"Two Centuries of Productivity Growth in Computing", Nordhaus 2007; excerpts:
"The present study analyzes computer performance over the last century and a half. Three results stand out. First, there has been a phenomenal increase in computer power over the twentieth century. Depending upon the standard used, computer performance has improved since manual computing by a factor between 1.7 trillion and 76 trillion. Second, there was a major break in the trend around World War II. Third, this study develops estimates of the growth in computer power relying on performance rather than components; the price declines using performance-based measures are markedly larger than those reported in the official statistics.
The usual way to examine technological progress in computing is either through estimating the rate of total or partial factor productivity or through examining trends in quality-adjusted prices. For such measures, it is critical to use constant-quality prices so that improvements in the capabilities of computers are adequately captured. The earliest studies, dating from around 1953, examined the price declines of mainframe computers and used computers. Early studies found annual price declines of 15 to 30 percent per year, and recent estimates find annual price declines of 25 to 45 percent. 2
Computers are such a pervasive feature of modern life that we can easily forget how much of human history existed with only the most rudimentary aids to addition, data storage, printing, copying, rapid communications, or graphics. The earliest recorded computational device was the abacus, but its origins are not known. The Darius Vase in Naples (dated around 450 BC ) shows a Greek treasurer using a table abacus, or counting board, on which counters were moved to record numbers and perform addition and subtraction. The earliest extant “calculator” is the Babylonian Salamis tablet (300 BC ), a huge piece of marble, which used the Greek number system and probably deployed stone counters. Analog devices developed during the first century BC , such as the Antikythera Mechanism, may have been used to calculate astronomical dates and cycles. 3 The design for the modern abacus appears to have its roots in the Roman hand-abacus, introducing grooves to move the counters, of which there are a few surviving examples. Counting boards looking much like the modern abacus were widely used as mechanical aids in Europe from Roman times until the Napoleonic era, after which most reckoning was done manually using the Hindu-Arabic number system. The earliest records of the modern rod abacus date from the thirteenth century in China (the suan-pan), and the Japanese variant (the modern soroban) came into widespread use in Japan in the nineteenth century. Improving the technology for calculations naturally appealed to mathematically inclined inventors. Around 1502 Leonardo sketched a mechanical adding machine; it was never built and probably would not have worked. The first surviving machine was built by Pascal in 1642, using interlocking wheels. I estimate that fewer than 100 operable calculating machines were built before 1800. 4 Early calculators were “dumb” machines that essentially relied on incrementation of digits. An important step in the development of modern computers was mechanical representation of logical steps. The first commercially practical information-processing machine was the Jacquard loom, developed in 1804. This machine used interchangeable punched cards that controlled the weaving and allowed a large variety of patterns to be produced automatically. This invention was part of the inspiration of Charles Babbage, who developed one of the great precursor inventions in computation. He designed two major conceptual breakthroughs, the “Difference Engine” and the “Analytical Engine.” The latter sketched the first programmable digital computer. Neither of the Babbage machines was constructed during his lifetime. An attempt in the 1990s by the British Museum to build the simpler Difference Engine using early-nineteenth-century technologies failed to perform its designed tasks. 5
The first calculator to enjoy large sales was the “arithmometer,” designed and built by Thomas de Colmar, patented in 1820. This device used levers rather than keys to enter numbers, slowing data entry. It could perform all four arithmetic operations, although the techniques are today somewhat mysterious. [The present author attempted to use a variant of the arithmometer but gave up after an hour when failing to perform a single addition.] The device was as big as an upright piano, unwieldy, and used largely for number crunching by insurance companies and scientists. 7 Contemporaneous records indicate that 500 were produced by 1865, so although it is often called a “commercial success,” it was probably unprofitable. 8
Table 1 shows an estimate of the cumulative production of computational devices (excluding abacuses and counting boards) through 1920. This tabulation indicates that fewer than 1,000 mechanical calculators were extant at the time of rise of the calculator industry in the 1870s, so most calculations at that time were clearly done manually. 9
It is difficult to imagine the tedium of office work in the late nineteenth century. According to John Coleman, president of Burroughs, “Bookkeeping, before the advent of the adding machine, was not an occupation for the flagging spirit or the wandering mind . . . . It required in extraordinary degree a capacity for sustained concentration, attention to detail, and a passion for accuracy.” 10
A 1909 report from Burroughs compared the speed of trained clerks adding up long columns of numbers by hand with that of a Burroughs calculator, as shown in Figure 1. These showed that the calculator had an advantage of about a factor of six, as reported: Ex-President Eliot of Harvard hit the nail squarely on the head when he said, “A man ought not to be employed at a task which a machine can perform.”
The Electrical Tabulating System, designed by Herman Hollerith in the late 1880s, saw limited use in hospitals and the War Department, but its first serious deployment was for the 1890 census. The Tabulator was unable to subtract, multiply, or divide, and its addition was limited to simple incrementation. Its only function was to count the number of individuals in specified categories, but for this sole function, it was far speedier than all other available methods. During a government test in 1889, the tabulator processed 10,491 cards in five and a half hours, averaging 0.53 cards per second.
The economics of the computer begins with a study by Gregory Chow. 12 He estimated the change in computer prices using three variables (multiplication time, memory size, and access time) to measure the performance of different systems over the period 1955– 1965. Many studies have followed in this tradition, and Jack Triplett provides an excellent recent overview of different techniques. 13 Overall, there are 253 computing devices in this study for which minimal price and performance characteristics could be identified. The full set of machines and their major parameters are provided in an appendix available online. 14
The bundle of computations performed by different systems evolves greatly over time. For the earliest calculators, the tasks involved primarily addition (say for accounting ledgers). To these early tasks were soon added scientific and military applications (such as calculating ballistic trajectories, design of atomic weapons, and weather forecasts). In the modern era, computers are virtually everywhere, making complex calculations in science and industry, helping consumers e-mail or surf the web, operating drones on the battlefield, producing images from medical scans, and combating electronic diseases. In all cases, I measure “computer power” as the number of times that a given bundle of computations can be performed in a given time; and the cost of computation as the cost of performing the benchmark tasks.
The data on manual calculations were taken from a Burroughs monograph, from estimates of Moravec, and from tests by the author. 22 The computational capabilities of the abacus are not easily measured because of the paucity of users in most countries today. One charming story reports a Tokyo competition in 1946 between the U.S. Army’s most expert operator of the electric calculator in Japan and a champion operator of the abacus in the Japanese Ministry of Postal Administration. According to the report, the addition contest consisted of adding 50 numbers each containing three to six digits. In terms of total digits added, this is approximately the same as the tests shown in Figure 1. The abacus champion completed the addition tasks in an average of 75 seconds, while the calculator champion required 120 seconds. They battled to a standoff in multiplication and division. The abacus expert won four of the five contests and was declared the victor. 23 This comparison suggests that, in the hands of a champion, the abacus had a computer power approximately four and a half times that of manual calculation. Given the complexity of using an abacus, however, it is unlikely that this large an advantage would be found among average users. We have reviewed requirements for Japanese licensing examinations for different grades of abacus users from the 1950s. These estimates suggest that the lowest license level (third grade) has a speed approximately 10 percent faster than manual computations. 24
able 5 shows a summary of the overall improvement in computing relative to manual calculations and the growth rates in performance. The quantitative measures are computer power, the cost per unit computer power in terms of the overall price level, and the cost of computation in terms of the price of labor. The overall improvements relative to manual computing range between two and 73 trillion depending upon the measure used. For the period 1850 (which I take as the birth of modern computing) to 2006, the compound logarithmic growth rate is around 20 percent per year. We now discuss the results in detail. Start with Figure 2, which shows the results in terms of pure performance—computing power in terms of computations per second. Recall that the index is normalized so that manual computation is one. Before World War II, the computation speeds of the best machines were between ten and 100 times the speed of manual calculations. There was improvement, but it was relatively slow. Figure 3 shows the trend in the cost of computing over the last century and a half. The prices of computation begin at around \$500 per MCPS for manual computations and decline to around \$6 x 10^–11 per MCPS by 2006 (all in 2006 prices), which is a decline of a factor of seven trillion.
Table 6 shows five different measures of computational performance, starting with manual computations through 2006. The five measures are computer power, cost per unit calculation, labor cost per unit calculation, cycles per second, and rapid memory. The general trends are similar, but different measures can differ substantially. One important index is the relative cost of computation to labor cost. This is the inverse of total labor productivity in computation, and the units are therefore CPS per hour of work. 25 Relative to the price of labor, computation has become cheaper by a factor of 7.3 × 10^13 compared to manual calculations.
Table 7 indicates modest growth in performance from manual computation until the 1940s. The average increase in computer productivity shown in the first three columns of the first row of Table 7—showing gains of around 3 percent per year—was probably close to the average for the economy as a whole during this period.
Statistical estimates of the decadal improvements are constructed using a log-linear spline regression analysis. Table 8 shows a regression of the logarithm of the constant-dollar price of computer power with decadal trend variables. The coefficient is the logarithmic growth rate, so to get the growth rate for a period we can sum the coefficients up to that period. The last column of Table 8 shows the annual rates of improvement of computer performance. All measures of growth rates are logarithmic growth rates. 26
Using decadal trend-break variables, as shown in Table 8, we find highly significant positive coefficients for the dummy variables beginning in 1945 and in 1985 (both indicating acceleration of progress). The only period when progress was slow (only 22 percent per year!) was during the 1970s. Table 7 uses a different methodology for examining subperiods. It shows a slowing in the 1960–1969 and 1970–1979 periods. We were unable to resolve the timing and cause of the slowdown in the 1960–1979 subperiod, and this is left as an open question.
In terms of the ideal measure described above, it is likely that standard measures of performance are biased downward. If we take an early output mix—addition only—then the price index changes very little, as discussed in the last paragraph. On the other hand, today’s output bundle was infeasible a century ago, so a price index using today’s bundle of output would have fallen even faster than the index reported here. Put differently, a particular benchmark only includes what is feasible, that is, tasks that can be performed in a straightforward way by that year’s computers and operating systems. Quantum chromodynamics is included in SPEC 2000, but it would not have been dreamt of by Kenneth Knight in his 1966 study. This changing bundle of tasks suggests that, if anything, the price of computation has fallen even faster than the figures reported here.
The price of supercomputing is generally unfavorable relative to personal computers. IBM’s stock model supercomputer, called “Blue Horizon,” is clocked at 1,700 Gflops and had a list price in 2002 of $50 million—about $30,000 per Gflop—which makes it approximately 34 times as expensive on a pure performance basis as a Dell personal computer in 2004.
Fifth, these results imply that there has been a rapid deepening of computer capital in the United States. Because of the growth in both the power and scope of computer power, the capital-labor ratio for computer capital has risen sharply. To provide an order-of-magnitude idea of the amount of capital deepening that has occurred, I estimate the amount of computer power available per hour of work. Using estimates of the number of machines and computer power per machine, I estimate that there was approximately 0.001 unit of (manual-equivalents of) computer power available per hour worked in 1900. That increased to about one unit of computer power per hour by the middle of the twentieth century. By 2005, computational power had increased to about 10^12 per hour worked. 37
At the same time, and as a sixth point, this enormous growth in computer power does not imply that there were correspondingly large increases in economic welfare all along the way. The rapid increase in productivity reflected an equally rapid decline in the cost of computation, and the decline was probably matched by a similar decline in the marginal productivity of computing."
"The present study analyzes computer performance over the last century and a half. Three results stand out. First, there has been a phenomenal increase in computer power over the twentieth century. Depending upon the standard used, computer performance has improved since manual computing by a factor between 1.7 trillion and 76 trillion. Second, there was a major break in the trend around World War II. Third, this study develops estimates of the growth in computer power relying on performance rather than components; the price declines using performance-based measures are markedly larger than those reported in the official statistics.
The usual way to examine technological progress in computing is either through estimating the rate of total or partial factor productivity or through examining trends in quality-adjusted prices. For such measures, it is critical to use constant-quality prices so that improvements in the capabilities of computers are adequately captured. The earliest studies, dating from around 1953, examined the price declines of mainframe computers and used computers. Early studies found annual price declines of 15 to 30 percent per year, and recent estimates find annual price declines of 25 to 45 percent. 2
Computers are such a pervasive feature of modern life that we can easily forget how much of human history existed with only the most rudimentary aids to addition, data storage, printing, copying, rapid communications, or graphics. The earliest recorded computational device was the abacus, but its origins are not known. The Darius Vase in Naples (dated around 450 BC ) shows a Greek treasurer using a table abacus, or counting board, on which counters were moved to record numbers and perform addition and subtraction. The earliest extant “calculator” is the Babylonian Salamis tablet (300 BC ), a huge piece of marble, which used the Greek number system and probably deployed stone counters. Analog devices developed during the first century BC , such as the Antikythera Mechanism, may have been used to calculate astronomical dates and cycles. 3 The design for the modern abacus appears to have its roots in the Roman hand-abacus, introducing grooves to move the counters, of which there are a few surviving examples. Counting boards looking much like the modern abacus were widely used as mechanical aids in Europe from Roman times until the Napoleonic era, after which most reckoning was done manually using the Hindu-Arabic number system. The earliest records of the modern rod abacus date from the thirteenth century in China (the suan-pan), and the Japanese variant (the modern soroban) came into widespread use in Japan in the nineteenth century. Improving the technology for calculations naturally appealed to mathematically inclined inventors. Around 1502 Leonardo sketched a mechanical adding machine; it was never built and probably would not have worked. The first surviving machine was built by Pascal in 1642, using interlocking wheels. I estimate that fewer than 100 operable calculating machines were built before 1800. 4 Early calculators were “dumb” machines that essentially relied on incrementation of digits. An important step in the development of modern computers was mechanical representation of logical steps. The first commercially practical information-processing machine was the Jacquard loom, developed in 1804. This machine used interchangeable punched cards that controlled the weaving and allowed a large variety of patterns to be produced automatically. This invention was part of the inspiration of Charles Babbage, who developed one of the great precursor inventions in computation. He designed two major conceptual breakthroughs, the “Difference Engine” and the “Analytical Engine.” The latter sketched the first programmable digital computer. Neither of the Babbage machines was constructed during his lifetime. An attempt in the 1990s by the British Museum to build the simpler Difference Engine using early-nineteenth-century technologies failed to perform its designed tasks. 5
The first calculator to enjoy large sales was the “arithmometer,” designed and built by Thomas de Colmar, patented in 1820. This device used levers rather than keys to enter numbers, slowing data entry. It could perform all four arithmetic operations, although the techniques are today somewhat mysterious. [The present author attempted to use a variant of the arithmometer but gave up after an hour when failing to perform a single addition.] The device was as big as an upright piano, unwieldy, and used largely for number crunching by insurance companies and scientists. 7 Contemporaneous records indicate that 500 were produced by 1865, so although it is often called a “commercial success,” it was probably unprofitable. 8
Table 1 shows an estimate of the cumulative production of computational devices (excluding abacuses and counting boards) through 1920. This tabulation indicates that fewer than 1,000 mechanical calculators were extant at the time of rise of the calculator industry in the 1870s, so most calculations at that time were clearly done manually. 9
It is difficult to imagine the tedium of office work in the late nineteenth century. According to John Coleman, president of Burroughs, “Bookkeeping, before the advent of the adding machine, was not an occupation for the flagging spirit or the wandering mind . . . . It required in extraordinary degree a capacity for sustained concentration, attention to detail, and a passion for accuracy.” 10
A 1909 report from Burroughs compared the speed of trained clerks adding up long columns of numbers by hand with that of a Burroughs calculator, as shown in Figure 1. These showed that the calculator had an advantage of about a factor of six, as reported: Ex-President Eliot of Harvard hit the nail squarely on the head when he said, “A man ought not to be employed at a task which a machine can perform.”
The Electrical Tabulating System, designed by Herman Hollerith in the late 1880s, saw limited use in hospitals and the War Department, but its first serious deployment was for the 1890 census. The Tabulator was unable to subtract, multiply, or divide, and its addition was limited to simple incrementation. Its only function was to count the number of individuals in specified categories, but for this sole function, it was far speedier than all other available methods. During a government test in 1889, the tabulator processed 10,491 cards in five and a half hours, averaging 0.53 cards per second.
The economics of the computer begins with a study by Gregory Chow. 12 He estimated the change in computer prices using three variables (multiplication time, memory size, and access time) to measure the performance of different systems over the period 1955– 1965. Many studies have followed in this tradition, and Jack Triplett provides an excellent recent overview of different techniques. 13 Overall, there are 253 computing devices in this study for which minimal price and performance characteristics could be identified. The full set of machines and their major parameters are provided in an appendix available online. 14
The bundle of computations performed by different systems evolves greatly over time. For the earliest calculators, the tasks involved primarily addition (say for accounting ledgers). To these early tasks were soon added scientific and military applications (such as calculating ballistic trajectories, design of atomic weapons, and weather forecasts). In the modern era, computers are virtually everywhere, making complex calculations in science and industry, helping consumers e-mail or surf the web, operating drones on the battlefield, producing images from medical scans, and combating electronic diseases. In all cases, I measure “computer power” as the number of times that a given bundle of computations can be performed in a given time; and the cost of computation as the cost of performing the benchmark tasks.
The data on manual calculations were taken from a Burroughs monograph, from estimates of Moravec, and from tests by the author. 22 The computational capabilities of the abacus are not easily measured because of the paucity of users in most countries today. One charming story reports a Tokyo competition in 1946 between the U.S. Army’s most expert operator of the electric calculator in Japan and a champion operator of the abacus in the Japanese Ministry of Postal Administration. According to the report, the addition contest consisted of adding 50 numbers each containing three to six digits. In terms of total digits added, this is approximately the same as the tests shown in Figure 1. The abacus champion completed the addition tasks in an average of 75 seconds, while the calculator champion required 120 seconds. They battled to a standoff in multiplication and division. The abacus expert won four of the five contests and was declared the victor. 23 This comparison suggests that, in the hands of a champion, the abacus had a computer power approximately four and a half times that of manual calculation. Given the complexity of using an abacus, however, it is unlikely that this large an advantage would be found among average users. We have reviewed requirements for Japanese licensing examinations for different grades of abacus users from the 1950s. These estimates suggest that the lowest license level (third grade) has a speed approximately 10 percent faster than manual computations. 24
able 5 shows a summary of the overall improvement in computing relative to manual calculations and the growth rates in performance. The quantitative measures are computer power, the cost per unit computer power in terms of the overall price level, and the cost of computation in terms of the price of labor. The overall improvements relative to manual computing range between two and 73 trillion depending upon the measure used. For the period 1850 (which I take as the birth of modern computing) to 2006, the compound logarithmic growth rate is around 20 percent per year. We now discuss the results in detail. Start with Figure 2, which shows the results in terms of pure performance—computing power in terms of computations per second. Recall that the index is normalized so that manual computation is one. Before World War II, the computation speeds of the best machines were between ten and 100 times the speed of manual calculations. There was improvement, but it was relatively slow. Figure 3 shows the trend in the cost of computing over the last century and a half. The prices of computation begin at around \$500 per MCPS for manual computations and decline to around \$6 x 10^–11 per MCPS by 2006 (all in 2006 prices), which is a decline of a factor of seven trillion.
Table 6 shows five different measures of computational performance, starting with manual computations through 2006. The five measures are computer power, cost per unit calculation, labor cost per unit calculation, cycles per second, and rapid memory. The general trends are similar, but different measures can differ substantially. One important index is the relative cost of computation to labor cost. This is the inverse of total labor productivity in computation, and the units are therefore CPS per hour of work. 25 Relative to the price of labor, computation has become cheaper by a factor of 7.3 × 10^13 compared to manual calculations.
Table 7 indicates modest growth in performance from manual computation until the 1940s. The average increase in computer productivity shown in the first three columns of the first row of Table 7—showing gains of around 3 percent per year—was probably close to the average for the economy as a whole during this period.
Statistical estimates of the decadal improvements are constructed using a log-linear spline regression analysis. Table 8 shows a regression of the logarithm of the constant-dollar price of computer power with decadal trend variables. The coefficient is the logarithmic growth rate, so to get the growth rate for a period we can sum the coefficients up to that period. The last column of Table 8 shows the annual rates of improvement of computer performance. All measures of growth rates are logarithmic growth rates. 26
Using decadal trend-break variables, as shown in Table 8, we find highly significant positive coefficients for the dummy variables beginning in 1945 and in 1985 (both indicating acceleration of progress). The only period when progress was slow (only 22 percent per year!) was during the 1970s. Table 7 uses a different methodology for examining subperiods. It shows a slowing in the 1960–1969 and 1970–1979 periods. We were unable to resolve the timing and cause of the slowdown in the 1960–1979 subperiod, and this is left as an open question.
In terms of the ideal measure described above, it is likely that standard measures of performance are biased downward. If we take an early output mix—addition only—then the price index changes very little, as discussed in the last paragraph. On the other hand, today’s output bundle was infeasible a century ago, so a price index using today’s bundle of output would have fallen even faster than the index reported here. Put differently, a particular benchmark only includes what is feasible, that is, tasks that can be performed in a straightforward way by that year’s computers and operating systems. Quantum chromodynamics is included in SPEC 2000, but it would not have been dreamt of by Kenneth Knight in his 1966 study. This changing bundle of tasks suggests that, if anything, the price of computation has fallen even faster than the figures reported here.
The price of supercomputing is generally unfavorable relative to personal computers. IBM’s stock model supercomputer, called “Blue Horizon,” is clocked at 1,700 Gflops and had a list price in 2002 of $50 million—about $30,000 per Gflop—which makes it approximately 34 times as expensive on a pure performance basis as a Dell personal computer in 2004.
Fifth, these results imply that there has been a rapid deepening of computer capital in the United States. Because of the growth in both the power and scope of computer power, the capital-labor ratio for computer capital has risen sharply. To provide an order-of-magnitude idea of the amount of capital deepening that has occurred, I estimate the amount of computer power available per hour of work. Using estimates of the number of machines and computer power per machine, I estimate that there was approximately 0.001 unit of (manual-equivalents of) computer power available per hour worked in 1900. That increased to about one unit of computer power per hour by the middle of the twentieth century. By 2005, computational power had increased to about 10^12 per hour worked. 37
At the same time, and as a sixth point, this enormous growth in computer power does not imply that there were correspondingly large increases in economic welfare all along the way. The rapid increase in productivity reflected an equally rapid decline in the cost of computation, and the decline was probably matched by a similar decline in the marginal productivity of computing."