Chapter 61: Endogenous Growth, Ideas, Human Capital, and Innovation. 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

More researchers, same long-run growth

Does a permanent increase in research labor raise technology growth for good?

Each new idea is harder to find as A grows when phi is below 1. A larger research workforce raises growth only until the knowledge stock catches up, so the policy changes the level of the technology path while the balanced growth rate stays tied to research labor growth.

$Equation, written in LaTeX: g_A=\frac{0.025(144)}{\sqrt{8{,}100}}=\frac{3.6}{90}=0.04$

$Equation, written in LaTeX: g_A^*=\frac{1(0.02)}{1-0.5}=0.04$

$Equation, written in LaTeX: g_A'=\frac{0.025(288)}{90}=0.08$

$Equation, written in LaTeX: 2^{\lambda/(1-\phi)}=2^{1/0.5}=4$

Scroll sideways for the whole equation

A is the knowledge stock, L_A research labor (growing at n = 2 percent), delta research productivity, lambda = 1 the return to research labor and phi the knowledge spillover. g_A is the proportional growth of A and g* its balanced rate. The path starts on the balanced path, so delta is set to make the starting growth equal g*.

Predict first. Doubling research labor at phi = 0.5: is long-run technology growth doubled?

Your prediction Yes, it stays at 8% No, it returns to 4%

Research labor multiple

Knowledge spillover phi

Figure: More researchers, same long-run growth. Technology growth over 150 years after research labor is multiplied by 2 with phi = 0.50. Growth starts at 8.00% and returns toward the balanced rate 4.00%; the long-run level of A is 4.00 times the old path. — Research labor multiple: x2, Knowledge spillover phi: 0.5
Constructed example: the chapter's hypothetical values (delta 0.025, L_A 144 and 288, A 8,100, phi 0.5, n 2 percent). A threefold increase and phi of 0.25 and 0.75 are added; for those phi values delta is recalibrated so the start is balanced.

Research labor L_A: 288
delta: 0.025
Impact g_A: 8.00%
Balanced g*: 4.00%
Long-run level multiple of A: 4.00

g* = 1 x 0.02 / (1 - 0.50) = 0.0400. With delta = 0.025000 the old path is balanced, and at A = 8,100 the impact rate is 0.025000 x 288 / 90.00 = 0.0800. The level multiple is 2^(1 / 0.50) = 4.00. Growth jumps to 8.00% and then decays back to 4.00%: the 2-fold research effort buys a permanently higher level, not faster long-run growth.

Worked steps

g* = lambda n / (1 - phi) = 0.02 / 0.50 = 0.0400
delta = g* x 8,100^0.50 / 144 = 0.040000 x 90.00 / 144 = 0.025000
Impact g_A = 0.025000 x 288 / 90.00 = 0.0800
Level multiple = 2^(1 / 0.50) = 4.00

Use the idea

When judging a research subsidy, ask whether it changes the growth of research effort or only its level; a one-time increase buys a level gain that is larger the stronger the spillover.

Where the conclusion applies

Research labor grows at 2 percent before and after the change, lambda = 1 and the economy starts on its balanced path. The model has no feedback from growth to research effort.

Check your understanding: With phi = 0.75 and research labor doubled, what is the long-run level multiple and the balanced rate?

2^(1 / 0.25) = 16, and g* = 0.02 / 0.25 = 0.08, or 8.00 percent.

Chapter 61 source: section "Semi-endogenous growth".

Demonstration 2 of 4

Creative destruction on a quality ladder

What happens to growth and to the incumbent when innovations arrive faster?

Each arrival raises quality by the step gamma, so expected log growth is lambda ln(gamma). The same arrival ends the current leader's profits, so the leader discounts at r plus lambda.

$Equation, written in LaTeX: g_A=0.10\ln(1.20)\approx0.018232$

$Equation, written in LaTeX: V=\frac{15}{0.05+0.10}=100$

$Equation, written in LaTeX: V'=\frac{15}{0.05+0.15}=75$

Scroll sideways for the whole equation

lambda is the yearly arrival rate of quality improvements, gamma the size of each step, pi = 15 the leader's yearly profit and r = 0.05 the interest rate. V is the value of being the current leader.

Predict first. When entrants innovate faster (lambda from 0.10 to 0.15), what happens to the incumbent's value?

Your prediction It rises It falls

Innovation arrival rate lambda

Quality step gamma

Figure: Creative destruction on a quality ladder. Left: expected growth rises with the arrival rate; at lambda 0.10 and step 1.20 it is 1.823 percent. Right: the leader's value falls with the arrival rate, to 100.00. — Innovation arrival rate lambda: 0.1, Quality step gamma: 1.2
Constructed example: the chapter's hypothetical component (gamma 1.20, lambda 0.10 and 0.15, profit 15, r 5 percent); lambda of 0.05 and 0.20 and steps of 1.10 and 1.30 are added.

ln(gamma): 0.182322
Expected growth: 1.823%
Leader value V: 100.00
Change in V vs lambda 0.10: 0.00

Growth = 0.10 x ln(1.20) = 0.10 x 0.182322 = 0.018232, or 1.823 percent a year. The leader's value is 15 / (0.05 + 0.10) = 100.00. Faster entry raises growth and shortens each leadership spell; the quality step changes growth but not V.

Worked steps

ln(1.20) = 0.182322
g = 0.10 x 0.182322 = 0.018232
r + lambda = 0.05 + 0.10 = 0.15
V = 15 / 0.15 = 100.00

Use the idea

A policy that speeds entry raises measured growth but lowers the reward to being the leader; count both effects when predicting research incentives.

Where the conclusion applies

Profit and the step size are held fixed when lambda changes; fiscal cost, business stealing and welfare are outside the calculation.

Check your understanding: At lambda = 0.15 and gamma = 1.20, what are growth and the leader's value?

0.15 x ln(1.2) = 0.15 x 0.182322 = 0.027348, or 2.735 percent; V = 15 / 0.20 = 75.

Chapter 61 source: section "Schumpeterian quality-ladder growth".

Demonstration 3 of 4

Weakest link production

Can rearranging workers, with no change in skill, raise total output?

Because qualities multiply, a high-quality worker is worth more beside another high-quality worker. Sorting raises total output by Bn(h - l)^2, and the same training upgrade pays more beside a better partner.

Equation, written in LaTeX: 120(0.90)(0.60)=64.8

Equation, written in LaTeX: 120(0.90)^2=97.2

Equation, written in LaTeX: 140.4-129.6=10.8

Equation, written in LaTeX: 120(0.90)(0.15)=16.2

Scroll sideways for the whole equation

Each two-task team produces Bn times the product of its workers' qualities, with Bn = 120. Two workers have quality h = 0.90 and two have the lower quality l.

Predict first. Moving from mixed to sorted teams with the same four workers: can total output rise?

Your prediction Yes No, total skill is unchanged

Low worker quality

Team assignment

Figure: Weakest link production. Bars for two sorted teams and their total, 140.4, with a dashed line at the other arrangement's total 129.6. — Low worker quality: 0.6, Team assignment: Sorted
Constructed example: the chapter's hypothetical teams (Bn 120, qualities 0.90 and 0.60, training of 0.15); low qualities 0.45 and 0.75 are added for comparison.

Mixed total: 129.6
Sorted total: 140.4
Sorting gain: 10.8
Training gain beside 0.90: 16.2
Training gain beside 0.60: 10.8

Mixed: 2 x 120 x 0.90 x 0.60 = 129.6. Sorted: 120 x 0.90^2 + 120 x 0.60^2 = 97.2 + 43.2 = 140.4. The gain from sorting is 120 x (0.90 - 0.60)^2 = 10.8 with no change in any skill. Raising a 0.60 worker by 0.15 adds 16.2 beside a 0.90 partner and 10.8 beside a 0.60 partner.

Worked steps

Mixed team = 120 x 0.90 x 0.60 = 64.8; two teams = 129.6
Sorted = 97.2 + 43.2 = 140.4
Gain = 140.4 - 129.6 = 10.8 = 120 x 0.30^2
Training: 120 x 0.90 x 0.15 = 16.2; 120 x 0.60 x 0.15 = 10.8

Use the idea

Where a single mistake ruins the product, place the best workers together and expect training to pay most where partners are already strong.

Where the conclusion applies

Two tasks per team, quality as a probability of success, fixed workers and no wage response. With additive production the sorting gain would be zero.

Check your understanding: With l = 0.45, what is the sorting gain?

120 x (0.90 - 0.45)^2 = 120 x 0.2025 = 24.3.

Chapter 61 source: section "O-ring production function".

Demonstration 4 of 4

Saving that raises growth forever

In an AK economy, does a higher saving rate change the growth rate or only the level?

With no diminishing returns to capital, the return to saving never falls, so a higher saving rate raises the growth rate permanently and the paths diverge.

Equation, written in LaTeX: g_y=(0.24)(0.25)-0.035-0.005=0.02

Equation, written in LaTeX: g_y'=(0.30)(0.25)-0.035-0.005=0.035

$Equation, written in LaTeX: y_{10}=50e^{0.02(10)}=50e^{0.20}\approx61.07$

$Equation, written in LaTeX: y_{10}'=50e^{0.035(10)}=50e^{0.35}\approx70.95$

Scroll sideways for the whole equation

Output per person is y = A k with A = 0.25. s is the saving rate, delta = 0.035 depreciation and n = 0.005 population growth. Growth of output per person is g_y = s A - delta - n.

Predict first. With saving at 0.30 instead of 0.24, will the gap between the two paths close over time?

Your prediction Yes, it closes No, it widens

Saving rate

Horizon in years

Figure: Saving that raises growth forever. Output paths from 50 over 10 years: baseline saving 0.24 grows at 2.0 percent to 61.07; saving 0.30 grows at 3.5% to 70.95. The shaded gap is 9.88. — Saving rate: 0.3, Horizon in years: 10
Constructed example: the chapter's hypothetical economy (A 0.25, s 0.24 and 0.30, delta 3.5 percent, n 0.5 percent, k0 200, horizon 10 years); saving of 0.18 and horizons of 20 and 40 years are added.

Growth at chosen s: 3.5%
Baseline growth: 2.0%
y at year 10: 70.95
Baseline y: 61.07
Gap: 9.88
Net accumulation at k = 200: 7.00

g = 0.30 x 0.25 - 0.035 - 0.005 = 0.0350. Starting from y0 = 50, y at year 10 is 50 e^(0.0350 x 10) = 70.95 against a baseline 50 e^(0.02 x 10) = 61.07, a gap of 9.88. At k = 200 net accumulation is 15.00 - 7 - 1 = 7.00. The gap keeps widening because the growth difference never shrinks in the AK model.

Worked steps

g = 0.30 x 0.25 - 0.035 - 0.005 = 0.0350
Investment at k = 200: 0.30 x 50 = 15.00; net = 15.00 - 7 - 1 = 7.00
y_T = 50 e^(0.3500) = 70.95
Baseline y_T = 50 e^(0.2000) = 61.07
Gap = 70.95 - 61.07 = 9.88

Use the idea

Treat a claim that a policy raises growth forever as a claim that returns to accumulation do not diminish, and check that assumption.

Where the conclusion applies

Constant returns to a broad capital stock and a constant growth difference by assumption. A concave production function would make the difference shrink during convergence.

Check your understanding: What is net accumulation at s = 0.30 and k = 200?

0.30 x 50 - 7 - 1 = 7, which is 3.5 percent of 200.

Chapter 61 source: section "AK endogenous-growth mechanism".

More researchers, same long-run growth

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Creative destruction on a quality ladder

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Weakest link production

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Saving that raises growth forever

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies