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Microeconomics I
Lesson 2. CT(I): Budget constraint
 Budget constraint
The budget constraint (bc) is a representation of the
set of opportunities open to the consumer. If we
have goods x and y , with prices px and p y , and the
consumer’s income is m , then the bc is
px x  p y y  m
(time dimension of bc)
The budget line (bl) is the line that limits the above
set. It represents the set of bundles  x, y  that costs
exactly m .
px x  p y y  m
Graphical representation
Solve above equation for y
m px
y

x
py py
This is a straight line with intercept  m p y  and
slope   px p y  .
2
y
m
py
Slope

px
py
A
6
B
4
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Properties
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m
px
x
-Given the two prices, the amount of money, m ,
determines the area of the budget set (the position of
the two intercepts). More m means more
opportunities open to the consumer.
-The slope of the bl measures the rate at which the
market is willing to “substitute” good x for good y .
Going from A to B we see that according to the bl,
obtaining 1 more unit of x means giving up 2 units
of y. The price of 1 unit of x is 2 units of y. Thus, the
slope measures the relative price of the two goods:
the price of x relative to the price of y.
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This can be seen more generally as follows:
px x  p y y  m
At A
(i)
At B
(ii) px  x  x   p y  y  y   m
Subtracting (i) from (ii) we see that these two
equations imply
px x  p y y  0
p y y   px x
p
y
 x
x
py
That is, the slope of the bl is the (negative) of the
ratio of prices.
 Changes in the budget line
a) Changes in income (holding prices constant)
m
y
x
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m
y
x
b) Changes in prices (holding income constant)
px
y
x
px and p y
y
x
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c) Equiproportional changes in income and prices
(all prices and income increase in the same
proportion)
y
x
The budget line remains unchanged.
Proof: Suppose all prices and income double. The
the new budget line will be
2 p x x  2 p y y  2m
Divide both sides of the equation by 2 and you get
p x x  p y y  m,
which is the original budget line.
 Applications
These changes in prices and the effects on the bl are
directly applicable to the analysis of the effects of
taxes and subsidies.
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Taxes
A tax of t € on good x has the effect of increasing the
price of x. The bl in this case is
 px  t  x  p y y  m ,
and therefore the effect is a rotation clockwise
around the vertical intercept like the one considered
above for the increase in the price of x.
m
py
Slope without tax 
px
py
Slope with tax 
m
 px  t 
px  t
py
m
px
The above tax is called a “unit” tax. It means that it
increases the price of x by a given quantity of € (say,
for example, 0.5 €).
The most usual form a tax on goods takes is that of
an “ad valorem” tax. In this case the tax is a
percentage of the price of the good. Lets us call this
tax  . Then, the bl is
px 1    x  p y y  m
If, for instance, the ad valorem tax is 10%, then 
equals 0.1, and the bl would be
px 1  0.1 x  p y y  m
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Subsidies
A subsidy is the opposite of a tax. A tax on a good
increases its price. A subsidy on good decreases its
price. Let us call the subsidy s. The the new bl is
 px  s  x  p y y  m
and the bl rotates anticlockwise around the vertical
intercept. As with taxes we have “unit” subsidies
and “ad valorem” subsidies.
Rationing
Suppose there is a law that says that the maximum
quantity of good x that can be consumed per unit of
time is 10 units.
Income and prices
Law
Both
10
10
Discounts
Good x becomes cheaper for all units in excess of 10
y

px
py

10
p x'
py
x
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Surcharge
Good x due to a tax becomes more expensive for all
units in excess of 10.

px
py

px  t
py
10
In these two last cases, the crucial thing in order to
draw correctly the bl, is to identify the
corresponding corner points. To illustrate how to do
this we complete this lesson with the solution to two
problems taken from the problem set.
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Lesson 2. Problem 9
m  200
If
0  x  10
If
x  10
px  4;
py  2
px  2 for units in excess of 10; p y  2
Budget line when purchase of x is less than or equal
to 10 units
4 x  2 y  200
y  100  2 x
Budget line when purchase of x is 10 units or more.
Here the thing to understand is that the consumer has
already made the purchase of x 0 units of x at p x0 and
that further purchases will be at px' . In general the
budget equation for these circumstances is
 m  px0 x0   px'  x  x0   p y y
Substituting in the numbers of the problem, we have
200  40  2( x  10)  2 y
160  2 x  20  2 y
180  2 x  2 y
y  90  x
10
y
100
90
A
80
10
50
100
Another way of finding out the second bl
At A we know two things
a) px  2 and p y  2
b) to make bundle (10, 80) affordable at these prices,
what income does the consumer need? (in other
words, what is the vertical intercept of the new bl?).
Let us call this intercept m . Then
x
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m  2 x  2 y
m  2(10)  2(80)  180
Therefore, 180  2 x  2 y
y  90  x
Repeat the problem at home, but suppose that after
10 units of x a per unit tax of 2 € is imposed on all
units in excess of 10.
Lesson 2. Problem 10
a) 200  4 x  2 y

y  100  2 x
y
100
50
x
b) If the subsidy was transferable, it would be like
money for the consumer. So then,
200  50  4 x  2 y  y  125  2 x
But it is not transferable. So the above budget line
does not apply until the consumer has not spent
entirely the subsidy on good x. That is, until the
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consumer has consumed 12.5 units of x. So the
complete budget equation is
if x  12.5 y  100
if x  12.5 y  125  2 x
y
125
100
x
12.5
c)
50
62.5
200  4 x  (2  1) y
200  4 x  3 y
100
200/3
50

y
200 4
 x
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