REVIEW OF CONSUMER THEORY
CHAPTER 1
2UTILITY FUNCTIONS
Review of Consumer Theory
 Describe how much “happiness” or “satisfaction” an individual
experiences from “consuming” goods – the benefit of consumption
 Marginal Utility
 The extra total utility resulting from consumption of a
small/incremental extra unit of a good
 Mathematically, the (partial) slope of utility with respect to that good
3UTILITY FUNCTIONS
Review of Consumer Theory
 Describe how much “happiness” or “satisfaction” an individual
experiences from “consuming” goods – the benefit of consumption
 Marginal Utility
 The extra total utility resulting from consumption of a
small/incremental extra unit of a good
 Mathematically, the (partial) slope of utility with respect to that good
 One-good case: u(c), with du/dc > 0 and d2u/dc2 < 0
 Recall interpretation: strictly increasing at a strictly decreasing rate
 Diminishing marginal utility
4UTILITY FUNCTIONS
Review of Consumer Theory
 Describe how much “happiness” or “satisfaction” an individual
experiences from “consuming” goods – the benefit of consumption
 Marginal Utility
 The extra total utility resulting from consumption of a
small/incremental extra unit of a good
 Mathematically, the (partial) slope of utility with respect to that good
 One-good case: u(c), with du/dc > 0 and d2u/dc2 < 0
 Recall interpretation: strictly increasing at a strictly decreasing rate
 Diminishing marginal utility
Alternative notation: du/dc OR u’(c) OR uc(c) OR u1(c)
5UTILITY FUNCTIONS
Review of Consumer Theory
 Describe how much “happiness” or “satisfaction” an individual
experiences from “consuming” goods – the benefit of consumption
 Marginal Utility
 The extra total utility resulting from consumption of a
small/incremental extra unit of a good
 Mathematically, the (partial) slope of utility with respect to that good
 One-good case: u(c), with du/dc > 0 and d2u/dc2 < 0
 Recall interpretation: strictly increasing at a strictly decreasing rate
 Diminishing marginal utility
 Two-good case: u(c1, c2), with ui(c1, c2) > 0 and uii(c1, c2) < 0 for
each of i = 1,2
 Utility strictly increasing in each good individually (partial)
 Diminishing marginal utility in each good individually
Alternative notation: du/dc OR u’(c) OR uc(c) OR u1(c)
6UTILITY FUNCTIONS
Review of Consumer Theory
 Describe how much “happiness” or “satisfaction” an individual
experiences from “consuming” goods – the benefit of consumption
 Marginal Utility
 The extra total utility resulting from consumption of a
small/incremental extra unit of a good
 Mathematically, the (partial) slope of utility with respect to that good
 One-good case: u(c), with du/dc > 0 and d2u/dc2 < 0
 Recall interpretation: strictly increasing at a strictly decreasing rate
 Diminishing marginal utility
 Two-good case: u(c1, c2), with ui(c1, c2) > 0 and uii(c1, c2) < 0 for
each of i = 1,2
 Utility strictly increasing in each good individually (partial)
 Diminishing marginal utility in each good individually
 Easily extends to N-good case: u(c1, c2, c3, c4,…, cN)
Alternative notation: du/dc OR u’(c) OR uc(c) OR u1(c)
7UTILITY FUNCTIONS
Review of Consumer Theory
 One-good case
c
u(c)
Slope (marginal utility) asymptotes to (but never
reaches…) zero
Example: u(c) = ln c or u(c) = sqrt(c)
8UTILITY FUNCTIONS
Review of Consumer Theory
 One-good case
 Two-good case
c
u(c)
Slope (marginal utility) asymptotes to (but never
reaches…) zero
Example: u(c) = ln c or u(c) = sqrt(c)
Example: u(c1, c2) = ln c1 + ln c2 or
u(c1, c2) = sqrt(c1) + sqrt(c2)
u(c1,c2)
9UTILITY FUNCTIONS
Review of Consumer Theory
 One-good case
 Two-good case
c
u(c)
Slope (marginal utility) asymptotes to (but never
reaches…) zero
Example: u(c) = ln c or u(c) = sqrt(c)
Example: u(c1, c2) = ln c1 + ln c2 or
u(c1, c2) = sqrt(c1) + sqrt(c2)
c1
u(c1,c2)
Viewed in
good-by-
good space
u(c1,c2)
10
UTILITY FUNCTIONS
Review of Consumer Theory
 One-good case
 Two-good case
c
u(c)
Slope (marginal utility) asymptotes to (but never
reaches…) zero
Example: u(c) = ln c or u(c) = sqrt(c)
Example: u(c1, c2) = ln c1 + ln c2 or
u(c1, c2) = sqrt(c1) + sqrt(c2)
c1
u(c1,c2)
c2
u(c1,c2)
Viewed in
good-by-
good space
u(c1,c2)
11
UTILITY FUNCTIONS
Review of Consumer Theory
Alternative views
Emphasizing
the contours Indifference Curve: the
set of all consumption
bundles that deliver a
particular level of
utility/happiness
12
UTILITY FUNCTIONS
Review of Consumer Theory
Alternative views
Emphasizing
the contours Indifference Curve: the
set of all consumption
bundles that deliver a
particular level of
utility/happiness
Viewing only
the contours
13
UTILITY FUNCTIONS
Review of Consumer Theory
 Marginal Rate of Substitution (MRS)
 Maximum quantity of one good consumer is willing to give up to obtain
one extra unit of the other good
 Graphically, the (negative of the) slope of
an indifference curve
 MRS is itself a function of c1 and c2
(i.e., MRS(c1, c2))
c1
c2
14
UTILITY FUNCTIONS
Review of Consumer Theory
 Marginal Rate of Substitution (MRS)
 Maximum quantity of one good consumer is willing to give up to obtain
one extra unit of the other good
 Graphically, the (negative of the) slope of
an indifference curve
 MRS is itself a function of c1 and c2
(i.e., MRS(c1, c2))
 MRS equals ratio of marginal utilities

 Using Implicit Function Theorem or alternatively totally differentiate utility
and set total differential to zero: dU!=0, which yields u1()dc1 + u2 ()dc2=0.
c1
c2
1 1 2
1 2
2 1 2
( , )
( , )
( , )
u c c
MRS c c
u c c

15
UTILITY FUNCTIONS
Review of Consumer Theory
 Marginal Rate of Substitution (MRS)
 Maximum quantity of one good consumer is willing to give up to obtain
one extra unit of the other good
 Graphically, the (negative of the) slope of
an indifference curve
 MRS is itself a function of c1 and c2
(i.e., MRS(c1, c2))
 MRS equals ratio of marginal utilities

 Using Implicit Function Theorem or alternatively totally differentiate utility
and set total differential to zero: dU!=0, which yields u1()dc1 + u2 ()dc2=0.
 Summary: whether graphically- or mathematically-formulated,
utility functions describe the benefit side of consumption
c1
c2
1 1 2
1 2
2 1 2
( , )
( , )
( , )
u c c
MRS c c
u c c

16
BUDGET CONSTRAINTS
The Graphics of Consumer Theory
 Describe the cost side of consumption
 One-good case (trivial): Pc = Y
 Assume income Y is taken as given by consumer for now…
17
BUDGET CONSTRAINTS
The Graphics of Consumer Theory
 Describe the cost side of consumption
 One-good case (trivial): Pc = Y
 Assume income Y is taken as given by consumer for now…
 Two-good case (interesting): P1c1 + P2c2 = Y
 Assume income Y is taken as given by consumer for now…
18
BUDGET CONSTRAINTS
The Graphics of Consumer Theory
 Describe the cost side of consumption
 One-good case (trivial): Pc = Y
 Assume income Y is taken as given by consumer for now…
 Two-good case (interesting): P1c1 + P2c2 = Y
 Assume income Y is taken as given by consumer for now…
Plotted in 2D-consumption-space
c1
c2
19
BUDGET CONSTRAINTS
The Graphics of Consumer Theory
 Describe the cost side of consumption
 One-good case (trivial): Pc = Y
 Assume income Y is taken as given by consumer for now…
 Two-good case (interesting): P1c1 + P2c2 = Y
 Assume income Y is taken as given by consumer for now…
Plotted in 2D-consumption-space
c1
c2
1 1 2 2c PP c Y 
Isolate c2 to
graph the budget
constraint
2 2 1 1c PP Yc  
1
2 1
2 2
Y
P
c
P
c
P
  
20
BUDGET CONSTRAINTS
The Graphics of Consumer Theory
 Describe the cost side of consumption
 One-good case (trivial): Pc = Y
 Assume income Y is taken as given by consumer for now…
 Two-good case (interesting): P1c1 + P2c2 = Y
 Assume income Y is taken as given by consumer for now…
Plotted in 2D-consumption-space
c1
c2
1 1 2 2c PP c Y 
Isolate c2 to
graph the budget
constraint
2 2 1 1c PP Yc  
1
2 1
2 2
Y
P
c
P
c
P
  
Slope = -P1/P2
21
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
22
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
c
u(c)
Y/P
23
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
c
u(c)
Y/P
24
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
 No decision to make here…
c
u(c)
Y/P
25
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
 No decision to make here…
 Two-good case
 How to optimally allocate Y across the two goods c1 and c2?
 A non-trivial decision problem…
c
u(c)
Y/P
c1
c2 Utility increasing
in the northeast
direction
26
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
 No decision to make here…
 Two-good case
 How to optimally allocate Y across the two goods c1 and c2?
 A non-trivial decision problem…
c
u(c)
Y/P
c1
c2 Utility increasing
in the northeast
direction
27
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
 No decision to make here…
 Two-good case
 How to optimally allocate Y across the two goods c1 and c2?
 A non-trivial decision problem…
c
u(c)
Y/P
c1
c2 Utility increasing
in the northeast
direction
Optimal choice
occurs at point of
tangency between
budget line and an
indifference curve Highest attainable
indifference curve
28
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
 No decision to make here…
 Two-good case
 How to optimally allocate Y across the two goods c1 and c2?
 A non-trivial decision problem…
c
u(c)
Y/P
c1
c2 Utility increasing
in the northeast
direction
Optimal choice
occurs at point of
tangency between
budget line and an
indifference curve Highest attainable
indifference curve
OPTIMALITY CONDITION:
At the optimal choice,
MRS = slope of budget line
29
CONSUMER OPTIMIZATION
The Graphics of Consumer Theory
 Consumer’s decision problem: maximize utility subject to budget
constraint – bring together both cost side and benefit side
 One-good case
 Trivially, choose c = Y/P
 No decision to make here…
 Two-good case
 How to optimally allocate Y across the two goods c1 and c2?
 A non-trivial decision problem…
c
u(c)
Y/P
c1
c2 Utility increasing
in the northeast
direction
Optimal choice
occurs at point of
tangency between
budget line and an
indifference curve Highest attainable
indifference curve
OPTIMALITY CONDITION:
At the optimal choice,
MRS = slope of budget line
ratio of marginal utilities = price ratio
30
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Consumer optimization a constrained optimization problem
 Maximize some function (economic application: utility function)…
 …taking into account some restriction on the objects to be maximized
over (economic application: budget constraint)
 Lagrange Method: mathematical tool to solve constrained
optimization problems
31
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Apply Lagrange tools to consumer optimization
 Objective function: u(c1,c2)
 Constraint: g(c1,c2) = Y – P1c1 – P2c2 = 0
32
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Apply Lagrange tools to consumer optimization
 Objective function: u(c1,c2)
 Constraint: g(c1,c2) = Y – P1c1 – P2c2 = 0
 Step 1: Construct Lagrange function
 Step 2: Compute first-order conditions with respect to c1, c2, λ
 1 2 1 2 1 1 2 2( , , ) ( , )L c c u c c Y Pc Pc    
33
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Apply Lagrange tools to consumer optimization
 Objective function: u(c1,c2)
 Constraint: g(c1,c2) = Y – P1c1 – P2c2 = 0
 Step 1: Construct Lagrange function
 Step 2: Compute first-order conditions with respect to c1, c2, λ
 1 2 1 2 1 1 2 2( , , ) ( , )L c c u c c Y Pc Pc    
2 1 2 2
1 1
1 1 1
2 2
2( , ) 0
( , ) 0
0
u c c
u c c P
P
Y Pc P c

 





34
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Apply Lagrange tools to consumer optimization
 Objective function: u(c1,c2)
 Constraint: g(c1,c2) = Y – P1c1 – P2c2 = 0
 Step 1: Construct Lagrange function
 Step 2: Compute first-order conditions with respect to c1, c2, λ
 1 2 1 2 1 1 2 2( , , ) ( , )L c c u c c Y Pc Pc    
1 1 2
1 1 2 1
2 1 2
2
2
( , ) 0
( , ) 0
0
u c c P
u c c
c
P
Y P P c


 



 
35
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Apply Lagrange tools to consumer optimization
 Objective function: u(c1,c2)
 Constraint: g(c1,c2) = Y – P1c1 – P2c2 = 0
 Step 1: Construct Lagrange function
 Step 2: Compute first-order conditions with respect to c1, c2, λ
 1 2 1 2 1 1 2 2( , , ) ( , )L c c u c c Y Pc Pc    
1 1 2 1
2 1 2 2
1 1 2 2
( , ) 0
( , ) 0
0
u c c P
u c c P
Y Pc P c


 
 
  
1)
2)
3)
36
LAGRANGE ANALYSIS
The Mathematics of Consumer Theory
 Apply Lagrange tools to consumer optimization
 Objective function: u(c1,c2)
 Constraint: g(c1,c2) = Y – P1c1 – P2c2 = 0
 Step 1: Construct Lagrange function
 Step 2: Compute first-order conditions with respect to c1, c2, λ
 Step 3: Solve (focus on eliminating multiplier from eqns 1 & 2)
 1 2 1 2 1 1 2 2( , , ) ( , )L c c u c c Y Pc Pc    
* *
1 1 2 1
* *
2 1 2 2
( , )
( , )
u c c P
u c c P

i.e., MRS = price ratio
OPTIMALITY CONDITION
1 1 2 1
2 1 2 2
1 1 2 2
( , ) 0
( , ) 0
0
u c c P
u c c P
Y Pc P c


 
 
  
1)
2)
3)
37
THE THREE MACRO (AGGREGATE) MARKETS
The Road Ahead
 Goods Markets
 Labor Markets
 Financial/Capital/Savings/Asset
Markets
 Will put micro-foundations under all three
c
P
labor
wage
capital
interest
rate