Download Unit 3.

Unit 3.
The Theory of Individual
Economic Behavior (Ch. 4)
Budget Constraint
 The maximum Q
combinations of goods that
can be purchased given one’s
income and the prices of the
goods.
Budget Constraint Variables
I (or M) =
the amount of income or money that a consumer
has to spend on specified goods and services.
X =
the quantity of one specific good or one specific bundle of
goods
Y =
the quantity of a second specific good or second specific
bundle of goods
Px =
the price or per unit cost of X
PY =
the price or per unit cost of Y
Budget Line Equation
• Income = expenses
• I = PxX+PYY
• Y = l/PY – (Px/PY)X
 straight line equation
 vert axis intercept = I/PY
 slope = dY/dX = -Px/PY
The Opportunity Set
Y
I/PY
Budget Line
PX
PY
I/PX
X
Budget Line: Axis Intercepts
& Slope
• Vertical Axis Intercept
=
=
I/PY
max Y (X = 0)
• Horizontal Axis Intercept
=
=
I/PX
max X (Y = 0)
• - Slope
= PX/PY
= ‘inverse’ P ratio
= X axis good P/Y axis good P
= Y/X
Budget Line Slope
Equation:
I Px
y

X
Py Py
¯Slope = ¯ dy  Px  inverse P ratio
dx
=
Py
rate at which y CAN be exchanged for x
(holding $ expenses constant)
e.g.
Px $10 2  y

 
Py $5 1  x
=> 2y can be exchanged for 1x
Changes in the Budget Line
• Changes in Income
-
Increases lead to a parallel,
outward shift in the budget line.
Decreases lead to a parallel,
downward shift.
Y
X
Changes in the Budget Line
• Changes in Price
-
-
A decrease in the price of good
X rotates the budget line counterclockwise.
An increase rotates the budget
line clockwise.
Y
New Budget Line for
a price decrease.
X
Your Preferences?
• Lunch
A:
B:
C:
1 drink, 1 pizza slice
1 drink, 2 pizza slices
2 drinks, 1 pizza slice
• Entertainment
A:
B:
C:
1 movie, 1 dinner
1 movie, 2 dinners
2 movies, 1 dinner
For each, indicate which of the following you prefer:
A vs B, B vs C, A vs C
Utility Concepts
• Utility:
satisfaction received from consuming goods
• Cardinal utility:
satisfaction levels that can be measured or specified with
numbers (units = ‘utils’)
• Ordinal utility:
satisfaction levels that can be ordered or ranked
• Marginal utility:
the additional utility received per unit of additional unit of an item
consumed (U/ X)
An Understanding of Concepts Related to
Indifference Curves and Utility Should Help
One:
1. Get along better with other people, by doing
things that increase their utility.
2. Make better business decisions that result in
improved customer satisfaction and, thus,
more sales.
3. Understand what motivates people and why
they behave the way they do, including how
people are likely to respond to economic
changes.
Utility Assumptions
1. Complete (or continuous)  can rank all
bundles of goods
2. Consistent (or transitive)  preference
orderings are logical and consistent
3. Consumptive (nonsatiation)  more of a
‘normal’ good is preferred to less
More of a Good is Preferred to
Less
The shaded area represents those combinations of X and Y that are
unambiguously preferred to the combination X*, Y*. Ceteris paribus,
individuals prefer more of any good rather than less. Combinations
identified by “?” involve ambiguous changes in welfare since they
contain more of one good and less of the other.
Indifference Curve Analysis
Indifference Curve
• A curve that defines the
combinations of 2 or more
goods that give a consumer
the same level of satisfaction.
Marginal Rate of Substitution
• The rate at which a consumer
is willing to substitute one good
for another and stay at the same
satisfaction level.
Investment Alternatives
Fund
Return
Safety
A
2.89%
Hi
B
6.59%
Med
C
7.29%
Low
1. Ida Dontcare is indifferent regarding all
three investment alternatives.
U(A) = U(B) = U(C)
2. Ralph Returnman prefers C over B and
prefers B over A.
U(C) > U(B) > U(A)
3. Sally Safetyfirst prefers A over B and
prefers B over C.
U(A) > U(B) > U(C)
MRS & MU
• MRS
= - slope of indifference curve
= -Y/ X
= the rate at which a consumer is willing to
exchange Y for 1more (or less) unit of X
U =
0 along given indiff curve
=
MUx(X)+MUY(Y) = 0
=
- Y/ X = MUx/MUY
=
- slope = inverse MU ratio
Indifference Curve Shape
- Slope = MRS = marginal rate of substitution
= Rate at which consumer is willing
to exchange y for x = ΔY/ΔX
Two ways to calculate:
1) Given utility function equation, derive inverse
MU ratio =
MU x
MU y
2) Given indifference curve equation,
derive ¯dy/dx directly.
e.g.
1)
MU x 2  y
u  2 x  1y 
 
MU y 1  x
 dy 2
2) y  U  2 x 

dx 1
=> Willing to exchange 2y for 1x
Types of Goods & Utility
Functions
1. Normal
2. Perfect Substitutes
3. Perfect Complements
Normal Goods
= goods for which a consumer’s willingness to
exchange one good for another varies
depending on Q’s of each
Represented by U = xαYB

 dy MU x x  1 y B
Y


 B 1 
dx
MU y B x y
BX
Perfect Substitutes
= goods for which a consumer is willing to
exchange one good for another at a constant
rate.
 Represented by U = αx + BY

Equation of indifferent curve = Y  U / B 
X
B
_
MU x 
 dy
 MRS 

dx
MU y B
(= a constant)
Perfect Complements
= goods that are used in fixed or constant
proportions with one another
Represented by U = min [αX, βY]
A consumer’s U = whichever is the least, αX
or βY
 too much of one good without more of the
other good will not increase one’s utility
 values where αX = βY lie along line (solve for
Y) where Y = (α/β)X
Non ‘Goods’ & Indifference
Curves
 1 Good and 1 ‘Neutral’
1 Good and 1 ‘Bad’
Utility Maximization
Words
Spend one’s income so as to get the most satisfaction
possible
Graph
Go to the highest indifference curve that is within reach of
the budget line
Math
Normal goods: point of tangency (equal slopes condition)
between budget line and highest attainable indifference
curve
Perfect substitutes: corner solution normally; if slope of
budget line flatter than slope of indifference curves => All
X; else => All Y
Perfect complements; pt. of intersection between budget
line and line through vertex pts of indifference curves
Consumer Equilibrium
(U Max)
• The equilibrium
consumption
bundle is the
affordable bundle
that yields the
highest level of
satisfaction.
Equal Slopes Condition
(for consumer equilibrium)
• MUX/MUY = PX/PY
• MUX/PX = MUY/PY
Consumer Equilibrium
(Perfect Substitutes)
Consumer Equilibrium
(Perfect Complements)
Changes in Price
• Substitute Goods
– An increase (decrease) in the price of good X
leads to an increase (decrease) in the consumption
of good Y.
• Complementary Goods
– An increase (decrease) in the price of good X
leads to a decrease (increase) in the consumption
of good Y.
Complementary Goods
Changes in Income
• Normal Goods
– Good X is a normal good if an increase (decrease) in
income leads to an increase (decrease) in its
consumption.
• Inferior Goods
– Good X is an inferior good if an increase (decrease)
in income leads to a decrease (increase) in its
consumption.
Normal Goods
Figure 4-13. An increase in the price of good X leads
to a substitution effect (A to B) and an income effect
(B to C).
Substitution effect – The movement along a given indifference curve that results
from a change in the relative prices of goods, holding real income constant.
Income effect – The movement from one indifference curve to another that results
from the change in real income caused by a price change.
Individual Demand Curve
• An individual’s
demand curve is
derived from each
new equilibrium
point found on the
indifference curve as
the price of good X
is varied.
Market Demand
• The market demand curve is the horizontal summation of
individual demand curves.
• It indicates the total quantity all consumers would
purchase at each price point.
A Classic Marketing Application
Variables:
• W =
• L =
•
•
•
•
P
Q
C
N
=
=
=
=
hrs/day worked (labored)
hrs/day leisured (happy)
Note: L = 24 – W
hourly wage or ‘pay’ rate
consumer good quantity
price per unit of Q
nonlabor income
Constraint:
 expenses = income
 CQ = N + PW
 Q = (N + 24P)/C –
(P/C)L
If C = 1,
Q = (N + 24P) - PL
Intertemporal Choice Model
• Inter  between; temporal  time pds
• Time pds  current (0) or next yr (1)
Variables:
C0 and C1 = Q of goods consumed
I0 and I1 = income levels
P = price of consumer goods (P0 = P1)
r = interest rate
• Objective (goal) = Max U = f(C0, C1)
• Constraint: PV of Income = PV of Expenses
Intertemporal Saving &
Borrowing Facts
• If you save an extra $ (i.e. reduce current
pd consumption by a $), you can
INCREASE future pd consumption by the
FV of the $.
• If you borrow a $ against your future
income (i.e. agree to pay back a $
principal and interest), you can
INCREASE current pd consumption by the
PV of the $.
Math Summary of Intertemporal
Choice Problem
• Max U (C0, C1)
• Subj. to PV of income = PV of expenses
I1
P1C1
I0 
 P0 C0 
(l  r )
(l  r )
I1
C1
I0 
 C0 
assume P0  P1  1
(l  r )
(l  r )
C1  I 0 (l  r )  I 1  (l  r ) C0
Intertemporal Choice Problem
Graph