the consumer problem and the budget constraint overheads

The Consumer Problem

and the Budget Constraint

Overheads

The fundamental unit of analysis

in consumption economics is the

individual consumer

The underlying assumption in

consumption analysis is that all

consumers possess a preference

ordering which allows them to rank

alternative states of the world.

The behavioral assumption in

consumption analysis is

that consumers make choices consistent

with their underlying preferences

The main constraint facing consumersin determining which goodsto purchase and consume is

This is called the budget constraint

the amount of income that they can spend


The consumer problem is to maximize

the consumer has to spend.

the satisfaction that comes from theconsumption of various goods

subject to the amount of income


Maximize satisfaction

subject to

income

Definition of the budget constraint

A consumer’s budget constraint identifies

which combinations of goods and services

the consumer can afford with a limited budget,

at given prices

Notation

Income - I

Quantities of goods - q1, q2, . . . qn

Prices of goods - p1, p2,. . . pn

Number of goods - n

Budget constraint with 2 goods

p1q1 p2q2 I

p1q1 p2q2 p3q3 pn qn I

Budget constraint with n goods

Example

Income = I = $1.20

q1 = Reese’s Pieces

p1 = price of Reese’s Pieces = $0.30

q2 = Snickers

p2 = price of Snickers = $0.20

Graphical Analysis of Budget Set

012345

0 1 2 3 4 5 6 7Snickers

Reese’s


012345

0 1 2 3 4 5 6 7q2

q1


012345

0 1 2 3 4 5 6 7q2

q1

4 Reese’s -- 0 Snickers

Cost = 4 x 0.30 + 0 x 0.20 = $1.20


012345

0 1 2 3 4 5 6 7q2

q1


Cost = 0 x 0.30 + 6 x 0.20 = $1.20


012345

0 1 2 3 4 5 6 7q2

q1


Cost = 2 x 0.30 + 3 x 0.20 = $1.20


012345

0 1 2 3 4 5 6 7q2

q1


Cost = 2 x 0.30 + 1 x 0.20 = $.80


012345

0 1 2 3 4 5 6 7q2

q1


Cost = 3 x 0.30 + 3 x 0.20 = $1.50


012345

0 1 2 3 4 5 6 7q2

q1

There are many different combinationsOnly some combinations are feasible


012345

0 1 2 3 4 5 6 7q2

q1

Some combinations exactly exhaust income


012345

0 1 2 3 4 5 6 7q2

q1

We say these points lie along the budget line


012345

0 1 2 3 4 5 6 7q2

q1

Or on the boundary of the budget set


012345

0 1 2 3 4 5 6 7q2

q1

Points inside or on the line are affordable

Graphical Analysis of Budget SetBudget Set

012345

0 1 2 3 4 5 6 7q2

q1

Points outside the line are not affordable

Slope of the Budget Constraint - q1 = h(q2)

p1 q1 p2q2 I

p1q1 I p2 q2

q1 Ip1

p2

p1q2

So the slope is -p2 / p1


012345

0 1 2 3 4 5 6 7q2

q1

0 Snickers -- 4 Reese’s

q2 = - 3


q1q1 = 2

Δq1

Δq2

2 3

23


012345

0 1 2 3 4 5 6 7q2

q1



q1 = 2

Δq1

Δq2

2 3

23

p2

p1

0.200.30

q2 = - 3

Numerical Example

I = $1.20, p1 = 0.30, p2 = 0.20

0.30q1 0.20q2 1.20

0.30q1 1.20 0.20q2

q1 1.200.30

0.200.30

q2

4 23q2

1

5 6 74321

2

3

4

5

Budget Constraint - 0.3q1 + 0.2q2 = $1.20

Affordable

Not Affordable

q1

q2

q1 4 23q2

0.3q1 1.2 0.2q2

1

5 6 74321

2

3

4

5


Affordable

Not Affordable

q2

q1 Double prices and incomeDouble prices and income

q1 4 23q2


0.6q1 2.4 0.4q2

1

5 6 74321

2

3

4

5


Affordable

q2

q1

Not Affordable

Double pDouble p11 from 0.3 to 0.6 from 0.3 to 0.6

q1 2 13q2


0.6q1 1.2 0.2q2

Just to review how to solveBudget Constraint - 0.6q1 + 0.2q2 = $1.20

0.60q1 1.20 0.20q2

q1 1.200.60

0.200.60

q2

q1 2 13

q2

1

5 6 74321

2

3

4

5


Affordable

q2

q1

Raise pRaise p22 from 0.2 to 0.3 from 0.2 to 0.3

Not Affordable

q1 4 q2


0.3q1 1.2 0.3q2

1

5 6 74321

2

3

4

5

q1

q2

Change in Income

Budget Constraint0 - 0.3q1 + 0.2q2 = $1.20


q1 2 23q2

0.3q1 0.6 0.2q2

Change in Price of Good 1 (price rises)


1

5 6 74321

2

3

4

5

q1

q2


Change in Price of Good 1 (price falls)



1

5 6 74321

2

3

4

5

q1

q2

Change in Price of Good 2 (price rises)



1

5 6 74321

2

3

4

5

q1

q2

The End

Graphical Analysis of Budget SetBudget Set

012345

0 1 2 3 4 5 6 7q2

q1

the consumer problem and the budget constraint overheads

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