CDAE 266 - Class 10Sept. 27

Last class: Result of Quiz 2 Project 1 2. Review of economic and business concepts

Today: 2. Review of economic and business concepts

Next class: 2. Review of economic and business concepts 3. Linear programming

Important dates: Project 1: due Tuesday, Oct. 2 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. 9 -- 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 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

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

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

QdQ

dM 44000

22400000,10 QQ

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

-- An example:

dQ

dTC

dQ

dTR

dQ

d

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: Demand function P = 7500 -3.75Q

TR = P*Q = (7500 – 3.75Q) Q = 7500Q - 3.75Q2

-- Derive the MR and MC

-- Set MR = MC and solve for Q*

dQ

dTRMR

dQ

dTCMC

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 + 1.25Q2

-- Demand function P = 7500 – 3.75Q -- TR = P*Q = (7500 – 3.75Q) Q = 7500Q - 3.75Q2

-- TR MR = 7500 - 7.5Q

-- TC MC = 1500 + 2.5Q

-- Set MR = MC 7500 - 7.5Q = 1500 + 2.5Q

-- Q* = 600

-- P = ? TC = ? TR = ? = ?

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

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

Class Exercise 4 (Thursday, Sept. 27)

1. 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?