Growth accounting and the Solow residual

Start from the Cobb–Douglas function and take logs. Products become sums:

Now differentiate with respect to time. The derivative of ln x is the growth rate of x, so

Read it as an accounting identity: output grew because capital grew, because labour grew, and because the economy got more out of both. The first two terms use observable data: growth of the capital stock, growth of labour input, and the income shares α and 1 − α (taken from the national accounts, since competitive pay equals the exponents). The last term is whatever is left, so it is computed, not measured:

Set the growth rates and the share, and read the split. The three presets are three different kinds of growth: one driven by accumulation, one by productivity, and one where inputs grew faster than output so the residual is negative.

That second example is the most common trap in this topic. With labour-augmenting technology, Y = K^α(AL)^1−α = A^1−αK^αL^1−α, so the “A” in the growth accounting equation is A^1−α and the measured residual is (1 − α)g_A. Always read the function before choosing the formula.

Growth theory cares most about output per worker. Subtract labour growth from both sides of the growth accounting equation:

Output per worker rises for two reasons: each worker has more capital (capital deepening), and the economy gets more from the same inputs (productivity). Capital deepening has diminishing returns, because more capital per worker adds less each time. That is why productivity is the long-run source of growth in per-worker terms, and the point on which the Solow model of Lesson 5 turns.

A labour-force puzzle

JNU 2017 gives two economies with constant unemployment rates (5% and 10%), output growth of 4% and 8%, and labour-productivity growth of 2% and 4%, and asks which has the faster-growing labour force. Output growth equals employment growth plus productivity growth, so employment grows 4 − 2 = 2% in A and 8 − 4 = 4% in B. With a constant unemployment rate, the labour force grows at the same rate as employment, so B has the faster-growing labour force. The unemployment rates themselves do not matter, because they are constant.

In 1957 Robert Solow applied the calculation to US data for 1909 to 1949 and found that the growth of capital per worker explained only about one-eighth of the rise in output per worker. The remaining seven-eighths was the residual, which he attributed to technical change. The finding changed growth economics: if accumulation is a small part of the story, the theory of growth needs a theory of technical progress.

But the residual is a computed number, and anything computed as a remainder collects every error in the other terms. It is often called a “measure of our ignorance”. It includes:

• Real technical progress and better organisation, what it is usually taken to be.
• Mismeasured inputs. If labour input is counted as people and not hours (CUET-PG 2020 asks: labour input is the number of workers employed times average hours worked), or if a better-educated workforce is counted the same as before, the improvement lands in the residual.
• Changes in how fully inputs are used, such as the business cycle and spare capacity.
• Embodied technical change. The calculation treats technology as disembodied, lifting all inputs equally. If progress is embodied in new machines (UGC NET 2025 Dec: technical improvements that can only be introduced into the productive system by new investment), then better capital looks like more capital or like a higher residual, depending on how it is measured.

Growth accounting became politically interesting in the 1990s, when it was applied to the fast-growing East Asian economies. Alwyn Young (1995) did careful accounting for Hong Kong, Singapore, South Korea and Taiwan and argued that their growth came mainly from the rapid growth of inputs: more of the population at work, more years of schooling, and a very high share of output invested. Productivity growth, the residual, was modest, unremarkable by comparison with richer countries. Paul Krugman (1994) drew the lesson in “The myth of Asia’s miracle”: growth built on accumulation alone is subject to diminishing returns and cannot go on at the same pace for ever, as the Soviet Union’s earlier growth had also shown.

Other economists disputed both the data and the interpretation, including whether a rapid rise in investment can be separated from the technology embodied in it. Growth accounting for China and India, for example, has been done by Bosworth and Collins (2008). The debate is not settled, and you should treat the numbers in any one study as that study’s estimates, not facts. What it taught is the method: separate accumulation from productivity before explaining a country’s growth.

Questions on the Solow residual, growth accounting with two and three inputs, labour input, and technical change, from CUET-PG, JNU, IIT JAM, GATE and UGC NET.