Profit Maximization

How much to produce

A competitive firm takes the output price \(p\) and input price (wage) \(w\) as given, and picks input \(x\) to maximize profit: \[ \pi(x) = p\,f(x) - w\,x. \] The first-order condition sets the value of the marginal product equal to the input’s price: \[ p\,f'(x) = w \qquad\Longleftrightarrow\qquad f'(x) = \frac{w}{p}. \] In words: hire until the extra output from one more unit of input, valued at the market price, just covers what that unit costs. Hiring past that point loses money on the margin; stopping short leaves profit on the table.

Geometrically, rewrite profit as an isoprofit line in (input, output) space: \[ y = \frac{\pi}{p} + \frac{w}{p}\,x. \] These are parallel lines of slope \(w/p\); higher lines mean more profit. The firm climbs to the highest isoprofit line still touching the production function — and at that point the line is tangent to \(f\), which is exactly \(f'(x)=w/p\).

The sim uses \(f(x)=A\sqrt{x}\) (diminishing marginal product), so the optimum is \(x^\* = \left(\dfrac{pA}{2w}\right)^{2}\).

Try it. Change the output price p and the wage w. The tangent isoprofit line slides to the best feasible level; the dot marks the profit-maximizing input and output. A higher price or lower wage → more hiring.

What to notice

  • The optimum is exactly where the isoprofit line is tangent to the production function — the slope of \(f\) (marginal product) equals \(w/p\).
  • Raise the price p (or cut the wage w) and the tangency slides right: the firm hires more. That rightward movement is the firm’s input demand / output supply response.
  • Because \(f\) has diminishing marginal product (it bends over), there’s a unique best input level — climbing further would require an isoprofit line too steep to stay tangent.