Cost-Minimising Input Mix Calculator

Problem → model → reason → result

What problem does this model solve?

Find the cost-minimising labour and capital mix for a Cobb-Douglas production target.

Why does the model apply?

The tangency condition equates output gained per currency unit across inputs. The result assumes positive exponents, divisible inputs and an interior solution.

What assumptions does it make?

The variables must describe the same market, firm, period or decision context and use consistent units. Any behavioural condition implied by the formula—such as other factors remaining unchanged—must be reasonable for the question being asked.

Formula

At the interior optimum K/L = (β/α)(w/r), combined with Q = A L^α K^β

Calculation and working

The calculator substitutes your inputs locally and displays the numerical result. Change one input at a time to test which relationship drives the result.

What does the result mean?

Interpret the result in the economic context named above, including its sign, magnitude, units and time period. A calculated relationship is not by itself evidence that one variable caused another.

Worked example

For Q = L^0.5K^0.5, target 100, wage $50 and capital rental $100, the cost-minimising mix is about 141.42 labour and 70.71 capital.

When does this model not apply?

Do not use the result when the input definitions, units or formula assumptions do not match the real situation. This is an educational model, not financial, investment, tax or policy advice; verify material decisions against primary data and professional guidance.