2. How to Derive a Demand Function
Fallaw Sowell
Tepper School of Business
Carnegie Mellon University
August 2006

3. Motivation An Alaskan oil pipeline is shutdown.
What will be the impact on the demand for “our” product?
It would be nice to have a function that gave the answer
-- quantity demanded for good i
-- per unit price for good j
-- consumer’s income to be spent
-- Demand Function

4. Basic Application of Derivatives
You have just learned to use derivatives to
determine critical values of a function
(typically maxima or minima).
Same basic approach is used to derive demand functions.
Make the connection between these today.
A preview of how these tools will be used.
More QSRP, Mini One Quant class(es), before …
problem sets in Managerial Economics during mini 2.

5. Where a Parabola Takes its Maximum
Find the value of X where the parabola takes its maximum.

6. Where a Parabola Takes its Maximum
Find the value of X where a parabola takes its maximum.
First derivative
First derivative set equal to zero.
One equation in one unknown.
The Solution.

7. Where a Parabola Takes its Maximum

8. The Common Structure Parabola Demand Function Objective Function
First Order
Conditions
one equation in
one unknown
N+1 equations in
N+1 unknowns
Solutions

9. Derive the Demand Function
New Math Tools to
Derive the Demand Function
Derivatives of a multivariate function.
Partial derivative – Derivative holding all the other
variables fixed.
(Section 17.2 in your book)
Optimal value of a multivariate function when a constraint
must be satisfied. Technique of Lagrange Multiplier.
(Section 17.8 in your book)
More QSRP, Mini One Quant class(es), before …
problem sets in Managerial Economics during mini 2.

10. Derive the Demand Function
New Economic Models to
Derive the Demand Function
Utility function – Gives the level of happiness for a
give bundle of goods. Typically denoted
-- quantity demanded for good i
-- per unit price for good i
-- consumer’s income to be spent
Budget Constraint – The consumer will spend all of
their income purchasing goods to consume.

11. The Basic Consumer Problem (N=2)
Go buy things to make yourself as happy as possible.
Select quantities of goods to maximize utility subject to
the constraint that you cannot spend more than your
income
max
s.t.
To solve this form the function.

12. The Objective Function
This function is called a Lagrangian.
The new variable is called the Lagrange multiplier.
Just like for the parabola, take derivatives with
respect to each of the variables
and set them equal to zero.
and

13. The First Order Conditions
Partial derivatives – derivative holding the other
variables constant.
Solve this system of 3 equations in 3 unknowns.

14. The Solutions Demand Functions – very useful in business.
Lagrange multiplier at the optimal consumption bundle.
Not that important in Economic business applications.
Very important in other business applications.

15. Example with CES Utility Function

16. First Order Conditions (FOC’s)
3 equations in 3 unknowns

17. The Demand Functions

18. Demand function for good

19. Cobb-Douglas Utility function
Additional Example
Cobb-Douglas Utility function

20. A quasilinear utility function
Additional Example
A quasilinear utility function