If consumers had unlimited incomes or if goods were free, there would be no problem of economizing. People could buy whatever they wanted and would have no problem of choice. But this is not generally the case. Consumers are constrained as to what bundles of goods they can purchase based on the market- determined prices of the goods and on their incomes. We now turn to an analysis of the income constraint faced by consumers.
Budget Lines
Consumers normally have limited incomes and goods are not free. Their problem is how to spend the limited income in a way that gives the maximum possible util- ity. The constraint faced by consumers can be illustrated graphically.
Suppose the consumer has a fixed income of $1,000, which is the maximum amount that can be spent on the two goods in a given period. For simplicity, as- sume the entire income is spent on X and Y. If the price of X is $5 per unit and the price of Y is $10 per unit, the amount spent on X ($5 3 X) plus the amount spent on Y ($10 3 Y) must equal the $1,000 income
$5X 1 $10Y 5 $1,000 Alternatively, solving for Y in terms of X,
Y 5 $1,000 ______
$10 2 $5 ____
$10 X 5 100 2 1 __ 2 X
Now try Technical Problem 5.
The graph of this equation, shown in Figure 5.5, is a straight line called the bud- get line. A budget line is the set of all combinations or bundles of goods that can be purchased at given prices if the entire income is spent. Bundles costing less than
$1,000 lie below AB. All bundles above budget line AB cost more than $1,000. Thus, the budget line divides the commodity space into the set of attainable bundles and the set of unattainable bundles.
To purchase any one of the bundles of X and Y on the budget line AB in Figure 5.5, the consumer spends exactly $1,000. If the consumer decides to spend all $1,000 on good Y and spend nothing on good X, 100 (5 $1,000y$10) units of Y can be purchased (point A in Figure 5.5). If the consumer spends all $1,000 on X and buys no Y, 200 (5 $1,000y$5) units of X can be purchased (point B).
In Figure 5.5, consumption bundles C, with 40X and 80Y, and D, with 120X and 40Y, represent two other combinations of goods X and Y that can be purchased by spending exactly $1,000, because (80 3 $10) 1 (40 3 $5) 5 $1,000 and (40 3 $10) 1 (120 3 $5) 5 $1,000.
The slope of the budget line, 21y2 (5 DYyDX), indicates the amount of Y that must be given up if one more unit of X is purchased. For every additional unit of X purchased, the consumer must spend $5 more on good X. To continue meeting the budget constraint, $5 less must be spent on good Y; thus the consumer must give up 1y2 unit of Y. To illustrate this point, suppose the consumer is currently
budget line The line showing all bundles of goods that can be purchased at given prices if the entire income is spent.
150
Quantity of Y
Quantity of X
40 50 100 120 150 200
100 80
50 40
0
A (0, 100)
C (40, 80)
Slope = –12–
D (120, 40)
B (200, 0) Blowup at D
D (120, 40)
E (121, 39.5)
12
10 –
20 1
F I G U R E 5.5 A Consumer’s Budget Constraint
purchasing bundle D but wishes to move to bundle E, which is composed of 1 more unit of X and 1y2 unit less of Y (see the blowup in Figure 5.5). Bundles D and E both cost $1,000 to purchase, but the consumer must trade-off 1y2 unit of good Y for the extra unit of good X in bundle E.
The rate at which the consumer can trade-off Y for one more unit of X is equal to the price of good X (here $5) divided by the price of good Y (here $10); that is,
Slope of the budget line 5 2PxyPy
where Px and Py are the prices of goods X and Y, respectively. In Figure 5.5, the slope of the budget line is 21y2, which equals 2$5y$10.
The relation between income (M) and the amount of goods X and Y that can be purchased can be expressed as
M 5 PxX 1 PyY
This equation can also be written in the form of a straight line:
Y 5 M ___ Py 2 __ Px Py X
The first term, MyPy , gives the amount of Y the consumer can buy if no X is pur- chased. As noted, 2PxyPy is the slope of the budget line and indicates how much Y must be given up for an additional unit of X.
The general form of a typical budget line is shown in Figure 5.6. The line AB shows all combinations of X and Y that can be purchased with the given
F I G U R E 5.6 A Typical Budget Line
A
Quantityof Y
Quantity of X
B Y = – XM
Py
— Px Py
— M
Py
—
M Px
—
F I G U R E 5.7 Shifting Budget Lines
Quantity of Y A
Quantity of X
240 160 200
Panel A — Changes in income R
F
Z B N
80 100 120
0
Quantity of Y A
Quantity of X
200
Panel B — Changes in price of X 100
250
C B D
0 125
income (M) and given prices of the goods (Px and Py). The intercept on the Y-axis, A, is MyPy; the horizontal intercept, B, is MyPx. The slope of the budget line is 2PxyPy. Shifting the Budget Line
If income (M) or the price ratio (PxyPy) changes, the budget line must change. Panel A of Figure 5.7 shows the effect of changes in income. Begin with the original budget line shown in Figure 5.5, AB, which corresponds to $1,000 income and prices of X and Y of $5 and $10, respectively. Next, let income increase to $1,200, holding the prices of X and Y constant. Because the prices do not change, the slope of the budget line remains the same (21y2). But since income increases, the vertical intercept (MyPy) increases (shifts upward) to 120 (5 $1,200y$10). That is, if the consumer now spends the entire income on good Y, 20 more units of Y can be purchased than was previously the case. The horizontal intercept (MyPx) also increases, to 240 (5 $1,200y$5). The result of an increase in income is, therefore, a parallel shift in the budget line from AB to RN. The increase in income increases the set of combinations of the goods that can be purchased.
Alternatively, begin once more with budget line AB and then let income decrease to
$800. In this case the set of possible combinations of goods decreases. The vertical and horizontal intercepts decrease to 80 (5 $800y$10) and 160 (5 $800/$5), respectively, causing a parallel shift in the budget line to FZ, with intercepts of 80 and 160. The decrease in income shrinks the set of consumption bundles that can be purchased.
Panel B shows the effect of changes in the price of good X. Begin as before with the original budget line AB and then let the price of X fall from $5 to $4 per unit.
Since MyPy does not change, the vertical intercept remains at A (100 units of Y).
However, when Px decreases, the absolute value of the slope (PxyPy) falls to 4y10
(5 $4y$10). In this case, the budget line becomes less steep. After the price of X falls, more X can be purchased if the entire income is spent on X. Thus the hori- zontal intercept increases from 200 to 250 units of X (5 $1,000y$4). In Panel B, the budget line pivots (or rotates) from AB to AD.
An increase in the price of good X to $8 causes the budget line to pivot backward, from AB to AC. The intercept on the horizontal axis decreases to 125 (5 $1,000y$8). When Px increases to $8, the absolute value of the slope of the line, PxyPy, increases to 8y10 (5 $8/$10). The budget line becomes steeper when Px
rises, while the vertical intercept remains constant.
Relation An increase (decrease) in income causes a parallel outward (inward) shift in the budget line.
An increase (decrease) in the price of X causes the budget line to pivot inward (outward) around the original vertical intercept.
Now try Technical Problem 6.