Capital per Worker and Labor Productivity
An economy can produce more output because it has more workers, because each worker has more productive resources to use, or because the production process itself has improved. This tutorial focuses on one specific source of higher output per worker: capital deepening, an increase in physical capital relative to the number of workers. A production-function diagram helps show both why this can raise labor productivity and why it is different from a productivity improvement.
Capital here means physical productive assets, such as machinery or buildings. The model treats the quantity of capital as comparable across periods and assumes that the number of workers is measured consistently. It abstracts from changes in hours worked, worker skills, capital quality, and the process by which investment is financed.
A Production Function Per Worker
A standard illustration uses a Cobb–Douglas production function:
where \(A>0\) represents the level of productivity in this model and \(\alpha\) is capital’s output elasticity. The function has constant returns to scale: multiplying both inputs, \(K\) and \(L\), by the same positive factor multiplies output by that factor. Dividing the production function by \(L\) gives the per-worker relationship:
This equation isolates the tutorial’s central comparison. If \(A\) is held fixed, more capital per worker raises output per worker. If \(A\) changes, the production relationship itself changes. The model is a simplified way to organize these effects; it does not assert that all economies have the same production function or that capital alone determines observed productivity.
What the Production-Function Diagram Shows
In a production-function diagram, put capital per worker \(k\) on the horizontal axis and output per worker \(y\) on the vertical axis. With \(A\) fixed, the curve \(y=A k^\alpha\) slopes upward: moving right along it represents capital deepening and corresponds to higher output per worker. The curve is concave, so its slope becomes smaller as \(k\) rises. A rise in productivity \(A\), by contrast, shifts the curve upward: at any given \(k\), output per worker is higher.
The slope of the curve is the marginal increase in output per worker from a small increase in capital per worker. Differentiating the production function gives:
The first inequality follows because \(A>0\), \(k>0\), and \(\alpha>0\). The second follows because \(\alpha-1<0\). Thus, in this model, capital deepening raises output per worker, but each additional unit of capital per worker adds less output than the preceding unit, other things equal. This is diminishing marginal product of capital per worker; it does not mean that total output per worker falls.
| Change in the model | Diagram interpretation | Effect at a given starting point |
|---|---|---|
| Higher \(k\), fixed \(A\) | Move right along the same curve | Higher \(y\), with a diminishing slope |
| Higher \(A\), fixed \(k\) | Shift the curve upward | Higher \(y\) at the same \(k\) |
| Higher \(k\) and higher \(A\) | Move right and use a higher curve | Both changes raise \(y\) in this model |
Deriving the Output Effect of Capital Deepening
The per-worker formula also provides a direct way to compare two dates. If productivity is unchanged, divide output per worker at the later date by output per worker at the earlier date. The fixed \(A\) cancels:
This derivation explains why output per worker does not generally increase by the same percentage as capital per worker. The capital ratio is raised to the power \(\alpha\), which is between zero and one. For small changes, the proportional-growth approximation is \(\Delta y/y \approx \alpha\,\Delta k/k\), with productivity fixed. That approximation is useful for small changes; for a sizeable change, use the exact ratio formula.
If productivity also changes, the comparison is:
This equation separates the two modeled sources of change. A capital-deepening calculation holds \(A\) constant. A productivity comparison holds \(k\) constant. If both change, the total effect combines them multiplicatively, so their percentage changes should not simply be added as exact values.
Worked Example: Exact Output Change from Capital Deepening
A hypothetical economy is represented by \(Y=8K^{0.4}L^{0.6}\). The number of workers is unchanged, while capital per worker rises from \(k_0=10\) to \(k_1=16\). Assume \(A=8\) and \(\alpha=0.4\) remain fixed. Calculate output per worker in each period and its percentage change.
Method. Apply \(y=A k^\alpha\) at both capital-per-worker levels, then compare the outputs. This keeps productivity fixed and isolates capital deepening.
Initially, \(y_0=8(10^{0.4})\). Since \(10^{0.4}\approx2.511886\), \(y_0\approx8(2.511886)=20.0951\).
Later, \(y_1=8(16^{0.4})\). Since \(16^{0.4}\approx3.031433\), \(y_1\approx8(3.031433)=24.2515\).
The output ratio can be checked using the derived formula: \(y_1/y_0=(16/10)^{0.4}=1.6^{0.4}=\exp(0.4\ln 1.6)\approx\exp(0.188001)=1.2068\).
The percentage change is \((1.206825-1)\times100\%\approx20.6825\%\), or 20.7% to one decimal place. The level check gives \(24.2515/20.0951\approx1.2068\), consistent with the ratio calculation.
When Capital and Productivity Both Change
The same equations help distinguish a movement along a curve from a shift of the curve when both inputs to the per-worker relationship change. One transparent approach is to change one factor at a time: first calculate the effect of new \(k\) holding \(A\) at its original value, then calculate the effect of new \(A\) at the new \(k\). The order used to describe the intermediate step affects how the total change is allocated between the two effects, but not the final output.
Worked Example: Separating Capital Deepening from a Productivity Improvement
A hypothetical production process follows \(y=A k^{0.5}\). Initially, \(A_0=2\) and \(k_0=25\). Later, \(A_1=2.2\) and \(k_1=36\). Find the initial and final output per worker, and separate the changes using the original productivity level first.
Initial output per worker. The square root of 25 is 5, so \(y_0=2\sqrt{25}=2(5)=10\).
Capital-deepening step. Hold \(A=2\) and change \(k\) to 36. The intermediate output is \(y_{\text{int}}=2\sqrt{36}=2(6)=12\). Capital deepening therefore raises output per worker by \(12-10=2\), or \(2/10=20\%\) relative to the initial level.
Productivity step. At \(k=36\), change \(A\) from 2 to 2.2. Final output per worker is \(y_1=2.2\sqrt{36}=2.2(6)=13.2\). This is an increase of \(13.2-12=1.2\), or \(1.2/12=10\%\) relative to the intermediate level.
Overall, output per worker rises from 10 to 13.2, an increase of \(3.2/10=32\%\). The two sequential percentage changes combine as \(1.20\times1.10=1.32\), not \(20\%+10\%=30\%\). The calculation identifies capital deepening as movement along the original curve and the increase in \(A\) as an upward shift.
From Output per Worker to Total Output
Capital deepening is defined using a ratio, so it is important to keep the worker count in view. Since \(K=kL\) and \(Y=yL\), an increase in total capital does not by itself establish that capital per worker has increased: the number of workers may also have changed. Conversely, with a fixed production function, more workers can raise total output even when capital per worker and output per worker remain unchanged.
Worked Example: Accounting for a Change in Workers
A hypothetical economy follows \(Y=3K^{0.5}L^{0.5}\). Initially it has \(K_0=400\) units of capital and \(L_0=100\) workers. Later it has \(K_1=625\) and \(L_1=125\). Assume the same \(A=3\) and \(\alpha=0.5\) in both periods. Calculate capital per worker, output per worker, and total output in each period.
Initial period. Capital per worker is \(k_0=400/100=4\). Output per worker is \(y_0=3\sqrt{4}=3(2)=6\). Total output is \(Y_0=y_0L_0=6(100)=600\).
Later period. Capital per worker is \(k_1=625/125=5\). Output per worker is \(y_1=3\sqrt{5}\approx3(2.236068)=6.7082\). Total output is \(Y_1=y_1L_1\approx6.708204(125)=838.5255\).
Capital per worker rises by \(5/4-1=25\%\), while output per worker rises by \(\sqrt{5}/2-1\approx11.8034\%\), or 11.8%. Total output rises by approximately \(838.5255/600-1=39.7542\%\), or 39.8%. The larger total-output increase also reflects the rise in the number of workers from 100 to 125; it is not all a rise in labor productivity.
Common Mistakes and Exam Traps
A common mistake is to treat the capital share parameter as an exact multiplier for any percentage change in capital per worker. In the first example, that shortcut gives \(0.4\times60\%=24\%\) output growth. It is a small-change approximation, not the exact answer for a 60% increase in \(k\). The exact calculation is \(1.6^{0.4}-1\approx0.2068\), or 20.7%.
Another trap is to confuse an increase in \(K\) with capital deepening. The relevant comparison is \(K/L\), not \(K\) alone. Finally, a higher \(y\) does not prove that capital deepening was the cause: in the model, a higher \(A\) can also raise output per worker. A sound calculation states which variable is changing and which is being held fixed.
Check Your Understanding
Work through each problem by identifying what changes and applying the per-worker production relationship. Answers are included for checking.
- A hypothetical economy has \(y=4k^{0.5}\). Capital per worker rises from 4 to 9, with productivity fixed. Find output per worker at each level and the percentage increase. (Answer: \(y\) rises from 8 to 12, an increase of 50%.)
- A production function is \(y=6k^{1/3}\). Capital per worker increases from 8 to 27. Calculate the initial and final output per worker and the percentage increase. (Answer: \(y\) rises from 12 to 18, an increase of 50%.)
- For \(y=2k^{0.25}\), capital per worker rises from 16 to 81 while \(A=2\) is fixed. Find the output-per-worker ratio and percentage increase. (Answer: the ratio is \(6/4=1.5\), so output per worker increases by 50%.)
- A hypothetical economy has \(y=A\sqrt{k}\). Initially \(A=3\) and \(k=4\); later \(A=3.6\) and \(k=9\). Calculate initial and final output per worker and the total percentage increase. (Answer: \(y\) rises from 6 to 10.8, an increase of 80%.)
- With \(A\) fixed in \(y=A k^{0.4}\), what ratio of capital per worker is required to raise output per worker by 12%? State the percentage increase in capital per worker to one decimal place. (Answer: \(k_1/k_0=1.12^{1/0.4}=1.12^{2.5}\approx1.3275\), so capital per worker must rise by approximately 32.8%.)