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 budget constraint and preferences to analyze consumer choices.

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Two goods

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

When the consumer purchases quantities <math>x\,</math> and <math>y\,</math>, his total spending is

<math>xp_X+yp_Y.\,</math>

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

<math>xp_X+yp_Y\leq W.\,</math>

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

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

When the consumer spends all his income we have

<math>xp_X+yp_Y=W.\,</math>

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

<math>\frac{p_X}{p_Y}\times\frac{p_Y}{p_X}=1\,</math>

unit of <math>X.\,</math>

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

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

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Many goods

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

<math>\sum_{i=1}^np_ix_i\leq W.</math>

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

<math>\sum_{i=1}^np_ix_i=W.</math>

In such case, to obtain an additional unit of good <math>i\,</math>, the consumer needs to give up a quantity <math>p_i/p_j\,</math> of say good <math>j.\,</math>

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