Budget Chapter

Consumption Choice Sets

  • Consumers must choose a consumption bundle; can be thought of as their “problem” with the “solution” being the optimal bundle
    • Consumption Choice Set: A collection of all consumption choices available to the consumer
    • Drawing graphs can be helpful to visualize the optimal way to pick a consumption bundle given a set of constraints
      • Constraints include anything that disallows you from consumption; other constraints include money, storage, time, etc.
      • Basic analysis will use money/income as the main constraint

Budget Constraints

  • A consumption bundle containing x1 units of commodity 1, x2 units of commodity, and so on up to xn units of commodity n is denoted by (x1, x2, …, xn)
  • Commoditive prices are exogenous (must be taken as given) and are given by p1, p2, …, pn
  • Q: When is (x1, x2, …, xn) affordable at p1, p2, …, pn?

  • A: When p1x1 + p2x2 + … + pnxn <= m, where the left side represents expenditure and the right side, m, represents the consumer’s income

  • Affordable budgets that are just affordable form the consumer’s budget constraint (BC)
Mathematically, the budget constraint can be written as follows:{(x1,x2,...,xn)∣p1x1+p2x2+...+pnxn=m,x1≥0,x2≥0,...,xn≥0}\text{Mathematically, the budget constraint can be written as follows:} \\ \{(x_1, x_2, ..., x_n) | p_1 x_1 + p_2 x_2 + ... + p_n x_n = m, x_1 \geq 0, x_2 \geq 0, ..., x_n \geq 0\}
  • Budget Set (B): The set of all affordable bundles
Mathematically, the budget set can be written as follows:B(p1,...,pn,m)={(x1,x2,...,xn)∣p1x1+p2x2+...+pnxn≤m,x1≥0,x2≥0,...,xn≥0}\text{Mathematically, the budget set can be written as follows:} \\ B(p_1, ..., p_n, m) = \{(x_1, x_2, ..., x_n) | p_1 x_1 + p_2 x_2 + ... + p_n x_n \leq m, x_1 \geq 0, x_2 \geq 0, ..., x_n \geq 0\}

Visualizing Budget Constraints and Budget Sets with Two Goods

  • With two goods, a graph for budget constraints and sets can be represented on two axes
    • The actual graph (line) is the budget constraint, and all consumption choice sets lie below the graph

image

For two goods where x1 is on the horizontal axis, the slope of the BC can be written as −p1p2\text{For two goods where } x_1 \text{ is on the horizontal axis, the slope of the BC can be written as } -\frac{p_1}{p_2}