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

Because the number of workers does not matter to the relationship between K/LK/L and Y/LY/L, we denote all quantities in per-worker terms. We refer to these in lowercase letters, so k=K/Lk = K/L and l=Y/Ll = 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)y = F(K/L, L/L) = F(k, 1) = f(k)

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

y=Y/L=KαL(1α)Ly = Y/L = \frac{K^\alpha L^{(1- \alpha)}}{L}
=KαL(α)=(KL)α=kα = K^\alpha L^{(- \alpha)} = (\frac{K}{L})^{\alpha} = k^{\alpha}

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

α\alpha

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