Quick lesson in some Mathematics used in

Managerial Economics

• Algebra• Derivatives (Marginal Analysis)

Algebra

Translating from implicit functions to explicit functions:

X + 2y – 4 = 0Solve for x or y

Given Qd = 150 – 5P, determine the price function

Rules of finding derivatives

• If a is a constant then da/dx = 0

• If a and b are constants and b≠ 0, then daxb/dx = baxb-1

dlnx/dx = 1/x

Maximization of a Function (one variable)First order condition: (necessary)For a function of one variable (Q) to

attain its maximum value (Q*) at some point, the derivative at that point (if it exists) must be 0

df/dQ (at Q*) = 0

Second order condition

• The second derivative (the derivative of what is already is a derivative) should be negative

• d2f/dQ2 < 0

• Global vs. Local maximum:If second derivative is negative at every point,

the Q* is a global maximum for every other value of Q, the optimizing variable will be smaller.

If second derivative is satisfied only near Q* then the point is a local maximum.

We might have to look at other values of Q where the first order conditions are satisfied to find the global maximum

Example

• Manager wants to maximize profit (Π)

• Π = 4Q – Q2 • df/dQ = 4 -2Q• df/dQ = 0 when Q =Q* = 2• Π = 4But how do you know that Π=4 is

the maximum? Check 2nd order condition:

• δ2f/δQ2 = -2 <0 maximum• Note that second derivative is

negative at every point, not just at Q*. This means Q=2 is a “global” maximum for this function.

• For every other value of Q, profits are smaller.

Functions of several variables (Partial derivaties)

2221

2121 ),( cXXbXaXXXfy

Given the following function:

δy/δX1 = 2aX1 + bX2

δy/δX2 = bX1 + CX2

Supply and Demand

Why?• Use supply and demand analysis to

– clarify the “big picture” (the general impact of a current event on equilibrium prices and quantities).

– organize an action plan (needed changes in production, inventories, raw materials, human resources, marketing plans, etc.).

The Business Map

Organization – Set of processes and network of transactions

Suppliers ----Organization----Customers

Suppliers are indirect competitors and collaborators to the organization and

Customers are potential competitors and collaborators

Competitors/collaborators or complementors

• Competitors – rivals (compete for resources and/or customers)

• “Complementors” – join forces and work together

Can competitors be “complementors” at the same time?

What does the term “industry” mean?

A collection of firms producing similar products (North American Industrial Classification System)

What about business/economics?Degree of substitutability (in

consumption) among products: A good book and a movie

Market Demand•Quantities of a good or service that people are ready (willing and able) to buy at various prices within some given time period, other factors held constant.

• Any item you are willing to buy must provide you with some benefits

• MB= benefit from additional unit of item

• Diminishing marginal benefit – each unit provides less benefit than the one before it

• Price you are willing to pay should decrease with quantity purchased

Market Demand Curve

Market demand is the sum of all the individual demands.

Law of Demand – The demand curve is downward

sloping.

Quantity

D

Price

What is Price?Could be absolute, relative, balance

or total

Absolute = Price of Product x (Px)

Could be real, specific or categorical

• Real = Px/IP (IP= index of prices of all products

• Specific = Px/Py (Py refers to price of product y)

• Categorical = Px/IPCat (IPCat = index of prices of products in a category)

Relative Price

Balance & Total• Balance = PPC/PRP

PPC = price paid by customers

PRP = price received by producers

Balance may be expressed as PPC-PRP

Total = Px + TC

TC = transaction costs

Market Demand• Changes in price result in changes

in the quantity demanded.– This is shown as movement along the

demand curve.• Changes in nonprice determinants

result in changes in demand.– This is shown as a shift in the

demand curve.

Change in Quantity Demanded

Price

Quantity

D0

4 7

6

A to B: Increase in quantity demanded

B

10A

Price

Quantity

D0

D1

6

7

D0 to D1: Increase in Demand

Change in Demand

13

Non-price Determinants of Demand

• Income– Normal good– Inferior good

• Prices of Related Goods– Prices of substitutes – Prices of complements

• Advertising and consumer tastes

• Population• Consumer expectations

Example

Determinants of demand for1.New homes?2.Washing machines in India3.Furniture in Nanaimo4.Pre-paid wireless telecom service

The Demand Function• A general equation representing the

demand curveQx

d = f(Px , PY , I, H,)

– Qxd = quantity demand of good X.

– Px = price of good X.– PY = price of a related good Y.

• Substitute good.• Complement good.

– M = income.• Normal good.• Inferior good.

– H = any other variable affecting demand.

Qxd = 1500 – 0.5Px + 0.25PY – 8Pz + 0.10I + 0.02Pop – 250Ay + 400Ax

Suppose PY = 5,900

Pz = 90

I = 55,000Pop = 10,000Ay = 15 (competitors advertising budget)

Ax = 10 (firm’s advertising budget)

Demand functionQx

d = 1500 – 0.5Px + 0.25(5900) – 8(90) + 0.10(55000) + 0.02(100000) – 250(15) + 400(10)

Qxd = 8205 - 0.5Px

Inverse Demand Function

• Price as a function of quantity demanded.

• Example:– Demand Function

• Qxd = 10 – 2Px

– Inverse Demand Function:• 2Px = 10 – Qx

d

• Px = 5 – 0.5Qxd

Consumer Surplus:

• The value consumers get from a good but do not have to pay for.

Consumer Surplus:The Continuous Case

Price $

Quantity

D

10

8

6

4

2

1 2 3 4 5

Valueof 4 units = $24Consumer

Surplus = $24 - $8 = $16

Expenditure on 4 units = $2 x 4 = $8

Consumer Surplus

– Demand Function• Qx

d = 5 – Px

– If P =2, what is company revenue? What is consumer surplus?

– P = 2 Q = 3. TR =6– Consumer surplus????

Market Supply Curve

• The supply curve shows the amount of a good that will be produced at alternative prices, other factors constant.

• Law of Supply – The supply curve is upward sloping.

Price

Quantity

S0

Non-price Determinants of Supply

• Input prices• Technology or

government regulations• Number of firms

– Entry – Exit

• Substitutes in production

• Taxes– Excise tax– Ad valorem tax

• Producer expectations

The Supply Function

• An equation representing the supply curve:

QxS = f(Px , PR ,W, H,)

– QxS = quantity supplied of good X.

– Px = price of good X.

– PR = price of a production substitute.

– W = price of inputs (e.g., wages).– H = other variable affecting supply.

Inverse Supply Function• Price as a function of quantity

supplied.• Example:

– Supply Function• Qx

s = 10 + 2Px

– Inverse Supply Function:• 2Px = 10 + Qx

s

• Px = 5 + 0.5Qxs

Change in Quantity Supplied

Price

Quantity

S0

20

10

B

A

5 10

A to B: Increase in quantity supplied

Price

Quantity

S0

S1

8

75

S0 to S1: Increase in supply

Change in Supply

6

Producer Surplus• The amount producers receive in excess of

the amount necessary to induce them to produce the good.

Price

Quantity

S0

Q*

P*

Market Equilibrium

• Balancing supply and demand

– QxS = Qx

d

• Steady-state

Price

Quantity

S

D

5

6 12

Shortage12 - 6 = 6

6

If price is too low…

7

Price

Quantity

S

D

9

14

Surplus14 - 6 = 8

6

8

8

If price is too high…

7

Comparative Static Analysis

• How do the equilibrium price and quantity change when a determinant of supply and/or demand change?

Applications of Demand and Supply

Analysis• Event: The WSJ reports that the prices

of PC components are expected to fall by 5-8 percent over the next six months.

• Scenario 1: You manage a small firm that manufactures PCs.

• Scenario 2: You manage a small software company.

Use Comparative Static Analysis to see the Big

Picture!

• Comparative static analysis shows how the equilibrium price and quantity will change when a determinant of supply or demand changes.

Scenario 1: Implications for a Small PC Maker

• Step 1: Look for the “Big Picture.”• Step 2: Organize an action plan

(worry about details).

Priceof

PCs

Quantity of PC’s

S

D

S*

P0

P*

Q0 Q*

Big Picture: Impact of decline in component prices on PC

market

Big Picture Analysis: PC Market

• Equilibrium price of PCs will fall, and equilibrium quantity of computers sold will increase.

• Use this to organize an action plan– contracts/suppliers?– inventories?– human resources?– marketing?– do I need quantitative estimates?

Scenario 2: Software Maker

• More complicated chain of reasoning to arrive at the “Big Picture.”

• Step 1: Use analysis like that in Scenario 1 to deduce that lower component prices will lead to– a lower equilibrium price for computers.– a greater number of computers sold.

• Step 2: How will these changes affect the “Big Picture” in the software market?

Priceof Software

Quantity ofSoftware

S

D

Q0

D*

P1

Q1

Big Picture: Impact of lower PC prices on the software market

P0

Big Picture Analysis: Software Market

• Software prices are likely to rise, and more software will be sold.

• Use this to organize an action plan.

Comparative Statics Analysis

• The short run is the period of time in which:– Sellers already in the market respond

to a change in equilibrium price by adjusting variable inputs.

– Buyers already in the market respond to changes in equilibrium price by adjusting the quantity demanded for the good or service.

Comparative Statics Analysis

• The rationing function of price is the change in market price to eliminate the imbalance between quantities supplied and demanded.

Short-run Analysis• An increase in

demand causes equilibrium price and quantity to rise.

Short-run Analysis

• A decrease in demand causes equilibrium price and quantity to fall.

Short-run Analysis

• An increase in supply causes equilibrium price to fall and equilibrium quantity to rise.

Short-run Analysis

• A decrease in supply causes equilibrium price to rise and equilibrium quantity to fall.

Comparative Statics Analysis

• The long run is the period of time in which:– New sellers may enter a market– Existing sellers may exit from a market– Existing sellers may adjust fixed factors of

production– Buyers may react to a change in equilibrium

price by changing their tastes and preferences or buying preferences

Comparative Statics Analysis

• The guiding or allocating function of price is the movement of resources into or out of markets in response to a change in the equilibrium price.

Long-run Analysis• Initial change:

decrease in demand from D1 to D2

• Result: reduction in equilibrium price and quantity, now P2,Q2

• Follow-on adjustment:– movement of resources

out of the market– leftward shift in the

supply curve to S2– Equilibrium price and

quantity now P3,Q3

Long-run Analysis• Initial change: increase

in demand from D1 to D2

• Result: increase in equilibrium price and quantity, now P2,Q2

• Follow-on adjustment:– movement of resources

into the market– rightward shift in the

supply curve to S2– Equilibrium price and

quantity now P3,Q3

Supply, Demand, and Price:

The Managerial Challenge

• In the extreme case, the forces of supply and demand are the sole determinants of the market price.– This type of market is “perfect competition”

• In other markets, individual firms can exert market power over their price because of their:– dominant size.– ability to differentiate their product through

advertising, brand name, features, or services