1. CDAE 266 - Class 10 Sept. 28 Last class: Result of problem set 1 2. Review of economic and business concepts Today: Result of Quiz 2 2. Review of economic and business concepts Next class: 3. Linear programming and applications Quiz 3 (sections 2.5 and 2.6) Reading: Basic Economic Relation

2. CDAE 266 - Class 10 Sept. 28 Important dates: Project 1 report due today Problem set 2 due Thursday, Oct. 5

3. Result of Quiz 2 N = 44Range = 4 –- 10Average = 8.62 1. PV, r and n  FVn 2. FVn, r and n  PV 3. Annual interest rate  effective annual interest rate 4. (a) Annual interest rate  effective annual interest rate (b) PV, r and n  FVn when interest is paid semiannually 5. Present value of a bond

4. 2. Review of Economics Concepts 2.1. Overview of an economy 2.2. Ten principles of economics 2.3. Theory of the firm 2.4. Time value of money 2.5. Marginal analysis 2.6. Break-even analysis

5. 2.5. Marginal analysis 2.5.1. Basic concepts 2.5.2. Major steps of using quantitative methods 2.5.3. Methods of expressing economic relations 2.5.4. Total, average and marginal relations 2.5.5. How to derive derivatives? 2.5.6. Profit maximization 2.5.7. Average cost minimization

6. 2.5.4. Total, average and marginal relations (1) General notations: P = price of a product (output) Q = quantity of a product (output) TR = P Q = Total revenue FC = total fixed costs VC = total variable costs TC = FC + VC = total costs AC = TC / Q = average cost  = TR - TC = total profit A  =  / Q = average profit

7. 2.5.4. Total, average and marginal relations (2) Marginal concepts: Marginal revenue (MR) = the change in total revenue (TR) when output quantity (Q) changes by one unit. Marginal cost (MC) = the change in total costs (TC) when output quantity (Q) changes by one unit. Marginal profit (M  ) = the change in total profit (  ) when output quantity (Q) changes by one unit.

8. 2.5.4. Total, average and marginal relations (3) An example Q MAQ MA 0 0 ------ 119 1919 252 3326 393 4131 4 136 4334 5 175 3935 6 210 3535 7 217 731 8 208 -926 10 190 ? ?

9. 2.5.4. Total, average and marginal relations (4) Graph the data (5) Relation between total profit (  ) and marginal profit (M  ) when M  > 0,  is increasing when M  < 0,  is decreasing when M  = 0,  reaches the maximum.

10. 2.5.5. How to derive derivatives? The first-order derivative of a function (curve) is the slope of the curve. (1) Constant-function rule (2) Power-function rule (3) Sum-difference rule (4) Examples

11. 2.5.6. Profit maximization (1) With a profit function ( relation between profit and output quantity ): (a) Profit function: (b) What is the profit-maximizing Q? -- A graphical analysis -- A mathematical analysis Set M  = 0 ==> Q* = 100 (c) Maximum profit = 10,000

12. 2.5.6. Profit maximization (2) With TR and TC functions: --  is at the maximum when M  = 0 -- Relations among M , MR and MC:  = TR - TC M  = MR - MC M  = 0 when MR = MC -- Graphical analysis (page 5 of the handout)  is at the maximum level when MR=MC

13. 2.5.6. Profit maximization (3) With TC and demand functions: -- Demand function: Relation between Q and P Example: Q = 2000 – 0.26667 P -- Derive TR function from a demand function Example: TR = PQ = 7500Q - 3.75Q 2 -- Derive the MR and MC -- Derive Q* be setting MR = MC

14. 2.5.6. Profit maximization (3) With TC and demand functions: -- An example from the handout: Demand: Q = 2000 – 0.26667 P Total cost: TC = 612500 + 1500Q + 1.25Q 2 -- TR = 7500Q - 3.75Q 2 -- MR = 7500 - 7.5Q -- MC = 1500 + 2.5Q -- Set MR = MC 7500 - 7.5Q = 1500 + 2.5Q -- Q* = 600 -- P = ? TC = ? TR = ?  = ?

15. Class Exercise 3 (Tuesday, Sept. 26) 1.Suppose a firm has the following total revenue and total cost functions: TR = 20 Q TC = 1000 + 2Q + 0.2Q 2 How many units should the firm produce in order to maximize its profit? 2. If the demand function is Q = 20 – 0.5P, what are the TR and MR functions?

16. 2.5.7. Average cost minimization (1) Relation between AC and MC: when MC < AC, AC is falling when MC > AC, AC is increasing when MC = AC, AC reaches the minimum level (2) How to derive Q that minimizes AC? Set MC = AC and solve for Q

17. 2.5.7. Average cost minimization (3) An example: TC = 612500 + 1500Q + 1.25Q 2 MC = 1500 + 2.5Q AC = TC/Q = 612500/Q + 1500 + 1.25Q Set MC = AC Q 2 = 490,000 Q = 700 or -700 When Q = 700, AC is at the minimum level

18. 2.6. Break-even analysis 2.6.1. What is a break-even? TC = TR or  = 0 2.6.2. A graphical analysis -- Linear functions -- Nonlinear functions 2.6.3. How to derive the beak-even point or points? Set TC = TR or  = 0 and solve for Q.

19. Break-even analysis: Linear functions Costs ($) Quantity FC TC TR B A Break-even quantity

20. Break-even analysis: nonlinear functions Costs ($) Quantity TC TR Break-even quantity 1Break-even quantity 2 

21. 2.6. Break-even analysis 2.6.4. An example TC = 612500 + 1500Q + 1.25Q 2 TR = 7500Q - 3.75Q 2 612500 + 1500Q + 1.25Q 2 = 7500Q - 3.75Q 2 5Q 2 - 6000Q + 612500 = 0 Review the formula for ax 2 + bx + c = 0 x = ? e.g., x 2 + 2x - 3 = 0, x = ? Q = 1087.3 or Q = 112.6

22. Class Exercise 4 (Thursday, Sept. 28) 1.Suppose a company has the following total cost (TC) function: TC = 200 + 2Q + 0.5 Q2 (a) What are the average cost (AC) and marginal cost (MC) functions? (b) If the company wants to know the Q that will yield the lowest average cost, describe how you could solve the problem mathematically (just list the step or steps and you do not need to solve it) 2.Suppose a company has the following total revenue (TR) and total cost (TC) functions: TR = 20 Q TC = 300 + 5Q How many units should the firm produce to have a break-even?