The firm's version of consumer theory: input mixes, returns to scale, and cost-minimizing bundles.
Warming up the engine…
The firm-side twin of consumer choice: isoquants map input combinations yielding equal output, isocost lines map equal spending, and cost minimization is their tangency.
At the optimum: MRTS = MPL/MPK = w/r
If a dollar of labor currently adds more output than a dollar of machinery, shift spending toward labor, keep going until the last dollar works equally hard in both directions. Returns to scale describe what happens when you scale ALL inputs together.
MRTS diminishes along an isoquant for the same reason MRS does, that's what makes the tangency an optimum.
Now prove you have it
Move the curve to where you think it lands, and get told exactly which part you got right.
Real-world scenarios on this model
Equilibrium at L 50, K 40.
Equilibrium: L* = 50.0, K* = 40.0
Current equations
The firm's version of consumer choice. The straight line shows every input mix a budget can buy at current wages and machine costs; the curve shows mixes that produce the same output. The cheapest way to hit an output target is where the isoquant just touches the isocost line.
Source: Varian, Intermediate Microeconomics, ch. 19-21