Cost-Minimizing Input Mix

Controls

Q = L^α × K^(1−α)

⚙ Production Parameters

Target Output (Q)

Labor Share (α)

💰 Input Prices

Wage Rate (w)

Rental Rate (r)

🎯 Find the Minimum Cost

Drag the isocost line until it just touches the isoquant. That tangency is the cost-minimizing input mix.

Total Cost Budget (C)

🎯 Keep Trying!

Slide the budget to find where the isocost line is tangent to the isoquant.

Optimal Labor

Optimal Capital

Min Cost

Cost per Unit

Optimality condition: Cost is minimized where the isocost line is tangent to the isoquant — the slopes are equal: w/r = MP_L/MP_K

Current Solution Values
Parameter	Value	Meaning

📘 Economic Interpretation

For a Cobb-Douglas production function Q = L^α K^1−α, the optimal input ratio is L/K = (α/1−α) × (r/w). When wages rise relative to rental rates, firms substitute capital for labor along the isoquant.