Consumer behaviour is a maximization problem. It means making the most of our limited resources to maximize our utility. As consumers are insatiable, and utility functions grow with quantity, the only thing that limits our consumption is our own budget. In general, the budget set represents all possible bundles of goods an individual can afford given their income and the prices of goods. When behaving rationally, an individual consumer should choose to consume goods at the point where the most preferred available indifference curve on their preference map is tangent to their budget constraint. The tangent point represents the amount of goods x and y the consumer should purchase to fully utilize their budget to obtain maximum utility. It is important to note that the optimal consumption bundle will not always be an interior solution. If the solution to the optimality condition leads to a bundle that is not feasible, the consumer's optimal bundle will be a corner solution which suggests the goods or inputs are perfect substitutes. A line connecting all points of tangency between the indifference curve and the budget constraint is called the expansion path. All two dimensional budget constraints are generalized into the equation: Where:
money income allocated to consumption
the price of a specific good
the price of all other goods
amount purchased of a specific good
amount purchased of all other goods
The equation can be rearranged to represent the shape of the curve on a graph: , where is the y-intercept and is the slope, representing a downward sloping budget line. The factors that can shift the budget line are a change in income, a change in the price of a specific good, or a change in the price of all other goods.
A production-possibility frontier is a constraint in some ways analogous to a budget constraint, showing limitations on a country's production of multiple goods based on the limitation of available factors of production. Under autarky this is also the limitation of consumption by individuals in the country. However, the benefits of international trade are generally demonstrated through allowance of a shift in the consumption-possibility frontiers of each trade partner which allows access to a more appealing indifference curve. In the "toolbox" Hecksher-Ohlin and Krugman models of international trade, the budget constraint of the economy is determined by the terms-of-trade as a downward-sloped line with slope equal to those TOTs of the economy..
Many goods
While low-level demonstrations of budget constraints are often limited to less than two good situations which provide easy graphical representation, it is possible to demonstrate the relationship between multiple goods through a budget constraint. In such a case, assuming there are goods, called for, that the price of good is denoted by, and if is the total amount that may be spent, then the budget constraint is: Further, if the consumer spends his income entirely, the budget constraint binds: In this case, the consumer cannot obtain an additional unit of good without giving up some other good. For example, he could purchase an additional unit of good by giving up units of good
