Semi-Endogenous Growth and Population Growth

Why Population Growth Matters in Semi-Endogenous Growth

In a semi-endogenous growth model, research creates new knowledge, but each additional unit of existing knowledge makes research less productive at the margin. This diminishing return changes the source of sustained growth: expanding the number of people available to research can offset the falling productivity of research effort. As a result, a permanent increase in the research share can raise the level of technology without permanently raising its long-run growth rate.

The distinction matters when assessing growth forecasts. A larger research workforce may accelerate knowledge accumulation during a transition, but the model’s balanced-growth rate is governed by population growth and the strength of diminishing returns. The analysis below focuses on that relationship and on how to calculate the level and growth path of knowledge.

Definition: Let $A(t)>0$ denote the stock of knowledge at time $t$, and let $L(t)>0$ denote population. A fraction $s_R$, where $0<s_R\leq 1$, works in research, so research labor is $H_A(t)=s_R L(t)$. The semi-endogenous research equation is $$ \dot A(t)=\delta A(t)^\phi H_A(t)^\lambda =\delta A(t)^\phi [s_R L(t)]^\lambda, $$ where $\dot A(t)$ is the change in knowledge per unit of time, $\delta>0$ is research productivity, $\lambda>0$ measures the contribution of research labor, and $\phi<1$ captures diminishing returns to the existing knowledge stock.

Assume continuous time, a constant research share $s_R$, and an exogenous constant population growth rate $n$, so that $\dot L/L=n$ and $L(t)=L_0e^{nt}$. Parameters $\delta,\lambda,\phi$ are constant. For the positive-growth balanced path derived below, take $n>0$. The condition $\phi<1$ is essential: it ensures that an expanding research workforce can offset diminishing returns to knowledge. The equation is not a claim that research effort always has this form in actual economies; it is a model with specific assumptions.

Deriving the Balanced-Growth Rate

The central calculation is easier after transforming the knowledge stock. Define $B(t)=A(t)^{1-\phi}$. Because $\phi<1$, this transformation is well-defined for positive $A$. Applying the chain rule gives

\dot B(t)=(1-\phi)A(t)^{-\phi}\dot A(t).

Substitute the research equation for $\dot A(t)$. The powers of $A$ cancel, leaving a growth equation in which the right-hand side depends only on research labor:

\dot B(t) =(1-\phi)A(t)^{-\phi}\delta A(t)^\phi[s_RL(t)]^\lambda =(1-\phi)\delta s_R^\lambda L_0^\lambda e^{\lambda nt}.

This cancellation explains why the transformation is useful: it removes the diminishing-returns term from the equation to be integrated. Integrating from time $0$ to $t$, with $B(0)=A_0^{1-\phi}$, yields

A(t)^{1-\phi} =A_0^{1-\phi} +\frac{(1-\phi)\delta s_R^\lambda L_0^\lambda}{\lambda n} \left(e^{\lambda nt}-1\right), \qquad n>0.

As time becomes large, the exponential term dominates the constant initial-stock term. Consequently, $A(t)^{1-\phi}$ grows asymptotically at rate $\lambda n$. Since $A$ is the $(1-\phi)$-power of this transformed quantity, knowledge grows at rate $\lambda n/(1-\phi)$. The same result follows by taking the growth rate of the original research equation: $g_A=\dot A/A=\delta A^{\phi-1}(s_RL)^\lambda$. On a balanced-growth path, $g_A$ is constant, so the changing population contribution $\lambda n$ must be offset by the changing knowledge term $(\phi-1)g_A$.

0=\lambda n+(\phi-1)g_A \qquad\Longrightarrow\qquad g_A=\frac{\lambda n}{1-\phi}.

Result: With $n>0$, constant $s_R$, $\lambda>0$, and $\phi<1$, the long-run balanced-growth rate of knowledge is $g_A=\lambda n/(1-\phi)$. It depends on population growth and the research and diminishing-returns parameters, but not on the constant research share $s_R$, the productivity level $\delta$, or the initial knowledge stock $A_0$.

The result does not say that research effort is irrelevant. The integrated path contains $s_R^\lambda$, so a larger research share raises knowledge along the transition and raises its asymptotic level. It does not alter the exponential growth rate when $s_R$ remains a constant share and the model parameters are unchanged. If knowledge is labor-augmenting technology in a standard production setting, its long-run growth rate also determines the growth rate of output per worker on a balanced-growth path. That interpretation does not change the research equation’s result.

Worked Example: Calculating the Long-Run Knowledge Growth Rate

A model has diminishing returns parameter $\phi=0.6$, research-labor elasticity $\lambda=0.4$, and population growing continuously at $n=0.015$ per year. The research share is held constant at $s_R=0.08$. Find the long-run growth rate of knowledge and explain whether the research share affects that rate.

Method. Use the balanced-growth formula, which applies because population growth is positive, $\lambda>0$, and $\phi<1$.

First calculate the numerator: $\lambda n=0.4(0.015)=0.006$. Next calculate the denominator: $1-\phi=1-0.6=0.4$. Therefore,

g_A=\frac{0.006}{0.4}=0.015.

The long-run knowledge growth rate is $0.015$, or 1.5% per year. The constant research share $s_R=0.08$ does not enter this formula. It affects the level of the knowledge path, not its asymptotic growth rate. Check: substituting $g_A=0.015$ into $\lambda n+(\phi-1)g_A$ gives $0.006-0.4(0.015)=0$, as required for a constant growth rate.

Common mistake avoided: Multiplying $n$ by the research share and reporting $s_Rn$ as the growth rate would give $0.12\%$. That incorrectly treats the number of researchers’ share as the model’s growth-rate formula; the correct expression is $\lambda n/(1-\phi)$.

How Research Effort Affects the Knowledge Path

The integrated solution makes the role of research effort explicit. For $n>0$, the coefficient on the exponential term is proportional to $s_R^\lambda$. At sufficiently long horizons, the initial-stock term becomes small relative to that exponential term. The asymptotic knowledge path can then be written as

A(t)\sim \left(\frac{(1-\phi)\delta s_R^\lambda L_0^\lambda}{\lambda n}\right)^{\!1/(1-\phi)} e^{\lambda n t/(1-\phi)}.

Thus the asymptotic level coefficient is proportional to $s_R^{\lambda/(1-\phi)}$. Holding the other parameters fixed, its elasticity with respect to the research share is $\lambda/(1-\phi)$. This is a level comparison, not a change in the exponential growth rate. During the transition, however, a higher research share can also change the current growth rate, since $g_A=\delta A^{\phi-1}(s_RL)^\lambda$ depends directly on research labor and on the current knowledge stock.

If population does not grow, the positive balanced-growth formula cannot be used to claim sustained positive growth. With $n=0$, research labor is constant when $s_R$ is constant, and the transformed equation becomes $\dot B=(1-\phi)\delta(s_RL_0)^\lambda$, a constant. Hence $B$ grows linearly with time, while $A=B^{1/(1-\phi)}$ grows at a declining proportional rate that tends to zero. The model therefore has no positive long-run growth rate in that case.

Worked Example: Solving a Knowledge Transition

Suppose $\phi=0.5$, $\lambda=0.5$, $\delta=0.01$, $n=0.02$ per year, $s_R=0.04$, $L_0=100$, and $A_0=100$. Find $A(10)$, the initial knowledge growth rate, and the long-run growth rate.

Method. Use the transformed solution for $A(t)^{1-\phi}$; with these parameters $1-\phi=0.5$, so the transformed stock is $\sqrt{A(t)}$. First, $\sqrt{s_R^\lambda L_0^\lambda}$ is not the required expression: compute $s_R^\lambda L_0^\lambda=(s_RL_0)^\lambda=\sqrt{4}=2$. Also, $A_0^{1-\phi}=\sqrt{100}=10$, and $\lambda n=0.5(0.02)=0.01$. The coefficient in the integrated solution is

\frac{(1-\phi)\delta(s_RL_0)^\lambda}{\lambda n} =\frac{0.5(0.01)(2)}{0.01}=1.

At $t=10$, $e^{\lambda nt}=e^{0.01(10)}=e^{0.1}\approx1.105170$. Therefore,

\sqrt{A(10)} =10+1(1.105170-1)=10.105170, \qquad A(10)=(10.105170)^2\approx102.114.

For the initial growth rate, use the original equation divided by $A$:

g_A(0) =\delta A_0^{\phi-1}(s_RL_0)^\lambda =0.01(100^{-0.5})(2) =0.01(0.1)(2) =0.002.

The initial rate is 0.2% per year. The balanced-growth rate is

g_A=\frac{0.5(0.02)}{1-0.5} =\frac{0.01}{0.5} =0.02,

or 2% per year. Knowledge has begun below its long-run rate and approaches that rate as the research workforce expands. Check: the integrated solution gives $A(10)>A_0$, consistent with positive research-driven accumulation. Common mistake avoided: applying the exponential growth rate $2\%$ directly to $A_0$ for ten years would give a constant-rate extrapolation, not the transition path; the growth rate here changes over time.

Worked Example: Comparing Permanent Research Shares

Use the parameters and initial conditions in the preceding calculation, but compare two economies: one with $s_R=0.04$ and another with $s_R=0.09$. Find each economy’s technology after ten years and compare their long-run growth rates.

Method. Apply the integrated knowledge equation separately to each research share. Here $\lambda=\phi=0.5$, $\delta=0.01$, $n=0.02$, $L_0=100$, and $A_0=100$. The initial transformed stock is $10$, and the exponential increment uses $e^{0.1}-1\approx0.105170$.

For $s_R=0.04$, $(s_RL_0)^\lambda=(4)^{0.5}=2$. The coefficient is $0.5(0.01)(2)/0.01=1$. Thus

\sqrt{A_{0.04}(10)} =10+1(0.105170)=10.105170, \qquad A_{0.04}(10)\approx102.114.

For $s_R=0.09$, $(s_RL_0)^\lambda=(9)^{0.5}=3$. The coefficient is $0.5(0.01)(3)/0.01=1.5$. Thus

\sqrt{A_{0.09}(10)} =10+1.5(0.105170)=10.157755, \qquad A_{0.09}(10)\approx103.180.

The higher research share produces a higher knowledge level at year 10. Yet both long-run growth rates are $\lambda n/(1-\phi)=0.5(0.02)/0.5=0.02$, or 2% per year. Check: $103.180>102.114$, while the growth-rate formula is identical for both cases. Common mistake avoided: treating a rise in $s_R$ as a permanent increase in the balanced-growth rate confuses a higher level path with a steeper long-run exponential rate.

Interpretation and Common Exam Traps

This model is called semi-endogenous because the long-run growth rate is not determined solely by a permanent choice of research effort. Population growth supplies an expanding research workforce, while diminishing returns to existing knowledge limit the growth that a fixed workforce can sustain. Unlike a model in which a constant research share directly determines sustained growth, here population growth is central to the balanced-growth rate.

A practical way to keep the calculations straight is to separate three questions:

What is the long-run growth rate? Use $g_A=\lambda n/(1-\phi)$, under the stated positive-population-growth assumptions.
What is the knowledge level at a particular date? Use the integrated equation, including $A_0$, $\delta$, $s_R$, and $L_0$.
What is growth at a particular date? Use $g_A(t)=\delta A(t)^{\phi-1}[s_RL(t)]^\lambda$, not automatically the long-run rate.

Common errors include using the level equation as though it were a growth-rate equation, omitting the denominator $1-\phi$, or assuming that research effort has no effect because it is absent from the long-run rate formula. Another trap is applying the positive-$n$ integrated formula at $n=0$, where it divides by zero. In that case, integrate the constant transformed-stock growth directly; do not infer a positive balanced-growth rate from the formula for $n>0$.

Key takeaway: In the semi-endogenous research model with diminishing returns to knowledge, $g_A=\lambda n/(1-\phi)$ when population grows at a constant positive rate. A constant research-share increase raises the knowledge path’s level and can affect transitional growth, but it does not change that long-run growth rate.

Check Your Understanding

Work through each problem by identifying whether it asks for a long-run rate, a current rate, or a technology level.

A research model has $\lambda=0.4$, $\phi=0.6$, and population growth $n=0.015$. Calculate the long-run knowledge growth rate. (Answer: $0.015$, or 1.5% per year.)
In another model, $\lambda=0.5$, $\phi=0.75$, and $n=0.02$. Calculate the balanced-growth rate and state whether changing a constant research share alone changes it. (Answer: $0.04$, or 4% per year; no, a constant research-share change affects the level path, not this rate.)
At a particular date, $\delta=0.02$, $A=1000$, $s_R=0.25$, $L=400$, $\lambda=0.5$, and $\phi=0.5$. Calculate the current knowledge growth rate. (Answer: $0.006325$, or approximately 0.6325% per year.)
A model has $\lambda=0.6$ and $\phi=0.4$. What population growth rate is required for a long-run knowledge growth rate of 3%? (Answer: $n=0.03$, or 3% per year.)
With $\lambda=0.35$ and $\phi=0.65$, population growth and all other parameters are fixed. Research share rises permanently by 20%. Find the proportional change in the asymptotic technology level coefficient and the change in the long-run growth rate. (Answer: the level coefficient rises by 20%, since $\lambda/(1-\phi)=1$; the long-run growth rate is unchanged.)

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