Why the Source of Productivity Growth Matters
An economy can become more productive because its workers use better technology, but a growth model must also say where that technology comes from. In some models, the productivity path is specified from outside the model. In others, productivity growth depends on choices made within the economy, such as devoting resources to research or education. This difference affects what the model can say about policy: a choice cannot change a variable that the model treats as externally fixed.
The Solow model with exogenous technological progress, developed in the earlier material on technology levels and balanced growth, is a useful benchmark. The comparison here concerns the model’s explanation of productivity growth, not whether technology matters: it matters in both cases. This distinction is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
“Endogenous” does not mean that every influence on technology must be explained in a single model. It means that the model includes a mechanism connecting choices to productivity growth. Likewise, “exogenous” does not mean that technology has no causes in reality. It means those causes are outside the model’s explanation.
The Exogenous-Technology Benchmark
Consider a standard Solow production model with labor-augmenting technology:
Here \(Y\) is output, \(K\) is physical capital, \(L\) is labor, and \(A\) is labor-augmenting productivity. The model assumes diminishing returns to capital when capital is measured per effective worker, and takes the saving rate \(s\), population growth rate \(n\), depreciation rate \(\delta\), and technology growth rate \(g_A\) as given. In the exogenous version, the path of \(A\) is not determined by firms’ or households’ research and education decisions.
In continuous time, define capital per effective worker as \(k=K/(AL)\), and output per effective worker as \(\tilde y=Y/(AL)\). With the production function above, \(\tilde y=f(k)=k^\alpha\). Capital per effective worker changes according to:
This equation assumes a constant saving rate, labor-force growth \(n\), technology growth \(g_A\), and depreciation \(\delta\). It says that investment per effective worker adds to \(k\), while population growth, technology growth, and depreciation dilute or wear down capital per effective worker. It is not a model of how research produces technology: \(g_A\) is given from outside.
Why Productivity Growth Passes Through to Output per Worker
The balanced-growth implication follows directly from the production function. Since \(K/L=A k\), output per worker is:
On a balanced growth path, \(k\) is constant. Therefore \(k^\alpha\) is constant too, and growth in output per worker comes from growth in \(A\). If \(A\) grows continuously at rate \(g_A\), output per worker also grows continuously at rate \(g_A\). This result describes the balanced path; during a transition, \(k\) can change, so output per worker need not grow at that same rate in every period.
Endogenous Progress: A Different Modeling Claim
An endogenous-technology model adds a mechanism that links deliberate choices to productivity growth. For example, a model might specify that research effort or education affects the growth rate of \(A\). In a general representation, this could be written as:
where \(R\) represents research effort, \(H\) represents an education or human-capital input, and \(G\) is a relationship specified by the model. This notation identifies the modeling idea; by itself it does not determine the function \(G\), how research is organized, or the size of any policy effect. Those conclusions require additional assumptions.
The central contrast is whether the model makes a choice relevant to the productivity-growth rate. If \(g_A\) is fixed independently of \(R\), changing \(R\) does not change \(g_A\) within that model. If the model specifies a relationship between \(R\) and \(g_A\), the choice can affect productivity growth, subject to the form and assumptions of that relationship.
| Question | Exogenous technological progress | Endogenous technological progress |
|---|---|---|
| Where is the technology-growth rate determined? | Outside the model | By a mechanism included in the model |
| Can a modeled research choice change \(g_A\)? | Not if \(g_A\) is specified as independent of that choice | It can, if the model links the choice to \(g_A\) |
| What does the model explain about long-run growth? | How a given technology path affects output and capital | How specified choices or incentives help determine technology growth |
| What must be known to assess a policy effect? | The policy’s effect on variables already modeled; \(g_A\) remains given | The modeled link from the policy-related choice to productivity growth, as well as any costs included |
This comparison does not establish that a particular research subsidy, education program, or other intervention will raise growth. That is a further question about the model’s mechanisms and assumptions. Nor does calling progress endogenous automatically imply a larger growth rate: the size and direction of the effect depend on the relationship specified.
Worked Examples
Worked Example: Comparing Externally Given Technology Paths
Two hypothetical economies have the same initial technology index, \(A_0=100\). In each economy, technology grows at a constant annual discrete rate. Economy X has an externally specified growth rate of 2.0% per year, while Economy Z has an externally specified rate of 2.4%. Find each technology index after eight years and the difference in their cumulative percentage increases from the common initial index.
Method. With a constant annual discrete growth rate \(g\), the level after \(t\) years is \(A_t=A_0(1+g)^t\). This is an exogenous comparison: the rates are given, not explained by a choice in either economy.
For Economy X:
Its cumulative percentage increase is \(17.1659381\%\), or \(17.17\%\) to two decimal places.
For Economy Z:
Its cumulative percentage increase is approximately \(20.892582\%\), or \(20.89\%\) to two decimal places. The difference between the unrounded cumulative increases is approximately \(20.892582-17.165938=3.726644\) percentage points, which rounds to 3.73 percentage points. The difference in technology levels is approximately \(120.892582-117.165938=3.726644\) index points.
Check and common mistake. Both final levels exceed 100, as positive growth requires. Subtracting the separately rounded increases, \(20.89-17.17\), gives 3.72, but that is not the correctly rounded difference of the unrounded values. The correct difference rounds to 3.73 percentage points.
Worked Example: How an Exogenous Rate Changes the Solow Steady State
A hypothetical economy follows the continuous-time Solow model with \(f(k)=\sqrt{k}\), saving rate \(s=0.25\), population growth \(n=0.01\) per year, depreciation \(\delta=0.05\) per year, and exogenous technology growth initially \(g_A=0.02\) per year. Calculate steady-state capital and output per effective worker. Then find their steady-state values if \(g_A\) rises to \(0.03\) per year, and state the corresponding balanced-growth rate of output per worker.
Method. At a steady state, \(\dot{k}=0\), so saving per effective worker equals break-even investment: \(s\sqrt{k}=(n+g_A+\delta)k\). For positive \(k\), divide by \(\sqrt{k}\) and solve:
Initially, the break-even rate is \(0.01+0.02+0.05=0.08\) per year. Thus:
When \(g_A=0.03\), the break-even rate is \(0.01+0.03+0.05=0.09\) per year. Therefore:
On the first balanced path, output per worker grows at 2% per year; on the second, it grows at 3% per year. The higher technology-growth rate raises long-run output-per-worker growth, but lowers steady-state capital and output per effective worker in this calculation because more investment is needed to maintain a unit of effective-worker capital.
Check and common mistake. The steady-state equation includes \(g_A\) in the break-even rate, but the balanced-growth rate of output per worker is \(g_A\), not \(n+g_A+\delta\). Confusing these rates would mix capital dilution with productivity growth.
Worked Example: Interpreting a Choice-Dependent Growth Assumption
A hypothetical economy begins with a productivity index \(A_0=100\). A model assumes that a deliberate research-and-education program raises the annual discrete growth rate of technology from 1.0% to 2.0%, and that the higher rate persists for five years. Under this assumption, calculate the two technology levels after five years and their difference. What does the calculation establish, and what does it not establish?
Method. Use discrete compounding separately for each stated rate. The assumed link from the program to the higher rate makes this a stylized endogenous-technology comparison; the compounding arithmetic alone does not validate that link.
At 1.0% per year:
At 2.0% per year:
The difference is \(110.40808032-105.10100501=5.30707531\) index points, or about 5.31 points. Relative to the common initial index of 100, the difference in cumulative increases is about 5.31 percentage points.
Interpretation and check. The calculation shows what follows if the program really changes the growth rate as assumed and the new rate persists for all five years. It does not show that the program causes a one-percentage-point increase, that the increase lasts indefinitely, or that the program is worthwhile after accounting for its costs. Treating the assumed policy effect as a result of the compounding calculation is the common mistake.
Policy Interpretation and Common Traps
The model’s classification determines which questions it can answer. In an exogenous-technology model, a policy may still affect saving, capital accumulation, or the transition toward a steady state if those channels are included. But if technology growth is fixed independently of the policy, the model cannot attribute a change in long-run \(g_A\) to that policy. In an endogenous model, such a claim is possible only if the model specifies a relevant connection between the policy-related choice and productivity growth.
Several distinctions are especially important:
- A higher technology level is not the same as a permanently higher growth rate. A one-time increase in \(A\) raises the level of productivity. A sustained increase in \(g_A\) changes the continuing rate at which that level grows.
- A growth-rate assumption is not a policy mechanism. Stating that a policy raises \(g_A\) is an assumption unless the model explains how the policy changes choices that determine \(g_A\).
- Balanced-path growth is not necessarily transition growth. While \(k\) adjusts after a change, output per worker reflects both the technology path and the changing capital-per-effective-worker ratio.
- Endogenous does not mean costless or automatically beneficial. Research or education can use resources that might otherwise be used elsewhere. A growth comparison alone does not measure the full welfare effect.
For an exam-style model comparison, first identify what is specified outside the model and what is determined by choices inside it. Then state the implication conditionally: if the model holds \(g_A\) fixed, the choice cannot change it through that model; if the model links the choice to \(g_A\), calculate or reason through the effect using that specified link.
Check Your Understanding
For each problem, identify the relevant modeling assumption before calculating or interpreting the result.
- A model specifies that technology grows at a constant 1.8% per year, independently of firms’ research spending. Classify the technology progress as exogenous or endogenous. If research spending rises but the model keeps the stated assumption, what happens to the model’s technology-growth rate? (Answer: Exogenous; it remains 1.8% per year.)
- A hypothetical economy has \(A_0=80\). Calculate its technology index after four years at a constant annual discrete growth rate of 1.0%, then at 2.0%, and find the difference between the resulting levels. (Answer: \(80(1.01)^4=83.2483\), \(80(1.02)^4=86.5946\); the difference is approximately 3.3463 index points.)
- In a continuous-time Solow model, \(f(k)=\sqrt{k}\), \(s=0.36\), \(n=0.02\), \(\delta=0.04\), and exogenous \(g_A=0.03\). Find \(k^*\), output per effective worker \(\tilde y^*\), and balanced-growth output per worker growth. (Answer: \(k^*=(0.36/0.09)^2=16\), \(\tilde y^*=4\), and output per worker grows at 3% per year on the balanced path.)
- With common initial technology \(A_0=200\), compare constant annual discrete growth rates of 1.5% and 2.5% over six years. Find the technology levels and the difference in cumulative percentage increases. (Answer: \(200(1.015)^6\approx218.6886\), \(200(1.025)^6\approx231.9386\); the difference is approximately 6.625 percentage points.)
- A model says an education policy raises the technology level immediately but specifies that the future technology-growth rate remains fixed and independent of education. Is future growth endogenous to education in this model? Distinguish the level effect from the growth-rate effect. (Answer: No. The modeled immediate level increase is not a change in the future growth rate; that rate remains exogenous and unchanged.)