CDAE Class 11 Oct. 2 Last class: 2. Review of economic and business concepts Today: 2. Review of economic and business concepts 3. Linear programming and applications Next class: 3. Linear programming Quiz 3 (sections 2.5 and 2.6) Important dates: Project 1: due today Problem set 2: due Tuesday, Oct. 9

Problem set 2 Problems 2.1., 2.2., 2.4., 2.5. and 2.8. from the reading “Basic Economic Relations” -- Due Tuesday, Oct Please use graphical paper to draw graphs -- Please staple all pages together before you turn them in -- Scores on problem sets that do not meet the requirements will be discounted Questions for Problem 2.4.: A.Construct a table showing Christensen's marginal sales per day in each state. B. If administrative duties limit Christensen to only 10 selling days per month, how should she spend them (i.e., how many days in each state)? C. Calculate Christensen's maximum monthly commission income.

One more application of TVM (Take-home exercise, Sept. 27) 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. B.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. (1)How do we help Mr. Zhang compare the two options? (2)If the annual interest rate is 12%, what is the difference in PV? (3)If the annual interest rate is 6%, what is the difference in PV? (4)At what interest rate, the two options are the same in PV?

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

2.5. Marginal analysis Basic concepts Major steps of using quantitative methods Methods of expressing economic relations Total, average and marginal relations How to derive derivatives? Profit maximization Average cost minimization

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

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

Class Exercise 4 (Thursday, Sept. 27) 1.Suppose a firm has the following total revenue and total cost functions: TR = 20 Q TC = Q + 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?

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

Average cost minimization (3) An example: TC = Q Q 2 MC = Q AC = TC/Q = /Q Q Set MC = AC Q 2 = 490,000 Q = 700 or -700 When Q = 700, AC is at the minimum level

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

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

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

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

Class Exercise 5 (Tuesday, Oct. 2) 1.Suppose a company has the following total cost (TC) function: TC = Q Q 2 (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 quadratic equation) 2.Suppose a company has the following total revenue (TR) and total cost (TC) functions: TR = 20 Q TC = Q How many units should the firm produce to have a break-even?

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?