Production functions visualised

Selected functions and their tangents
Selected functions and their tangentsEach selected function is drawn on shared axes with its tangent at the chosen input.0246802468Capital per worker kOutput per worker f(k)
0.57.5
0.502.00

AK

f(k)=Akf(k)=Ak
f′(k)=Af'(k)=A
f′′(k)=0f''(k)=0
Domain
k ≥ 0
Minimum
f(0) = 0, attained
Maximum
None; output rises without bound

What to notice. A straight line with constant marginal product A. Raising A rotates the line upward.

Why it matters in AK. With saving rate s, depreciation δ and population growth n, per-worker capital grows at sA − (δ + n).

Cobb-Douglas

f(k)=Ak1/3f(k)=Ak^{1/3}
f′(k)=A3k−2/3f'(k)=\tfrac{A}{3}k^{-2/3}
f′′(k)=−2A9k−5/3f''(k)=-\tfrac{2A}{9}k^{-5/3}
Domain
k ≥ 0
Minimum
f(0) = 0, attained
Maximum
None; output rises without bound

What to notice. Increasing and strictly concave. The marginal product falls toward zero.

Where it appears. Diminishing returns give the Solow and Ramsey models a steady state.

CES

f(k)=A(13k+23)2f(k)=A\left(\tfrac13\sqrt{k}+\tfrac23\right)^2
f′(k)=A9(1+2/k)f'(k)=\tfrac{A}{9}\left(1+2/\sqrt{k}\right)
f′′(k)=−A9k−3/2f''(k)=-\tfrac{A}{9}k^{-3/2}
Domain
k ≥ 0
Minimum
f(0) = 4A/9 = 0.44, attained
Maximum
None; output rises without bound

What to notice. Increasing and strictly concave. The marginal product falls toward A/9 = 0.11, not zero.

Where it appears. With easy substitution, capital alone sustains growth when sA/9 > δ + n.