1. Managerial Economics & Business Strategy Chapter 1 The Fundamentals of Managerial Economics

2. Derivative RULES Constants n ALWAYS ZERO!!! What is the derivative of 10? –Zero Power Functions

3. More Rules Sums and Differences n The derivative of the sum (difference) is equal to the sum (difference) of the derivatives of the individual terms n If U=g(x) and V=h(x) then…

4. And more… Products n The derivative of the product of two expressions is equal to the first term multiplied by the derivative of the second, PLUS the second term times the derivative of the first

5. And more… Quotient n Denominator MULTIPLIED by derivative of the numerator MINUS numerator MULTIPLIED by the derivative of the denominator ALL DIVIVED BY the denominator squared

6. Divide by 10X

7. Total Revenue TR = 7Q – 0.01Q 2 What is the Marginal Revenue function? n MR = 7 - 0.02Q TC = 100 – 8Q + 10Q 2 What is the Marginal Cost function? n MC = -8 + 20Q

8. Partial Derivatives When an equation has MANY independent variables we may want to isolate the impact of a single variable on dependent variable Q = -100P + 5I + P s + 2N n What is the impact of a change in the number of customers on Q? Holding all other factors constant –Remember the derivative of a constant is ZERO

9. Maximum and Minimum Values Our goal is to find the OPTIMAL points n Marginal analysis and derivatives help us to find these points TR = 7Q – 0.01Q 2  What Q will maximize TR? Set equal to ZERO to find revenue-maximizing Q

10. What price and output will maximize profit for the firm? Given : P = 172 – 10Q  demand function TC = 100 + 65Q + Q 2  = TR – TC What is TR??? Price * Quantity TR = 172Q – 10Q 2  = TR – TC = 172Q – 10Q 2 – (100 + 65Q + Q 2 ) = -100 + 107Q – 11Q 2 Find the derivative, set equal to zero, and Q *

11. Is this a Maximum or a Minimum???

12. How do you know?? Take the second derivative n Rate of change of the change Maximum Value: Minimum Value:

13. So what was it??

14. Quadratic Formula  Yuck When you have a quadratic functional form aX 2 + bX + c = 0 You need the quadratic formula to solve for X

15. Let’s try it What is the profit-maximizing level of output for a firm with TR = 50Q TC = 100 + 60Q – 3Q 2 + 0.1Q 3 ???

16. Plug into the quadratic formula to get Q*

18. Which is it?? Second derivative must be NEGATIVE Q* = 18.166 is the maximum n Profit-maximizing level of output

19. Key Functions we will be using Five key functions n Demand Linear n Total Revenue Quadratic n Production Cubic n Total cost Cubic n Profit Cubic

20. $ Q Demand P=a-bQ (or Q = a-bP)

21. $ Q Total Revenue TR=a+bQ-cQ 2 a=0

22. $ Q Production (short run) TP=a+bL+cL 2 -dL 3 a=0

23. $ Q Cost (short run) TC=a+bQ-cQ 2 +dQ 3

24. $ Q Profit  = a-bQ+cQ 2 -dQ 3 a<0

25. Why is this important? The more data that can be obtained The more mathematics can be used The more precise we can be The closer we can get to maximized profits