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: 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?