Cost Minimization

This scene finds the cheapest input mix for a target output.

The cheapest mix is where the isoquant touches an isocost line. When a factor price changes, that spot slides. You use less of the pricier input and more of the cheaper one. The lowest cost then becomes the cost function below.

For example, baking 100 🍞 bread loaves, 💵 wages rise but ⚙️ machine prices stay put. The cheapest mix then uses fewer 🧑 workers and more machines.

min⁡zw⋅zs.t.f(z)≥q\min_z w\cdot z \quad \text{s.t.} \quad f(z) \ge q
c(w,q)=w1z1(w,q)+w2z2(w,q)c(w,q) = w_1 z_1(w,q) + w_2 z_2(w,q)

Part of these learning paths

Examples & Exercises

With w1=w2=1w_1 = w_2 = 1, q=8q = 8, α=β=1/2\alpha = \beta = 1/2, find the conditional factor demands z(w,q)z(w, q) and the cost function value c(w,q)c(w, q) from the first-order condition.

w1w2=MRTS=z2z1  ⇒  z1=z2\frac{w_1}{w_2} = MRTS = \frac{z_2}{z_1} \;\Rightarrow\; z_1 = z_2
8=z11/2z21/2=z1  ⇒  z(w,q)=(8,8),c(w,q)=168 = z_1^{1/2} z_2^{1/2} = z_1 \;\Rightarrow\; z(w,q) = (8, 8), \quad c(w, q) = 16
w=(1,1)w = (1, 1), q=8q = 8 — the symmetric solution.
Show proof

  1. At an interior solution the first-order condition is price ratio = MRTS: 1=z2/z11 = z_2/z_1, so z1=z2z_1 = z_2.

  2. Substituting into the production function gives 8=z11/2z11/2=z18 = z_1^{1/2} z_1^{1/2} = z_1, so z=(8,8)z = (8, 8) and c=1⋅8+1⋅8=16c = 1 \cdot 8 + 1 \cdot 8 = 16. This is the point where the isocost z1+z2=16z_1 + z_2 = 16 touches the isoquant in the figure.

When w1w_1 rises from 11 to 22 (holding w2=1w_2 = 1), how do conditional factor demands change?

Compute numerically and verify the tangency condition w1/w2=MRTSw_1/w_2 = MRTS.

z1=[8(12)1/2]=42≈5.657z_1 = \left[8\left(\tfrac{1}{2}\right)^{1/2}\right] = 4\sqrt{2} \approx 5.657
z2=2z1=82≈11.314,c=162≈22.627z_2 = 2 z_1 = 8\sqrt{2} \approx 11.314, \quad c = 16\sqrt{2} \approx 22.627
w1w_1 up to 22 — substituting away from the dearer input.
Show proof

  1. Plugging into the formula: z1=[8⋅(1/2)1/2]=42≈5.657z_1 = [8 \cdot (1/2)^{1/2}] = 4\sqrt{2} \approx 5.657 and z2=2z1≈11.314z_2 = 2 z_1 \approx 11.314. From the symmetric (8,8)(8, 8), the firm cuts the dearer z1z_1 and raises the cheaper z2z_2 — substitution in conditional factor demand.

  2. Check: f(5.657,11.314)=5.657×11.314=8f(5.657, 11.314) = \sqrt{5.657 \times 11.314} = 8 ✓ and MRTS=z2/z1=2=w1/w2MRTS = z_2/z_1 = 2 = w_1/w_2 ✓. Cost rises to 2⋅5.657+11.314=162≈22.632 \cdot 5.657 + 11.314 = 16\sqrt{2} \approx 22.63.

6. Production