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Chapter 2 NAME
Budget Constraint
Introduction. These workouts are designed to build your skills in de-
scribing economic situations with graphs and algebra. Budget sets are a
good place to start, because both the algebra and the graphing are very
easy. Where there are just two goods, a consumer who consumes x1units
of good 1 and x2units of good 2 is said to consume the consumption bun-
dle,(x1,x
2). Any consumption bundle can be represented by a point on
a two-dimensional graph with quantities of good 1 on the horizontal axis
and quantities of good 2 on the vertical axis. If the prices are p1for good 1
and p2for good 2, and if the consumer has income m, then she can afford
any consumption bundle, (x1,x
2), such that p1x1+p2x2≤m. On a graph,
the budget line is just the line segment with equation p1x1+p2x2=m
and with x1and x2both nonnegative. The budget line is the boundary
of the budget set. All of the points that the consumer can afford lie on
one side of the line and all of the points that the consumer cannot afford
lie on the other.
If you know prices and income, you can construct a consumer’s bud-
get line by finding two commodity bundles that she can “just afford” and
drawing the straight line that runs through both points.
Example: Myrtle has 50 dollars to spend. She consumes only apples and
bananas. Apples cost 2 dollars each and bananas cost 1 dollar each. You
are to graph her budget line, where apples are measured on the horizontal
axis and bananas on the vertical axis. Notice that if she spends all of her
income on apples, she can afford 25 apples and no bananas. Therefore
her budget line goes through the point (25,0) on the horizontal axis. If
she spends all of her income on bananas, she can afford 50 bananas and
no apples. Therfore her budget line also passes throught the point (0,50)
on the vertical axis. Mark these two points on your graph. Then draw a
straight line between them. This is Myrtle’s budget line.
What if you are not told prices or income, but you know two com-
modity bundles that the consumer can just afford? Then, if there are just
two commodities, you know that a unique line can be drawn through two
points, so you have enough information to draw the budget line.
Example: Laurel consumes only ale and bread. If she spends all of her
income, she can just afford 20 bottles of ale and 5 loaves of bread. Another
commodity bundle that she can afford if she spends her entire income is
10 bottles of ale and 10 loaves of bread. If the price of ale is 1 dollar per
bottle, how much money does she have to spend? You could solve this
problem graphically. Measure ale on the horizontal axis and bread on the
vertical axis. Plot the two points, (20,5) and (10,10), that you know to
be on the budget line. Draw the straight line between these points and
extend the line to the horizontal axis. This point denotes the amount of
6BUDGET CONSTRAINT (Ch. 2)
ale Laurel can afford if she spends all of her money on ale. Since ale costs
1 dollar a bottle, her income in dollars is equal to the largest number of
bottles she can afford. Alternatively, you can reason as follows. Since
the bundles (20,5) and (10,10) cost the same, it must be that giving up
10 bottles of ale makes her able to afford an extra 5 loaves of bread. So
bread costs twice as much as ale. The price of ale is 1 dollar, so the price
of bread is 2 dollars. The bundle (20,5) costs as much as her income.
Therefore her income must be 20 ×1+5×2 = 30.
When you have completed this workout, we hope that you will be
able to do the following:
•Write an equation for the budget line and draw the budget set on a
•Graph the effects of changes in prices and income on budget sets.
•Understand the concept of numeraire and know what happens to the
•Know what the budget set looks like if one or more of the prices is
•See that the idea of a “budget set” can be applied to constrained
NAME 7
2.1 (0) You have an income of $40 to spend on two commodities. Com-
modity 1 costs $10 per unit, and commodity 2 costs $5 per unit.
(b) If you spent all your income on commodity 1, how much could you
(c) If you spent all of your income on commodity 2, how much could
below.
0246
8
2
4
6
x1
x2
8
Blue Line
Red Line
Black Line
Black Shading
Blue
Shading
(d) Suppose that the price of commodity 1 falls to $5 while everything else
equation. 5x1+5x2=30.Use black ink to draw this budget
line.
(f) On your diagram, use blue ink to shade in the area representing com-
modity bundles that you can afford with the budget in Part (e) but could
not afford to buy with the budget in Part (a). Use black ink or pencil to
(e).
2.2 (0) On the graph below, draw a budget line for each case.
8BUDGET CONSTRAINT (Ch. 2)
(a) p1=1,p2=1,m= 15. (Use blue ink.)
(d) p1=p2,m=15p1. (Use pencil or black ink. Hint: How much of
x 2
20
15
10
5
051015 20
x1
Red Line
Blue Line = Pencil Line
Black Line
2.3 (0) Your budget is such that if you spend your entire income, you
can afford either 4 units of good xand 6 units of good yor 12 units of x
and 2 units of y.
(a) Mark these two consumption bundles and draw the budget line in the
graph below.
04812
16
4
8
12
x
y
16
NAME 9
(c) If you spent all of your income on x,howmuchxcould you buy?
(d) If you spent all of your income on y,howmuchycould you buy?
(e) Write a budget equation that gives you this budget line, where the
(f) Write another budget equation that gives you the same budget line,
2.4 (1) Murphy was consuming 100 units of Xand 50 units of Y.The
price of Xrose from 2 to 3. The price of Yremained at 4.
(a) How much would Murphy’s income have to rise so that he can still
2.5 (1) If Amy spent her entire allowance, she could afford 8 candy bars
and 8 comic books a week. She could also just afford 10 candy bars and
4 comic books a week. The price of a candy bar is 50 cents. Draw her
0 8 16 24 32
8
16
24
Candy bars
Comic books
32
12
10 BUDGET CONSTRAINT (Ch. 2)
2.6 (0) In a small country near the Baltic Sea, there are only three
commodities: potatoes, meatballs, and jam. Prices have been remark-
ably stable for the last 50 years or so. Potatoes cost 2 crowns per sack,
meatballs cost 4 crowns per crock, and jam costs 6 crowns per jar.
(a) Write down a budget equation for a citizen named Gunnar who has
(b) The citizens of this country are in general very clever people, but they
are not good at multiplying by 2. This made shopping for potatoes excru-
ciatingly difficult for many citizens. Therefore it was decided to introduce
a new unit of currency, such that potatoes would be the numeraire. A
(c) In terms of the new currency, what is the price of jam? 3
(d) What would Gunnar’s income in the new currency have to be for him
to be exactly able to afford the same commodity bundles that he could
180.Is Gunnar’s budget set any different than it was before the change?
2.7 (0) Edmund Stench consumes two commodities, namely garbage and
punk rock video cassettes. He doesn’t actually eat the former but keeps
it in his backyard where it is eaten by billy goats and assorted vermin.
The reason that he accepts the garbage is that people pay him $2 per
sack for taking it. Edmund can accept as much garbage as he wishes at
that price. He has no other source of income. Video cassettes cost him
$6 each.
(a) If Edmund accepts zero sacks of garbage, how many video cassettes
NAME 11
(b) If he accepts 15 sacks of garbage, how many video cassettes can he
(d) Draw Edmund’s budget line and shade in his budget set.
0 5 10 15 20
5
10
15
Video cassettes
Garbage
20
Budget Line
Budget Set
2.8 (0) If you think Edmund is odd, consider his brother Emmett.
Emmett consumes speeches by politicians and university administrators.
He is paid $1 per hour for listening to politicians and $2 per hour for
0255075
100
25
50
75
Politician speeches
Administrator speeches
100
