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ECON 520 (2011 Fall)
Kansas State University
Zijun Luo
Page
1
of
9
LECTURE 2:
Budget Constraint and Consumer’s Utility
Maximization Problem
Study tips for the lecture
In a twogood world, understand why consumers will always use up all their income
when deriving and drawing a budget line.
It is very helpful that you can quickly work out changes in budget line when there are
changes in prices of the two goods and income.
The analysis of consumers’ optimum (UMP) is a standard section of any intermediate
microeconomics textbook, but it is also the most difficult one. The double
development (both graphically and mathematically) of the analysis are intended to
provide ample opportunity for you to learn the subject deeper.
Related to last chapter, it is very instructive to remember the solution to the UMP
under some special cases of preferences (indifference curves).
1. Budget Set and Budget Line
We all face constraints, even though we have plenty of money, we still have limited
time to consume. In this section, we try to model this situation. We will call this constraint a
“budget constraint”. It is not just about money budget.
You can apply the same analysis to
time budget/constraint. You will see an example of time constraint in Homework Two
.
Definition of Budget Set:
A
competitive
budget set is the set of all feasible consumption
baskets for the consumer who faces market prices
p
and has income
I.
1
Mathematically speaking, for a market that has only two goods,
X
and
Y
, we have
for a typical consumer, the following equation to be satisfied (page 94/93)
X X
YY
PQ
I
+≤
in which
X
P
and
Y
P
are prices of
X
and
Y
, respectively;
X
Q
and
Y
Q
are quantity of
X
and
Y
, respectively. “Feasible” in the above definition means the total expenditure of a
1
See Andreu MasColell, Michael D. Whinston, and Jerry R. Green,
Microeconomic Theory
, page 21,
Definition 2.D.1.
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View Full DocumentECON 520 (2011 Fall)
Kansas State University
Zijun Luo
Page
2
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9
consumer (
X X
YY
PQ
+
in this twogood case) must be less or equal to his or her total
income.
For example, if the price of banana is 1 dollar per pound, the price of apple is 2
dollars per pound, and you have 8 dollars income. Then the following baskets are
feasible/affordable to you:
Basket A: 4 pounds of apples;
Basket B: 7 pounds of bananas;
Basket C: 3 pounds of apples and 1 pound of bananas;
Basket D: 2 pounds of apples and 4 pounds of bananas;
Basket E: 1 pound of apples and 5 pounds of bananas.
Among these baskets, basket A and basket D use up all your budget, while the rest of
the baskets does not. On the other hand, the following baskets are not feasible/affordable:
Basket F: 6 pounds of apples;
Basket G: 4 pounds of apples and 2 pounds of bananas;
Basket H: 3 pounds of apples and 4 pounds of bananas.
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 Spring '11
 Schneider
 Utility

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