Why Balanced Growth Matters
An economy can have a stable capital-output relationship even while its total production and capital stock keep expanding. The key is to distinguish growth in the aggregate from growth per worker, and to account for both population growth and improvements in labor-augmenting technology. A balanced-growth path describes a situation in which the economy’s variables grow at consistent rates, rather than remaining fixed in levels.
This tutorial derives those rates in the Solow model. The central result is that, along a balanced-growth path, output and capital per worker grow at the rate of labor-augmenting technical progress, while aggregate output and aggregate capital grow at the combined rate of population growth and technical progress. The model’s effective-worker formulation, introduced in the earlier material on labor-augmenting progress and effective workers, makes the distinction precise.
Model and Growth-Rate Conventions
Use the continuous-time Solow model with a constant saving rate \(s\), depreciation rate \(\delta\), population growth rate \(n\), and labor-augmenting technology growth rate \(g\). Thus, \(\dot L/L=n\) and \(\dot A/A=g\), where a dot denotes a time derivative. Production has constant returns to scale and can be written \(Y=F(K,AL)\). Dividing by effective labor gives \(Y/(AL)=f(k)\), where \(f\) is the production function in per-effective-worker terms.
The model assumes that saving finances investment, capital depreciates at rate \(\delta\), and \(s,n,g,\delta\) are constant. For the usual steady-state result, \(f\) is increasing and concave, with diminishing marginal product of capital. The saving rate and depreciation assumption govern the level of the steady state; the growth rates derived below follow from the balanced-growth condition and the growth of labor and technology.
The effective-worker capital stock changes according to investment per effective worker minus the investment needed to replace depreciated capital and equip growing effective labor. Its equation of motion is
The term \((n+g+\delta)k\) is break-even investment in effective-worker units. At a steady state \(k^*\), \(\dot k=0\), so actual investment just offsets depreciation and the dilution of capital across expanding effective labor:
This condition determines the constant value of capital per effective worker. It does not mean that aggregate capital is constant: effective labor itself is growing.
Deriving Growth on the Balanced-Growth Path
The derivation uses two identities. First, aggregate capital is \(K=ALk\). Second, aggregate output is \(Y=AL f(k)\). On a balanced-growth path, \(k=k^*\) is constant, and consequently \(f(k^*)\) is constant as well. Since \(A\) grows at rate \(g\) and \(L\) at rate \(n\), both \(K\) and \(Y\) grow with the product \(AL\).
On the path, \(k=K/(AL)=k^*\) and \(\tilde y=Y/(AL)=f(k^*)\). Their growth rates are zero.
Because \(K/L=A k^*\) and \(Y/L=A f(k^*)\), both capital per worker and output per worker grow at the rate of \(A\), namely \(g\).
Because \(K=ALk^*\) and \(Y=AL f(k^*)\), their growth rates are the sum of the growth rates of \(A\) and \(L\), namely \(g+n\).
The aggregate result also follows directly by differentiating the identities. For example, \(Y=AL f(k^*)\) gives
The final term is zero because \(k^*\), and therefore \(f(k^*)\), is constant. Similarly, differentiating \(Y/L=A f(k^*)\) gives \(\frac{d}{dt}\ln(Y/L)=g\). This is why per-worker growth is not \(n+g\): population growth changes the number of workers, not output per worker.
Worked Applications
Worked Example: Finding a Cobb-Douglas Balanced-Growth Path
Consider a continuous-time economy with \(Y=K^{1/3}(AL)^{2/3}\). The saving rate is \(s=0.24\), population grows at \(n=0.02\), labor-augmenting technology grows at \(g=0.015\), and capital depreciates at \(\delta=0.045\). Find the steady-state capital and output per effective worker, then state all four balanced-growth rates. Use the method of solving the effective-worker steady-state condition before calculating growth rates.
Step 1: Write production per effective worker. Dividing production by \(AL\) gives \(\tilde y=f(k)=k^{1/3}\).
Step 2: Calculate break-even investment. The total rate that dilutes or wears down effective-worker capital is \(n+g+\delta=0.02+0.015+0.045=0.08\). Thus the steady-state equation is \(0.24k^{1/3}=0.08k\).
Step 3: Solve for \(k^*\). For positive \(k\), divide by \(0.08k^{1/3}\) to obtain \(k^{2/3}=0.24/0.08=3\). Raising both sides to the power \(3/2\) gives \(k^*=3^{3/2}=3\sqrt{3}\approx5.1962\).
Step 4: Find output per effective worker. \(\tilde y^*=(k^*)^{1/3}=(3\sqrt{3})^{1/3}=\sqrt{3}\approx1.7321\). As a check, saving per effective worker is \(0.24(1.7321)\approx0.4157\), while break-even investment is \(0.08(5.1962)\approx0.4157\), agreeing up to rounding.
Step 5: State the growth rates. Capital and output per effective worker are constant, so their growth rates are zero. Capital per worker and output per worker each grow at \(g=1.5\%\) per year. Aggregate capital and output each grow at \(n+g=0.02+0.015=3.5\%\) per year.
Method and trap avoided. Solving in effective-worker units separates the steady-state levels from the growth rates. A common error is to call \(k^*\) “capital per worker”; it is capital per effective worker, so it is constant even though \(K/L\) grows at 1.5%.
Worked Example: Distinguishing a Transition From Balanced Growth
In a hypothetical economy with Cobb-Douglas production, output per worker is \(Y/L=A k^{1/3}\), where \(k=K/(AL)\). Over one year, technology rises by 3% and capital per effective worker rises by 8%. Calculate the exact annual growth in output per worker using the stated discrete annual rates. Explain why the result is not the balanced-growth rate.
Step 1: Use the per-worker production identity. The technology factor is \(1.03\), while the effective-worker capital factor is \(1.08\). Thus the output-per-worker factor is \(1.03(1.08)^{1/3}\).
Step 2: Evaluate the factor. The cube root is \((1.08)^{1/3}\approx1.0259856\). Multiplying gives \(1.03(1.0259856)\approx1.0567652\).
Step 3: Convert the factor to a percentage growth rate. Subtract one: \(1.0567652-1=0.0567652\), or approximately 5.68% when rounded to two decimal places. The factor is greater than one, as expected when both technology and effective-worker capital increase.
Interpretation and trap avoided. This is a transition because \(k\) is growing, not constant. On a balanced-growth path, \(k\) does not grow, so output per worker grows at the technology rate alone. Using the shortcut \(3\%+(1/3)(8\%)=5.6667\%\) misses the small compounding difference in the exact discrete annual calculation; it is a first-order approximation, not the exact rate.
Worked Example: Exact Aggregate Growth With Annual Rates
A hypothetical economy is on a balanced-growth path. Labor grows by 2% per year and labor-augmenting technology grows by 3% per year, both stated as discrete annual rates. Calculate the exact annual growth rates of aggregate output, output per worker, capital per worker, and aggregate capital. Use growth factors rather than adding percentages.
Step 1: Apply the balanced-growth relationships. Output per worker and capital per worker grow with technology, so each has annual growth factor \(1.03\), or growth rate 3%.
Step 2: Combine labor and technology for aggregates. The aggregate factor is \(1.02\times1.03=1.0506\). Therefore aggregate output and aggregate capital each grow by \(1.0506-1=0.0506\), or 5.06%.
Step 3: Check the distinction. Aggregate growth exceeds per-worker growth because there are more workers as well as more output and capital per worker. The calculation \(2\%+3\%=5\%\) is the continuous-time result, or an approximation for small discrete rates. It is not the exact discrete annual answer, which includes the cross-product \(0.02\times0.03=0.0006\), equal to 0.06 percentage points.
Method and trap avoided. With discrete annual growth rates, multiply the growth factors before subtracting one. Adding the stated percentages directly would understate aggregate growth by 0.06 percentage points.
Continuous Rates and Discrete Rates
The model equation above is written in continuous time. In that convention, growth rates add exactly: if \(A\) grows at rate \(g\) and \(L\) at rate \(n\), then \(AL\), aggregate capital, and aggregate output on the balanced-growth path grow at rate \(n+g\). This follows from differentiating a product.
For annual discrete growth rates, the exact calculation is different. If labor grows by \(n\) and technology by \(g\) over a year, their factors are \(1+n\) and \(1+g\). Their product is
Consequently, the exact discrete aggregate growth rate is \(n+g+ng\), not merely \(n+g\). For small rates, the cross-product \(ng\) is often small, which is why adding the rates can be a useful approximation. State which convention the problem uses before calculating.
What the Result Does and Does Not Say
Balanced growth concerns growth rates, not equality of levels across economies. The steady-state condition \(s f(k^*)=(n+g+\delta)k^*\) determines the capital stock per effective worker consistent with the model’s parameters. The growth-rate result then describes how the main aggregates evolve if the economy is on that path. Neither result says that aggregate output is constant.
The growth rate of output per worker on the path is tied to exogenous labor-augmenting technical progress. It does not depend on the Cobb-Douglas capital share in the way a transitional calculation may. For instance, away from the path in the second example, a change in \(k\) contributes to output-per-worker growth through the production function. On the balanced-growth path, \(k\) is fixed, so that contribution is zero.
- Do not confuse the units. \(K/(AL)\) is capital per effective worker; \(K/L\) is capital per worker.
- Do not assign population growth to per-worker quantities. Population growth contributes to aggregate growth, not directly to output growth per worker.
- Do not treat the steady-state level condition as a growth-rate formula. Saving, depreciation, and diminishing returns affect \(k^*\); the balanced-growth rates follow from \(n\), \(g\), and the constant effective-worker quantities.
- Do not mix rate conventions. In continuous time, aggregate rates add exactly. For discrete annual rates, combine factors for an exact result.
These distinctions are useful when interpreting growth accounting and long-run data: a rising aggregate can reflect more workers, greater output per worker, or both. The balanced-growth framework identifies the model’s long-run rate for each component without confusing population expansion with improvements in living standards per worker.
Check Your Understanding
For each problem, identify whether the rates are continuous or discrete before applying the balanced-growth relationships.
- A continuous-time economy has labor growth \(n=0.012\) and technology growth \(g=0.018\). On its balanced-growth path, state the growth rates of aggregate output, output per worker, aggregate capital, and capital per worker. (Answer: aggregate output 3.0%; output per worker 1.8%; aggregate capital 3.0%; capital per worker 1.8%.)
- In a discrete annual model, labor grows by 1% and technology by 4%. Find the exact growth rates of aggregate output and output per worker on the balanced-growth path. (Answer: aggregate output \(1.01(1.04)-1=5.04\%\); output per worker 4%.)
- A continuous-time Cobb-Douglas Solow economy has \(f(k)=k^{1/2}\), \(s=0.25\), \(n=0.01\), \(g=0.02\), and \(\delta=0.07\). Find \(k^*\), output per effective worker at the steady state, and the growth rate of aggregate output. (Answer: \(n+g+\delta=0.10\), \(0.25\sqrt{k}=0.10k\), so \(k^*=6.25\); \(\tilde y^*=2.5\); aggregate output grows at 3%.)
- In a discrete annual transition with \(Y/L=A k^{0.4}\), technology rises by 2% and capital per effective worker by 6%. Find the exact output-per-worker growth rate, rounded to two decimal places. (Answer: \(1.02(1.06)^{0.4}-1\approx0.0441\), or 4.41%.)
- A discrete annual economy on a balanced-growth path has population growth of 1.5% and technology growth of 2.5%. Find the exact annual growth rates of aggregate output, aggregate capital, output per worker, and capital per worker. (Answer: aggregate output and aggregate capital grow by \(1.015(1.025)-1=4.0375\%\), or 4.04%; output per worker and capital per worker grow by 2.5%.)