budget constraint notes

1. The budget constraint and choice

2. The budget constraint
• The basic concept is really straightforward:
– The consumer has a limited income (I) to purchase
different goods
– Each type of good has a defined price (p) per unit
– We assume that the consumer does not save and
spends all his income
• This possibility will be examined later

3. The budget constraint
• The general budget constraint for n goods is:
• If we only look at 2 goods (Same simplification
as last week), it can be expressed as:
n
i i
i 1
I p x

 
1 1 2 2I p x p x 

4. The budget constraint
• Imagine the following “student entertainment
budget”
– You have 50 €
– The price of a meal is 10 €
– The price of a cinema ticket is 5 €
• Your budget constraint is:
50 5 tickets 10 meals   
1 1 2 2I p x p x 

5. The budget constraint
Meals
Cinema
max
mealx 
max
meal
meal
I
x
p
 Maximum amount of
meals you can buy
Diagram in “consumption space”

6. The budget constraint

max
cin.
cin.
I
x
p

Maximum amount of
cinema tickets you
can buy

max
cin.x Cinema
Meals
max
mealx

7. The budget constraint


max
cin.x Cinema
Meals
max
mealx
Budget constraint

8. The budget constraint


max
cin.x Cinema
Meals
max
mealx
1 1 2 2I p x p x 
2
1 2
1 1
pI
x x
p p
 
The budget constraint is
Dividing by p1 and rearranging:
cin.
meal cin.
meal meal
pI
x x
p p
 
slope
intercept

9. The budget constraint
Any bundle within the budget
constraint is affordable , but
not all the budget is spent
(C,D).
Any bundle beyond the
budget constraint cannot be
afforded (H,G).
C
H
D
G
Any bundle on the budget
constraint is affordable and
ensures all the budget is
spent (E,F).


max
cin.x Cinema
Meals
max
mealx
F
E

10. The budget constraint


max
cin.x Cinema
Meals
max
mealx
Budget constraint
Budget set

11. The budget constraint
• The position of the budget constraint depends
on
• The income of the agent (I)
• The price of the two goods (p1 and p2)
2
1 2
1 1
pI
x x
p p
 

12. The budget constraint


max
cin.x Cinema
Meals
max
mealx
Effect of a fall in income (I)

13. The budget constraint


max
cin.x Cinema
Meals
max
mealx
Effect of a rise in income (I)

14. The budget constraint


max
cin.x Cinema
Meals
max
mealx
Increase in the price of cinema
tickets

15. The budget constraint


max
cin.x Cinema
Meals
max
mealx
Decrease in the price of cinema
tickets

16. Lets think about these things
• Budget constraint equation
• Slope of budget line – Px / Py
• budget line is of 45 ° if???
• Shifts in line can be caused by???
• Shows purchasing power of indivisuals
• Budget line and what it represents

17. Questions
A budget constraint
a.shows the prices that a consumer chooses to pay
for products he consumes.
b.shows the purchases made by consumers.
c.shows the consumption bundles that a consumer
can afford.
d.represents the consumption bundles that give a
consumer equal satisfaction.

18. A consumer that doesn't spend all of her
income
a.would be at a point outside of her budget
constraint.
b.would be at a point inside her budget constraint.
c.must not be consuming positive quantities of all
goods.
d.must be consuming at a point where her budget
constraint touches one of the axes.

19. The following diagram shows a budget
constraint for a particular consumer.
If the price of x is $10, what is the price of y?
a.$15
b.$25
c.$35
d.$70

20. Which of the
graphs in
the figure
reflects a
decrease in
the price of
good X only?
a.graph (a)
b.graph (b)
c. graph (c)
d.graph (d)

21. Which of the
graphs in
the figure
reflects an
increase in
the price
of good Y
only?
a. graph (a)
b.graph (b)
c. graph (c)
d.graph (d)

22. Which of the
graphs in
the figure
could
reflect a
decrease in
the prices
of both
goods?
a. graph (a)
b.graph (b)
c. graph (c)
d.graph (d)

23. • A halving of the prices good A and good B has
the same effect on the budget line as doubling
the income. Is this sentence true or false?
Show it by using the equation of the budget
line.
• What about the opposite? Doubling the prices
good A and good B has the same effect on the
budget line as halving of the income.

24. Suppose the price of pizza is $10, the price of cola is $1, and the
consumer’s income is $50. In addition, suppose the consumer’s
budget constraint measures pizza on the horizontal axis and cola
on the vertical axis.
1) If the price of cola doubles to $2, then the
a. budget constraint intersects the vertical axis at 25 colas.
b. slope of the budget constraint rises to -2.
c. budget constraint intersects the vertical axis at 100 colas.
d. budget constraint shifts inward in a parallel fashion.
2) If the consumer's income rises to $60, then the budget line for
pizza and cola would
a. now intersect the horizontal axis at 6 pizzas and the vertical axis at
60 colas.
b. not change.
c. now intersect the horizontal axis at 4 pizzas and the vertical axis at
16 colas.
d. rotate outward along the cola axis.

25. Answer the following questions based on the table. A consumer is
able to consume the following bundles of rice and beans when the
price of rice is $2 and the price of beans is $3.
RICE BEANS
12 0
6 4
0 8
a. How much is this consumer's income?
b. Draw a budget constraint given this information. Label it B.
c. Construct a new budget constraint showing the change if the price
of rice falls $1. Label this C.
d. Given the original prices for rice ($2) and beans ($3), construct a
new budget constraint if this consumer's income increased to $48.
Label this D.

26. Draw a budget constraint that is consistent with the
following prices and income.
Income = 200
PY = 50
PX = 25
a. Demonstrate how your original budget constraint
would change if income increases to 500.
b. Demonstrate how your original budget constraint
would change if PY decreases to 20.
c. Demonstrate how your original budget constraint
would change if PX increases to 40.

27. Evaluate the following statement, "Warren
Buffet is the second richest person in the
world. He doesn't face any constraint on his
ability to purchase commodities he wants."

28. The budget constraint and choice
The budget constraint
The optimal consumer choice
Income and substitution effects

29. The optimal consumer choice
• This requires bringing in the two elements of the
theory
– The indifference curves, which show how agents rank the
different bundles
– The budget constraint, which shows which bundles are
affordable, and which are not
• Both of these are defined over the “consumption
space”, so they can be superposed easily

30. The optimal consumer choice


max
cin.x Cinema
Meals
max
mealx
Which is the best bundle ?

F
Optimal bundle

C

D

E

B

A

31. The optimal consumer choice


max
cin.x Cinema
Meals
max
mealx
The budget constraint is
tangent to the
indifference curve at F

F
Definition of the
MRS at F !!!

32. The optimal consumer choice
• The optimal bundle is on the tangency between the
budget constraint and the indifference curve.
• This means that for the optimal bundle the slope of
the IC is equal to the slope of the budget constraint
MRS = ratio of prices

33. The optimal consumer choice
• This condition gives a central result of consumer
theory:
• The optimal bundle is the one which equalises the
marginal utility per € spent
– If you were to receive an extra € of income, your
marginal utility will be the same regardless of where
you spend it
2 2
1
2
1 21
1mU p
MRS
mU p
mU mU
p p
    

34. The optimal consumer choice


max
cin.x Cinema
Meals
max
mealx
Example of optimal choice with concave preferences

F

G

The optimal solution is a
“corner solution”

35. The budget constraint and choice
The budget constraint
The optimal consumer choice
Income and substitution effects

36. Income and substitution effects
• Consumer theory is used to understand how choice
is affected by changes in the environment
• These can be complex, and the theory helps to
isolate these different effects
• The separation of income and substitution effects is a
good illustration of the concept of “ceteris paribus”
– Each variable is isolated and analysed separately from the
others

37. Income and substitution effects
• 1: A change in real income
– A previously affordable bundle (A) is no
longer affordable
• 2: A relative price change
– The slope of the budget constraint
changes, and meals become relatively
cheaper


max
cin.x Cinema
Meals
max
mealx
An increase in the price of cinema tickets has 2 effects :

A

38. Income and substitution effects
• Fall in the consumption of cinema
• Increase in the consumption of meals
• Question: How can we separate the
effect of the change in real income from
the effect of the change in relative
prices ?


max
cin.x Cinema
Meals
max
mealx

A

B
Effect of an increase in the price of cinema tickets on
consumer choice

39. Income and substitution effects
• Parallel to the new budget constraint
• Tangent to the original IC
• There is only a single curve that satisfies
these two requirements
• This gives an imaginary optimal bundle
(Im)


max
cin.x Cinema
Meals
max
mealx

A

B
In order to separate the 2 effects, we add an imaginary
budget constraint

Im

40. Income and substitution effects
• From A to Im, real income is held
constant
– We are still on the same indifference
curve, so utility is the same
• The change of bundle is due entirely to
the change in relative price
• This is the substitution effect


max
cin.x Cinema
Meals
max
mealx

A

B
The substitution effect

Im

41. Income and substitution effects
• From Im, to B, relative prices are held
constant
– The two budget constraints are parallel,
so the slope is the same
• The change of bundle is due entirely to
the fall in income.
• This is the income effect


max
cin.x Cinema
Meals
max
mealx

A

B
The income effect

Im

42. Income and substitution effects
• By combining the two, one gets the
overall effect
• One can see that the interaction is
different for the two goods
– The 2 effects can work against each
other, or add up
– Depending on the relative strength of
the effects, this can lead to increases or
falls in consumption


max
cin.x Cinema
Meals
max
mealx

A

B
The overall effect

Im

43. Income and substitution effects
• This type of approach is fundamental to micro-economic
analysis
– Any price change is always accompanied by income and
substitution effects.
• So this helps understand the effects of taxation, shocks to
prices, taste changes, etc.
– Look at the complex effects of oil price increases on
consumption
• Price change ⇒ Complex change in bundle
– Clearly, this will also help understand how demand curves
are built (next week)