ECON 410 — Interactive Module A

Production Function Explorer · Mankiw Ch. 3 §3.1
Section 2 · Module A

Adjust the sliders to set total factor productivity A, capital share α, capital stock K, and labor L. Watch real GDP Y change live.

Y = A · Kα · L(1−α)
1.00
0.30
100
200
Output Y
203.16
Output per worker (Y/L)
1.02
Capital per worker (K/L)
0.50
Capital share α·Y
61.0
Labor share (1-α)·Y
142.2
Concept check
In Cobb-Douglas, capital income = α·Y and labor income = (1−α)·Y. With α = 0.30, capital earns 30% of output and labor earns 70%. This matches the U.S. labor share of ≈70% historically (now closer to 60%).

Constant returns to scale: F(zK, zL) = z·F(K,L). Pick z and compare F(zK, zL) to z·Y.

F( zK , zL ) = z · F(K, L)
2.0
Using A=1, α=0.30, K=100, L=200 from Tab 1
F(K, L)
203.16
F(zK, zL)
406.32
z · F(K, L)
406.32
CRS holds?
✓ YES
Why this matters
CRS lets us replicate: doubling K and L exactly doubles Y. Per-worker output Y/L depends only on K/L, the capital-to-labor ratio. This is the foundation of the Solow growth model (Section 4).

Marginal products: hold one factor fixed and vary the other. Watch each curve flatten — that's diminishing marginal product. Both panels are independent of Tab 1's sliders.

Marginal Product of Labor (MPL)
Hold K fixed, vary L
MPL = ∂Y/∂L = (1−α) · A · Kα · L(−α) = (1−α)·Y/L
1.00
0.30
100
200
Production function Y(L) with K held fixed. The slope at the red dot IS the MPL there. Increasing K shifts the entire curve UP.
MPL at L=100
0.700
MPL at L=200
0.569
MPL at L=300
0.503
MPL at current L · = Real wage W/P
0.569
Labor side — concept check
Each new worker is less productive than the last (less capital to work with). Competitive real wage W/P = MPL. Try doubling K: the MPL curve shifts up — every worker becomes more productive.
Marginal Product of Capital (MPK)
Hold L fixed, vary K
MPK = ∂Y/∂K = α · A · K(α−1) · L(1−α) = α·Y/K
1.00
0.30
200
100
Production function Y(K) with L held fixed. The slope at the red dot IS the MPK there. Increasing L shifts the entire curve UP.
MPK at K=50
0.487
MPK at K=100
0.396
MPK at K=200
0.322
MPK at current K · = Real rental R/P
0.396
Capital side — concept check
Each new unit of capital is less productive than the last (less labor to operate it). Competitive real rental R/P = MPK. Doubling L shifts the MPK curve UP — every machine becomes more productive when there are more workers to operate it.
Income distribution check (Euler's theorem)
MPK·K + MPL·L = ? · Y = ? · Difference = 0.00 (must always be 0 under CRS)

Four quick questions to lock in the production function intuition.

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