PAM 200 – Microeconomics - Week 3 Lecture Notes
I.
MATHEMATICS OF TANGENCY BETWEEN BUDGET CONTRAINTS AND
INDIFFERENCE CURVES
As was shown graphically in the previous set of notes, for interior solutions to maximizing utility,
the consumer will select a point along the budget constraint that is tangent to an indifference curve.
Doing so will ensure that the consumer is on the indifference curve with the highest achievable level
of utility.
The point of this section is to determine a mathematical way of identifying this maximum utility
point.
Doing so will require three steps: 1) Finding the slope of the budget constraint; 2) finding the
slope of the indifference curve; 3) Finding the point on the budget constraint where the slope of the
budget constraint is tangent to the highest utility indifference curve.
1)
Finding the Slope of the Budget Constraint
For a budget constraint of the form
X
P
Y
P
M
X
Y
+
=
, in the Week 1 Notes, it was demonstrated
that the slope of the budget constraint is:
Y
X
P
P
X
Y
Slope
−
=
Δ
Δ
=
.
2)
Finding the Slope of an Indifference Curve
Since we are now using mathematics, it is appropriate to represent a consumer’s utility with a
mathematical expression of the form:
(
)
X
Y
U
U
,
=
.
This simply means that the amount of utility,
U, is a function of the amount of good Y and the amount of good X that is consumed.
Recall from the Week 2 Notes that marginal utility is the change in utility from a change in the
amount of goods consumed.
We can update the definition for marginal utility to include two goods:
Y and X.
That is,
Y
U
MU
and
X
U
MU
Y
Y
X
X
Δ
Δ
=
Δ
Δ
=
,
where
is the marginal utility from one additional unit of X, and
is the marginal utility of
one additional unit of Y.
By rearranging each of the two expressions we obtain expressions for the
change in utility from a change in the amount of X or Y.
That is,
X
MU
Y
MU
Y
MU
U
and
X
MU
U
Y
Y
X
X
Δ
⋅
=
Δ
Δ
⋅
=
Δ
Summing these two expressions yields an expression for the full change in utility if both X and Y are
changed simultaneously.
Y
MU
X
MU
U
U
U
Y
X
Y
X
Δ
⋅
+
Δ
⋅
=
Δ
+
Δ
=
Δ

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