One Grain, Two Grains, Four Grains

One Grain, Two Grains, Four Grains

Once upon a time, a peasant invented the game of chess. The king was so enthralled by the game that he offered the peasant to name any reward of his choice. The peasant responded with the humble request that the king place a grain of rice on the first square of the chessboard, two grains on the second square, four on the third, and so forth, by continually doubling the amount until he had reached the 64th and last square of the board. The king readily accepted, laughing at the peasant’s naïveté. The king never arrived at the last square. If he had he would have owed the peasant more than nine quintillion grains or 230 billion tons, which is equivalent to 340 years of modern global rice production.

The story, although apocryphal, illustrates the importance of growth rates. While a single grain of rice may seem negligible, compounded 64 times, it becomes an unimaginable quantity. The corollaries with economic prosperity are obvious. When evaluating the prosperity of an economy, the most important figure is not the current level of production, but rather its growth rate. There is a nifty mathematical trick known as the rule of 70. Divide 70 by a growth rate to find how long it takes for the initial amount to double. For example, if an economy grows at a rate of 3.5 percent per year, it will double in size in 20 years. If an economy grows 7 percent per year, it will quadruple in size in the same span of time. Seemingly trivial differences in growth rates result in prodigious differences in magnitude over time.

Naturally, one of the most important questions for macroeconomic theorists and policymakers is how to maximize economic growth. One of the major breakthroughs in macroeconomics originated at our very own Amherst College. In 1928, Charles Cobb, a mathematician at Amherst College, and Paul Douglas, a visiting lecturer, published a study detailing the Cobb-Douglass production function, which relates total production in an economy to the amount of labor and capital (goods used in the production of other goods and services). Working with the Cobb-Douglas framework, Robert Solow developed one of the most important models in macroeconomic theory to explain long-run economic growth, for which he was awarded the Nobel Prize in 1987.

The Solow model offers several valuable insights. For example, it shows that growth in living standards is partially driven by capital deepening, i.e., increases in the amount of capital per worker. This makes sense. The more ovens bakers have, the more cakes they can produce, and the more income they can earn. What is disconcerting, however, is what the Solow model does not show. Only a fraction of growth in living standards is actually due to capital deepening. The majority of the growth is actually due to what economists term total factor productivity. Thus arises one of the most problematic holes in macroeconomic theory.

Economists often equate total factor productivity to technology. This can be somewhat misleading. Total factor productivity cannot be measured directly. It may change when tangible technology, such as computers, develops or improves but also changes when new ideas, such as a new manufacturing processes or managerial systems, are conceived. A more accurate way of describing total factor productivity is as the Solow residual. The Solow residual refers to the growth that is not accounted for by known factors, namely increases in labor and capital. In a word, total factor productivity represents growth that occurs due to factors that are not precisely known.

Therefore, most of economic growth is caused by increases in total factor productivity, and increases in total factor productivity cannot be measured or precisely explained. The implication is clear and broadly pessimistic. Economist theorists do not really know what causes economies to grow at the rates they do, and economic policy makers do not really know how to make economies grow faster.

Various economists have attempted to expand on the Solow model. For example, the Romer model explains total factor productivity growth endogenously as a function of the fraction of the population devoted to research and development, and the productivity of research and development (which is essentially the productivity of productivity) and the size of the population. All of these appendages, however, produce anomalies and are not particularly compelling.

Considering the tremendous importance of growth rates, it is perturbing that there exist few satisfactory and conclusive answers. It is worth noting that the rampant economic growth enjoyed by the United States and much of the world in the past half-century is the exception rather the rule relative to historic growth rates. There is nothing to guarantee that the American economy will continue to grow faster or even as fast as it has in the past. Whether the economy grows two or three percent this year may seem trivial. Whether it grows two or thee percent next year may also seem trivial. But so does a grain of rice. Even two grains of rice.