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UNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
Example Problems
Example (1)
In the problem below, find the demand curve for X and the Engel curve for X.
max
U = 2*ln(3X)
X
subject to:
P
x
X ≤ I, where P
x
is the perunit price of X, and I is income allocated to purchase of X.
Note that both P
X
and I are parameters.
Converting the problem into an equivalent Lagrangian problem, and introducing the new choice variable
λ
into the problem as a Lagrangian multiplier:
max
L = 2*ln(3X) +
λ
(I  P
x
X)
X,
λ
subject to:
nothing
F.O.C.'s:
(1)
0
Px
3
)
X
3
1
(
2
X
L
=
λ
⋅

⋅
⋅
=
∂
∂
(2)
0
X
Px
I
L
=
⋅

=
λ
∂
∂
The FOC equations provide us with two equations in two unknowns, X and
λ
.
Solve these two equations
for the solution values of X and
λ
.
There are several ways to work the algebra to solve these two
equations.
One way is illustrated below:
Solve FOC (2) for X:
I  P
x
X = 0
I = P
x
X
X
*
= I/P
x
(Note: we typically use a star "*" superscript to denote a solution value.)
Note: We know that the equation above gives the solution for X because X is alone on one side of
the equation, and everything else on the other side of the equation is a parameter.
At this point, we have the demand curve for X.
The
demand curve for X
is:
X
*
= I/P
x
,
where X
*
and P
X
are allowed to vary, and I is held constant.
We also have the Engel curve for X.
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 Fall '08
 Dumas
 Economics, Microeconomics

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