Budget constraint

thumb|250px|Budget constraint, where <math>A=\frac{m}{P_y}</math> and <math>B=\frac{m}{P_x}</math>

In economics, a budget constraint represents all the combinations of goods and services that a consumer (or other decision-maker) can purchase given current prices and a given level of income or wealth. In consumer theory, the budget constraint and a preference map (or system of indifference curves) are the basic tools used to analyse consumer choice. In the standard two-good case, the budget constraint can be represented graphically as a straight line showing the trade-off between the two goods. If <math>x</math> and <math>y</math> denote the quantities of two goods, with prices <math>P_x</math> and <math>P_y</math>, and <math>m</math> denotes income, the budget line is given by

:<math>P_x x + P_y y = m.</math>

Solving for <math>y</math> yields

:<math>y = \frac{m}{P_y} - \frac{P_x}{P_y}\,x,</math>

where <math>m/P_y</math> is the vertical intercept (the maximum amount of <math>y</math> the consumer can buy if <math>x=0</math>) and <math>-P_x/P_y</math> is the slope of the budget line, representing the opportunity cost of one more unit of <math>x</math> in terms of <math>y</math> forgone.

Uses in consumer theory

Individual choice

thumb|right|An individual maximises utility at the tangency point (Qx, Qy).

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.

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 <math>n</math> goods, denoted <math>x_i</math> for <math>i = 1, \dots, n</math>, with corresponding prices <math>p_i</math>. Let <math>W</math> be the total amount that may be spent. The budget constraint is then

:<math>\sum_{i=1}^n p_i x_i \leq W.</math>

If the consumer spends the entire amount <math>W</math>, the budget constraint binds:

:<math>\sum_{i=1}^n p_i x_i = W.</math>

In this case, obtaining an additional unit of good <math>x_i</math> requires reducing consumption of other goods. For example, an additional unit of <math>x_i</math> can be purchased by giving up <math>p_i/p_j</math> units of good <math>x_j</math>, 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.

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.

Other applications

International economics

thumb|right|Point X is unobtainable given the current "budget" constraints on production.

In international economics, an economy's production-possibility frontier (PPF) describes the combinations of two goods that can be produced using available factors of production. Under autarky, the PPF also limits aggregate consumption. With international trade, however, the country can specialise in production and trade at given terms of trade, so that its consumption-possibility frontier (CPF) lies outside the PPF. The CPF in this case plays a role analogous to a budget constraint for the economy as a whole, with a slope determined by the relative price <math>P_x/P_y</math> of exports to imports in standard Heckscher–Ohlin and New Trade Theory models.

Soft budget constraint

A soft budget constraint describes a situation in which an organisation expects that, if it runs a deficit or faces financial distress, an external actor will cover its shortfall ex post. The external actor is typically a government, higher-level public authority or parent organisation. Because decision-makers do not fully internalise the cost of failure, the budget constraint they face is considered soft compared with a hard budget constraint, under which persistent deficits would lead to bankruptcy, closure or some other credible termination of support.

The term was introduced in the late 1970s by Hungarian economist János Kornai to analyse the behaviour of state-owned enterprises in socialist economies, especially in centrally planned systems. In his account, socialist firms could normally expect to be rescued through subsidies, tax relief, preferential credit, administered prices or the forgiveness of arrears rather than being allowed to fail. This recurring pattern of bailouts created what Kornai called a "soft budget constraint syndrome": firms invested too much, economised too little on costs and lobbied for additional resources instead of adjusting output or employment, contributing to chronic shortages and macroeconomic imbalance.

Subsequent literature has generalised the concept beyond centrally planned economies to transition and market economies. A budget constraint may be soft whenever creditors or sponsors cannot credibly commit to withhold support ex post, for example because the organisation is politically important, systemically significant or provides essential services. The expectation of rescue generates moral hazard: managers may undertake projects that would be rejected under a hard budget constraint and may delay restructuring in the hope of further external assistance. Theoretical work interprets soft budget constraints as a dynamic commitment problem for the sponsor, closely related to other issues of time inconsistency in public finance and industrial policy.