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 budget constraint and preferences to analyze consumer choices.

Two goods

---

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

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

$xp\_X+yp\_Y.\backslash ,$

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

$xp\_X+yp\_Y\backslash leq\; W.\backslash ,$

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

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

When the consumer spends all his income we have

$xp\_X+yp\_Y=W.\backslash ,$

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

$\backslash frac\{p\_X\}\{p\_Y\}\backslash times\backslash frac\{p\_Y\}\{p\_X\}=1\backslash ,$

unit of $X.\backslash ,$

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

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

Many goods

---

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

$\backslash sum\_\{i=1\}^np\_ix\_i\backslash leq\; W.$

Like before, if the consumer spends his income entirely, the budget constraint binds:

$\backslash sum\_\{i=1\}^np\_ix\_i=W.$

In such case, to obtain an additional unit of good $i\backslash ,$, the consumer needs to give up a quantity $p\_i/p\_j\backslash ,$ of say good $j.\backslash ,$

Economics