# Economic Elasticity Problem

Gace814

Budgets 27

shaded triangle in Figure 2.3, while the corresponding budget set is the vol- ume of the tetrahedron formed by the budget surface and the three axes. The three dots along the axes show consumption bundles where all the income is spent on that good; at any point on the budget surface, the consumer spends all her income on some combination of the three goods.

In general, if there are n goods where x i denotes the ith good and pi its price, the budget constraint is

p1 x 1 + p2 x 2 + . . . + pn x n ≤ m. (2.4)

2.3 Changes in Prices or Income

By changing prices of goods or income one at a time, we can see how budgets are affected. The simplest way to see these consequences is to begin with equation (2.3) and to observe what happens to the intercept and the slope of the budget line as we change each price or income in isolation.

5 8

10

x 2

x 10

– 1.25 x 1 x 15

10

0

5

–1

x 2

5

10

0

x 2

3

6

– 2

Figure 2.4 Budgets after changes in p1, p2 and m

Starting with the budget in Figure 2.2 where p1 = $2, p2 = $1, and m = $10, consider each of the following changes. In Figure 2.4, the new positions of the budget lines after each of the changes are shown with magenta arrows.

(a) Suppose p1 decreases to $1.25 per unit. This leaves the vertical inter- cept unchanged but makes the slope of the budget –1.25, as depicted ¶b in the left panel.

. = B== 7 EA 2 = E= A = 3A = EA - 8 D .MAD A ? - M D= ?= 4 M= 1: C /= D =: C = D IM= E DA: M = AD A , 20

/ = = E M

/ A?

P M

D= ?=

- DD

A? =

= =

28 Chapter 2

(b) Suppose p2 increases to $2 per unit. This lowers the vertical intercept and flattens the slope of the budget from –2 to –1, as shown in the center panel.

(c) Suppose m decreases to $6. This decreases both the horizontal and ver- tical intercepts but leaves the slope unchanged, as shown in the right panel.

2.3.1 Endowment budgets

Ms. i comes into the world with 4 apples and 3 bananas. We refer to this initial amount of the two goods as i’s individual endowment and write this commodity bundle as ωi = (4, 3).5 Suppose the price of an apple is $1 and each banana is priced at $2. Then we may think of the value of an individ- ual’s endowment as ($1 × 4) + ($2 × 3) = $10 and refer to this as i’s income, since this is the amount of money she would have if she sold all her apples and bananas. With this income of $10, we can draw her budget line with slope –0.5 which is shown in Figure 2.5 with a thin blue line. If the price of an apple remains at $1 but the price of a banana falls to $1, then i’s income becomes $7 and the the budget line is steeper with a slope of –1, i.e., the bud- get line pivots around the individual endowment point as the relative price ratio changes.

0

3

5

7

apples

bananas

4 107

–1

ωi

– 0.5

Figure 2.5 Budgets with endowments

5The Greek letter ω is read as ‘omega’.

. = B== 7 EA 2 = E= A = 3A = EA - 8 D .MAD A ? - M D= ?= 4 M= 1: C /= D =: C = D IM= E DA: M = AD A , 20

/ = = E M

/ A?

P M

D= ?=

- DD

A? =

= =