Part IIA, Paper 1 Consumer and Producer Theory

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Part IIA, Paper 1Consumer and Producer Theory

• Lecture 2
• Direct and Indirect Utility Functions
• Flavio Toxvaerd

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Todays Outline

• Indifference curves
• Marginal rates of substitution
• Marshallian demand functions
• Types of goods
• Indirect utility function
• Consumer surplus and welfare
• Some mathematical results (Envelope Thm.)
• Roys Identity

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Utility Function

• Recall from last lecture given Axioms of
  Choice, continuity and local non-satiation - a
  consumers preference ordering can be represented
  by a utility function
• Simplifying Assumption The domain consists of
  only two types of commodities, types 1 and 2
• A specific consumption bundle, x, will be
  represented by a vector x (x1,x2) and we can
  write the utility function as u(x1,x2)

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Indifference Curves

• Indifference curves show combinations of
  commodities for which utility is constant

Slope of Indifference Curve Marginal Rate of
Substitution
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Utility Maximisation
First order conditions
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Utility Maximisation
Eliminating the Lagrange Multiplier (?) gives
MRS Ratio of Prices
Slope of Indifference Curve Slope of Budget
Line
and
Budget Line
Note, with more than two commodities have FOC
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Utility Maximisation
To solve move along budget line, until point of
tangency with the indifference curve
x2
x2
A
D
C
B
x1
x1
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Demand Function
When the indifference curves are strictly convex
the solution is unique, say x, where x1
x1(p1,p2,m) , x2 x2(p1,p2,m) giving demand
as a function of prices and income
Marshallian Demand Functions
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Goods
x2
Price expansion path
x1
p1
Demand Curve
p1
x1
x1(p1,p2,m)
x1(p1,p2,m)
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Goods
? lt 0 ? gt 0 ? gt 1
Graphically Engel Curve
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Practice

• Problem 1
• Show that the Marshallian Demand Function is
  homogenous degree zero. (So consumers never
  suffer from Money Illusion)
• Problem 2
• Show that consumers purchase decisions are
  unaffected by any monotonic transformation of the
  utility function.
• (Hint A monotonic transformation can be
  represented by a strictly increasing function
  f(.). Use the chain rule to show that the MRS
  remains unaffected)

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Convex Indifference Curves

• Convex indifference curves means that the
  (absolute value of the) slope of the indifference
  curve is decreasing as x1 increases
• That is Diminishing MRS
• Problem 3
• Show that diminishing marginal utilities is
  neither a necessary nor a sufficient condition
  for convex indifference curves

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Example Cobb-Douglas Utility
First Order Conditions
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Example Cobb-Douglas Utility
Eliminating ? gives
Substitution into the budget constraint gives the
solution
With Cobb-Douglas Utility the consumer spends a
fixed proportion of income on each commodity
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Indirect Utility Function
It is often useful to consider the utility
obtained by a consumer indirectly, as a function
of prices and income rather than the quantities
actually consumed
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Properties of Indirect Utility Fn

• Property 1 v(p,m) is non-increasing in prices
  (p), and non-decreasing in income (m).
• Proof Diagramatically, it is clear that any
  increase in prices or decrease in income
  contracts the affordable set of commodities
  as nothing new is available to the consumer
  utility cannot increase
• Property 2 v(p,m) is homogeneous degree zero.
• Proof No change in the affordable set, or in
  preferences

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Properties of Indirect Utility Fn
These two are General Properties and NOT reliant
on additional restrictions such as convexity of
indifference curve, more is better etc.
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Direct and Indirect Utility
We will see that direct and indirect utility
functions are closely related - and that any
preference ordering that can be represented by a
utility function can also be represented by an
indirect utility function. This means we are free
to use whichever specification we please
For Example If the price of commodity 1 changes
from, say, p1a to p1b, we may want to use the
indirect utility function to measure the change
in consumer welfare
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Mathematical Digression

• The Envelope Theorem

Proof See Varian Microeconomic Analysis, p. 502.
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Application 1

• Marginal utility of Income

The marginal utility of income is given by the
Lagrange multiplier
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Application 2

• Roys Identity

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Consumer Surplus and Welfare
We saw earlier that, following a price change,
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Consumer Surplus and Welfare
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Summary

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

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Readings

• Texts
• Varian, Intermediate Economics (7th ed.) chapters
  4, 5, 6, 14.
• Varian, Microeconomic Analysis, chapter 7