ECON 410 · Section 2 · Module D — Deeper Dive Interactive Lab

Production · Factor Markets · Loanable Funds · Practice — Mankiw Ch.3

Returns to scale — interactive checker

Enter exponents for Y = A·K^α·L^β; the tool checks F(zK, zL) vs z·F(K, L) and classifies returns to scale.
0.30
0.70
1.00
Y = 1.00 · K^0.30 · L^0.70

Check F(zK, zL) vs z·F(K, L)

2.0

With K = 100, L = 100:

F(K, L)100.00
z · F(K, L)200.00
F(zK, zL)200.00
Ratio F(zK,zL) / [z·F(K,L)]1.000

Returns to scale

CRS
α + β = 1.00 → Constant returns to scale. F(zK, zL) = z · F(K, L) exactly.

Quick rule for Cobb-Douglas

  • α + β = 1 → Constant returns to scale
  • α + β > 1 → Increasing returns
  • α + β < 1 → Decreasing returns
Try these:
  • Set α = 0.5, β = 0.5 → CRS (this is √(KL))
  • Set α = 0.25, β = 0.75 → CRS
  • Set α = 1, β = 1 → Increasing returns (α + β = 2)
  • Set α = 0.3, β = 0.5 → Decreasing returns (α + β = 0.8)

Optimal hiring — labor demand simulator

Pick a wage W/P. The tool computes MPL at each L and tells you whether to hire more or stop.

Settings

6.0
Discrete schedule from Mankiw textbook

Result

Optimal labor L*
5
At L* = 5, MPL = 6 = W/P. Hiring stops here — any more workers cost more than they produce.

Labor demand table

Labor LOutput YMPLDecision when W/P = 6

Rule: hire one more worker iff MPL ≥ W/P. The optimum L* is where MPL = W/P.

Cobb-Douglas factor share explorer

Watch how MPL, MPK, factor incomes, and shares move as you change K, L, α, and A. Verify Euler's theorem live.
100
100
0.30
1.00
Y = 1.00 · K^0.30 · L^0.70

Output and marginal products

Output Y100.00
MPL = (1−α)·Y/L0.700
MPK = α·Y/K0.300

Income distribution

Labor income MPL·L70.00
Capital income MPK·K30.00
MPL·L + MPK·K (should = Y)100.00 ✓

Factor shares

Labor share = MPL·L / Y = 0.700 = (1−α). Capital share = α = 0.300.

Move K, L, A around — labor share stays at (1−α). Only α changes the share. This is the famous CONSTANT factor share property.

Loanable funds — crowding-out simulator

Move G, T, MPC, or investment-demand parameters. See how S and I curves shift and what happens to r*.
2500
2000
0.67

Fixed: Y = 8,000 (set by factor markets). C = 1,000 + MPC·(Y−T). I = 1,200 − 100r.

Saving decomposition

Disposable income Y − T6000
Consumption C5020
Private saving Spriv980
Public saving Spub = T − G−500
National saving S480

Equilibrium

Real interest rate r*
7.20%
Equilibrium I*480
Crowding out from baseline0

Loanable funds market

S (vertical) and I (downward-sloping). Equilibrium at r* where they intersect.

Try these scenarios:
  • Reset to baseline (G=2500, T=2000, MPC=0.67) — note r* and I*.
  • Raise G to 3000. r* should rise sharply. ΔI ≈ −ΔG (full crowding out).
  • Reset, then drop T to 1500. r* rises, but less. (Partial crowding out — depends on MPC.)
  • Try MPC = 0.95 with the same tax cut. Crowding out gets worse. Why?

Practice quiz — Section 2 v3

10 questions in the style of practice-exam material. Get instant feedback after each answer.
Score: 0/10