Unformatted text preview: Chapter 4 Consumer
Choice 1 Chapter Four Overview
1. The Budget Constraint
2. Consumer Choice
3. Duality
4. Some Applications
5. Revealed Preference Chapter Four 2 Key Definitions
Budget Set:
• The set of baskets that are affordable
Budget Constraint:
• The set of baskets that the consumer may purchase given the limits of the available income. Budget Line:
• The set of baskets that one can purchase when spending all available income. PxX + PyY = I
Y = I/Py – (Px/Py)X
Chapter Four 3 The Budget Constraint
Assume only two goods available: X and Y
• Price of x: Px ; Price of y: Py • Income: I
Total expenditure on basket (X,Y): PxX + PyY The Basket is Affordable if total expenditure does not exceed total Income: PXX + PYY ≤ I
Chapter Four 4 A Budget Constraint Example
Two goods available: X and Y
I = $10
Px = $1
Py = $2
All income spent on X → I/Px
units of X bought
All income spent on Y → I/Py
units of X bought Budget Line 1: 1X + 2Y = 10
Or
Y = 5 – X/2
Slope of Budget Line = ‐Px/Py = ‐1/2 Chapter Four 5 A Budget Constraint Example
Y I/PY= 5 Budget line = BL1 A • ‐PX/PY = ‐1/2
B •C • I/PX = 10
Chapter Four X
6 Budget Constraint
• Location of budget line shows what the income level is.
• Increase in Income will shift the budget line to the right.
– More of each product becomes affordable • Decrease in Income will shift the budget line to the left.
– less of each product becomes affordable
Chapter Four 7 A Budget Constraint Example
Y Shift of a budget line I = $12
PX = $1
PY = $2 If income rises, the budget line shifts parallel to the right (shifts out) 6 Y = 6 ‐ X/2 …. BL2
5 If income falls, the budget line shifts parallel to the left (shifts in) BL2
BL1
10
Chapter Four 12 X
8 Budget Constraint
• Location of budget line shows what the income level is.
• Increase in Income will shift the budget line to the right.
– More of each product becomes affordable • Decrease in Income will shift the budget line to the left.
– less of each product becomes affordable
Chapter Four 9 A Budget Constraint Example
Y Rotation of a budget line
If the price of X rises, the budget
line gets steeper and the
horizontal intercept shifts in I = $10
PX = $1
BL1 PY = $3 6
5 If the price of X falls, the budget
line gets flatter and the
horizontal intercept shifts out Y = 3.33 ‐ X/3 …. BL2
3.3
3
BL2 10
Chapter Four X 10 A Budget Constraint Example
Two goods available: X and Y
I = $800
Px = $20
Py = $40
All income spent on X → I/Px
units of X bought
All income spent on Y → I/Py
units of X bought Budget Line 1: 20X + 40Y = 800
Or
Y = 20 – X/2
Slope of Budget Line = ‐Px/Py = ‐1/2 Chapter Four 11 A Budget Constraint Example Chapter Four 12 Consumer Choice
Assume:
Only non‐negative quantities
"Rational” choice: The consumer chooses the
basket that maximizes his satisfaction given the
constraint that his budget imposes. Consumer’s Problem:
Max U(X,Y) Subject to: PxX + PyY < I Chapter Four 13 Interior Optimum
Interior Optimum: The optimal consumption basket is
at a point where the indifference curve is just tangent
to the budget line.
A tangent: to a function is a straight line that has the
same slope as the function…therefore…. MRSx,y = MUx/MUy = Px/Py
“The rate at which the consumer would be willing to
exchange X for Y is the same as the rate at which they
are exchanged in the marketplace.”
Chapter Four 14 Interior Consumer Optimum
Y
B • Preference Direction • Optimal Choice (interior solution) IC
C
• BL 0
Chapter Four X
15 Interior Consumer Optimum Chapter Four 16 Interior Consumer Optimum
• U (X,Y) = XY and MUx = Y while MUy = X
• I = $1,000
• PX = $50 and PY = $200
• Basket A contains (X=4, Y=4)
• Basket B contains (X=10, Y=2.5)
• Question: • Is either basket the optimal choice for the consumer? Basket A: MRSx,y = MUx/MUy = Y/X = 4/4 = 1
Slope of budget line = ‐Px/Py = ‐1/4 Basket B: MRSx,y = MUx/MUy = Y/X = 1/4
Chapter Four 17 Interior Consumer Optimum
Y 50X + 200Y = I 2.5 • 0 10 U = 25 Chapter Four X
18 Equal Slope Condition
MUx/Px = MUy/Py
“At the optimal basket, each good gives equal bang for the buck”
Now, we have two equations to solve for two unknowns (quantities of X and Y in the optimal basket):
1. MUx/Px = MUY/PY
2. PxX + PyY = I
Chapter Four 19 Contained Optimization
What are the equations that the
optimal consumption basket must
fulfill if we want to represent the
consumer’s choice among three
goods? • MUX / P = MU / P is an example of “marginal reasoning” to maximize
Y
Y
X
• PX X + P Y = I reflects the “constraint”
Y Chapter Four 20 Contained Optimization
U(F,C) = FC
PF = $1/unit
PC = $2/unit
I = $12 Solve for optimal choice of food
for optimal choice of food
and clothing
Chapter Four 21 Some Concepts
Corner Points: One good is not being consumed at all – Optimal basket lies on the axis
Composite Goods: A good that represents the collective expenditure on every other good except the commodity being considered Chapter Four 22 Some Concepts Chapter Four 23 Some Concepts Chapter Four 24 Some Concepts Chapter Four 25 Some Concepts Chapter Four 26 Some Concepts Chapter Four 27 Duality
The mirror image of the original (primal) constrained
optimization problem is called the dual problem.
Min PxX + PyY
(X,Y) subject to: U(X,Y) = U*
where: U* is a target level of utility. If U* is the level of utility that solves the
primal problem, then an interior
optimum, if it exists, of the dual
problem also solves the primal problem.
Chapter Four 28 Optimal Choice
Y
Example: Expenditure Minimization • Optimal Choice (interior solution) U = U*
Decreases in
expenditure level PXX + PYY = E* 0
Chapter Four X
29 Optimal Choice
Y Example: Expenditure Minimization
50X + 200Y = E 25 = XY (constraint)
Y/X = 1/4 (tangency condition) 2.5 • 0 10
Chapter Four U = 25
X
30 Revealed Preference
Suppose that preferences are not
known. Can we infer them from
purchasing behavior? If A purchased, it must be preferred
to all other affordable bundles Chapter Four 31 Revealed Preference
Suppose that preferences are “standard” – then:
All baskets to the Northeast of A must be preferred to A.
This gives us a narrower range over which indifference curve must lie
This type of analysis is called revealed preference analysis. Chapter Four 32 ...
View
Full
Document
This note was uploaded on 09/15/2011 for the course ECON 300 taught by Professor Zh during the Spring '11 term at SUNY Albany.
 Spring '11
 zh

Click to edit the document details