1. Finance 510: Microeconomic Analysis
Optimization

2. Don't Panic!

3. Functions
Optimization deals with functions. A function is simply a mapping from one space to another. (that is, a set of instructions describing how to get from one location to another)
Is the range
Is a function
Is the domain

4. Functions
For any
and
Note: A function maps each value of x to one and only one value for y

5. For example
For
Range
Domain

6. 20
For
Range
Y =14 5
Domain 5
X =3

7. 20 5 5 Here, the optimum occurs at x = 5 (y = 20) Range Domain5
Optimization involves finding the maximum value for y over an allowable range.

8. 5 10 What is the solution to this optimization problem?
There is no optimum because f(x) is discontinuous at x = 5

9. 12 6 What is the solution to this optimization problem?
There is no optimum because the domain is open (that is, the maximum occurs at x = 6, but x = 6 is NOT in the domain!) 6

10. 12 What is the solution to this optimization problem?
There is no optimum because the domain is unbounded (x is allowed to become arbitrarily large)

11. Necessary vs. Sufficient Conditions
Sufficient conditions guarantee a solution, but are not required
Necessary conditions are required for a solution to exist
Gas is a necessary condition to drive a car
A gun is a sufficient condition to kill an ant

12. The Weierstrass Theorem
The Weierstrass Theorem provides sufficient conditions for an optimum to exist, the conditions are as follows:
is continuous
over the domain of
The domain for
is closed and bounded

13. Derivatives Formally, the derivative of is defined as follows:
All you need to remember is the derivative represents a slope (a rate of change)

14. Slope =

15. Example:

16. Useful derivatives Linear Functions Exponents Logarithms Products
Composites

17. Practice Makes Perfect…

18. Unconstrained maximization
Strictly speaking, no problem is truly unconstrained. However, sometimes the constraints don’t “bite” (the constraints don’t influence the maximum)
First Order Necessary Conditions If
is a solution to the optimization problem or
then

19. An Example
Suppose that your company owns a corporate jet. Your annual expenses are as follows:
You pay your flight crew (pilot, co-pilot, and navigator a combined annual salary of $500,000.
Annual insurance costs on the jet are $250,000
Fuel/Supplies cost $1,500 per flight hour
Per hour maintenance costs on the jet are proportional to the number of hours flown per year.
Maintenance costs (per flight hour) = 1.5(Annual Flight Hours)
If you would like to minimize the hourly cost of your jet, how many hours should you use it per year?

20. An Example Let x = Number of Flight Hours
First Order Necessary Conditions

21. An Example
Hourly Cost ($)
Annual Flight Hours

22. How can we be sure we are at a minimum?
Secondary Order Necessary Conditions If
is a solution to the maximization problem
then If
is a solution to the minimization problem
then

23. The second derivative is the rate of change of the first derivative
Slope is decreasing
Slope is increasing

24. An Example Let x = Number of Flight Hours
First Order Necessary Conditions
Second Order Necessary Conditions
For X>0

25. Choose the level of advertising AND price to maximize sales
Multiple Variables
Suppose you know that demand for your product depends on the price that you set and the level of advertising expenditures.
Choose the level of advertising AND price to maximize sales

26. Partial Derivatives
When you have functions of multiple variables, a partial derivative is the derivative with respect to one variable, holding everything else constant
Example (One you will see a lot!!)

27. Multiple Variables
First Order Necessary Conditions

28. Multiple Variables
(2)
(1)
(1)
(2) 40 50

30. Again, how can we be sure we are at a maximum?

31. Recall, the second order condition requires that
For a function of more than one variable, it’s a bit more complicated…

32. Actually, its generally sufficient to see if all the second derivatives are negative…

33. Constrained optimizations attempt to maximize/minimize a function subject to a series of restrictions on the allowable domain
To solve these types of problems, we set up the lagrangian
Function to be maximized
Constraint(s)
Multiplier

34. Once you have set up the lagrangian, take the derivatives and set them equal to zero
First Order Necessary Conditions
Now, we have the “Multiplier” conditions…

35. Constrained Optimization
Example: Suppose you sell two products ( X and Y ). Your profits as a function of sales of X and Y are as follows:
Your production capacity is equal to 100 total units. Choose X and Y to maximize profits subject to your capacity constraints.

37. Constrained Optimization
Multiplier
The first step is to create a Lagrangian
Constraint
Objective Function

38. Constrained Optimization
First Order Necessary Conditions
“Multiplier” conditions
Note that this will always hold with equality

39. Constrained Optimization

40. The Multiplier
Lambda indicates the marginal value of relaxing the constraint. In this case, suppose that our capacity increased to 101 units of total production.
Assuming we respond optimally, our profits increase by $5

42. Another Example
Suppose that you are able to produce output using capital (k) and labor (l) according to the following process:
The prices of capital and labor are
and
respectively.
Union agreements obligate you to use at least one unit of labor.
Assuming you need to produce
units of output, how would
you choose capital and labor to minimize costs?

43. Minimizations need a minor adjustment…
To solve these types of problems, we set up the lagrangian
A negative sign instead of a positive sign!!

44. Inequality Constraints
Just as in the previous problem, we set up the lagrangian. This time we have two constraints.
Holds with equality
Doesn’t necessarily hold with equality

45. First Order Necessary Conditions

46. Case #1:
Constraint is non-binding
First Order Necessary Conditions

47. Case #2:
Constraint is binding
First Order Necessary Conditions

48. Constraint is Binding
Constraint is Non-Binding

49. Try this one…
You have the choice between buying apples and oranges. You utility (enjoyment) from eating apples and bananas can be written as:
The prices of Apples and Bananas are given by
and
Maximize your utility assuming that you have $100 available to spend

50. (Objective)
(Income Constraint)
(You can’t eat negative apples/oranges!!)
Objective
Non-Negative Consumption Constraint
Income Constraint

51. First Order Necessary Conditions
We can eliminate some of the multiplier conditions with a little reasoning…
You will always spend all your income
You will always consume a positive amount of apples

52. Case #1: Constraint is non-binding
First Order Necessary Conditions

53. Case #1: Constraint is binding
First Order Necessary Conditions