We have shown that the per-worker production function is a single-variable function for $Y/L$, which we can obtain from the intersection of the original production function and the $L=1$ plane, plotted in red.

Because the number of workers does not matter to the relationship between $K/L$ and $Y/L$, we denote all quantities in per-worker terms. We refer to these in lowercase letters, so $k = K/L$ and $l = Y/L$ designate output and capital per worker respectively.

We can now re-write the per-worker production function as:

$y = F(K/L, L/L) = F(k, 1) = f(k)$

We recall that for a Cobb-Douglas production function, $Y = K^\alpha L^{(1- \alpha)}$ and hence:

$y = Y/L = \frac{K^\alpha L^{(1- \alpha)}}{L}$
$= K^\alpha L^{(- \alpha)} = (\frac{K}{L})^{\alpha} = k^{\alpha}$

Hence, we have derived the per-worker production function as $y = f(k) = k^{\alpha}$. Let's see how this function changes shape as we vary $\alpha$.

$\alpha$

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