Applied Economics for Business Management
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Transcript of Applied Economics for Business Management
Applied Economics for Business Management
Lecture outline:Introduction to courseMath reviewIntroduction to consumer behavior
Introduction
Applied economics for business management involves investigating both consumer and producer behavior. The course description states that the theory of the consumer, firm and market is developed. Often in agricultural economics programs, applied economics is covered in two courses – a course in applied production analysis and a course in applied price analysis. We will try to cover both topics in one course.
Introduction
The first half of the course will concentrate on consumer behavior and the second half of the course will be on producer behavior. But before we start on consumer behavior, we will do a quick math review. This will not be a review of calculus per se, but will provide an overview of optimization.
Math Review
The primary uses of mathematics in the study of production and price analyses are two fold:(i) to find extreme values of functions e.g., maximum values of certain functions (e.g., utility, profit, etc.) and minimum values of certain functions (e.g., costs, expenditures, etc.).(ii) to study under which conditions economic optima (maxima and minima) hold.
Math Review
Example of (ii): (consumer equilibrium for utility maximization)
(producer equilibrium for profit maximization)
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Math Review
There are two general types of optimization problems: - unconstrained optimization - constrained optimization
Math Review
Unconstrained optimization(i) Simplest case: single argument functions with one explanatory variable(ii) General case:multiple argument functions
with n explanatory variables
Math Review
Suppose you had these two single argument functions:
Math Review
What do we observe from these 2 graphs?(i) the peaks and troughs occur where the slope of the function is zero (where critical points occur)(ii) the slopes are positive and negative
to the left and right of the critical points
Math Review
By using derivatives, we can solve for critical values and determine if these critical values are relative maxima or relative minima.
(critical value) (critical value)
Math Review
First Derivative Test:What is the function doing around the critical value?
relative max relative min
Math Review
Second Derivative Test:Another test to determine whether critical values are relative maximum(s)/minimum(s)(i) Relative maximum: second derivative is negative or concave down(ii) Relative minimum:second derivative is positive or concave up
Math Review
For the previous example: Critical value is a relative max
Critical value is a relative min
Math Review
Procedurefor optimizing a single argument function(1) Given , find (2) Set and solve for critical value(s) (3) Either use the First Derivative Test or Second Derivative Test to verify whether the critical value(s) are relative max or relative min or neither.
Math Review
Example:Let
(critical value)
Math Review
Using the Second Derivative Test:is a relative min
Math Review
Another example:
(critical values)
Math Review
Second Derivative Test:is a relative maxis a relative min
Unconstrained optimization of multiple argument functions:
Let’s take an example:Take partial derivatives and set equal to zero:
Unconstrained optimization of multiple argument functions:
Solving these two equations simultaneously:
Unconstrained optimization of multiple argument functions:
Distributing -10
Combining like terms
Subtracting -28 to both sides
Dividing both sides by -28
Unconstrained optimization of multiple argument functions:
So and or are the critical values. However, we don’t know if these critical values represent a relative max or relative min or neither.
Math Review
Before we investigate the second order or sufficient condition for relative extrema, we should briefly discuss the concept of higher order partial derivatives and their notation.
Math Review
GivenFirst order partial derivatives:
Math Review
Second order direct partial derivatives:
Math Review
Second order cross partial derivatives:
If and are continuous functions, then by Young’s Theorem .See Silberberg (pages 68 – 70) for proof.
Unconstrained optimization of multiple argument functions:
Returning to the example:Recall the critical values were .
Unconstrained optimization of multiple argument functions:
Also recall the following derivatives:
Unconstrained optimization of multiple argument functions:
Second order direct partials:
Unconstrained optimization of multiple argument functions:
Second order cross partials:Shows that symmetry condition holds
Unconstrained optimization of multiple argument functions:
Using the criteria for optimization with single argument functions, we are tempted to conclude that if and critical values represent a relative max Unfortunately, the second order conditions for multiple argument functions is not that simple. Because the sign of the second order direct partials only insure an extremum in the dimension or the dimension, but not the dimension.
Saddle Point
If the second order conditions rested solely on the signs of the second order direct partials, you could get cases such as the saddle point.See the example on saddle point:
The intersection of the , , and shows a minimum in the space and a maximum in the space.
Saddle Point
The point being: the second order condition for multiple argument functions is not so simple.For this case, we have to set-up and then evaluate a Hessian determinant.
Where
Unconstrained optimization of multiple argument functions:
So the cookbook procedure for optimizing multiple argument functions are:(i) Take first order partial derivatives(ii) Set first order partial derivatives equal to zeroand solve simultaneously for critical values(iii) Take second order direct and cross partial derivatives(iv) Evaluate the Hessian determinant
Determinants
Square matrix: Number of rows and columns are equal
Review of determinants: Associated with any square matrix A, there is a scalar quantity called the determinant of A and written: or |A| If A is n x n, then |A| is said to be of order n.(So n is the dimension of the square matrix)
Determinants
Determinants are defined as follows:
(1 x 1) matrix
Determinants
Determinants
(iii) for n>2, the determinant of an n x n matrix may be defined in terms of determinants of (n - 1) x (n - 1) submatrices as follows:(a) the minor of an element of A is the determinant of the remaining matrix by deleting the i th row and j th column
Determinants
The minor of (formed by deleting the 1st row and 2nd column) or the element 2 in the 1st row and 2nd column is:
Determinants
(b) Cofactor of is written in terms of its assigned minor.
Determinants
(c) The determinant of an n x n matrix is defined as the sum of the product of theelements of any row or column of A and their cofactors
Determinants
La Place Transformation by any row or any column.By any row:1st row2nd row3rd row
Determinants
By any column:1st column2nd column3rd column
Determinants
Find |A|.You can use the La Place Transformation by expanding on any row or column.
Determinants
First, expand by 1st row:Find cofactors:
Determinants
Finding cofactors, continued:
Determinants
Determinants
Or criss-cross method (Chiang)
Determinants
For your own practice, expand by the 3rd row:
Determinants
Determinants
Second Order or Sufficient Condition
Rules for second order or sufficient condition for multiple argument functions: Let Form Hessian determinant consisting of second order direct and cross partials:
Second Order or Sufficient Condition
The first principal minor is defined by deleting all rows and columns except the first row and first column. So, First principal minor
Second Order or Sufficient Condition
The second principal minor is formed by the first and second rows and columns and deleting all other rows and columns So,
Second principal minor
Second Order or Sufficient Condition
The third principal minor: You can use the La Place Transformation procedure or the criss-cross method shown by Chiang to solvefor
Second Order or Sufficient Condition
Fourth, fifth, etc. principal minors follow this same pattern.The second order or sufficient condition for a relative max is:
(alternating signs)
Second Order or Sufficient Condition
The second order or sufficient condition for a relative min is:
Second Order or Sufficient Condition
Recall the example:Earlier we found the critical value to be Is this critical value a relative max or min?
Second Order or Sufficient Condition
Use second order or sufficient condition:
represents a relative max