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 €


   Your budget constraint is:
                   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.
             can buy


                                  
                                        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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