1 / 31

Economic Concepts and Linear Programming: Evaluating Investment Options

This class review covers economic and business concepts, with a focus on linear programming and applications. The example of Mr. Zhang's investment options in Canada is used to demonstrate how to compare and calculate present values at different interest rates.

tulley
Download Presentation

Economic Concepts and Linear Programming: Evaluating Investment Options

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CDAE 266 - Class 12 Oct. 4 Last class: 2. Review of economic and business concepts Today: 3. Linear programming and applications Quiz 3 (sections 2.5 and 2.6) Next class: 3. Linear programming Important date: Problem set 2: due Tuesday, Oct. 9

  2. Mr. Zhang in Beijing plans to immigrate to Canada and start a business in Montreal and the Canadian government has the following two options of “investment” requirement: A. A one-time and non-refundable payment of $120,000 to the Canadian government. A payment of $450,000 to the Canadian government and the payment (i.e., $450,000) will be returned to him in 4 years from the date of payment. How do we help Mr. Zhang compare the two options? If the annual interest rate is 12%, what is the difference in PV? If the annual interest rate is 6%, what is the difference in PV? At what interest rate, the two options are the same in PV? One more application of TVM(Take-home exercise, Sept. 27)

  3. 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

  4. 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

  5. 2.5.6. Profit maximization (4) Summary of procedures (a) If we have the total profit function: Step 1: Take the derivative of the total profit function  marginal profit function Step 2: Set the marginal profit function to equal to zero and solve for Q* Step 3: Substitute Q* back into the total profit function and calculate the maximum profit (b) If we have the TR and TC functions: Step 1: Take the derivative of the TR function  MR Step 2: Take the derivative of the TC function  MC Step 3: Set MR=MC and solve for Q* Step 4: Substitute Q* back into the TR and TC functions to calculate the TR and TC and their difference is the maximum total profit

  6. 2.5.6. Profit maximization (4) Summary of procedures (c) If we have the demand and TC functions Step 1: Demand function  P = … Step 2: TR = P * Q = ( ) * Q Then follow the steps under (b) on the previous page

  7. Suppose a firm has the following total revenue and total cost functions: TR = 20 Q TC = 1000 + 2Q + 0.2Q2 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? Class Exercise 4 (Thursday, Sept. 27)

  8. 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

  9. 2.5.7. Average cost minimization (3) An example: TC = 612500 + 1500Q + 1.25Q2 MC = 1500 + 2.5Q AC = TC/Q = 612500/Q + 1500 + 1.25Q Set MC = AC Q2 = 490,000 Q = 700 or -700 When Q = 700, AC is at the minimum level

  10. 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.

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

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

  13. 2.6. Break-even analysis 2.6.4. An example TC = 612500 + 1500Q + 1.25Q2 TR = 7500Q - 3.75Q2 612500 + 1500Q + 1.25Q2 = 7500Q - 3.75Q2 5Q2 - 6000Q + 612500 = 0 Review the formula for ax2 + bx + c = 0 x = ? e.g., x2 + 2x - 3 = 0, x = ? Q = 1087.3 or Q = 112.6

  14. 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, how will you solve the problem mathematically (list the steps and you do not need to solve the equation) 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? Class Exercise 5 (Tuesday, Oct. 2)

  15. 3. Linear programming & applications 3.1. What is linear programming (LP)? 3.2. How to develop a LP model? 3.3. How to solve a LP model graphically? 3.4. How to solve a LP model in Excel? 3.5. How to do sensitivity analysis? 3.6. What are some special cases of LP?

  16. 3.1. What is linear programming (LP)? 3.1.1. Two examples: Example 1. The Redwood Furniture Co. manufactures tables & chairs. Table A on the next page shows the resources used, the unit profit for each product, and the availability of resources. The owner wants to determine how many tables and chairs should be made to maximize the total profits.

  17. Table A (example 1): --------------------------------------------------------------- Unit requirements Resources ---------------------- Amount Table Chair available --------------------------------------------------------------- Wood (board feet) 30 20 300 Labor (hours) 5 10 110 =====================================Unit profit ($) 6 8 ---------------------------------------------------------------

  18. 3.1. What is linear programming (LP)? 3.1.1. Two examples: Example 2. Galaxy Industries (a toy manufacture co.) 2 products: Space ray and zapper 2 resources: Plastic & time Resource requirements & unit profits: Table B on the next page.

  19. Table B (example 2): --------------------------------------------------------------- Unit requirements Resources ---------------------- Amount Space ray Zapper available --------------------------------------------------------------- Plastic (lb.) 2 1 1,200 Labor (min.) 3 4 2,400 =====================================Unit profit ($) 8 5 ---------------------------------------------------------------

  20. 3.1. What is linear programming (LP)? 3.1.1. Two examples: Example 2. Galaxy Industries: Additional requirements (constraints): (1) Total production of the two toys should be no more than 800. (2) The number of space ray cannot exceed the number of zappers plus 450. Question: What is the optimal quantity for each of the two toys?

  21. Management is seeking a production schedule that will maximize the company’s profit.

  22. Linear programming (LP) can provide intelligent solution to such problems

  23. 3.1. What is linear programming (LP)? 3.1.2. Mathematical programming: (1) Linear programming (LP) (2) Integer programming (3) Goal programming (4) Dynamic programming (5) Non-linear programming …… ……

  24. 3.1. What is linear programming (LP)? 3.1.3. Linear programming (LP): (1) A linear programming model: A model that seeks to maximize or minimize a linear objective function subject to a set of linear constraints. (2) Linear programming: A mathematical technique used to solve constrained maximization or minimization problems with linear relations.

  25. 3.1. What is linear programming (LP)? 3.1.3. Linear programming (LP): (3) Applications of LP: -- Product mix problems -- Policy analysis -- Transportation problems …… ……

  26. 3.2. How to develop a LP model? • 3.2.1. Major components of a LP model: • (1) A set of decision variables. • (2) An objective function. • (3) A set of constraints. • 3.2.2. Major assumptions of LP: • (1) Variable continuity • (2) Parameter certainty • (3) Constant return to scale • (4) No interactions between decision • variables

  27. 3.2. How to develop a LP model? • 3.2.3. Major steps in developing a LP • model: • (1) Define decision variables • (2) Express the objective function • (3) Express the constraints • (4) Complete the LP model • 3.2.4. Three examples: • (1) Furniture manufacturer • (2) Galaxy industrials • (3) A farmer in Iowa

  28. Table A (example 1): --------------------------------------------------------------- Unit requirements Resources ---------------------- Amount Table Chair available --------------------------------------------------------------- Wood (board feet) 30 20 300 Labor (hours) 5 10 110 =====================================Unit profit ($) 6 8 ---------------------------------------------------------------

  29. Step 1. Define the decision variables Two variables: T = number of tables made C = number of chairs made Step 2. Express the objective function Step 3. Express the constraints Step 4. Complete the LP model Develop the LP model

  30. Example 2. Galaxy Industries (a toy manufacturer) 2 products: Space ray and zapper 2 resources: Plastic & time Resource requirements & unit profits (Table B) Additional requirements (constraints): (1) Total production of the two toys should be no more than 800. (2) The number of space ray cannot exceed the number of zappers plus 450.

  31. Table B (example 2): --------------------------------------------------------------- Unit requirements Resources ---------------------- Amount Space ray Zapper available --------------------------------------------------------------- Plastic (lb.) 2 1 1,200 Labor (min.) 3 4 2,400 =====================================Unit profit ($) 8 5 ---------------------------------------------------------------

More Related