Intermediate Microeconomics

1
Intermediate Microeconomics

• Utility Theory

2
Utility

• A complete set of indifference curves tells us
  everything we need to know about any individuals
  preferences over any set of bundles.
• However, our goal is to build a model that is
  useful for describing behavior.
• While indifference curves are often sufficient
  for this, they are somewhat cumbersome.
• Therefore, we will often think of individual
  preferences in terms of Utility

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Utility

• Utility is a purely theoretical construct defined
  as follows
• If an individual strictly prefers bundle
  A-q1a,q2a,..,qna to another bundle
  B-q1b,q2b,..,qnb, then an individual is said to
  get a higher level of utility from bundle A
  than bundle B.
• If an individual is indifferent between a bundle
  A-q1a,q2a,..,qna and another bundle
  B-q1b,q2b,..,qnb, then an individual is said to
  get the same level of utility from bundle A
  than bundle B.
• How is utility related to happiness?

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Utility Function

• A utility function U is just a mathematical
  function that assigns a numeric value to each
  possible bundle such that
• If an individual strictly prefers bundle
  A-q1a,q2a,..,qna to another bundle
  B-q1b,q2b,..,qnb, then U(q1a,q2a,..,qna) gt
  U(q1b,q2b,..,qnb)
• If an individual is indifferent between a bundle
  A-q1a,q2a,..,qna and another bundle
  B-q1b,q2b,..,qnb, then U(q1a,q2a,..,qna)
  U(q1b,q2b,..,qnb)
• We can often think of individuals using goods as
  inputs to produce utils, where production is
  determined by utility function.
• So how do utility functions relate to
  Indifference curves? (hill of utility)

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Constructing a Utility Function

• Consider an individuals preferences over
  consumption bundles containing different numbers
  of hours of watching college (qc) and pro
  football (qp).
• The person always prefers a bundle that allows
  him to watch more total hours of games to fewer.
• Person likes college and pro games equally,
  meaning he would always be willing to trade a
  bundle with one fewer hr of college game for a
  bundle with one more hour of pro game and vice
  versa.
• What will this persons indifference curves
  over bundles containing these two goods look like?

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Constructing a Utility Function

• What is a utility function that captures this
  individuals preferences?
• How do we get Indifference Curves from this
  utility function?
• What is Marginal Rate of Substitution (MRS)?
• Would this utility function be a good
  approximation for your preferences over beer and
  pizza?

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Other Commonly Used Utility Functions

• Two commonly used forms for utility functions
  are
• Quasi-linear Utility
• U(q1,q2) f(q1) q2 for some concave function
  f.
• Examples?
• Cobb-Douglas Utility
• U(q1,q2) q1a q2b for some positive a and
  b.
• Examples?
• Do they exhibit Diminishing MRS?
• To understand what types of situations they would
  be appropriate for, let us look deeper at MRS.

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Marginal Utility

• In economics, we are generally interested in
  trade-offs at the margin.
• How does a consumer value a little more of a
  particular good?
• Consider the ratio of the change in utility (?U)
  associated with a small increase in q1 (?q1),
  holding the consumption of other goods fixed, or
• Ex U(q1,q2) q1q2
• At bundle 1,4, with ?q1 0.5
• What happens when ?q1 gets really small?

q2 4
?q10.5
u6
u4
1 1.5 q1
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Marginal Utility

• Marginal Utility of a good 1 (MU1) - the
  rate-of-change in utility from consuming more of
  a given good, or
• MU1 - the partial derivative of the utility
  function with respect to good 1
• So what is experession for marginal utility of
  good 1 given utility functions at some
  bundle-q1, q2?
• U(q1,q2) q1 q2
• U(q1,q2) q10.5 q20.5

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Ordinal Nature of Utility

• However, there is a major constraint with this
  concept of marginal utility as a way to measure
  how much someone values a little more of a
  good, and it has to do with ordinal nature of
  utility.
• Consider again the person who saw college
  football and pro football as perfect substitutes.
• As we saw, his preferences over these two goods
  were captured by
• U(qc,qp) qc qp
• Could his preferences also be captured by
  U(qc,qp) 5(qc qp) ?

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Ordinal Nature of Utility

• Utility function is constructed to summarize
  underlying preferences.
• Therefore, no new information in utility
  function.
• So generally more than one utility function can
  capture a given set of preferences.
• Since preferences were strictly ordinal, so must
  be the utility function.
• Utility level of one bundle is only meaningful in
  as much as it is higher, lower, or the same as
  another bundle.
• How much higher isnt informative.
• This also means Marginal Utility isnt
  necessarily very informative in and of itself.

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Marginal Utility

• Instead, consider the following thought exercise
  
• Suppose we increase individuals q1 by a little
  bit (?q1),
• How much q2 would he be willing to give up for
  this more q1?
• For small (?q1), individuals change in utility
  will be approximately
• ?q1 MU1(q1,q2)
• Therefore, we would have to decrease some ?q2
  large enough such that
• ?q1MU1(q1,q2) ?q2MU2(q1,q2) 0
• or
• What does this mean if both ?q2 and ?q1 are small?

q2 4
?q1
?q2
u4
1 q1
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Marginal Utility and MRS

• Therefore, MRS is both
• The slope of an indifference curve at a
  particular point, and
• The negative ratio of marginal utilities at that
  particular point.
• Should this be surprising?

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Ordinal Nature of Utility and MRS

• Recall again that the following utility functions
  both capture the same underlying preferences
• u(qq,q2) q1 q2
• v(qq,q2) 5(q1 q2)
• As discussed previously, issues regarding
  preferences revolve around an individuals
  willingness-to-trade one good for more of the
  other.
• Therefore, one way to confirm whether or not two
  utility functions represent the same underlying
  preferences is to determine whether they result
  in the same MRS at every bundle.
• Is this true with above utility functions?

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Interpreting MRS Equations

• So consider a Cobb-Douglas utility function
  U(q1,q2) q10.5q2.
• MU1 ?
• MU1 ?
• MRS?
• So how do we interpret this expression for MRS?
• What is MRS at 4,4?
• What is MRS at 9, 1?

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Comparing Utility functions

• Recall again
• Quasi-linear utility function
• U(q1,q2) q1a q2 (for 0 lt a lt 1)
• Cobb Douglas utility function
• U(q1,q2) q1aq2b (for a, b gt 0)
• What is expression for MRS under each
  specification?
• What is key difference between specifications?

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Comparing Utility functions

• Quasi-linear

• Cobb-Douglas

q2
q2
q1
q1
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Cobb-Douglas vs. Quasi-linear Utility Functions

• For comparisons between bundles with the
  following two goods, would Quasi-linear or
  Cobb-Douglas utility function be more
  appropriate?
• pizza and beer?
• composite good vs. pencils?
• composite good vs. clothes?

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Marginal Utility and MRS

• So this is primary benefit of modeling
  preferences in terms of utility
• It allows us to be very precise in describing a
  given individuals MRS, or his willingness to
  trade off one good for more of another.
• Recall our discussion of MRS in the context of
  indifference curves.
• We could only describe MRS at any given point by
  approximating the slope of Indifference curve.
• With utility function, we can easily calculate
  MRS at any given bundle.
• Given MRS is key to thinking about an
  individuals willingness to make trade-offs, this
  precision will be important.