CDAE 266 - Class 10Sept. 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

CDAE 266 - Class 10Sept. 28

Important dates: Project 1 report due today Problem set 2 due Thursday, Oct. 5

Result of Quiz 2N = 44 Range = 4 –- 10 Average = 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

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

2.5.4. 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

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.

2.5.4. Total, average and marginal relations

(3) An example Q M A 0 0 --- ---

1 19 19 19

2 52 33 26

3 93 41 31

4 136 43 34

5 175 39 35

6 210 35 35

7 217 7 31

8 208 -9 26

10 190 ? ?

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.

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

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: TR = PQ = 7500Q - 3.75Q2

-- Derive the MR and MC

-- Derive Q* be setting MR = MC 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

-- TR = 7500Q - 3.75Q2

-- MR = 7500 - 7.5Q

-- MC = 1500 + 2.5Q

-- Set MR = MC

7500 - 7.5Q = 1500 + 2.5Q

-- Q* = 600

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

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 + 0.2Q2

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?

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

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

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.

Break-even analysis: Linear functionsCo

sts

($)

Quantity

FC

TC

TR

B

A

Break-even quantity

Break-even analysis: nonlinear functionsCo

sts

($)

Quantity

TCTR

Break-even quantity 1 Break-even quantity 2

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

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?