Why Population Size Can Affect Growth
In a research-based growth model, workers devoted to research help create new knowledge, and existing knowledge makes research more productive. That feedback can make the rate of knowledge growth depend on how many people are doing research. If the research share of the population is held constant, a larger population also means more researchers. The model can therefore predict a scale effect: larger economies grow faster per person, not merely produce more in total.
This is a feature of a particular model structure, not a general rule that follows from population growth alone. To interpret the prediction, it is important to keep population size, research labor, and the research share distinct.
A Simple Research-Based Model
Consider a continuous-time economy with a knowledge stock \(A(t)>0\), population \(L(t)\), and research labor \(L_A(t)\). The knowledge production equation is
Here \(\dot A(t)\) is the rate at which knowledge changes. The parameter \(\delta>0\) measures research productivity. For dimensional consistency, suppose research labor is counted in blocks of 100,000 workers; then \(\delta\) is measured per research block per year. The equation assumes that knowledge is nonrival, that each research block contributes in proportion to the existing knowledge stock, and that there are no diminishing returns to the number of researchers or to the stock of knowledge in this equation.
Suppose a constant fraction \(s_R\), with \(0<s_R<1\), of the population works in research. Then \(L_A=s_R L\). Dividing the knowledge equation by \(A\) gives the instantaneous proportional growth rate:
This is the source of the scale effect. If \(\delta\) and \(s_R\) are unchanged, doubling population doubles research labor and therefore doubles the model’s knowledge growth rate. The prediction concerns the growth rate, not just the amount of knowledge or the size of total output.
Why the Growth-Rate Formula Holds
The result comes directly from converting the level equation for knowledge into a proportional growth rate. The following steps make the argument explicit.
The model specifies \(\dot A=\delta L_A A\), so the change in knowledge is proportional to both the research workforce and the existing knowledge stock.
Because \(A>0\), dividing both sides by \(A\) is valid: \(\dot A/A=\delta L_A A/A=\delta L_A\). By definition, \(\dot A/A\) is the instantaneous proportional growth rate of knowledge.
If \(L_A=s_R L\), then \(g_A=\delta s_R L\). With \(\delta\) and \(s_R\) fixed, a population change scales the growth rate in the same proportion.
For constant \(L\), \(s_R\), and \(\delta\), the knowledge equation can also be solved over an interval of length \(t\):
This solution follows because the proportional growth rate is constant, so the logarithm of knowledge changes at a constant rate: \(d(\ln A)/dt=\delta s_R L\). Integrating from \(0\) to \(t\) gives \(\ln A(t)-\ln A(0)=\delta s_R L t\); exponentiating yields the stated expression.
From Knowledge Growth to Output per Worker
To connect the result to living standards, suppose output is \(Q=A L_Y\), where \(L_Y=(1-s_R)L\) is labor in production. This simplified production relationship assumes output is proportional to knowledge and production labor. With a constant research share, output per worker is
Since \(1-s_R\) is constant, \(q\) grows at the same proportional rate as \(A\): \(g_q=g_A\). Thus, in this model, a larger population with the same research share implies faster growth in output per worker. By contrast, total output \(Q\) also depends on the number of workers. Its growth rate need not be confused with growth in output per worker.
The connection between \(A\) and \(q\) depends on the production relationship just stated. The scale result for \(g_A\) comes from the knowledge production equation; it does not, by itself, establish that every economy’s measured output per person grows at that rate.
Worked Example: Doubling Population at a Fixed Research Share
Two hypothetical economies have the same research productivity, \(\delta=0.03\) per research block per year, where one block is 100,000 researchers. Each allocates 2% of its population to research. Economy A has 10 million people and Economy B has 20 million. Find each model-implied annual knowledge growth rate and the five-year knowledge growth factor. Assume the populations and research shares stay constant for five years.
Method. Calculate research labor first, express it in 100,000-worker blocks, and then use \(g_A=\delta L_A\). The rates are continuous-time growth rates.
For Economy A, research labor is \(0.02(10{,}000{,}000)=200{,}000\) workers, or 2 blocks. Therefore,
The annual knowledge growth rate is 6% in continuous-time terms. Over five years, knowledge is multiplied by approximately 1.3499, an increase of about 34.99%.
For Economy B, research labor is \(0.02(20{,}000{,}000)=400{,}000\) workers, or 4 blocks. Therefore,
Economy B’s annual knowledge growth rate is 12%, and its five-year knowledge factor is approximately 1.8221. Under the stated output relationship and constant research share, output per worker has the same five-year factors. The result illustrates the scale effect: doubling population, with all else fixed, doubles the model’s growth rate. The common mistake avoided here is to treat the 6% rate as a discrete annual percentage and calculate \(1.06^5\); the model specifies a continuous-time rate, so the correct factor is \(\exp(g_A t)\).
Population Size Is Not Research Intensity
The model’s formula also shows when the scale prediction does not follow. Growth depends on \(L_A\), the number of research workers, not on population in isolation. If population increases while research labor stays fixed, then the research share falls and the model’s knowledge growth rate does not change. A larger population produces a higher rate only under an additional condition, such as a constant research share.
Worked Example: Same Research Workforce, Different Population
Two hypothetical economies use the same model, with \(\delta=0.02\) per research block per year. Economy C has 15 million people, of whom 2% work in research. Economy D has 30 million people, of whom 1% work in research. Compare their research labor and knowledge growth rates.
Method. Compute the research workforce in people before comparing the economies. Do not infer the workforce from population alone.
For Economy C, \(L_A=0.02(15{,}000{,}000)=300{,}000\), or 3 blocks. Its growth rate is \(g_{A,C}=0.02(3)=0.06\), or 6% in continuous-time terms.
For Economy D, \(L_A=0.01(30{,}000{,}000)=300{,}000\), also 3 blocks. Its growth rate is \(g_{A,D}=0.02(3)=0.06\), also 6%.
The economies have different populations but the same research workforce, so the model predicts the same knowledge growth rate. If the output relationship and research shares remain constant, they also have the same growth rate of output per worker. The calculation avoids the mistake of assuming that a population twice as large must grow twice as fast regardless of its research allocation.
Interpreting the Prediction and Its Limits
The scale effect is a consequence of specific assumptions. In particular, the equation makes the return to additional research labor proportional and allows existing knowledge to raise research productivity without imposing a limit in the equation. Keeping the research share fixed then makes the number of researchers rise in proportion to population. Together, these conditions generate a strong prediction: larger populations have higher per-person growth rates.
That prediction can conflict with observed growth patterns. Across countries and periods, population size by itself does not reliably predict faster growth in output per person. A small economy may grow rapidly while a much larger one grows slowly; the model’s equation does not explain such differences unless its assumptions or inputs differ. This does not establish that research is unimportant. It indicates that population size alone is not a sufficient account of growth.
Several features of real research activity can weaken the simple prediction. A larger population need not devote the same fraction of its labor to research. Research effort can have diminishing returns, and not every research project produces equally useful knowledge. The model also abstracts from the time needed to train researchers, the allocation of research across fields, and the possibility that knowledge is shared across national borders. These observations are reasons to assess the model’s assumptions; they are not additional results implied by its equation.
Worked Example: How Much Research Labor Preserves Growth?
A hypothetical economy has 40 million people, and 1.5% of them work in research. Let \(\delta=0.025\) per research block per year, with one block equal to 100,000 researchers. A population change reduces the economy to 32 million people. How many researchers, and what research share, are needed to preserve its original knowledge growth rate? Also compare four-year knowledge factors if the research share were instead kept at 1.5% after the population change. Assume constant population and research allocation within each four-year calculation.
Method. First find the initial research workforce and growth rate. Then preserve the growth rate by preserving \(L_A\), since \(\delta\) is unchanged.
Initially, \(L_A=0.015(40{,}000{,}000)=600{,}000\), or 6 blocks. The initial growth rate is \(g_A=0.025(6)=0.15\), or 15% in continuous-time terms. To preserve that rate after the population change, the economy still needs 600,000 researchers. The required share is
If the economy instead keeps its research share at 1.5%, then research labor becomes \(0.015(32{,}000{,}000)=480{,}000\), or 4.8 blocks. The resulting growth rate is \(g_A=0.025(4.8)=0.12\), or 12%. Over four years, the original rate gives a factor of \(\exp(0.15\cdot4)=\exp(0.60)\approx1.8221\); the reduced rate gives \(\exp(0.12\cdot4)=\exp(0.48)\approx1.6161\). Preserving the research workforce preserves the model’s growth rate, whereas preserving the research share does not. The common mistake avoided is to assume that the same research share always preserves growth when the population changes.
Common Exam Trap: Confusing a Rate Change With a Percentage-Point Change
Suppose the model predicts 6% continuous-time knowledge growth for one population and that population doubles while the research share and \(\delta\) stay fixed. A tempting but incorrect calculation is to add two percentage points and report 8%. The formula does not say to add an amount based on the population change. It says \(g_A=\delta s_R L\), so doubling \(L\) doubles \(g_A\):
The correct model-implied rate is 12% in continuous-time terms. Another common error is to call the resulting difference in growth rates a difference in output levels. The model predicts a different rate of change in knowledge; over time, that rate difference causes the knowledge levels to diverge.
Check Your Understanding
For each problem, use the stated research-block units and treat growth rates as continuous-time rates.
- A hypothetical economy has 24 million people, a research share of 2.5%, and \(\delta=0.015\) per 100,000-researcher block per year. Find its annual knowledge growth rate and its three-year knowledge growth factor. (Answer: 9% continuous-time growth; factor \(\exp(0.27)\approx1.3100\).)
- Economy A has 12 million people and a research share of 3%; Economy B has 24 million people and a research share of 1.5%. If \(\delta=0.04\) per 100,000-researcher block per year, calculate each growth rate and explain whether population size alone predicts a difference. (Answer: Both have 360,000 researchers, or 3.6 blocks, and both grow at 14.4% continuous-time; population size alone does not distinguish their rates.)
- A hypothetical economy has 18 million people and a research share of 2%. With \(\delta=0.02\) per 100,000-researcher block per year, what research share would preserve its growth rate if population increased to 27 million and the number of researchers were held fixed? (Answer: 1.3333% approximately; the fixed workforce is 360,000 researchers, giving a 7.2% continuous-time growth rate.)
- Two hypothetical economies have populations of 25 million and 50 million. Each allocates 1.2% of its population to research, and \(\delta=0.025\) per 100,000-researcher block per year. Find their knowledge growth rates and two-year knowledge factors. (Answer: The first has 300,000 researchers and grows at 7.5%, with factor \(\exp(0.15)\approx1.1618\); the second has 600,000 researchers and grows at 15%, with factor \(\exp(0.30)\approx1.3499\).)