1. Mathematics for Economists [Section A – Part 1 / 4]
Instructor: Annika M. Mueller, Ph.D., The Wang Yanan Institute for Studies in Economics
Main Reference: C. P. Simon and L. Blume (1994), Mathematics for Economists
Further Reading: W. Nicholson and C. Snyder (2011), Microeconomic Theory

2. Optimization
Economic models often based on assumption that economic agents (individuals, firms) are seeking to find the optimal value of some function, for example
consumers maximize utility
firms maximize profit
Cover mathematics common to these problems

3. Maximization of a Function of One Variable
Classic example: Profit maximization of a firm 
Profit maximum (*) occurs at quantity q* * q*
 = f(q)
Quantity

4. Profit Maximization (cont'd)
q* will likely be reached through an adjustment process
an increase from q1 to q2 leads to a rise in profits  * 2 q2
 = f(q) 1
Quantity q1 q*

5. Profit Maximization (cont'd)
If output is increased beyond q*, profit will decline
an increase from q* to q3 leads to a decline of profits  * 3 q3
 = f(q)
Quantity q*

6. Derivatives
The derivative of  = f(q) is the limit of /q for very small changes in q
The value of this ratio depends on the value of q1

7. Value of a Derivative at a Point
The evaluation of the derivative at the point q = q1 can be denoted
In our previous example,

8. Second Order Conditions
The first order condition (d/dq) is a necessary condition for a maximum, but it is not a sufficient condition 
If the profit function was u-shaped,
the first order condition would result
in q* being chosen and  would
be minimized * q*
Quantity

9. Second Order Conditions
The first order condition is a necessary condition for a maximum, but not a sufficient condition c
If the function was u-shaped (think: cost
function), the first order condition results
in q* being chosen, minimizing c. c* q*
Quantity

10. Second Order Conditions
In order for q* to be the optimum for a profit function,
and
Therefore, at q*, d/dq must be zero.

11. Second Derivatives
The derivative of a derivative is called a second derivative
The second derivative can be denoted by

12. Second Order Condition
The second order condition to represent a (local) maximum is

13. Functions of Several Variables
Most functions that economic agents are trying to optimize depend on more than just one variable
trade-offs must be made
The dependence of one variable (y) on a series of other variables (x1,x2,…,xn) is denoted by

14. Partial Derivatives
The partial derivative of y with respect to x1 is denoted by
It is understood that in calculating the partial derivative, all of the other x’s are held constant
“How do changes in one variable affect some outcome when other influences are held constant (ceteris paribus)”

15. Partial Derivatives
A more formal definition of the partial derivative is

16. Partial Derivatives Units of measurement matter:
if q represents the quantity of gasoline demanded (measured in billions of gallons) and p represents the price in dollars per gallon, then q/p will measure the change in demand (in billions of gallons per year) for a dollar per gallon change in price

17. Second-Order Partial Derivatives
The partial derivative of a partial derivative is called a second-order partial derivative

18. Young’s Theorem
Under general conditions, the order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matter

19. Use of Second-Order Partials
Second-order partials play an important role in many economic theories
One of the most important is a variable’s own second-order partial, fii
shows how the marginal influence of xi on y (y/xi) changes as the value of xi increases
a value of fii < 0 indicates diminishing marginal effectiveness

20. Total Differential Suppose that y = f(x1,x2,…,xn)
If all x’s are varied by a small amount, the total effect on y will be

21. First-Order Condition for a Maximum (or Minimum)
A necessary condition for a maximum (or minimum) of the function f(x1,x2,…,xn) is that dy = 0 for any combination of small changes in the x’s
The only way for this to be true is if
A point where this condition holds is
called a critical point

22. Finding a Maximum Suppose that y is a function of x1 and x2
y = - (x1 - 1)2 - (x2 - 2)2 + 10
y = - x12 + 2x1 - x22 + 4x2 + 5
First-order conditions imply that OR

23. The Envelope Theorem
The envelope theorem concerns how the optimal value for a particular function changes when a parameter of the function changes
This is easiest to see by using an example

24. The Envelope Theorem Suppose that y is a function of x
y = -x2 + ax
For different values of a, this function represents a family of inverted parabolas
If a is assigned a specific value, then y becomes a function of x only and the value of x that maximizes y can be calculated

25. The Envelope Theorem
Optimal Values of x and y for alternative values of a

26. The Envelope Theorem As a increases, the maximal value
for y (y*) increases

27. y* = -(x*)2 + a(x*) = -(a/2)2 + a(a/2)
The Envelope Theorem
To calculate the slope of the function, we must solve for the optimal value of x for any value of a
dy/dx = -2x + a = 0
x* = a/2
Substituting, we get
y* = -(x*)2 + a(x*) = -(a/2)2 + a(a/2)
y* = -a2/4 + a2/2 = a2/4

28. The Envelope Theorem Therefore,
dy*/da = 2a/4 = a/2 = x*
But, we can save time by using the envelope theorem
for small changes in a, dy*/da can be computed by holding x at x* and calculating y/ a directly from y

29. The Envelope Theorem Holding x = x*
y/ a = x
Holding x = x*
y/ a = x* = a/2
This is the same result found earlier

30. The Envelope Theorem
The envelope theorem states that the change in the optimal value of a function with respect to a parameter of that function can be found by partially differentiating the objective function while holding x (or several x’s) at its optimal value

31. The Envelope Theorem
The envelope theorem can be extended to the case where y is a function of several variables
y = f(x1,…xn,a)
Finding an optimal value for y would consist of solving n first-order equations
y/xi = 0 (i = 1,…,n)

32. The Envelope Theorem
Optimal values for theses x’s would be determined that are a function of a
x1* = x1*(a)
x2* = x2*(a)
xn*= xn*(a) .

33. y* = f [x1*(a), x2*(a),…,xn*(a),a]
The Envelope Theorem
Substituting into the original objective function yields an expression for the optimal value of y (y*)
y* = f [x1*(a), x2*(a),…,xn*(a),a]
Differentiating yields

34. The Envelope Theorem
Because of first-order conditions, all terms except f/a are equal to zero if the x’s are at their optimal values
Therefore,