Constrained Optimization

Calculus 3 · Axiom Academy

One workshop, one budget. Squeeze the most output out of what you can afford — and meet the shadow price, the true value of one more dollar. Your workshop turns labor-hours and machine-hours into product, but you only have a 300 daily budget to spend on them — labor runs 5 /hr, machines 10 /hr. Three moves: find the best mix , see why it's best , and price one more dollar . Every dollar not on labor goes to machines — so a single slider slides you along the whole budget line. Output is (labor-hours × machine-hours). Slide it: output climbs, peaks, then falls. There's a sweet spot. Here's the picture behind the peak. Raise an output curve Q = k toward the budget line. Too low and it cuts through with budget to spare; too high and it never reaches. At exactly k = 450 it just touches — one point of tangency. That touch is the optimum, where lines up with . Price one more dollar — the shadow price The multiplier isn't bookkeeping — it's a price . Here : at the optimum, each extra 1 of budget buys about 3 more units of output. Drag extra budget and watch the best-possible output climb along the -line — the marginal value of loosening the constraint.

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