From Aggregate Production to Output Per Worker

Why Express Production Per Worker?

An aggregate production function describes total output using economy-wide inputs. In the Solow model, it is useful to restate that relationship per worker: this makes it possible to compare production economies of different sizes without confusing scale with productivity. The key step is not a new production assumption about individual workers. It is an algebraic consequence of constant returns to scale in the aggregate production function.

Let \(K\) denote the aggregate capital stock and \(L\) the number of workers. Assume \(L>0\), and write total output as \(Y=F(K,L)\). The production function is held fixed while the inputs are transformed; technology and any other parameters are not changing in this derivation.

Definition: Capital per worker is \(k=K/L\), and output per worker is \(y=Y/L\). A production function \(F(K,L)\) has constant returns to scale (CRS) in capital and labor if, for every scale factor \(\lambda>0\), \(F(\lambda K,\lambda L)=\lambda F(K,L)\). This is also called homogeneity of degree one.

The word “constant” here refers to the response of total output when every input is scaled by the same factor. If both capital and labor double, CRS says total output doubles. It does not say that output stays fixed, that output per worker is constant as capital per worker changes, or that each input contributes the same amount.

Deriving the Intensive Production Function

The derivation uses homogeneity to scale the labor input to one. Because \(L>0\), choose the scale factor \(\lambda=1/L\). Applying CRS to the original input pair \((K,L)\) gives

F\left(\frac{K}{L},1\right)=\frac{1}{L}F(K,L).

Since \(Y=F(K,L)\), the right-hand side is \(Y/L=y\), while \(K/L=k\). Define \(f(k)=F(k,1)\), the output produced when the labor input is one and capital is \(k\). Therefore,

y=\frac{Y}{L}=F\left(\frac{K}{L},1\right)=f(k).

This is the intensive production function: under CRS, output per worker depends on capital per worker alone, provided there are no additional inputs or fixed conditions that break the assumed homogeneity. The derivation does not require a particular algebraic form for \(F\). It requires the CRS property and the stated inputs.

Start with aggregate production.
Write \(Y=F(K,L)\), with \(L>0\).

Normalize labor to one.
Scale both inputs by \(1/L\), so the inputs become \(K/L=k\) and \(L/L=1\).

Apply homogeneity.
CRS gives \(F(K/L,1)=F(K,L)/L=Y/L\).

Name the per-worker function.
Set \(f(k)=F(k,1)\), yielding \(y=f(k)\).

Worked Forms of the Intensive Function

The intensive function can be derived for any CRS production form by evaluating the aggregate function at labor equal to one. Common production functions make the transformation especially transparent.

Worked Example: Cobb-Douglas Production per Worker

A hypothetical economy has aggregate production \(F(K,L)=12K^{1/3}L^{2/3}\). It has \(K=64\) units of capital and \(L=8\) workers. Derive its intensive production function, calculate output per worker and total output, and check the result directly.

Method. The exponents sum to \(1/3+2/3=1\), so this Cobb-Douglas function is homogeneous of degree one in \(K\) and \(L\). Set labor equal to one to find \(f(k)\):

f(k)=F(k,1)=12k^{1/3}1^{2/3}=12k^{1/3}.

Next calculate capital per worker:

k=\frac{K}{L}=\frac{64}{8}=8.

Since \(8^{1/3}=2\), output per worker is

$$ y=f(8)=12(2)=24. $$

The implied aggregate output is \(Y=yL=24(8)=192\). Check by substituting directly into the aggregate function: \(64^{1/3}=4\) and \(8^{2/3}=4\), so \(F(64,8)=12(4)(4)=192\). Dividing by \(L=8\) gives \(192/8=24\), agreeing with the intensive calculation.

Common mistake avoided. The exponents must be checked before treating the function as CRS. Here they sum to one, so dividing aggregate production by \(L\) removes the overall scale and leaves a function of \(K/L\).

More generally, if \(F(K,L)=A K^\alpha L^{1-\alpha}\), with \(A>0\) and \(0<\alpha<1\), its exponents sum to one. Substitution gives \(f(k)=A k^\alpha\). This form is convenient, but the intensive representation does not depend on Cobb-Douglas production.

Worked Example: A Leontief Production Function

A hypothetical workshop produces according to \(F(K,L)=\min(4K,6L)\), where output is the smaller of the two input-capacity terms. It has \(K=15\) units of capital and \(L=8\) workers. Find the intensive production function and compute output per worker and total output.

Method. First check CRS. For \(\lambda>0\),

F(\lambda K,\lambda L) =\min(4\lambda K,6\lambda L) =\lambda\min(4K,6L) =\lambda F(K,L).

Thus the function is homogeneous of degree one. At labor equal to one, its intensive form is

f(k)=F(k,1)=\min(4k,6).

For the given inputs, \(k=K/L=15/8=1.875\). The two per-worker capacity terms are \(4k=4(1.875)=7.5\) and \(6(1)=6\). Hence \(y=\min(7.5,6)=6\), and total output is \(Y=yL=6(8)=48\).

Check using aggregate inputs: \(4K=4(15)=60\), while \(6L=6(8)=48\). Therefore \(F(15,8)=\min(60,48)=48\), and \(Y/L=48/8=6\), as before.

Common mistake avoided. Do not automatically use the capital term as output. The minimum defines production here, so the binding capacity is \(6L\), not \(4K\).

Why Homogeneity Matters

Homogeneity is what allows the ratio \(K/L\) to summarize the relevant input combination. Scaling the economy changes \(K\) and \(L\) in the same proportion but leaves \(k=K/L\) unchanged. Under CRS, it also scales \(Y\) proportionally, leaving \(y=Y/L\) unchanged. Thus two economies with the same capital per worker have the same output per worker if they share the same CRS production function and other relevant conditions.

A useful way to test a proposed function \(F\) is to compute \(F(\lambda K,\lambda L)\) and compare it with \(\lambda F(K,L)\). For a differentiable two-input function, a standard equivalent condition for degree-one homogeneity is that the powers in a Cobb-Douglas form sum to one. Merely seeing capital and labor in the formula does not establish CRS.

Worked Example: Detecting When the Per-Worker Reduction Fails

Suppose a hypothetical production function is \(F(K,L)=K^{1/2}L^{1/4}\). Determine whether it is CRS. Then compare output per worker in two cases with the same capital per worker: \(K=4,L=1\), and \(K=64,L=16\).

Method. The exponents sum to \(1/2+1/4=3/4\), not one. More directly,

F(\lambda K,\lambda L) =(\lambda K)^{1/2}(\lambda L)^{1/4} =\lambda^{3/4}F(K,L),

which is not \(\lambda F(K,L)\) for general \(\lambda\). This function is not CRS in \(K\) and \(L\), so the intensive reduction to \(y=f(k)\) alone is not justified.

In the first case, \(k=4/1=4\), \(Y=4^{1/2}1^{1/4}=2\), and \(y=Y/L=2\). In the second case, \(k=64/16=4\) again, but \(Y=64^{1/2}16^{1/4}=8(2)=16\), so \(y=16/16=1\). The capital-per-worker ratio is the same, but output per worker differs.

The algebra shows why: dividing the aggregate function by labor gives

\frac{Y}{L} =\frac{K^{1/2}L^{1/4}}{L} =\left(\frac{K}{L}\right)^{1/2}L^{-1/4} =k^{1/2}L^{-1/4}.

The remaining factor \(L^{-1/4}\) means that \(k\) alone does not determine \(y\). The calculation avoids the common mistake of assuming every production function can be rewritten in per-worker form using only \(K/L\).

Assumptions and Common Traps

The conclusion \(y=f(k)\) is precise but conditional. It follows for a two-input production function that is homogeneous of degree one in capital and labor, with the technology and other conditions fixed. If production also depends on a fixed quantity of land, an unscaled resource, or another input that is not included in the normalization, output per worker may depend on more than \(k\). If labor is measured in hours rather than workers, the definitions and units must be applied consistently: capital per hour and output per hour are not the same quantities as capital and output per worker.

A frequent algebra error is to write \(F(K,L)/L=F(K/L,L)\). That expression does not follow from CRS: it scales capital and labor inconsistently. For example, the correct CRS step is

\frac{F(K,L)}{L}=F\left(\frac{K}{L},1\right),

because both inputs are divided by \(L\). The labor input on the right must therefore be \(L/L=1\), not the original \(L\). A second trap is to conclude that CRS means \(y\) is constant. CRS says \(y=f(k)\); unless \(f\) is itself constant, changing capital per worker can change output per worker.

Key takeaway: With \(Y=F(K,L)\), \(K/L=k\), \(Y/L=y\), and CRS in \(K\) and \(L\), homogeneity gives \(y=F(k,1)=f(k)\). Verify the degree of homogeneity before using this intensive form.

This derivation is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It establishes the per-worker production relationship; it does not by itself describe how capital per worker changes over time.

Check Your Understanding

For each problem, check both the input ratio and whether the production function has the required homogeneity.

A hypothetical economy has \(F(K,L)=7K^{1/4}L^{3/4}\), with \(K=256\) and \(L=16\). Derive \(f(k)\), then find \(k\), \(y\), and \(Y\). (Answer: \(f(k)=7k^{1/4}\), \(k=16\), \(y=14\), \(Y=224\).)
A production process has \(F(K,L)=\min(3K,5L)\), with \(K=12\) and \(L=6\). Derive the intensive form and identify the binding capacity. (Answer: \(f(k)=\min(3k,5)\), \(k=2\), \(y=5\), \(Y=30\); the labor capacity binds.)
Let \(F(K,L)=4K^{1/2}L^{1/2}\). Find \(y\) and \(Y\) for \(K=25,L=4\). If both inputs are then tripled, find the new \(k,y,Y\). (Answer: initially \(k=6.25\), \(y=10\), \(Y=40\); after tripling, \(k=6.25\), \(y=10\), \(Y=120\).)
Consider \(F(K,L)=K^{3/5}L^{3/5}\). Check its degree of homogeneity. Compare output per worker at \((K,L)=(1,1)\) and \((K,L)=(4,4)\), which have the same \(k\). (Answer: degree \(6/5\), so it is not CRS; \(k=1\) in both cases, while \(y=1\) and \(y=4^{1/5}\), respectively.)
A proposed production function is \(F(K,L)=2K+6L\), with \(K=10\) and \(L=5\). Derive its intensive form and verify aggregate output by both methods. (Answer: \(f(k)=2k+6\), \(k=2\), \(y=10\), and \(Y=50\) by either method.)

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