Part IIA, Paper 1 Consumer and Producer Theory

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Part IIA, Paper 1Consumer and Producer Theory

• Lecture 1
• Consumer Choice and the Utility Function
• Flavio Toxvaerd

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Information

• Email fmot2_at_cam.ac.uk
• Office hour By appointment
• Office Room number 59, 4th floor
• Handouts/slides available at
• http//people.pwf.cam.ac.uk/fmot2/Teaching.htm

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General Overview
Consumer and Producer Theory AIM To understand
the theoretical structure underpinning the
analysis of consumer and producer theory, and to
be able to apply and manipulate the theory in
alternative settings
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Lecture Structure

• Consumer choice and the utility function
• Direct and indirect utility functions
• Duality, Slutsky equation and revealed preference
• Welfare and project choice
• Different domains Intertemporal choice, labour
• Production functions and profit maximisation
• Cost functions and duality
• Applications Monopoly pricing

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Core Texts

• Varian H. Intermediate Microeconomics A modern
  approach (6th/7th edition), W.W. Norton Co,
  2003/2006
• Katz and Rosen Microeconomics (3rd edition)
  Nicholson Microeconomic Theory (8th edition)
• Perloff Microeconomics (2nd edition)
• Pindyck Rubinfeld Microeconomics (5th edition)
  
• Gravelle H. R. Rees Microeconomics (2nd
  edition)
• Varian H. Microeconomic Analysis (3rd edition)
• Jehle G. P. Reny Advanced Microeconomic Theory
  (2nd edition).

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Todays Outline

• Define and analyse a consumers preferences
• Assumptions on preferences
• Representation From preferences to utility
• Ordinality and cardinality

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Consumer Theory

• A theory of consumer behaviour requires three
  steps
• Identify the consumers preferences
• Identify constraints facing the consumer
• Identify the consumers choice, given these
• Rationality Consumers choose most preferred
  option, given constrains.

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Consumer Theory

• A consumers preference ordering can be
  represented by a (real valued) utility function
  u(.) if, for any two bundles x and y,
• x y if and only if u(x) gt u(y)
• Question What restrictions over preferences are
  required for the preference ordering to be
  represented by a utility function?

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The Utility Function

• Any function is a mapping from a domain to a
  range

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The Mapping

• Specifically A function is a mapping where, for
  every element in the domain there exists one, and
  only one, element in the range
• Axiom of choice Completeness
• Any two elements can be compared
• Axiom of choice Reflexivity
• Any element is at least as good as itself

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The Range

• The real line is adopted as the range of the
  utility function, with higher numbers
  representing more preferred options
• Axiom of Choice Transitivity
• If an apple is preferred to an orange and an
  orange is preferred to a banana, then an apple is
  preferred to a banana

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The Basic Axioms
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Conditions, conditions

• Completeness, Reflexivity and Transitivity are
  necessary conditions for the preference ordering
  to be represented by a utility function
• But are they also sufficient conditions? That is,
  if a consumers preference ordering satisfies
  these three conditions can it be represented by a
  utility function?
• NO!

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Lexicographic Preferences

• A consumer has lexicographic preferences if, no
  matter what, he prefers more of one commodity
  (say commodity x) than less but, given the same
  amount of x, prefers more of another commodity
  (say y) to less
• These preferences satisfy the three axioms of
  choice identified, but cannot be represented by a
  utility function

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Further Restriction

• Continuity Small changes in the quantity of one
  commodity can be compensated by small changes
  in the quantity of another commodity
• More formally For every element in the domain,
  the set of weakly preferred elements and the
  set of weakly less preferred elements are
  closed

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Theorem

• Gerard Debreu (1954)
• A consumers preference ordering can be
  represented by a (continuous) utility function if
  and only if the preferences ordering is complete,
  reflexive, transitive and continuous
• Furthermore, the preferences can be represented
  graphically by continuous indifference curves

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Uniqueness

• Given these restrictions - will the utility
  function describing preferences be unique?
  No!
• If u(x) is a function representing a consumers
  preferences then any monotonic transformation of
  that function will also represent the consumers
  preferences
• E.g. ?u(x)?, logu(x), ?u(x) (when ? gt0)
• Lesson Utility functions are ordinal, not
  cardinal

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Well-Behaved Preferences

• Common to additionally assume that
• More is better (monotonicity)
• So indifference curves downward sloping, thin
  and all elements of the domain are good
• (marginal utility is positive)
• Strictly convex preferences
• So indifference curves strictly convex and
  solution to the maximisation problem is unique
• (utility function is strictly quasi-concave)

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An Indifference Curve
x2
x x x
x
x
x
x1
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The Domain

• The domain of the utility function will depend on
  the problem we wish to analyse
• We will consider a number of different domains in
  this course, namely
• Commodity space
• Consumption/income space
• Intertemporal consumption/income space
• State (of the world) space (really!)

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Homo Economicus

• Somebody whose preferences can be represented by
  a utility function, and whose choices can be
  modeled as the solution to the utility
  maximisation problem
• Does this imply greed? Or selfishness?

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Utility Maximisation

• If consumer choice is to be represented in the
  (constrained) utility maximisation problem
• max u(x) s.t. x ? C ? X
• then we need everything that influences the
  consumers choice to be included in the domain
• we also need the process by which the
  constraint is imposed to have no impact on the
  choice

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Things Outside the Domain

• Things not explicitly included in the domain will
  have no impact on the consumers decision if
• they have no impact on the preference ordering
• they remain unchanged (ceteris paribus
  assumption)

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Common Assumption

• The domain of a consumers utility function only
  includes goods/resources allocated explicitly to
  that consumer

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Summary

• Have taken a first look at consumer preferences
• Under some assumptions, preferences may be
  represented by utility functions
• This step immensely eases formal analysis

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Readings

• Varian Intermediate Economics (7th ed.) chapter
  3.
• Debreu G. (1954) Representation of a Preference
  Ordering by a Numerical Function in R. Thrall
  et al. Decision Processes. Reprinted in G. Debreu
  (1983) Mathematical Economics.

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Next Time

• Indifference curves
• Marginal rates of substitution
• Marshallian demand functions
• Types of goods
• Indirect utility function
• Consumer surplus and welfare