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Demand Curves:
To solve analytically for Demand Curves use the following steps:
1. Set up a constrained optimization problem using the specified utility function and a general budget line
of the form I  p
1
x
1
 p
2
x
2
. Note I , p
1
, and p
2
are left as constants.
2. Solve the constrained optimization problem for the optimal x*
1
and x*
2
using one of the mathematical
techniques we have established. Your results will be in terms of I , p
1
, and p
2
and are called your Demand
Functions.
3. (Optional) To more easily plot your Good 1 Demand Curve, solve your optimal x
1
solution for p
1
. Plug
in points for x
1
and plot the corresponding values of p
1
. To more easily plot your Good 2 Demand Curve,
solve your optimal x
2
solution for p
2
. Plug in points for x
2
and plot the corresponding values of p
2
.
Question 1:
Assume an individual’s preferences can be represented by the CobbDouglas utility function:
U(x1, x2) = x
1
1/2
x
2
1/2
Solve for her x
1
and x
2
demand functions then plot her Good 1 demand curve.
Step 1: Set up a constrained optimization problem using the specified utility function and a general budget line
of the form
I  p1x1  p2x2
. Note I , p1, and p2 are left as constants.
Step 2: Solve the constrained optimization problem for the optimal x* 1 and x*2 using one of the mathematical
techniques we have established.
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View Full Document Step 3: Plot your Good 1 Demand Curve, solve your optimal x1 solution for p1. Plug in points for x1 and plot
the corresponding values of p1. To more easily plot your Good 2 Demand Curve, solve your optimal x2 solution
for p2. Plug in points for x2 and plot the corresponding values of p2.
Law of Demand: demand curves should be downward sloping. In other words, if we are consuming our
optimal constrained bundles, if the price of a good goes down we will purchase more of the good. We can
directly check this by calculating the derivative of a good’s demand function and checking the sign of the
derivative with respect to the price of the good.
NOTE: The Law of Demand holds when the derivative is negative. If
, the Law of Demand holds.
Question 2:
Assume we have determined that an individual’s demand functions are of the form:
x
1
= Ip
2
/ p
1
2
+p
2
and
x
2
= Ip
1
/ p
2
2
+p
1
Do both Goods 1 and 2 follow the Law of Demand?
Step 1: Calculate the derivative for x
1
.
Is the derivative negative?
The Law of demand
does / does not
hold for Good 1.
Step 2: Calculate the derivative for x
2
.
Is the derivative negative?
The Law of demand
does / does not
hold for Good 2.
Elasticity:
Elasticity
A,B
:
•
Ɛ
A,B
•
“Elasticity of A with respect to B”
•
“B Elasticity of A”
•
The percent change in A that results from a percent change in B.
Demand Elasticity
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This note was uploaded on 01/05/2012 for the course ECON 410 taught by Professor Codrin during the Fall '07 term at UNC.
 Fall '07
 Codrin
 Utility

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