Why Inputs Can Be More Productive Together
A firm may install more advanced equipment, train its workers, or use improved production methods. Each change can raise output. But the gain from one change may depend on what else is in place: equipment can be more useful when workers have the skills to operate it, and those skills can be more productive when workers have effective tools and methods. This interaction is called complementarity.
The focus here is the production-side relationship among physical capital, worker skills, and technology. It is not a claim that every machine works better with every skill or that every technology raises every worker’s productivity. Complementarity is a property of a specified production setting or model, and real-world conclusions require evidence about the setting being studied.
A Simple Model of Capital, Skills, and Technology
To make the interaction precise, use a stylized production model:
Here, \(Y\) is output; \(K\) is the quantity of physical capital; \(H\) is skill-adjusted labor input; and \(A\) is a technology or production-efficiency index. A higher \(H\) represents more effective labor input—for example, the same number of workers with greater relevant skills. The model treats \(K\), \(H\), and \(A\) as positive, measurable inputs or indexes, and assumes the production relationship is unchanged while a comparison is made except for the specified input changes.
This particular function is a teaching model, not a universal law. It has constant returns to scale in \(K\) and \(H\): multiplying both by the same factor multiplies output by that factor, when \(A\) is held fixed. It also gives positive marginal products and positive cross-partials for the three factors. Other production relationships can imply different interactions.
Worked Example: Capital and Skills in Combination
A hypothetical workshop is represented by \(Y=2\sqrt{KH}\), with output measured in production units, capital \(K\) in equipment units, and skill-adjusted labor \(H\) in labor units. Initially, \(K=100\) and \(H=25\). Compare the initial output with output after (a) capital alone rises to 144, (b) skill-adjusted labor alone rises to 36, and (c) both rise to those new levels. Use the same technology index in every case.
Method. Substitute each pair of inputs into the production function. This holds the technology index fixed and isolates the separate and combined input effects.
Initial output is:
With capital alone increased:
The capital change raises output by \(120-100=20\) units, or \(20/100=20\%\) of initial output. With skill-adjusted labor alone increased:
This change also raises output by 20 units, or 20%. With both inputs increased:
Output rises by \(144-100=44\) units, or 44%. The two separate output gains sum to \(20+20=40\), which is 4 units less than the combined gain. Equivalently, the combined output is \(144\), rather than \(120+120-100=140\), the level implied if the separate gains simply added.
Check. The capital-only and skill-only calculations each give 120, while applying both input changes gives 144. The extra 4 units are the interaction in this comparison. A common mistake is to report the combined gain as 40 units by adding the two individual gains; the function shows that the combined gain is 44 units.
Why the Model Implies Complementarity
The production function defines the marginal product of capital as the partial derivative of output with respect to \(K\), holding \(H\) and \(A\) fixed. The marginal product of skill-adjusted labor is defined in the same way. Differentiating gives:
For positive \(K\), \(H\), and \(A\), both marginal products are positive. To test whether capital and skill-adjusted labor are complements, differentiate the marginal product of capital with respect to \(H\). The same result follows by differentiating \(MP_H\) with respect to \(K\):
\(MP_K=(A/2)H^{1/2}K^{-1/2}\), with \(A\) and \(K\) fixed for the derivative with respect to \(H\).
\(\partial MP_K/\partial H=(A/4)H^{-1/2}K^{-1/2}\).
Since \(A>0\), \(H>0\), and \(K>0\), \((A/4)H^{-1/2}K^{-1/2}>0\). Thus, more skill-adjusted labor raises the marginal product of capital in this model.
This derivation explains why the two inputs reinforce one another here: \(MP_K\) rises with \(H\), rather than merely remaining unchanged. The marginal products also depend positively on \(A\). For example, \(MP_K=(A/2)\sqrt{H/K}\) increases when \(A\) increases, with \(K\) and \(H\) held fixed. Technology therefore raises the modeled marginal product of capital and of skill-adjusted labor.
The cross-partials make these claims explicit:
These positive signs apply within this model and for positive inputs. A positive cross-partial is a local statement about how marginal products change; it does not, by itself, establish the size of an actual productivity gain or prove that a particular investment is worthwhile.
Worked Example: Separating Three Input Improvements
A hypothetical production line uses the same function, \(Y=A\sqrt{KH}\). Initially, \(A=2\), \(K=100\), and \(H=25\). A proposed operating change would raise the inputs to \(A=2.5\), \(K=144\), and \(H=36\). Calculate initial output, the output with each one input changed separately, and output after all three changes. Then compare the combined gain with the sum of separate gains.
Method. First calculate the baseline, then change one input at a time while holding the other two at baseline values. This prevents attributing a combined effect to any one input.
Initial output is \(Y_0=2\sqrt{100(25)}=2(50)=100\). If only capital changes, output is \(2\sqrt{144(25)}=2(60)=120\), a gain of 20. If only skill-adjusted labor changes, output is \(2\sqrt{100(36)}=2(60)=120\), also a gain of 20. If only technology changes, output is \(2.5\sqrt{100(25)}=2.5(50)=125\), a gain of 25.
The sum of these separate gains is \(20+20+25=65\). Now apply all three changes:
The combined gain is \(180-100=80\), which exceeds the sum of separate gains by \(80-65=15\) output units. This 15-unit difference is the aggregate interaction among the changes relative to the baseline; it is not a separate input or an independently observed productivity measure.
Check. The three one-input output levels are 120, 120, and 125; the all-input output is 180. The separate gains sum to 65, while the combined gain is 80. A common mistake is to add the three percentage changes to predict the combined percentage increase. The production function instead multiplies the effects: the output ratio is \(180/100=1.8\), an 80% increase.
Marginal Returns Depend on the Production Setting
Complementarity has a practical interpretation: the return to an additional unit of one input can be higher when other complementary inputs are already available. This is a statement about marginal product, not necessarily about profit or the value of an investment. To determine profitability, a decision-maker would also need costs, output prices, and other relevant conditions.
Worked Example: How Skills and Technology Change the Marginal Product of Capital
A hypothetical firm uses \(Y=A\sqrt{KH}\). Compare the output gain from increasing capital from \(K=100\) to \(K=110\) in three settings: (1) \(A=2,H=25\), (2) \(A=2,H=100\), and (3) \(A=3,H=100\). Keep the stated \(A\) and \(H\) fixed within each comparison. Report output gains to two decimal places.
Method. Calculate output at both capital levels in each setting, then subtract. This finite change is the actual output difference for a 10-unit increase in capital; it need not equal the marginal product at one point times 10.
Setting 1. At \(K=100\), output is \(2\sqrt{100(25)}=2(50)=100\). At \(K=110\), output is \(2\sqrt{110(25)}=2\sqrt{2750}\), which rounds to 104.88. The gain, rounded to two decimals, is \(104.88-100=4.88\).
Setting 2. At \(K=100\), output is \(2\sqrt{100(100)}=2(100)=200\). At \(K=110\), output is \(2\sqrt{110(100)}=2\sqrt{11000}\), which rounds to 209.76. The gain, rounded to two decimals, is \(209.76-200=9.76\).
Setting 3. At \(K=100\), output is \(3\sqrt{100(100)}=3(100)=300\). At \(K=110\), output is \(3\sqrt{110(100)}=3\sqrt{11000}\), which rounds to 314.64. The gain, rounded to two decimals, is \(314.64-300=14.64\).
The same 10-unit capital increase produces gains of 4.88, 9.76, and 14.64 output units across the three settings. In the model, greater skill-adjusted labor raises the marginal product of capital, and a higher technology index scales it up further. The marginal product formula confirms the direction: \(MP_K=(A/2)\sqrt{H/K}\). For instance, at \(K=100\), it is \(0.50\) in setting 1, \(1.00\) in setting 2, and \(1.50\) in setting 3.
Check. The direct output calculations and the marginal-product formula agree on the ordering: setting 3 has the highest marginal product, followed by setting 2 and then setting 1. A common mistake is to compare the output levels across settings instead of comparing each setting’s output change. The relevant quantities here are the within-setting gains from \(K=100\) to \(K=110\).
How to Use Complementarity Carefully
For a production comparison, first identify what is held fixed and what changes. A change in output after capital, skills, and technology all change cannot be assigned to capital alone from the output comparison. One-input-at-a-time calculations or an explicit production model can show how the modeled effects differ, but they do not establish that the model captures every real-world cause of productivity.
The earlier material on physical capital quality, human capital, and labor composition helps distinguish the inputs being represented. Here, \(K\) refers to physical capital, while \(H\) summarizes effective labor input that reflects worker skills. \(A\) represents the model’s technology or efficiency index. Keeping these concepts distinct helps avoid counting the same improvement twice—for example, recording a training-related change both as higher \(H\) and again as a separate increase in \(A\) without a clear measurement basis.
The model also illustrates why separately calculated gains should not automatically be added. With complementary inputs, changing one input can alter the marginal product of another. In the examples, the production function makes the interaction measurable. In a different production function, inputs could be less complementary, unrelated at the margin, or substitutes. The sign and size of the interaction must come from the specified model or suitable evidence; they should not be assumed just because inputs are used together.
Check Your Understanding
For each problem, use the stated production function and units. These are hypothetical production settings.
- A workshop has \(Y=3\sqrt{KH}\), with \(K=64\) and \(H=25\). Find initial output and output if \(H\) alone doubles to 50. (Answer: Initial output \(=120\); after the increase, output \(=3\sqrt{3,200}\approx169.71\).)
- A plant has \(Y=2\sqrt{KH}\), initially \(K=81,H=16\). Calculate output if only \(K\) rises to 121, output if only \(H\) rises to 25, and output if both changes occur. How much does the combined gain exceed the sum of the separate gains? (Answer: Baseline \(=72\); capital-only \(=88\); skills-only \(=90\); both \(=110\); excess \(=4\) output units.)
- For \(Y=4\sqrt{KH}\), calculate \(MP_K\) at \(K=100,H=25\), then at \(K=100,H=100\). State how the change in \(H\) affects the marginal product of capital. (Answer: \(MP_K=1\) initially and \(MP_K=2\) after the increase in \(H\); it doubles.)
- A line has \(Y=A\sqrt{KH}\), initially \(A=1.5,K=100,H=36\). All three rise to \(A=1.8,K=121,H=49\). Calculate initial output, final output, and percentage output growth. (Answer: Initial output \(=90\); final output \(=138.60\); output growth \(=54\%\).)
- A firm uses \(Y=2\sqrt{KH}\). Compare the gain from raising \(K\) from 100 to 121 when \(H=25\) and when \(H=49\), holding \(A=2\). Round gains to two decimals. (Answer: With \(H=25\), gain \(=10\); with \(H=49\), gain \(=14\); the higher skill-adjusted labor level raises the gain by \(4\) output units.)