Budget Constraint

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\,

p_X\,

and the price of

Y\,

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
:

rac{p_X}{p_Y}	imesrac{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.\,

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:
:

\sum_{i=1}^np_ix_i\leq W.

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

\sum_{i=1}^np_ix_i=W.

In such case, to obtain an additional unit of good

i\,

, the consumer needs to give up a quantity

p_i/p_j\,

of say good

j.\,

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