GDP per capita was only $4400, but Finland's was $20,100. Cuba's per capita GDP was $3100, but Mexico's was $8400.

3. In spite of convergence within the OECD, overall economic performance has diverged.

   Even if attention is confined to noncommunist-ruled economies, there still has been enormous divergence in relative output-per-worker levels over the past 100 years. Since 1870, the ratio of riches to poorest economies has increased sixfold. In 1870 two-thirds of all countries had GDP per capita levels between 60 and 160 percent of the average. Today the range that includes two-thirds of all countries extends from 35 to 280 percent of the average.
   --DeLong, Macroeconomics, p. 139

This divergence is not necessarily what one might expect. In fact, we would expect the following phenomena to promote convergence.

• Labor Mobility

  People who live in poor countries will want to move to rich countries, in order to take advantage of higher wages. This should increase the supply of labor in rich countries, bringing down average income there. It should reduce the supply of labor in poor countries, and with less excess labor the average incomes of workers should rise.

• Capital Mobility.

  Businesses in rich countries will want to take advantage of cheap labor overseas. They will move their plants and equipment to those countries. This should help to raise labor incomes in poor countries while holding down labor incomes in rich countries.

• Information Mobility.

  Information and knowledge to enhance productivity should flow from countries that have that knowledge to countries that need such knowledge. As poor countries take advantage of ideas developed in rich countries, income disparity should narrow.

Review

1. Assume that in 1947, North Korea and South Korea had identical GDP of $500 per year. Over the next 50 years, what was the average annual growth rate in North Korea under Communism? What was the average growth annual growth rate of South Korea under capitalism?
2. Where are differences in per capita income relatively large? Where are such differences relatively small?

"I don't believe you're really an economist," snarled the time share salesman for Spinnaker Resort of Hilton Head Island, S.C., as my family and I left without buying. Having just drawn for us a trash can to illustrate where our vacation rental payments were going, he no doubt felt deserving of the Nobel Prize.

Meanwhile, I had determined that the deal was a loser, based on his figures and an economic formula for the profitability of buying vs. renting.

profitability = rental rate + appreciation rate - interest cost

When people go to the beach for a week, they typically rent the place where they stay. People who spend a lot of time at the beach might choose to buy a place. The idea of time-sharing is that instead of buying or renting a beach condo, you buy a week at a beach condo. Every year, you can go to the beach and stay in the condo for the week that you own it.

Isn't it always better to buy than to rent? After all, if you buy, you "own" something, while if you rent, you do not. Well, if you have to pay $500,000 to buy something, and you could rent it for a nickel a year, would you still buy it? No. In fact, the decision to rent or buy depends on prices, rents, and other factors that go into the profitability formula.

A major advantage of owning something is that when you are finished using it, you have something of value. The value of a piece of property will depend on the rate at which the price appreciates. That is why the appreciation rate is in the formula.

When you own the place where you are staying, you do not have to pay rent. Therefore, you can add in the rental rate (the ratio of the rent to the purchase price) to the profitability calculation.

The main disadvantage of buying is that you have to tie up cash (or borrow funds). The interest cost on these funds has to be subtracted in the profitability calculation.

Here is how I used the salesman's figures in the formula.

1. For the rental rate, I noted that the rent for our vacation was $1200 for that week. His time share cost $11,900 to buy--call this $12,000. Simply dividing one by the other would have given a 10 percent rental rate, which would have been very nice.

   However, it is important to adjust this calculation for fees. The timeshare charged a maintenance fee of $433.25 per year, a membership fee of $200 per year, a publication subscription fee of $67 per year, and another fee of $93 per year, which I believe was a processing fee of some sort (I cannot be entirely sure that I have this straight, because when I tried to take the piece of paper on which I had written this down, the salesman tore it out of my hand).

   The fees add up to $793, so every year instead of saving $1200 in rent, we would save $1200 - $793, or $407. This is the net rental value of the timeshare. Dividing $407 by the price of the time share, $11,900, gives 3.4 percent for the rental rate, which is the first figure required by the formula.

2. When he was giving his pitch, he used the classic timeshare salesman assumption that rent and prices will go up by 10 percent per year, so I used that for the appreciation rate.

3. He said that the financing rate for us would be 17.9 percent, so I used that for the interest cost.

When I put all of these figures into the formula, the net result was:

profitability = 3.4 + 10.0 - 17.9 = - 4.5%

The negative number means that compared with renting, buying this time share would cost 4.5 percent more per year. To illustrate the economic value of this timeshare, you should draw an even bigger trash can.

This quick calculation has some flaws. For example, I have assumed implicitly that the fees will go up at the same rate as rents. They could go up by more, or by less.

Also, our rental condo and his timeshare were not exactly identical. They were very close in terms of square footage and furnishings, but there were some differences:

• "I would never stay on a public beach," the salesman said, pointing out the value of the 24-hour security at the timeshare. "Anybody with a machine gun can go in there." Mercy! We thought: not only were we staying at the public beach, but we were letting our children swim without flak jackets!

• Offsetting the value of the security guard was the fact that our rental was closer to the beach. We were right on the beach, while the timeshare was, according to the salesman, "A 4-minute walk to the beach." We later measured the distance as 7/10 of a mile, which takes 10-15 minutes to walk.

• While perhaps a case can be made that beachfront rents will go up because such land is limited, I don't see that degree of scarcity in property 7/10 of a mile away. So, I don't think the 10 percent annual appreciation is something I would bank on.

• Finally, I need to mention one thing. We were vacationing in August, when children are out of school. The timeshare price of $11,900 was for mid-May, when I imagine you could rent an equivalent place for less than the cost of the annual fees. The price for August? $17,900.

Another way to see what a bad deal this was would be to add up the price of all 52 weeks and compare it to the price of a condo. The total price for all the weeks came to about $600,000. My guess is that the condo did not cost more than $200,000. And on top of that $400,000 in profit for the timeshare company come all those lovely annual fees.

I don't want to generalize and say that all timeshare salesman are sleazebags, only the ones that I've met. Nor do I mean to criticize people who buy timeshares. I'm sure there are some happy owners. However, the economics are very unfavorable for the buyer.

The formula is something that an economist might use to determine the value of a capital asset. A capital asset is something that will last for a long time, such as a house, a factory, or a truck. A textbook example of a capital asset is a fruit-bearing tree (most economists love fruit-bearing trees, but I'm allergic to the ones near where I live).

An asset will yield "rents" (the fruit from the tree) and will enjoy price appreciation (I may be able to sell the tree for more than the original price I paid). The formula for determining whether or not it is profitable to buy the fruit tree is

profitability = rental rate + appreciation rate - interest cost

What I mean by profitability is the expected annual profit, expressed as a percent of the price of the asset. The asset could be a house, some shares of stock or of a mutual fund, or our fruit tree.

If a house costs $100,000 and the profitability is 1.5 percent, this means that every year I save 1.5 percent of $100,000, or $1500, by buying the house rather than renting. If the profitability is -1.0 percent per year, then I could save $1000 per year by renting rather than buying. If profitability is close to 0, this would say that buying and renting are economically equivalent.

The rental rate is the ratio of the first year rent to the purchase price. The first-year rent for a house would be the rent on an equivalent house. The "rent" from shares of common stock would be the dividends from the stock. The rent from the fruit tree is the proceeds from selling the fruit.

The appreciation rate is the rate at which the price increases, expressed as an annual percentage rate. Much of this price increase could be due to general inflation. In the late 1970's, inflation in the U.S. reached 10 percent per year and over. More recently, inflation has been closer to 2.5 percent per year.

Some of the price increase may be specific to the particular market. In housing, over long periods of time prices go up at the same rates as rents in an area. However, over short periods of time, housing prices can run up quickly or go into decline.

The interest cost is the cost of financing the asset purchase. With housing, most people think of this as the mortgage interest rate. With stocks or mutual funds, many individuals do not borrow. However, they could have put their money in CD's or bonds and earned interest, and it is this foregone interest (or "opportunity cost") that should be used as interest cost. Whether we borrow to buy the tree or finance the tree with our own funds, there is an interest cost to tying up our money in the tree.

Here is a way to look at the cash flows involved in buying $100,000 fruit tree, and then selling the tree after three years. The assumptions are:

1. We borrow the entire $100,000 to pay for the tree.
2. The interest rate is 12 percent per year.
3. The first year, the fruit from the tree is worth $7000. This is the "rent."
4. Each year, the rent and the price of a fruit tree go up by 6 percent per year.

The business has a negative cash flow. The "rent" from the fruit trees is less than the interest cost. Thus, the business gets more in debt each year, so that the interest cost keeps rising. However, if you include the increase in the value of the fruit tree as income, the business has a profit.

Incidentally, a major league baseball franchise is like this fruit tree. The "rent" is equal to revenues minus operating expenses, which is not enough to cover interest costs. So the baseball owner's cash flow is negative, but nonetheless the franchise appreciates in value. As long as the value appreciates by more than the negative cash flow, the business is worth owning.

Review

Suppose that we are considering buying a baseball team for $100 million. We will have to borrow money at a 10 percent interest rate. Annual revenues are $70 million, and annual expenses are $65 million.

1. Assume that revenues and expenses go up at a rate of 3 percent per year. The price of a franchise also goes up at 3 percent per year. Is this a worthwhile investment?
2. Assume that revenues, expenses, and the price of a franchise go up at a rate of 6 percent per year. Is this a worthwhile investment?
3. At what rate should revenues, expenses, and the price of a franchise go up to make this a break-even investment?

We want to talk about economic growth. However, first we need to introduce some of the standard concepts used by economists. This lesson looks at capital and related concepts.

Josh plans to have a lawn-mowing business during the five-month mowing season. There are plenty of people willing to pay $25 each time their lawn is mowed. Josh needs a lawnmower and also a pickup truck to haul the lawnmower to the jobs. The lawnmower and the pickup truck are capital goods. Capital goods include office buildings, factory equipment, airplanes, and other durable (long-lived) goods. Capital goods are contrasted with consumer goods and services, such as food, haircuts, and movie tickets. Some of the differences are listed in the table below.

There are 100 days in the lawn mowing season. Josh can mow 8 lawns a day. So his total revenue will be ($25 per lawn)(8 lawns per day)(100 days) = $20,000.

A new lawnmower costs $600 and a new pickup truck costs $25,000. But Josh still thinks he can make money.

Josh can lease the pickup truck for the mowing season for $3,000, and he can lease the lawnmower for $300. The total lease cost of $3300 is the rental cost of capital for Josh's lawn-mowing business.

Why does it cost $300 to lease the lawnmower? Because leasing the lawnmower is like borrowing money to buy it and then selling it after five months.

Suppose that Josh borrowed $600 at a simple interest rate of 1 percent per month for five months. That means that after five months, Josh would owe 5 percent in interest, or $30 in interest, in addition to the $600 principal. That is, a bank lends Josh $600 today, and he has to pay back $630 in five months.

What if Josh does not borrow the $600? Instead, he takes the money out of a savings account, where it could earn interest at the rate of one percent a month. Either way, the interest cost of the money is $30. This is true whether Josh pays the interest on a loan or foregoes the interest on a savings account.

To Think About: Is the interest rate at which Josh can borrow likely to be higher than the rate that he can earn on savings? Why? Does that mean that it is cheaper for Josh to finance his lawnmower with savings than with borrowing?

When Josh sells the lawnmower in five months, he can get $330 for it. The difference between the purchase price of $600 and the $330 he can get for the lawnmower after five months of use is called depreciation. Capital goods depreciate because of physical wear and tear. Another type of depreciation comes from technological change. As newer models come out (think of computers), older models become less valuable. For a lawnmower, most of the depreciation reflects wear and tear, rather than technological change.

If Josh gets his lawnmower by borrowing, buying, and selling, then he pays $630 (principal plus interest) and gets back $330, for a net cost of $300. This happens to be the same cost as leasing the lawnmower. If the leasing cost were $200, then it would be cheaper for Josh to lease. Conversely, if the leasing cost were $400, Josh would find it cheaper to buy a lawnmower and then sell it.

Josh is going to mow 800 lawns and get $20,000 in revenue. He can lease his capital equipment for $3300. Suppose that fuel and other expenses (marketing, billing, bookkeeping, and so on) come to $1700. That means that he has cash income of $15,000. Is that profit?

Suppose that Josh puts 8 hours a day into his business, for a total of 800 hours for the season. In that case, on average, Josh is getting just under $20 an hour. He should not consider all of this to be profit.

Suppose that Josh could earn $22 an hour working in an office. If you include the value of his time, ($22)(800) = $17,600, the lawn mowing business loses $2600. In other words, he would be better off working in an office than starting his lawn-mowing business.

On the other hand, suppose that Josh's best alternative is to make $12 an hour. Now, the lawn mowing business is a better choice. But to calculate his true profit he should subtract $12 an hour from his proceeds. Multiplying $12 an hour by 800 hours gives $9600 in salary. Subtracting this from $15,000 gives $5400 in economic profit.

Even if Josh does not call the $9600 salary, economists would call it his opportunity cost. Opportunity cost is what you have to give up in order to get something. In his textbook Hidden Order, David Friedman writes (p. 32),

The cost of an A on a midterm for one of my students may be three parties, a night's sleep, and breaking up with his current significant other. The cost of living in my house is not only taxes, maintenance, and the like; it also includes the interest I could collect on the money I would have if I sold the house to someone else...

Different people would realize different profits from the lawn mowing business, because of differences in opportunity cost. A surgeon would have a high opportunity cost, which translates into a loss if the surgeon were to mow lawns. An unskilled worker whose alternative is flipping burgers would have a low opportunity cost. For the unskilled worker, mowing lawns would be profitable.

An entrepreneur like Josh may be able to earn much more from his business than he could get working for someone else. This additional income is economic profit. A fancy term for it is Ricardian rent. A company's Ricardian rent is its income minus all of the opportunity costs, including the rental cost of capital and the value of the owners' time. A company's accounting profits, on the other hand, do not net out all opportunity costs. For example, if Josh does not pay himself a salary, then his reported profits will be $15,000 even though his Ricardian rent is only $5400.

Suppose that there is a surgeon who is more skilled at mowing lawns than Josh. In fact, the surgeon can mow a lawn in half the time that it takes Josh to mow a lawn. Should the surgeon mow the lawn herself, or should she pay Josh to mow her lawn?

The surgeon definitely should pay Josh to mow her lawn. He will charge her $25. Even if it only takes her half an hour to mow her lawn, had she spent that time doing surgery she probably would have earned about $1000.

The surgeon can mow her lawn in half the time that it takes Josh to mow her lawn. We say that she has an absolute advantage in lawn mowing. However, her comparative advantage is in doing surgery. She is better off spending her time doing surgery, and then trading some of her income earned as a surgeon to someone else to mow her lawn.

There is a sense in which all market activity reflects comparative advantage. If there were no such thing as comparative advantage, you would do everything for yourself. However, because there is comparative advantage, people tend to specialize in their work and trade for the goods and services that they consume.

Your parents may be better than you at both folding laundry and loading the dishwasher. But that does not mean that the best way to handle the chores is for them to do all the work while you watch TV! If you are a lot worse at folding laundry but only a little worse at loading the dishwasher, then you have a comparative advantage in taking care of the dishes.

Suppose that Josh wants to expand his business and mow more lawns. He could lease another lawnmower. This might enable him to work more steadily, because if one lawnmower runs out of gas or requires maintenance, he can use the other lawnmower. But leasing another lawnmower will not enable Josh to double the number of lawns that he mows.

The fact that doubling the number of lawnmowers will not double the number of lawns one person can mow is an illustration of the law of diminishing returns. When a production process requires many inputs (or "factors"), adding more of one input usually results in a less-than-proportionate increase in output.

Suppose that Josh keeps just one lawnmower, but he tries to double the number of lawns he mows by working longer. He will get tired, and he will find that working eight more hours does not enable him to mow eight more lawns in a day. That is another illustration of the law of diminishing returns.

Constant Returns to Scale

In theory, if you double all inputs in a production process, you should be able to double the output. That is called constant returns to scale.

In practice, businessmen tend to look at scale in terms of increases in some inputs but not in others. If they can double output without having to double all inputs, they say that there are economies of scale. For example, if Josh can double the number of lawns mowed by adding another worker and another lawnmower--but without having to add another pickup truck--then his business has economies of scale.

In general, suppose that you can produce x units of output with a given set of inputs. Now, suppose we double the level of inputs, and ask whether or not we get 2x units of output. We describe the returns to scale as follows:

There is an argument to be made that any business ought to have constant returns to scale if you can identify all inputs and increase them proportionately. However, in practice, there are inputs, such as managerial supervision, that are nearly impossible to increase proportionately. For example, suppose that Josh's inputs consist of four workers (including himself), four lawnmowers, and one pickup truck. If he doubles all of those inputs, then the second pickup truck will have to go to work at a different neighborhood, and it will be more difficult for Josh to supervise the workers. Thus, Josh's business is likely to exhibit diminishing returns once he has to use more than one pickup truck.

Diminishing returns arise when an important factor or input is fixed, in that it cannot be increased along with other factors. Economists believe that just about any business is subject to diminishing returns at some point. However, many businesses have increasing returns at normal levels of output.

Suppose that there are two types of lawnmowers--economy and deluxe. With the economy lawnmower, Josh can mow 800 lawns in a season. With the deluxe lawnmower, he can mow 840 lawns in a season. At $25 a lawn, the deluxe lawnmower is worth $1000 more for a season. If the economy lawnmower costs $300 to lease and the deluxe lawnmower costs $800 to lease, he should go for the deluxe lawnmower. However, if the deluxe lawnmower costs $1500 to lease, Josh should stick with the economy lawnmower.

The decision of which lawnmower to use is an example of substitution. If the price is right, Josh will substitute the deluxe lawnmower for the economy lawnmower.

Another type of substitution involves capital and labor. Suppose that Josh runs his business with four workers (including himself) and four lawnmowers. What would make him use five workers and three lawnmowers, or vice-versa?

Suppose that a worker needs a lawnmower to be productive, and that lawnmowers sometimes break down. It might be worthwhile for Josh to have spare lawnmowers and fewer workers, so that workers never have to sit idle because of mechanical failure.

Alternatively, suppose that lawnmowers are very costly to lease, but that labor is inexpensive. Josh might want to have more workers than lawnmowers, so that when one worker is taking a break that worker's lawnmower is used by another worker. That way, lawnmowers never sit idle.

Josh's decisions about substitution between capital and labor will depend on two general factors.

1. Technology.

   Different combinations of workers and lawnmower will produce different levels of output in terms of lawns mowed. The results of different combinations of labor and capital constitute the technology that is available to Josh.

2. Relative prices.

   The wages of workers and the leasing cost of different lawnmowers help determine whether Josh wants to use more or less of either input. A higher hourly wage rate would lead Josh to use less labor and more capital. A higher leasing cost would lead him to use less capital and more labor. The ratio of the leasing cost to the wage rate is called a relative price.

Another important concept is called the elasticity of substitution. In economics, an elasticity is a measure of the amount by which quantities will adjust to a change in price. When the quantity adjustment is large, we say that the relationship is very elastic. When the quantity adjustment is small, we say that it is inelastic, which means not very elastic.

When the elasticity of substitution is high, it means that a small change in relative prices will cause a large change in the inputs used. For example, suppose that there are two brands of lawnmowers of similar quality that cost about the same. A small drop in the leasing cost of one brand would cause everyone with a lawnmowing business to switch to that brand. The elasticity of substitution between similar lawnmowers will be very high.

On the other hand, consider the elasticity of substitution between capital and labor. A small drop in the price of lawnmowers probably is not going to cause Josh to replace two workers with two lawnmowers. The elasticity of substitution between workers and lawnmowers within his business is likely to be low.

We introduce some mathematical notation to describe Josh's lawnmowing business. In particular, let

K be the number of lawnmowers
L be the number of workers
Y be the number lawns per day that can be mowed using those inputs

(To keep things simple, we will omit the need for the pickup truck and other inputs.) As economists, we are particularly interested in the ratio of output per worker, or Y/L. This ratio, called labor productivity, is likely to depend on the ratio of capital (lawnmowers) per worker, or K/L. We say that there is a production function

Y/L = f(K/L)

This can be read as "the ratio of output to labor is a function of the ratio of capital to labor." If we raise the number of lawnmowers per worker, we increase output per worker. That is because the more lawnmowers we have, the more backup we have in case a lawn mower breaks down. More lawnmowers per worker means less time that workers have to spend idle while waiting for a working lawnmower.

The properties of returns to scale and substitution depend on the characteristics of the function, f(). Mathematically, the simplest function is a constant multiplied by the capital/labor ratio. That is, the simplest function would be something like Y/L = 8K/L. This would say that the number of lawns each worker mows is equal to 8 times the ratio of lawnmowers to worker.

Unfortunately, this constant function cannot be realistic. It says that if you start with one lawnmower per worker, then each worker can mow 8 lawns, which might be accurate. But then if you have two lawnmowers per worker, the constant function would say that each worker can then mow 16 lawns. If you have ten lawnmowers per worker, then each worker can mow 80 lawns! This is absurd. The problem is that a constant function violates the law of diminishing returns.

To obtain diminishing returns, we can use a function with an exponent that is less than one. For example, we could have

Y/L = 8(K/L)^0.25

If this were the production function, then with one lawnmower per worker each worker can again mow eight lawns. However, now if we were to have two lawnmowers per worker, instead of doubling productivity we only increase it to 9.5 lawns per worker. This smaller increase is more realistic. That is, it is more realistic to estimate that doubling the ratio of lawnmowers to workers results in some increase in productivity, but the increase is quite a bit less than double.

An extreme form of diminishing returns is to set the exponent equal to zero, so that we have something like Y/L = 8, regardless of the number of lawnmowers, as long as there is at least one lawnmower per worker. This type of production function, called a fixed-coefficient production function, gives us no opportunities to substitute capital for labor. This is very unrealistic for a complex economy. Nonetheless, environmentalists have on many occasions made predictions that we will run out of resources (such as oil, or fresh water), and these predictions are based implicitly on a model of zero substitution. Because that model is not realistic, the predictions have been badly off base.

We want to keep the concept of a production function as we change perspective from looking at Josh's lawn mowing busines to looking at the economy as a whole. For the economy as a whole, however, there are many types of output besides lawn mowing services. Also, there are many types of capital goods besides lawnmowers and pickup trucks--including office buildings, factory equipment, airplanes, and other durable goods. Finally, there are many types of labor, from unskilled workers to brain surgeons.

Economists use a process called aggregation to come up with a single measure of output that summarizes all of the different goods and services produced in the economy. Think of aggregation as taking a weighted average of lawns mowed, apples bought, movies rented. The weights are closely related to the relative prices of the goods. That is, an expensive surgery will have a higher weight in the aggregation process than an inexpensive pen. The aggregate measure of output is called real gross domestic product, or real GDP.

Similarly, economists take a weighted average of the number of lawnmowers, office buildings, and other capital goods to arrive at a measure of the aggregate stock of capital. One complicating factor in measuring the capital stock is computing the rate of depreciation. A drill press purchased in 1995 will have lost some of its value by now. A computer purchased in 1995 will have lost nearly all of its value by now.

In theory, economists could contruct a weighted average measure of labor input, in which a brain surgeon gets higher weight than someone with only a high school education. However, we leave the measure of labor unweighted. We do so because we are interested in average output per worker (unweighted), in order to compare across countries and to measure improvement over time.

Economists also draw a distinction between stocks and flows. In this case, a stock does not refer to the securities traded on the stock market. It means any quantity that is measured at a snapshot at a point in time. In contrast, a flow measures a quantity used or produced within a period of time. Josh could mow the Millers' lawn five times over the course of the season. We would say that the Millers' lawn represents a stock of one lawn. However, Josh's mowing represents a flow of five units of lawn mowing services.

Output and labor input are flows. The aggregate measure of real GDP is output per year. When we talk about labor input, we talk about number of workers per year. We can measure labor productivity by dividing output per year by the number of workers employed in that year. Alternatively, we can measure labor input as total hours worked, and divide this into output to obtain output per hour.

The aggregate measure of capital is a stock. We measure the capital stock as of a point in time, such as the end of 2001.

The stock of capital is related to the flow of investment. The change in the aggregate stock of capital goods between the end of 2000 and the end of 2001 is equal to the amount of capital goods produced in 2001 minus the depreciation of the capital stock that was in place at the end of 2000. Both production and depreciation of capital goods are flows. In algebraic terms, we can write

K₂₀₀₁ - K₂₀₀₀ = I₂₀₀₁ - dK₂₀₀₀

Using the Production Function to Choose K/L

Suppose that we have a business where workers are paid $100 a day and each unit of capital equipment costs $10 a day to lease. Suppose that the production function is

Y/L = 6(K/L)^0.2

where Y is the number of units of output that we have to produce each day. If we typically sell 25 units of output per day, how many workers and how many units of capital should we use?

Suppose we try using one worker. Then Y/L has to at least equal 25. If we use 100 units of capital, then Y/L is only 15. It turns out that with one worker we need 1255 units of capital to bring Y/L up to 25. Using one worker and 1255 units of capital costs $100 + (1255)($10) = $12,650.

Next, we try using two workers. Now, we need to bring Y/L up to 25/2 = 12.5 It turns out that we need 79 units of capital to do this. Using 2 workers and 79 units of capital costs (2)($100) + (79)($10) = $990.

Next, we try using three workers. With 3 workers, we would need 16 units of capital. This combination costs 3($100) + 16($10) = $460.

Next, we try using four workers. This requires 5 units of capital, for a cost of 4($100) + 5($10) = $450.

If we use five workers, we know that the cost will be at least $500. (Why?)

Economists have measured large differences in GDP per capita over time and across countries. Our first impulse is to interpret this data using the production function, which relates per capita output to the capital/labor ratio. If the exponent is 0.25, then this function is

(Y/L) = (K/L)^0.25

Suppose that we are interested in the percentage difference in per capita GDP between two points in time or between two countries. Mathematically, percentage differences behave rather like logarithms. If we were to take the logs of both sides of the production function, we would have

log(Y/L) = .25 log(K/L)

Thinking of this as an equation in percentage changes, it says that for every one percentage point difference in the capital-labor ratio, we should get a .25 percentage point difference in output per worker. Conversely, if we observe that one country has 10 percent higher output per worker than another country, then we would expect the more productive country to have 40 percent more capital per worker.

In theory, differences in the capital-labor ratio should explain all of the differences in output per worker. There is nothing else in the equation.

The capital-labor ratio certainly is important. Countries increase this ratio through capital accumulation. This means that a large share of output goes to investment, which helps to increase the stock of capital. DeLong has a chart in his text which demonstrates that most of the countries with high rates of labor productivity have investment shares of more than twenty percent of output. Conversely, the majority of low-productivity countries have investment rates below twenty percent.

However, differences in the capital-labor ratio can explain no more than half of differences in output per worker. This is true whether you are trying to explain output per worker over time in one country or you are trying to explain differences in output per worker across different countries.

Another way of putting this is that the differences in output per worker are larger than what you would predict on the basis of the capital-labor ratio. In the United States, growth in output per worker has been faster than what one have predicted based on the increase in the capital-labor ratio. Moreover, the difference between per capita output in the U.S. and that in other countries is larger than what one would predict on the basis of differences in the capital-labor ratio.

This phenomenon of unexplained differences in output per worker was first discovered in the 1950's, and dubbed "the residual." The residual is so important that we need to find a place for it in the production function. DeLong's Macroeconomics textbook calls it E, the efficiency of labor. Using this formulation, the production function is

(Y/L) = (K/L)^0.25E^0.75

Efficiency of Labor and Growth Accounting

This new construct, the efficiency of labor, gives us another element in the equation. Growth in output per worker is explained as a weighted average of the growth in capital per worker and growth in the efficiency of labor. Taking logs of both sides of the new production function gives

log(Y/L) = .25 log(K/L) + .75 log(E)

Now, we have an equation that says that economic growth is a weighted average of growth in the capital-labor ratio and growth in the efficiency of labor. Keep in mind that the efficiency of labor is not a number you can look up in the Economic Report of the President or other compendium of government statistics. It is whatever is needed to enable a production function to fit the observed data on output per worker and capital per worker.

Having coined the term "efficiency of labor," economists are obligated to produce some analysis of what determines it. Some plausible factors include:

• education per worker
• knowledge
• economic, political, and social systems

Of these factors, the only one that has a ready scale of measurement is education. In fact, some of the differences in the efficiency of labor across time and across countries can be explained by differences in the average years of schooling per worker. However, education does not explain enough to make us comfortable that it is the overwhelming factor that determines E.

Knowledge is an important factor in explaining differences in E over time. We simply know things today that we did not know years ago. For example, even if we lost all of our medical equipment and our doctors, we would still know much more about sanitation and health than people did hundreds of years ago.

Some of our knowledge is scientific and technical. Other knowledge is more prosaic. When you start a new job, you typically are given a formal orientation, company manuals, and help from senior employees who through trial and error have learned better ways of doing the work. All of this knowledge, from abstract science to everyday experience, contributes to E.

Some knowledge is in the public domain, and some knowledge is proprietary. Most scientific knowledge is available to anyone who can understand it. However, other knowledge, from the formula for Coke to the source code for Microsoft software, is considered a secret by its corporate owners.

Because most knowledge is in the public domain, knowledge does not provide a promising explanation for variations in E across countries. Even proprietary knowledge is not limited to a single country. For example, Coke has manufacturing plants throughout the world, so that its secret formula is used by workers everywhere.

When we attempt to explain differences in the efficiency of labor in different countries, economists almost inevitably are forced to focus on differences in economic, political, and social systems. The contrast that DeLong draws between output per worker in neighboring pairs of Communist and non-Communist countries certainly underlines this issue.

The production function provides a framework for accounting for growth. It leads to an approach that subdivides growth into two components--the capital-labor ratio and the efficiency of labor.

The efficiency of labor is constructed indirectly, based on the residual that results from trying explain differences in output per worker on the basis of differences in the capital-labor ratio. Economists believe that the efficiency of labor is affected by education, knowledge, and the social system.

If the capital-labor ratio were the only determinant of growth, then economic performance would be easy to manipulate. Communism and other forms of dictatorship would not necessarily fail, because an economy controlled by the government is at least as capable of setting aside output for investment as is an economy that permits a large private sector. The solution to underdevelopment would be simple, because transfers of capital to underdeveloped countries would be sufficient to raise their standards of living.

The fact that social systems matter for economic growth makes the problem of underdevelopment far more complex. It also means that non-economic variables impinge on economic performance.

Communism is not the only social system that fails. Ralph Peters, a retired Lieutenant Colonel formerly with U.S. Army Intelligence, identified seven "failure factors" that he says characterize poorly-performing states.

• Restrictions on the free flow of information.
• The subjugation of women.
• Inability to accept responsibility for individual or collective failure.
• The extended family or clan as the basic unit of social organization.
• Domination by a restrictive religion.
• A low valuation of education.
• Low prestige assigned to work.

Some of the poorest African nations suffer from nearly all of these characteristics. Many Arab countries have these problems, and apart from oil wealth their economies are primitive as a result.

China does not have many of the characteristics of failed states, but it does have strong restrictions on the flow of information. Peters believes that this will create challenges for China, either leading to reduced economic growth or to a revolution that changes the government's policies restricting information flows.

Bruce Bueno de Mesquita and Hilton L. Root argue that autocracy plays a critical role in underdevelopment. They differentiate between a government that has a broad power base (an inclusive government) and a government with a narrow power base (an autocracy). They show that the leader of an autocracy stays in power longer if there is less economic development. Autocrats channel foreign aid to their constituents and supporters, which stabilizes the regime but hurts the country.

Richard Roll and John Talbott compared economic performance in countries before and after major political changes. They found that

When countries undertake a democratic change such as deposing a dictator, they enjoy a dramatic spurt in economic growth, which persists for at least two decades. In contrast, an anti-democratic event is followed by a reduction in growth.

Many social scientists have noticed that democracy and economic growth tend to be linked. However, in theory this might all be due to relationship in which economic growth causes democracy. However, the Roll-Talbott approach shows that there is a causal relationship running from the political system to economic growth.

Another perspective on social systems is provided by looking at systems that succeed. Physicist David Brin argues that successful systems are characterized by rules that allow new ideas to emerge and compete with old ideas, with the better ideas winning.

Consider four marvels of our age -- science, democracy, the justice system and fair markets...for years, rules have been fine-tuned in each of these fields of endeavor, to reduce cheating and let quality or truth win much of the time. By harnessing human competitiveness, instead of suppressing it, these "accountability arenas" nourished much of our unprecedented wealth and freedom.

Brin notes that each of these arenas permits vigorous opposition and debate. In each case, there is a well-accepted process for settling conflicts and arriving at resolution. The result is that new ideas and methods are generated, sifted, and evaluated. With the best ideas surviving, improvement is continual.

Progress requires failure as well as success. If an inefficient company is not allowed to fail, then there will be fewer resources available to successful companies.

There is a political impulse to "protect jobs." As industries become more efficient, this is difficult. For example, farm productivity keeps increasing, so that the number of people that can be fed per farmer rises. The economy needs fewer farmers, and people can be employed more productively in other occupations. However, this creates political pressure to "save the family farm." As a result, the United States and most European countries spend enormous amounts on subsidies to farmers. Most tariffs and other trade restrictions are motivated by political pressure to keep people working in industries where they are no longer are needed.

Economists hold strong opinions about the proper role of government. They do not all agree, but in most cases economists want government to play a role that promotes growth. Some of the major issues include:

1. Property rights.

   Owners of private property will tend to take actions to enhance its long-term value. They will save and invest for the future. They will adopt useful innovations. Private property can be threatened either by excessive government regulation or inadequate law enforcement. Government must be strong enough at defining and protecting property rights, without going overboard with complex and unnecessary regulations.

2. Research and development.

   Pure scientific research tends to benefit a broad set of people. If the social benefits exceed the benefits to any one company, then research may be under-funded if it is left to the private sector. Moreover, if we rely on the private sector to undertake research, firms may want to keep the results out of the public domain. For these reasons, most economists favor government support for scientific research. However, they view market competition as better at sorting out the application of research to solving human problems. Thus, economists would propose that pure research be supported by the government, but applied research and development would best be done by the private sector. This raises the question of where to set the boundary. Economist Paul Romer says

   In the next century we’re going to be moving back and forth, experimenting with where to draw the line between institutions of science and institutions of the market.

3. Education.

   Economists are convinced that education helps to promote growth. However, this does not mean that all economists support public schools. Many economists believe that the government should provide vouchers and let parents choose schools. This would make it easier for weaker schools to fail and encourage innovative improvements in education.

4. Anti-trust policy

   A monopoly will tend to resist innovation. However, economists disagree about whether particular firms are harmful monopolists. There were prominent economists on both sides of the Microsoft anti-trust case, in which one of the key questions is whether or not Microsoft's actions will discourage innovation in the future.

   Some economists believe that it is impossible for a private company to maintain a monopoly by itself. From their point of view, monopolies are sustainable only to the extent that they are granted or protected by the government. If that is the case, then anti-trust policy does not have to be very aggressive.

Some political, economic, and social systems are more conducive to growth than others. To succeed, a system must reward education, work, and successful innovation. It must not insulate people and companies from their own mistakes and failures. In order to fail, a society must repress the natural desires of people to learn and to improve themselves.

Some of the more important ideas about economic growth are based on mathematical models. This lesson looks at some of these.

Before 1800, technological progress was relatively slow. The result was that output per worker hardly increased at all, but population grew. In 1798, Thomas Malthus wrote an essay on population that presented a pessimistic picture of economic growth. He said that when food is ample, population grows exponentially. Because there are diminishing returns to labor in food production, exponential population growth leads to starvation, and population falls again.

Here is a numerical example of a two-equation Malthusian model.

[food production] Y_t = 1000 + L_t

[population growth] L_t = 600 + 100*(Y_t-1/L_t-1)²

This generation's food production, Y_t, increases linearly with this generation's labor supply (population). However, the next generation's labor supply increases with the square of this generation's ratio of food to population.

You can solve these two equations for values of Y and L that will be stable. These are called the equilibrium values. In this case, they are 2000 for Y and 1000 for L. If L is 1000, then according to the food production equation, Y will be 2000. If Y is 2000, then population will grow to be 1000.

What happens if we start out with 2000 units of food, but disease causes the population to fall to 900? You can use the calculator below to see what happens if population starts out too low or too high. If you click on "calculate" the economy will move forward in time one generation. Keep clicking on "calculate" and you will see Y and L oscillate back and forth until they converge to their equilibrium values. You can try starting out with different values of Y and L and see the convergence process from different starting points.

Next, suppose that we get better technology in food production, so that the food production equation becomes

Y_t = 1200 + L_t

What happens to the equilibrium values of Y, L, and Y/L? Use the calculator below to find out. Keep clicking on calculate until the values stop changing.

What is the equilibrium level of food? What is the equilibrium level for the population? What is the equilibrium ratio of food to population?

At first, with the population at 1000, the technological improvement brings food production to 2200, and the ratio of food to population rises to 2.200. However, in the final equilibrium, because of population increases, the ratio of food to population is only 2.136. This is the Malthusian effect by which population growth dissipates technological advances. In fact, prior to 1800, the Malthusian effect was so strong that there was very little progress in average output per capita; instead, nearly all of the inventions and technological advances until 1800 served primarily to increase population.

When the Industrial Revolution broke out of the Malthusian trap, economies began to accumulate capital goods. In order to accumulate capital, you have to save. This means that you cannot consume all of your output.

Start with a constant level of output, Y, and no growth. Suppose that capital depreciates at a rate of 5 percent per year. In order to keep the level of capital constant we have to replace 5 percent of the capital stock each year. This means that saving, S, must equal 5 percent of the capital stock.

[1] S = .05K

We think of the saving rate, s, as the ratio of savings to income, S/Y. Writing equation [1] in terms of s, we have

[2] s = S/Y = .05(K/Y)

where all we did was divide the previous equation by Y on both sides. What this equation says is that in order to maintain constant output, we need a savings rate that equals the rate of depreciation times the capital/output ratio. If we want to have high labor productivity, we need a high ratio of capital to output, and therefore we need a high saving rate. Thus, we expect to find a relationship between countries with high saving rates and countries with high productivity, and this is indeed what Brad DeLong found when he indicated that countries with high productivity tend to have saving rates over 20 percent.

Suppose that we want the capital stock to grow at a rate of 2 percent per year. In that case, we need

[3] S = .05K + .02K

Or, in terms of s and K/Y, we need

[4] s = .05(K/Y) + .02(K/Y)

If we use the symbol d to represent depreciation, the symbol k to stand for the capital-output ratio, and the symbol x to stand for the growth rate of capital, then we can write

[5] s = dk + xk

To see how the saving rate affects the growth rate of capital, we can solve [5] for x, the growth rate of capital.

[6] x = s/k - d

If the labor force is growing at a rate n, then the capital/labor ratio will grow at the rate of x-n. For example, suppose that the saving rate s is .25 (i.e., 25 percent), the capital/output ratio k is 2.5, the depreciation rate d is .05, and the growth rate of the labor force n is .01. Then we have

[7] x - n = s/k - d - n = .25/2.5 - .05 -.01 = .04

which says that the capital/labor ratio grows at 4 percent per year. (In a moment, when I discuss balanced growth, I will argue that this is not a reasonable long-term growth rate for the capital/labor ratio.)

We are interested in the growth rate of labor productivity, Y/L. To look at productivity, we return to the production function that we used in the growth accounting lesson.

[8] (Y/L) = (K/L)^0.25E^0.75

where E is the efficiency of labor. When we took logs of both sides, we obtained an equation for the growth rate of productivity. If y is the growth rate of output and n is the growth rate of the labor force, then the growth rate of productivity is y-n. Letting g be the symbol for the growth rate of E, the efficiency of labor, we have

[9] y - n = 0.25(x - n) + .75g

When we made numerical assumptions in equation [7], we found that x-n = .04. Plugging this into equation [9] and assuming that the growth rate of the efficiency of labor, g, is .02, we have

[10] y - n = 0.25(.04) + .75(.02) = .025

Thus, the assumptions about saving rate, depreciation, and so forth imply growth in labor productivity of 2.5 percent per year.

Economists define a balanced growth path as a path along which capital and output grow at the same rate. The alternatives to a balanced growth path are not sustainable. If capital grows more slowly than output, then the capital stock will eventually drop to zero. If capital grows more quickly than output, then the share of output that you set aside for capital goods will increase until you reach the point where the amount available for consumption is zero.

Looking at equation [9], the only way that x and y can be equal is if

[11] g = x - n

That is, for balanced growth, the growth rate of the efficiency of labor must be matched by the growth rate of capital minus the growth rate of the labor force.

The requirement for balanced growth implies that there is only one sustainable ratio of capital to ouput. That is, there is only one ratio of capital to output, k that is consistent with a balanced growth path. Using equations [11] and [7] we have

[12] g = x - n = s/k - d - n

We can solve this equation for a balanced-growth value for k, given the other parameters. Using s = .25, g = .02, n = .01 and d = .05, we have

[13] k = s/(g + n + d) = .25/(.02+.01+.05) = 3.125

Therefore, the balanced-growth capital-output ratio is 3.125. If the capital-output ratio happens to be above this level, the savings rate is not high enough to maintain it, and the ratio will tend to fall back to 3.125. Conversely, if the capital-output ratio happens to start out below the balanced-growth level, the savings rate is high enough to generate capital accumulation until the ratio rises back to 3.125. Back at equation [10] when we computed labor productivity growth, we had assumed earlier an arbitrary capital-output ratio of 2.5. Now, we know that this is not a balanced-growth ratio given the saving rate, depreciation rate, and other assumed parameters. Using the balanced-growth ratio of 3.125 in equation [7] gives

[7'] x - n = .25/3.125 - .05 - .01 = .02

Putting this into [10], we have

[10'] y - n = .25(.02) + .75(.02) = .02

What we have found is that on a balanced growth path, output per worker and capital per worker grow at the same rate as the efficiency of labor. In our example, this is 2 percent per year.

Let us review what we have learned from mathematical growth models.

For the Malthusian model:

1. Whenever the food supply expands, population grows exponentially.

2. The economy has an equilibrium in which population stays constant.

3. If the equilibrium is disturbed, population will oscillate. A small generation enjoying a high ratio of food to population will reproduce excessively, leading to a large generation with a low ratio of food to population. This large generation will reproduce minimally, leading to a small generation, etc. As the magnitude of the oscillations diminishes (if indeed they do dampen), the economy goes back to its equilibrium.

4. Increases in productivity will lead to less than proportionate increases in output per worker. Instead, population expansion and diminishing returns will dissipate much of the technological improvement.

For the balanced-growth model of capital accumulation:

1. The growth rate of the economy is equal to the growth rate of the efficiency of labor. Capital per worker and output per worker both grow at this rate.

2. The savings rate affects the level of productivity and the level of the capital/output ratio. The higher the savings rate, the higher the capital/output ratio and the higher the level of productivity.

From growth theory, we have learned that for an entire country, the following factors are important in determining the level of well-being.

1. The savings rate, which determines the country's ability to accumulate capital

2. The growth rate of the efficiency of labor, which in turn depends on

   • Education
   • Cumulative Knowledge
   • Adaptive Social Institutions

For individuals, these same factors affect relative well-being. For example, young people generally tend to be better off than preceding generations, because as society accumulates knowledge, this adds to wealth. Historically, it took hundreds of years for this accumulation of knowledge to have a noticeable effect. Now, you can see the effect within a generation. Even if your parents are in the top half of the wealth distribution and you wind up in the bottom half, you are almost sure to enjoy better health care, better technology products, and a higher standard of living in general.

For over 100 years, from the time of Karl Marx until the latter part of the 20th century, economists looked at capital accumulation as the main factor in economic growth and individual wealth. In Marxist economics, it is capitalists who save and accumulate the economy's capital. They become wealthier and wealthier, while workers stay miserable until they finally get fed up and launch the Communist revolution.

The view that saving leads to wealth is not wrong. However, saving is not the only road to wealth, for a nation or for an individual. In fact, one irony is the fact that most people living under Communist dictatorships are worse off than ordinary workers under capitalism, because Communist dictatorships do not do well at adapting to advances in knowledge.

Marx's jargon of "class struggle" continues to permeate political dialogue. Marx saw the struggle as taking place between the capitalist class of savers and the working class getting by on subsistence wages. Today, people talk about a number of supposed victim classes: women, gays, and ethnic minorities are spoken of using the "class struggle" jargon, even though the original economic basis for Marxist classes--savers vs. workers--does not apply to these victim classes.

In the twentieth century, particularly in the United States, poverty has been receding. Fewer and fewer people face the squalor that was typical 150 years ago, and that is still typical in some regions of the world. Most Americans live well above subsistence levels. In fact, researchers have found that saving takes place among Americans of all income groups (there are also people at all income levels who try to live beyond their means).

Differences in well-being reflect more than just differences in capital accumulation. Two hundred years ago, when the efficiency of labor was growing slowly, inherited wealth and the lack thereof played an important role in determining people's station. With the acceleration in the rate of technological change, your inherited financial capital matters relatively less and your personal earnings power and saving rate matter relatively more.

The growth rate of your personal "efficiency of labor" will be a big factor in determining your place in the distribution of well-being. If you make good use of your education and you adapt to readily take advantage of the technologies that emerge over the next 30 years, you will be rich. If you fail to do so, then you will gradually slip to a lower place in the wealth distribution.

Statisticians collect three measures of economic well-being.

1. Income is the amount of money that an individual or a household earns in a year. Income is a flow.

2. Consumption is the value of goods and services that an individual or a household consumes in a year. Consumption is a flow.

3. Wealth is the value of the assets of an individual or a household at a point in time. Wealth is a stock.

Economists have issues with using income as a measure of well-being.

• Income has a transitory component. Some years, people earn windfalls, due to unusually large bonuses or high profits from personal businesses. In other years, people earn less than usual, because they might be laid off part of the year or they may own a business that does poorly that year.

• Income also has a "life-cycle" component, meaning that it depends on where you are in the life cycle. A graduate student may have a low income, but once she completes her degree her income likely will take a leap. A retired person may have a low income, but he has sufficient wealth to sustain a lavish lifestyle.

Wealth also has some shortcomings as a measure of well-being. Statistical measures of wealth count only financial assets, without taking an individual's earning power into account. A new graduate of medical school may have no wealth (in fact, she could be carrying a large debt on a student loan), but her prospects for future earnings may be bright. In general, younger people have less wealth than what they will be able to accumulate later in their lives.

People seem to make consumption decisions more on the basis of long-term income and wealth than on the basis of current income and wealth. Therefore, it makes sense to focus on consumption as an indicator of how people view their economic circumstances. Using consumption as a measure, economists tend to find that poverty in the United States is shrinking.

For example, W. Michael Cox and Richard Alm, in Myths of Rich & Poor, present information on the ownership of durable goods in 1994 by households whose income was below the official poverty line of around $13,000 per year. On page 15, table 1.2, they compare this to the ownership of those same types of durable goods by all households in 1971.

Looking at the table, it seems reasonable to say that a "poor" household in 1994 was at least as well off as an average household in 1971. This is without taking into account the fact that a majority of poor households have microwave ovens, VCR's, and cable television hookups, none of which were available to the average household in 1971.

Cox and Alm examine a large study of income dynamics undertaken by the University of Michigan. It tracked income of specific households from 1975 through 1991. As Cox and Alm report (p. 73),

Those who started in the bottom 20 percent in 1975 had an inflation-adjusted gain of $27,745 in average income by 1991. Among workers who began in the top fifth, the increase was just $4,354. The rich may have gotten a little richer, but the poor have gotten much richer.

The University of Michigan data suggest that low income is largely a transitory experience for those willing to work...Nearly a quarter of those in the bottom tier in 1975 moved up the next year and never again returned. By contrast, long-term hardship turned out to be rare: Less than 1 percent of the sample remained in the bottom fifth every year from 1975 to 1991.

Cox and Alm argue that if one counts as poor only households that remain below the poverty line for at least two years, then the poverty rate is 4 percent, rather than the 13 percent that was reported at the time. It may be that true poverty among the able-bodied and able-minded (meaning people who are not substance abusers or otherwise incapacitated by mental illness) has been essentially eradicated in this country.

If you compare people at a single point in time in terms of either income or wealth, then disparities stand out. Today, the top-to-bottom ratio of income or wealth is larger than ever. Some economists would downplay this fact, and instead focus on absolute levels of well-being.

However, people seem to care about relative economic standing as well as their absolute standing. For example, Reason's Ronald Bailey cites a fascinating experiment conducted by British economists Daniel John Zizzo and Andrew Oswald. First, the researchers placed subjects in a gambling game. Then, as Bailey reports,

At the conclusion of the gambling sessions, each player was given the chance to spend his own money to anonymously "burn" some of the cash won by his fellow participants. It was made clear that there was no prospect that burning his fellow player’s winnings would in any way make him richer. In fact, if he chose to burn another player’s money, he had to pay between 2 cents and 25 cents for each dollar subtracted from the other player’s take.

Zizzo and Oswald found that nearly two-thirds of players happily paid for the privilege of impoverishing their fellow participants.

This suggests that a political platform of "soak the rich" will have support. In fact, one consequence of the increased dispersion in incomes is that in the United States the income tax is focused on the upper end of the income distribution.

Since the 1960's, the share of income accounted for by the top fifth of households is up somewhat. More important has been the increase in all levels of income. The average real income of people in the second fifth of households today exceeds the average real income of people in the top twenty percent in the 1960's. See the following table, which comes from the census report on income distribution, in dollars of constant purchasing power.

The combination of a large rise in overall income and a slight increase in the share at the top means that households earning over $100,000 now account for something close to three-fourths of all income. If we think of "rich" in absolute terms ($100,000 per year in household income, adjusted for inflation) rather than in relative terms (the top 20 percent), the "rich" now earn enough income to fund both baseline government functions plus programs to help the poor.

We do not need the middle class to pay taxes any more. In fact, with income taxes, the middle-class taxpayer is on the road to extinction. Data from the U.S. Treasury compiled by Daniel Mitchell for the Heritage Foundation show that the bottom 50 percent of the income distribution accounts for only 4.2 percent of tax revenues, as shown in the follwing table:

If the distribution of income were static, then these data would suggest that most people are unaffected by tax cuts, because they pay so little in taxes already. However, keep in mind that the distribution of income is fluid, so that people who are in the bottom 50 percent one year may be in the top ten percent the next year.

1. Individual well-being is affected by the same factors that determine economic growth, including the level of cumulative knowledge when the person is born as well as the person's saving rate, education, and ability to adapt to new technology.

2. Income in any given year is not a reliable indicator of an individual's wealth or poverty. A large percentage of today's "poor" people own durable goods that are as good or better than those of the average household thirty years ago. Most people with low income one year will do much better in other years. Only about 4 percent of the population has an income below the poverty line for two years or more.

3. Income disparities, which may lead to resentment, are widening. This shows up clearly in data on Federal tax revenues, where very little is collected from people in the bottom 50 percent of the income distribution in any given year.

1. A fishing business uses boats and fishermen. A new boat costs $130,000 to purchase. The interest rate is 15 percent per year. At the end of one year, you can sell a used boat for $117,000. If buying and re-selling the boat costs the same as leasing, how much should it cost to lease the boat?

2. What should the leasing cost be if the interest rate is 20 percent per year?

3. "I built my deck myself. I'm more efficient at it than a contractor." Comment.

4. You and your roommate prepare meals together. It takes you one hour to cook and one hour to clean up after a meal. It takes your roommate 20 minutes to cook and 30 minutes to clean up after a meal. Are you better off taking turns, where you cook and clean one night and your roommate cooks and cleans the next night, or should one of you always cook and one of you always clean? What economic concept does this illustrate?

5. For each situation below, state whether it illustrates increasing returns to scale, decreasing returns to scale, or constant returns to scale.

   • To get two cups of cooked rice, start with two cups of water and one cup of uncooked rice. To get four cups of cooked rice, start with 3-1/2 cups of water and 2 cups of rice.

   • The school can add four students to the economics class without requiring another teacher.

   • The school cannot increase enrollment by 20 percent without increasing faculty and staff by at least 25 percent.

   • You are using a recipe for chocolate chip cookies. The recipe says that if you want to triple the number of cookies, you just triple all of the ingredients.

6. A school can use blackboards and chalk as well as whiteboards and markers. Will the elasticity of substitution between blackboards and chalk be higher or lower than the elasticity of subtitution between blackboards and whiteboards? Explain how a drop in the price of blackboards might affect the use of chalk, whiteboards, and markers.

Variable Input Levels

So far, profit maximization for Cool and Slick has actually been simple. We assumed that there would be exactly 2000 hours of Cool's technical input and 2000 hours of Slick's sales input. All we had to do is allocate them between two goods. We do this based on how much of each good can be produced and sold with different combinations of their input, and based on the relative prices of those two goods.

The problem becomes more complicated if we allow for variable hours. Perhaps one of them will work less than 2000 hours and take a pay cut, or work more and take a pay increase. Or we could allow them to hire additional technical and sales staff.

Suppose that we focus on the consulting business. Suppose that the cost of a salesperson works out to $20 an hour, and the cost of a technical person works out to $40 an hour. Each sales person can land 3 consulting contracts per 200 hours of work. The productivity of technical workers is as follows:

The reason that profits start to decline after output increases beyond a certain amount is old friend, the law of diminishing returns. If you compare the first two columns, you will see that it starts to require more and more technical staff to complete three additional contracts. There are diminishing returns to adding technical staff.

Incidentally, there is a famous essay by Frederick Brooks on productivity in computer programming, which illustrates the law of diminishing returns. The essay is called "the mythical man-month." Brooks pointed out that a project manager will say that a project requires 2 people for 4 months, or 4 man-months. However, if you try to get the job done in one month with 8 people, it does not work. To an economist, this indicates that there are diminishing returns. As you add a person to a project for a month, you get less than one man-month's worth of results.

1. "The lecture notes that I made up for this course are a stock. The number of lectures that I give per week are a flow." Comment.

2. Suppose that the production function for mowing laws is

   Y/L = 16K - 2L

   where Y is output, L is labor input, and K is capital input. If capital costs $50 an hour to rent and labor costs $10 an hour, what is the lowest-cost combination of labor and capital that can produce 25 units of output? Assume that labor has to be an integer (1, 2, 3, etc.) but that capital can be any positive rational number, including fractions.

3. Suppose that from now on, India's per capita GDP grows at a 5 percent annual rate, and in the United States per capita GDP grows at a 1.5 percent annual rate. Which economy will have a higher GDP per capita in 20 years? in 50 years? in 100 years?

4. Suppose that the production function for an economy is

   (Y/L) = (K/L)^0.25E^0.75

   where Y/L is labor productivity, K/L is the capital-output ratio, and E is the efficiency of labor.

   • If the capital-output ratio rises from 3.0 to 3.3, what will be the approximate percentage increase in labor productivity?
   • If the efficiency of labor falls from $10,000 to $9500, what will be the approximate percentage decrease in labor productivity?

1. The number of telephone subscribers is growing most rapidly in poor countries with competitive telephone industries. It is growing least rapidly in countries with state-run telephone monopolies. Is this consistent with what you have studied so far about growth? Explain

2. Many Internet businesses have failed. What does this say about the Internet as a catalyst for economic growth?

3. In an economy with a depreciation rate of 4 percent (.04), population growth of 2 percent, a savings rate of 30 percent, and growth in the efficiency of labor of 1 percent,

   • What is the growth rate of capital per worker that will keep the economy on a balanced growth path? What will be the growth rate of output per worker?

   • What is the capital-output ratio along the balanced growth path?

4. What evidence exists to show that poverty is decreasing?