Budget constraint

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A Budget Constraint represents the combinations of goods and services that a consumer can purchase given current prices and his income. Consumer theory uses the concepts of a budget constraint and a preference ordering to analyze consumer choices. Both concepts have a ready graphical representation in the two-good case.

[edit] Two goods

Consider a world of two goods, called $X\,$ and $Y\,$ , which can be purchased in quantities denominated by $x\,$ and $y\,$ , respectively. Let the price of $X\,$ be $p_X\,$ and the price of $Y\,$ be $p_Y\,$ . Finally, let the income of the consumer be denoted by $W\,$ .

When the consumer purchases quantities $x\,$ and $y\,$ , his total spending is

$xp_X+yp_Y.\,$

The budget constraint states that total spending cannot exceed his revenue:

$xp_X+yp_Y\leq W.\,$

The graphical representation of the budget constraint is the budget line which represents the maximum quantity of $Y\,$ the consumer can purchase for any given quantity of $x\,.$

The maximum quantity of $y\,$ that can be purchased (i.e., if $x=0\,$ ) is $W/p_Y\,$ . The maximum quantity of $x\,$ that can be purchased (i.e., if $y=0\,$ ) is $W/p_X\,$ .

When the consumer spends all his income we have

$xp_X+yp_Y=W.\,$

In this case, in order to obtain an additional unit of $X,\,$ the consumer needs to give up a certain amount of $Y.\,$ This amount is exactly $p_X/p_Y.\,$ Why? Because by giving up one unit of $Y\,$ the consumer saves $p_Y\,$ units of his income which buy $p_Y/p_X\,$ units of $X.\,$ Thus the consumer needs to do this operation exactly $p_X/p_Y\,$ times, obtaining in the end

$\frac{p_X}{p_Y}\times\frac{p_Y}{p_X}=1\,$

unit of $X.\,$

The number $p_X/p_Y\,$ is the number of units of $Y\,$ that he needs to give up and the number $p_Y/p_X\,$ is the number of units of $X\,$ that can be purchased for each $Y.\,$

This can be seen through an example. Suppose $p_X=10\,$ and $p_Y=5\,$ (think of dollars for instance.) If the consumer gives up one unit of $Y\,$ he saves 5 which purchase only 1/2 of $X\,$ (Notice that 1/2 is exactly $p_Y/p_X\,$ .) In order to obtain exactly one unit of $X\,$ the consumer needs to give up 2 units of $Y\,$ which saves exactly 10 (i.e., the price of $X\,$ .) Observe that 2 is exactly $p_X/p_Y.\,$

[edit] Many goods

Suppose there are $n\,$ goods called $X_i\,$ for $i=1,\dots,n.\,$ Let the price of goods $i\,$ be denoted by $p_i.\,$ The budget constraint writes as before: