Cost Minimization

The cheapest way to make a given output

Before a firm can decide how much to produce, it needs to know the cheapest way to produce any given output. That’s the cost-minimization problem: pick inputs \((x_1,x_2)\) to minimize spending \(w_1 x_1 + w_2 x_2\) subject to hitting a target output \(f(x_1,x_2)=y\).

The picture is the dual of consumer choice:

  • The isoquant for output \(y\) plays the role of the indifference curve — all input mixes that make \(y\).
  • An isocost line \(w_1 x_1 + w_2 x_2 = C\) plays the role of the budget line — all mixes costing the same \(C\), with slope \(-w_1/w_2\).

Minimizing cost means finding the lowest isocost line that still touches the isoquant — and that’s a tangency: \[ \frac{MP_1}{MP_2} = \frac{w_1}{w_2}. \] The marginal product per dollar is equalized across inputs: \(MP_1/w_1 = MP_2/w_2\). If one input gave more output per dollar, the firm would substitute toward it. The tangency bundle gives the conditional input demands — how much of each input to use for output \(y\) at prices \((w_1,w_2)\).

Try it. Change the wage of input 1 and the output target. The tangency slides along the isoquant — when input 1 gets pricier, the firm substitutes toward input 2. Raising output pushes the isoquant (and cost) out.

What to notice

  • Cost is minimized at the tangency — the lowest (blue) isocost that still reaches the isoquant. The grey dashed line reaches the same output but wastes money.
  • Raise w₁ and the tangency slides down-right along the isoquant: the firm substitutes toward input 2, now relatively cheaper. That substitution is exactly conditional input demand responding to prices.
  • Raise the output target and the isoquant shifts out — minimum cost rises. Tracing minimum cost against \(y\) is how we build the firm’s cost curve.