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ResearchMicroeconomics / Consumer theory / Foundations

The budget constraint

Foundations02 / NOTE

Foundations note — the budget line, why its slope is a pure relative price, and the difference between a parallel shift and a pivot.

A foundations note — keeping the building blocks sharp.

Preferences say what a consumer wants; the budget constraint says what they can afford. With prices p_x, p_y and money income m, affordable bundles satisfy

p_x·x + p_y·y ≤ m        (the budget set)
p_x·x + p_y·y = m        (the budget line — its frontier)

Under monotonic preferences the optimum always sits on the line: any leftover income could always buy a little more of something.

Rearrange to see the geometry:

y = m/p_y − (p_x/p_y)·x
  • The slope is −p_x/p_y. Its magnitude, the relative price, is the rate at which the market lets you swap y for x — one more unit of x costs you p_x/p_y units of y. Crucially, this reflects prices only, not tastes. (The tastes side is the MRS; the two meet only at the optimum.)
  • The intercepts m/p_x and m/p_y are the most of each good you could buy by spending everything on it.

How the line moves is where the classic error lives:

  • Income change. Raising m scales both intercepts equally, so the line shifts outward and parallel — same slope, more of everything affordable. Lowering m shifts it inward.
  • A single price change. Raising p_x shrinks only the x-intercept, so the line pivots about the y-intercept, becoming steeper; lowering p_x pivots it outward. A price change is a rotation, not a parallel shift.

A subtler property: scale all of p_x, p_y, m by the same factor and the budget line does not move at all. Demand is homogeneous of degree zero in prices and income — only relative prices and real income (purchasing power) matter, never the nominal money amounts. This is why we can normalise a price or income to 1 without loss, and why pure inflation that lifts everything together changes nothing real. Mistaking nominal for real is the money-illusion error.

Common slips: reading the slope as a taste parameter (it is pure price); thinking a price change shifts the line in parallel (only income does — a price change pivots it); and forgetting the degree-zero homogeneity that makes proportional changes neutral.