budgetset

The budget set in consumer theory is the collection of all bundles of goods that a consumer can afford given current prices and income. Formally, with a price vector p = (p1, ..., pn) where pi ≥ 0 and income m ≥ 0, the budget set is B(p, m) = { x ∈ R^n_+ : p · x ≤ m }, where p · x = ∑ pi xi and R^n_+ denotes nonnegative quantities.

Key properties include that the budget set is convex and downward closed: if x is affordable, any bundle with smaller quantities of goods is also affordable. The boundary p · x = m is the budget line (or budget frontier) in the nonnegative orthant, with interior points p · x < m and feasible bundles on or inside the line. The set is nonempty when m ≥ 0. If all prices pi are strictly positive, B(p, m) is compact (closed and bounded). If some pi = 0, the set becomes unbounded in the corresponding good’s direction.

Changes in income and prices affect the budget set: increasing m expands B, while changes in p rotate or reshape the boundary. The budget set itself is independent of preferences, but the consumer’s chosen bundle depends on the utility function and optimization over B.

Connections to demand and expenditure: the Marshallian ( uncompensated) demand is the utility-maximizing bundle x* ∈ B(p, m). The Hicksian (compensated) approach uses a budget that exactly achieves a target utility, relating to the expenditure function e(p, u) and the compensated budget set { x ∈ R^n_+ : p · x ≤ e(p, u) }.