Budget constraint

${\displaystyle P_{x}x+P_{y}y=m.}$

Solving for ${\displaystyle y}$ yields

${\displaystyle y=\frac{m}{P_y}-\frac{P_x}{P_y}x,}$

where ${\displaystyle m/P_y}$ is the vertical intercept (the maximum amount of ${\displaystyle y}$ the consumer can buy if ${\displaystyle x=0}$) and ${\displaystyle -P_x/P_y}$ is the slope of the budget line, representing the opportunity cost of one more unit of ${\displaystyle x}$ in terms of ${\displaystyle y}$ forgone. Similar constraints appear in models of labour–leisure choice, intertemporal consumption, firm behaviour and international trade.

Uses in consumer theory

Individual choice

In microeconomic consumer theory, the budget constraint is combined with a description of preferences to study individual utility maximisation. For a given income and price vector, the budget set (all bundles on or below the budget line) represents all the consumption bundles an individual can afford.

A common assumption is that preferences are well behaved: they are complete, transitive and monotonic, so that "more is better", and that indifference curves are downward-sloping and convex. Under these conditions, the individual's most preferred affordable bundle typically lies at a point where an indifference curve is tangent to the budget line. This tangency point represents the quantities of the two goods that maximise utility subject to the budget constraint.

However, the optimal bundle need not always be an interior solution. If the tangency point implied by the first-order conditions lies outside the feasible set, the optimum will instead be a corner solution, at which the consumer consumes only one of the goods, as in the case of perfect substitutes. As income varies, the locus of interior tangency points is called the expansion path.

Introductory treatments typically assume a linear budget constraint as above. In more realistic settings, budget constraints may be kinked or non-linear because of taxes, subsidies, quantity discounts, rationing, welfare benefits or other institutional features; these cases can also be analysed with the same basic tools, but the geometry and optimality conditions may be more complex.

In labour economics, the same two-good framework is used to model the trade-off between leisure and consumption. A worker is assumed to have a fixed time endowment that can be allocated between hours of work and hours of leisure; labour income equals the wage rate times hours worked, plus any non-labour income. The wage can then be interpreted as the price of leisure: taking one more hour of leisure reduces labour income, and hence consumption possibilities, by one hour’s wage. The resulting labour–leisure budget line, together with preferences over consumption and leisure, underlies the standard derivation of an individual labour supply curve.

Many goods

While introductory presentations of budget constraints are often limited to two goods because they are easier to illustrate graphically, the idea extends straightforwardly to an arbitrary number of goods. Suppose there are ${\displaystyle n}$ goods, denoted ${\displaystyle x_i}$ for ${\displaystyle i=1,\dots ,n}$, with corresponding prices ${\displaystyle p_i}$. Let ${\displaystyle W}$ be the total amount that may be spent. The budget constraint is then

${\displaystyle \sum _{i=1}^n{p}_i{x}_i\le W.}$

If the consumer spends the entire amount ${\displaystyle W}$, the budget constraint binds:

${\displaystyle \sum _{i=1}^n{p}_i{x}_i=W.}$

In this case, obtaining an additional unit of good ${\displaystyle x_i}$ requires reducing consumption of other goods. For example, an additional unit of ${\displaystyle x_i}$ can be purchased by giving up ${\displaystyle p_i/p_j}$ units of good ${\displaystyle x_j}$, holding total expenditure fixed. Under standard monotonic preferences, the optimal consumption bundle will lie on the boundary of the budget set (i.e. with the constraint binding), just as in the two-good case.

Borrowing and lending

Budget constraints can also link consumption across time through borrowing and lending. In a simple two-period setting with interest rate ${\displaystyle r}$, the intertemporal budget constraint can be written as

${\displaystyle c_1+\frac{c_2}{1+r}\le y_1+\frac{y_2}{1+r},}$

where ${\displaystyle c_t}$ and ${\displaystyle y_t}$ denote consumption and income in period ${\displaystyle t}$. Borrowing in period 1 allows a consumer to increase current consumption at the cost of reduced consumption in period 2, while saving (lending) has the opposite effect.

According to behavioural economics, intertemporal choices involving borrowing and lending may also be affected by present bias, in which individuals place disproportionate weight on current consumption relative to future consumption. Models with present-biased preferences often distinguish between sophisticated individuals, who are aware of their present bias, and naïve individuals, who are not.

Non-linear budget constraints

In many applications, budget constraints are not linear. Progressive income taxes, social insurance contributions, welfare benefits, means-tested benefits, quantity discounts and other institutional features often generate piecewise linear or more complex budget sets, with kinks or notches where effective marginal tax rates change. In labour supply models, for example, the budget line relating hours of work and disposable income typically has different slopes in different ranges of hours because of tax brackets and benefit withdrawal rules. These non-linearities can lead to non-convex budget sets and make optimisation and empirical analysis more complicated, prompting a specialised literature on econometric methods for kinked and non-linear budget constraints.