 1.2BanerjeeChpt2.2CompetitiveBudgets.pdf

24 Chapter 2

0

3

x1

x2

4

A

X

Figure 2.1 Commodity space

or even irrational amounts (e.g., √

2 units of good 2). Then anypoint in X is a commodity bundle and the entire non-negative orthant in Figure 2.1 is the commodity space X.1

2.2 Competitive Budgets

Let p1 denote the per-unit price of the first good and x1 the quantity that the consumer purchases. Analogously, the price of one unit of good 2 is p2 and the quantity purchased is x2.2 In this section, we assume that the per-unit prices of the two goods, p1 and p2, are fixed and given to the consumer by the market. She may buy as many units as she desires at these prices but is unable to influence p1 or p2 through her purchases (e.g., through discounts for bulk purchases).

Denote the consumer’s income by m and assume that this too is a fixed amount. Then the consumer’s budget constraint or budget set is given by

p1 x1 + p2 x2 ≤ m (2.1)

which expresses the idea that the expenditure on good 1 (p1 x1) and the ex- penditure on good 2 (p2 x2) should not exceed the consumer’s income. In

1If a good is an indivisible or discrete, it can only be purchased in whole units. When both goods are discrete, the commodity space is a grid of dots where each dot is a commodity bundle with coordinates whose values are either zero or a whole number.

2Sometimes we will refer to the goods as xand yinstead of 1 and 2. In that case, per-unit prices will be written as px and py.

/: C 8: B : 3 M =B:M 4B B . E / BE=B .II : A 7 ME = 5 M 2 D 0 M :E AMMI D M :E I M EB = = M:BE : MB -= 31,

0 :M = ? =

0 IP

B AM

Q 7

ME =

. EE

B AM

=

Budgets 25

0 x1

x2

(2, 2)

(2, 6)

(5, 4)

(0, 10)

(5, 0)

–2

m/p2

m/p1 Figure 2.2 Budget line and set

other words, this combination of goods (x1, x2) is affordable with the con- sumer’s income. Such a budget constraint is called a competitive budget because this embodies the notion of a consumer who purchases the goods in perfectly competitive markets at fixed per-unit prices.

When all of the consumer’s disposable income is spent on these two goods, we replace the inequality in (2.1) with an equality and refer to the resulting equation

p1 x1 + p2 x2 = m (2.2)

as a budget line.3 To illustrate, suppose p1 = \$2, p2 = \$1, and m = \$10. Then the budget line (shown in Figure 2.2 as a blue straight line) can be drawn simply by calculating its endpoints as follows. If the consumer were to spend all of her income of \$10 on good 1, she can purchase m/ p1 = 10/2 = 5 units which yields the bundle (5, 0) on the horizontal axis; similarly if she spent all of her \$10 on good 2, she can purchase m/ p2 = 10/1 = 10 units which yields the bundle (0, 10) on the vertical axis. Joining these two end- points yields the budget line showing other combinations of x1 and x2 that can be purchased at the current prices while spending the consumer’s entire income.

The budget constraint or budget set then consists of all bundles on the budget line or inside the shaded triangle in Figure 2.2. If a bundle lies in the interior of the budget set — say, (2, 2) — the consumer’s income is not spent in its entirety and she has some savings. Likewise, a bundle that lies on the

3When the goods are referred to as xand y, the budget line is given by pxx+ pyy= m.

/: C 8: B : 3 M =B:M 4B B . E / BE=B .II : A 7 ME = 5 M 2 D 0 M :E AMMI D M :E I M EB = = M:BE : MB -= 31,

0 :M = ? =

0 IP

B AM

Q 7

ME =

. EE

B AM

=

26 Chapter 2

budget line, such as (2, 6), uses up all of the consumer’s income, while a bundle like (5, 4), which is outside the budget set, is unaffordable.

By rearranging the terms in equation (2.2), we may write

x2 = m p2

− p1 p2

x1, (2.3)

which is the equation of a straight line with vertical intercept m/ p2 and slope –p1/ p2. So a competitive budget line (in the case of two goods) will always be a straight line with a slope given by the negative of the ratio of the two prices,4 while a competitive budget set will comprise a triangle that includes the budget line and all the points to its southwest bounded by the axes (since goods cannot be consumed in negative amounts).

2.2.1 Three goods or more

It is easy to extend the idea of a budget to three or more goods. In the case of three goods labeled as 1, 2 and 3, the budget line is

p1 x1 + p2 x2 + p3 x3 = m,

where pi xi is the expenditure on the ith good, i = 1, 2, 3.

x1 x2

x3

(0, 10, 0)

(0, 0, 10)

(10, 0, 0)

Budget surface

Figure 2.3 Budget surface with three goods

For instance, if m = \$20 and all three goods are priced at \$2 (i.e., p1 = p2 = p3 = \$2) then the budget “line” is actually a surface shown by the

4The price of the good on the horizontal axis is always in the numerator of this ratio.

/: C 8: B : 3 M =B:M 4B B . E / BE=B .II : A 7 ME = 5 M 2 D 0 M :E AMMI D M :E I M EB = = M:BE : MB -= 31,

0 :M = ? =

0 IP

B AM

Q 7

ME =

. EE

B AM

=