The Ohio State University
Department of Economics
Econ 501.02—Prof. James Peck
Equations for the Midterm
Axiom 1: Preferences are complete: for any two bundles, A and B, exactly
one of the following is true. (1) A is preferred to B,
A
P
B
, (2) B is preferred to
A,
B
P
A
,or(3)Aisind
i
f
erent to B,
A
I
B
.
Axiom 2: Preferences are re
f
exive:
A
I
A
.
Axiom 3: Preferences are transitive:
A
P
B
and
B
P
C
⇒
A
P
C
.
Axiom 4: Preferences are continuous. If
A
P
B
,andi
f
C
is su
ﬃ
ciently close
to
B
,then
A
P
C
.
Axioms 1-4 allow preferences to be represented graphically by
indi
f
erence
curves
, and by a utility function,
u
.
A
P
B
if and only if
u
(
x
A
,y
A
)
>u
(
x
B
B
)
.
Axiom 5: More is better.
Starting with the bundle
A
=(
x
A
A
)
,th
en
increasing any of the goods in
A
yields a new bundle that is preferred to
A
.
Axiom 6: All indi
f
erence curves exhibit diminishing
marginal rates of sub-
stitution.
The MRS is the absolute value of the slope of the indi
f
erence curve.
The indi
f
erence curve
f
attens as you move along the indi
f
erence curve to the
right.
The marginal rate of substitution equals the ratio of marginal utilities:
MRS
yx
=
∂u
∂x
∂y
.
The utility maximization problem is
max
u
(
x, y
)
subject to
:
p
x
x
+
p
y
y
=
M
The
Lagrangean approach transforms a constrained optimization prob-
lem into an unconstrained problem of choosing x, y, and the Lagrange multiplier,
λ,
to maximize
L
=
u
(
x, y
)+
λ
[
M
−
p
x
x
−
p
y
y
]
.
By di
f
erentiating L with respect to x, y, and
λ
, and setting the derivatives
equal to zero, the resulting
F
rst order conditions are:
−
λp
x
=0
,
−
λp
y
,
and
M
−
p
x
x
−
p
y
y
.
x
∗
(
p
x
,p
y
,M
)
and
y
∗
(
p
x
y
)
are known as the
generalized demand func-
tions
.
1