Transcript lecture 2

Quick lesson in some
Mathematics used in
Managerial Economics
• Algebra
• Derivatives (Marginal Analysis)
Algebra
Translating from implicit functions to
explicit functions:
X + 2y – 4 = 0
Solve 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=4
But 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)
Given the following function:
y  f ( X 1 , X 2 )  aX 12  bX 1 X 2  cX 22
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.
Price
D
Quantity
What is Price?
Could be absolute, relative, balance or
total
Absolute = Price of Product x (Px)
Relative Price
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)
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
A to B: Increase in quantity demanded
10
A
B
6
D0
4
7
Quantity
Change in Demand
Price
D0 to D1: Increase in Demand
6
D1
D0
7
13
Quantity
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 for
1. New homes?
2. Washing machines in India
3. Furniture in Nanaimo
4. Pre-paid wireless telecom service
The Demand Function
• A general equation representing the demand
curve
Qxd = 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 = 55,000
Pop = 10,000
Ay = 15 (competitors advertising budget)
Ax = 10 (firm’s advertising budget)
•  Demand function
Qxd = 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 – Qxd
• 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 $
10
Consumer
Surplus =
$24 - $8 =
$16
Value
of 4 units = $24
8
6
Expenditure on 4 units =
$2 x 4 = $8
4
2
D
1
2
3
4
5
Quantity
Consumer Surplus
– Demand Function
• Qxd = 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
S0
Quantity
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
• Qxs = 10 + 2Px
– Inverse Supply Function:
• 2Px = 10 + Qxs
• Px = 5 + 0.5Qxs
Change in Quantity
Supplied
Price
A to B: Increase in quantity supplied
S0
B
20
A
10
5
10
Quantity
Change in Supply
S0 to S1: Increase in supply
Price
S0
S1
8
6
5
7
Quantity
Producer Surplus
• The amount producers receive in excess of
the amount necessary to induce them to
produce the good.
Price
S0
P*
Q*
Quantity
Market Equilibrium
• Balancing supply and
demand
– QxS = Qxd
• Steady-state
If price is too low…
Price
S
7
6
5
D
Shortage
12 - 6 = 6
6
12
Quantity
If price is too high…
Surplus
14 - 6 = 8
Price
S
9
8
7
D
6
8
14
Quantity
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).
Big Picture: Impact of decline in
component prices on PC market
Price
of
PCs
S
S*
P0
P*
D
Q0
Q*
Quantity of PC’s
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?
Big Picture: Impact of lower PC
prices on the software market
Price
of Software
S
P1
P0
D*
D
Q0 Q1
Quantity of
Software
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