Chapter 60: Growth Accounting, Accumulation, and Convergence. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Two economies, one steady state

Does capital per effective worker rise or fall, and where does it settle?

Investment per effective worker rises with k but at a falling rate, while break-even investment rises in proportion. Below k* the curve lies above the line and capital deepens; above k* the line lies above the curve and capital thins. Both starting points head to the same k*.

$Equation, written in LaTeX: k^*=(\frac{0.24}{0.06})^{1/0.60}=4^{5/3}\approx10.079.$

$Equation, written in LaTeX: y^*=(10.079)^{0.40}\approx2.520.$

$Equation, written in LaTeX: \dot{k}\approx0.418-0.240=0.178>0.$

$Equation, written in LaTeX: \dot{k}\approx0.728-0.960=-0.232<0.$

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k is capital per effective worker and y = k^0.40 output per effective worker. s is the saving rate and n + g + delta = 0.06 the break-even rate. k-dot = s k^0.40 - 0.06 k is the net change in k; k* is the steady state where it is zero.

Predict first. With the book's saving rate, does an economy starting at k = 16 grow or shrink its capital?

Your prediction It grows It shrinks

Saving rate s

Starting capital per effective worker

Figure: Two economies, one steady state. Solow diagram with the investment curve 0.24 k^0.40 and the break-even line 0.06k crossing at k* = 10.079. At k0 = 4 investment is 0.418 and break-even 0.240. — Saving rate s: 0.24, Starting capital per effective worker: 4
Constructed example: the chapter's hypothetical economy (alpha 0.40, s 0.24, n + g + delta 0.06, starting k of 4 and 16); saving rates 0.12 and 0.36 and a start at the steady state are added. Constructed teaching numbers, not data from any real economy.

Steady state k*: 10.079
Steady output y*: 2.520
Output at k0: 1.741
Investment at k0: 0.418
Break-even at k0: 0.240
k-dot: 0.178
k-dot as share of k0: 4.45%
Capital: grows

With s = 0.24 the target is k* = 10.079. At k0 = 4 investment is 0.24 x 1.7411 = 0.4179 and break-even investment is 0.06 x 4 = 0.2400, so k-dot = 0.4179 - 0.2400 = 0.178 (4.45% of k0): investment exceeds break-even needs, so capital per effective worker rises toward k*.

Worked steps

k* = (0.24 / 0.06)^(1/0.60) = 4^(5/3) = 10.079
y* = 10.079^0.40 = 2.520
Output at k0: 4^0.40 = 1.7411
Investment = 0.24 x 1.7411 = 0.4179
Break-even = 0.06 x 4 = 0.2400
k-dot = 0.4179 - 0.2400 = 0.178, or 4.45% of k0

Use the idea

Compare investment per effective worker with the break-even amount before reading fast growth as permanent: a positive gap shows a transition toward a target, not a higher growth rate.

Where the conclusion applies

Cobb-Douglas output with alpha = 0.40, fixed saving and break-even rates, and saving that turns fully into productive capital. Thresholds or increasing returns could create more than one steady state.

Check your understanding: At k = 16 with s = 0.24, what is k-dot as a share of k?

Investment 0.24 x 3.0314 = 0.7275, break-even 0.06 x 16 = 0.9600, so k-dot = -0.232 and -0.232 / 16 is -1.45 percent.

Chapter 60 source: section "Solow-Swan convergence mechanism".

Demonstration 2 of 4

The capital stock that maximizes consumption

Which maintained capital stock gives the most consumption, and is it the one with the most output?

Consumption is the vertical gap between output and the break-even line. The gap is widest where the output curve's slope, the MPK, equals the slope of the line. Past that point each extra unit of capital costs more upkeep than the output it adds.

$Equation, written in LaTeX: 0.40k^{-0.60}=0.08.$

$Equation, written in LaTeX: k_{GR}=5^{1/0.60}\approx14.620.$

$Equation, written in LaTeX: c_{GR}\approx2.924-1.170=1.754.$

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k is steady-state capital per effective worker, f(k) = k^0.40 output, and n + g + delta the break-even rate. Consumption is c(k) = f(k) - (n + g + delta) k. MPK = 0.40 k^(-0.60) is the marginal product of capital; k_GR is the golden-rule stock.

Predict first. Does the economy with the highest output (k = 25) also have the highest consumption?

Your prediction Yes No

Steady-state capital k

Break-even rate n + g + delta

Figure: The capital stock that maximizes consumption. Output k^0.40, the break-even line 0.08k and the consumption curve beneath them. At k = 14.620 the gold bar shows consumption 1.754; the consumption curve peaks at the golden rule k = 14.620 with 1.754. — Steady-state capital k: 14.62, Break-even rate n + g + delta: 0.08
Constructed example: the chapter's hypothetical f(k) = k^0.40 with n + g + delta = 0.08 and its stocks 9, 14.620 and 25; break-even rates 0.06 and 0.10 are added. Constructed teaching numbers, not data from any real economy.

Output: 2.924
Break-even investment: 1.170
Consumption: 1.754
Saving rate needed: 0.400
MPK: 0.080
Golden-rule k: 14.620
Golden-rule consumption: 1.754
Position: at the golden rule

At k = 14.620 output is 2.9240 and break-even investment 0.08 x 14.620 = 1.1696, leaving consumption 2.9240 - 1.1696 = 1.754. With n + g + delta = 0.08 the golden-rule stock is 14.620 with consumption 1.754. This k is at the golden rule: MPK equals the break-even rate (difference 0.000 at three decimals), so no steady-state stock gives more consumption.

Worked steps

Golden rule: 0.40 k^(-0.60) = 0.08, so k^0.60 = 5.000 and k_GR = 14.620
Output = 14.620^0.40 = 2.9240
Break-even = 0.08 x 14.620 = 1.1696
c = 2.9240 - 1.1696 = 1.754
Saving rate = 1.1696 / 2.9240 = 0.400
MPK = 0.40 x 14.620^(-0.60) = 0.080

Use the idea

To judge whether an economy saves too little or too much for sustainable consumption, compare the social marginal product of capital with depreciation plus effective-labor growth.

Where the conclusion applies

Steady states only: the consumption given up while building capital, or gained while running it down, is not counted. The chosen stock 14.620 is the golden rule only when n + g + delta = 0.08.

Check your understanding: At k = 25 with n + g + delta = 0.08, what are consumption and the MPK?

3.6239 - 2.0000 = 1.624; MPK = 0.40 x 25^(-0.60) = 0.058, below 0.08.

Chapter 60 source: section "Golden rule of accumulation".

Demonstration 3 of 4

Same income, different destinations

Two regions with the same income: will they grow at the same rate?

Predicted growth is proportional to the log distance from the region's own target. Raising the target widens the gap and speeds the transition; raising lambda speeds every region in proportion to its gap.

$Equation, written in LaTeX: g_E=0.15\ln(\frac{30{,}000}{15{,}000})=0.15\ln(2)\approx0.1040,$

$Equation, written in LaTeX: g_W=0.15\ln(\frac{18{,}000}{15{,}000})\approx0.02735,$

$Equation, written in LaTeX: g_W'=0.15\ln(\frac{27{,}000}{15{,}000})\approx0.08817,$

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Income is per effective worker. g is approximate log growth over the interval, y* the region's conditioned target, y0 = 15,000 its starting income and lambda the speed of adjustment. One log point is 0.01.

Predict first. Two regions start at 15,000; one heads for 18,000 and the other for 30,000. Do they grow at the same rate?

Your prediction Yes, same start means same growth No, the region with the higher target grows faster

Target income y*

Convergence parameter lambda

Figure: Same income, different destinations. Bars of predicted growth for targets 18,000, 27,000 and 30,000 from a common start of 15,000 at lambda 0.15: 2.73, 8.82, 10.40 log points; the bar for 30,000 is highlighted. — Target income y*: 30,000, Convergence parameter lambda: 0.15
Constructed example: the chapter's hypothetical East and West (start 15,000, targets 30,000, 18,000 and 27,000, lambda 0.15); lambda values 0.10 and 0.20 are added. Constructed teaching numbers, not data from any real economy.

Starting income: 15,000
Target income: 30,000
Log gap: 0.6931
Predicted growth: 0.1040
Growth in log points: 10.40

Both regions start at 15,000. With target 30,000 and lambda = 0.15, predicted growth is 0.15 x 0.69315 = 0.1040, or 10.40 log points, because ln(2.0) = 0.69315. Across the three targets growth runs from 2.73 to 10.40 log points: equal starting income does not mean equal growth when destinations differ.

Worked steps

Ratio = 30,000 / 15,000 = 2.0
Log gap = ln(2.0) = 0.69315
g = 0.15 x 0.69315 = 0.1040
In log points: 10.40

Use the idea

Before predicting catch-up from low income, ask what each economy's own target is: saving, schooling and demography set the destination.

Where the conclusion applies

A fixed lambda and fixed targets over the interval. The calculation does not claim that any policy produces a given target.

Check your understanding: With target 27,000 and lambda 0.15, what is predicted growth?

0.15 x ln(1.8) = 0.15 x 0.58779 = 0.0882, about 8.82 log points.

Chapter 60 source: section "Conditional convergence".

Demonstration 4 of 4

What the residual measures

How much of output growth is left after measured inputs, and how fragile is that remainder?

The residual is whatever measured inputs do not explain. Any revision that raises measured input growth, or shifts weight toward the faster-growing input, shrinks it one for one.

Equation, written in LaTeX: 0.35(9)+0.65(1.5)=3.15+0.975=4.125

$Equation, written in LaTeX: \ln(1.07)-0.35\ln(1.09)-0.65\ln(1.015)\approx0.027819,$

$Equation, written in LaTeX: \ln(1.07)-0.35\ln(1.09)-0.65\ln(1.025)\approx0.021446,$

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Output grows 7 percent and capital services 9 percent; labor services grow at the chosen rate. alpha is the capital weight and 1 - alpha the labor weight. The residual is output growth minus weighted input growth, in percentage points; the exact version uses log changes.

Predict first. If a revised series shows labor services grew faster, does measured productivity rise or fall?

Your prediction It rises It falls

Measured labor growth

Capital weight alpha

Figure: What the residual measures. Stacked bar splitting 7 percent output growth into capital 3.150, labor 0.975 and residual 2.875 points, beside the exact log residual of 2.78 log points. — Measured labor growth: 1.5%, Capital weight alpha: 0.35
Constructed example: the chapter's hypothetical production account (alpha 0.35, output 7, capital 9, labor 1.5 and revised 2.5 percent); labor growth 3.5 percent and weights 0.25 and 0.45 are added. Constructed teaching numbers, not data from any real economy.

Capital contribution: 3.150
Labor contribution: 0.975
Measured inputs: 4.125
Residual (points): 2.875
Exact log residual (log points): 2.78

Measured inputs explain 0.35 x 9 + 0.65 x 1.5 = 4.125 points of the 7 percent output growth, leaving a residual of 2.875 points (2.78 log points on exact logs). Faster measured labor growth or a larger capital weight moves part of the remainder into inputs without anything changing in production.

Worked steps

Capital: 0.35 x 9 = 3.150
Labor: 0.65 x 1.5 = 0.975
Inputs = 3.150 + 0.975 = 4.125
Residual = 7 - 4.125 = 2.875
Exact: ln(1.07) - 0.35 ln(1.09) - 0.65 ln(1.015) = 0.027819

Use the idea

Before reading a residual as technical progress, rebuild it with alternative labor-quality, capital-service and factor-share assumptions and report how much it moves.

Where the conclusion applies

Constant returns, factor shares used as output elasticities and correctly measured output. Markups, utilization changes or omitted inputs all end up in the residual.

Check your understanding: With labor growth 2.5 percent and alpha 0.35, what is the approximate residual?

7 - (3.15 + 1.625) = 7 - 4.775 = 2.225 points.

Chapter 60 source: section "Solow residual".

Two economies, one steady state

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

The capital stock that maximizes consumption

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Same income, different destinations

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

What the residual measures

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies