# The budget constraint and choice by icecube

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The budget constraint
and choice

The problem of limited resources
and its effect on choice
The budget constraint and choice

   Last week:
   We saw that preferences can be represented by
utility functions ...
   That indifference curves can be used to map a
utility function into “consumption space”
   But we still don’t know how consumers choose
amongst the different bundles...
   This week:
   We introduce the concept of a budget,
   This is the 2nd half of consumer theory
The budget constraint and choice

The budget constraint

The optimal consumer choice

Income and substitution effects
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
The budget constraint

   The general budget constraint for n
goods is:
n
I   pi x i
i 1

   If we only look at 2 goods (Same
simplification as last week), it can be
expressed as:

I  p1 x1  p 2 x 2
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 €

I  p1 x1  p 2 x 2
50  5  tickets 10  meals
The budget constraint

Diagram in “consumption space”

Meals
I      Maximum amount of
x   max

x   max
meal   
meal                           p meal   meals you can buy

Cinema
The budget constraint

Meals

x max
meal   

Maximum amount of                       I
cinema tickets you   x    max
cin.    
p cin.


max
x cin.                Cinema
The budget constraint

Meals

x max
meal   

Budget constraint


max
x cin.    Cinema
The budget constraint

The budget constraint is I  p1 x1  p 2 x 2
Dividing by p1 and rearranging:
Meals

x max                                      I p2
meal      intercept                x1      x2
p1 p1

slope

I         p cin.
x meal                       x cin.
p meal       p meal

max
x cin.               Cinema
The budget constraint

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

Meals
max
Budget set
x   meal   

Budget constraint


max
x cin.    Cinema
The budget constraint

   The position of the budget constraint
depends on
I p2
x1      x2
p1 p1

   The income of the agent (I)

   The price of the two goods (p1 and p2)
The budget constraint

Effect of a fall in income (I)

Meals

x max
meal   


max
x cin.    Cinema
The budget constraint

Increase in the price of cinema
tickets
Meals

x max
meal   


max
x cin.   Cinema
The budget constraint and choice

The budget constraint

The optimal consumer choice

Income and substitution effects
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
The optimal consumer choice

Which is the best bundle ?

Meals

x max
meal   
A           Optimal bundle
C               


D                           B
       F                   


E


max
x cin.   Cinema
The optimal consumer choice

The budget constraint is
tangent to the
indifference curve at F
Meals

x max
meal   

Definition of the
F

MRS at F !!!


max
x cin.   Cinema
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
The optimal consumer choice

   This condition gives a central result of
consumer theory:
mU 2    p2       mU1 mU 2
MRS                     
mU1     p1        p1   p2

   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
The optimal consumer choice

Example of optimal choice with concave preferences

Meals                           The optimal solution is a
x max    
“corner solution”
meal


F



G


max
x cin.   Cinema
The budget constraint and choice

The budget constraint

The optimal consumer choice

Income and substitution effects
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
Income and substitution effects

An increase in the price of cinema tickets has 2 effects :
   1: A change in real income
Meals                            A previously affordable bundle (A)
is no longer affordable
x max
meal   

   2: A relative price change
   The slope of the budget
constraint changes, and meals
A           become relatively cheaper



max
x cin.      Cinema
Income and substitution effects

Effect of an increase in the price of cinema tickets on
consumer choice
Meals                           Fall in the consumption of cinema
max                         Increase in the consumption of
x   meal                        meals
   Question: How can we separate
the effect of the change in real
             income from the effect of the
B         A       change in relative prices ?



max
x cin.    Cinema
Income and substitution effects

In order to separate the 2 effects, we add an imaginary
budget constraint
   Parallel to the new budget
Meals                                constraint
   Tangent to the original IC
x max
meal   

Im
               There is only a single curve that
satisfies these two requirements

B                A      This gives an imaginary optimal
       bundle (Im)


max
x cin.     Cinema
Income and substitution effects

The substitution effect
   From A to Im, real income is held
constant
Meals
   We are still on the same
x   max
                                indifference curve, so utility is the
meal
Im
same
               The change of bundle is due
entirely to the change in relative

A       price
B
      This is the substitution effect


max
x cin.       Cinema
Income and substitution effects

The income effect
   From Im, to B, relative prices are
held constant
Meals
   The two budget constraints are
x   max
                                  parallel, so the slope is the same
meal
Im            The change of bundle is due
                  entirely to the fall in income.
                     This is the income effect
B                  A



max
x cin.      Cinema
Income and substitution effects

The overall effect
   By combining the two, one gets
the overall effect
Meals
   One can see that the interaction
max
x   meal                                is different for the two goods
Im               The 2 effects can work against
                     each other, or add up
   Depending on the relative

B                 A           strength of the effects, this can
           lead to increases or falls in
consumption


max
x cin.      Cinema
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)

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