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
Income and Substitution effect
• A Δ in the price of a product generates an “income” and “substitution” effect.
An increase in price of x • motivates customers to demand more
of a substitute y• Reduces real income or purchasing
power reducing customer purchases.• Income effect may increase, decrease
or not affect demand (normal, inferior, neutral good)
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 customersPRP = price received by producers
Quite useful also to express Balance as PPC-PRPFocus on just “price” ignores factors affecting
profitability: Qty discounts to whsalers, rebates, rewards
to distributors, shipping, insurance, taxes.Strips the “price” of possible distortions
and shows what producers actually pocket
Total Price = Px + TCTC = 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.Parking at VIU?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????
Customer value created by a Product
• 2 products x and y• Y is the best feasible alternative to x• Customer benefit of x =$6, Px=3, • Customer benefit of y =$10, Py= 8
• Willingness to pay for x = benefits of x – (benefits of y – Py) =4
• Customer value of x (consumer surplus of x) = willingness to pay for x – Px = benefits of x – (benefits of y – Py) – Px = 1
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 controls (ceilings) and its effect in different markets
Price control reduces incentive to produceScarcity of x creates excess demand for x
and increases the effective price paidThe higher price leads to increased
demand for substitutesIncreased demand for substitute y
increases the priceOld consumers of substitute are hurtProduction of substitute increasesPrice control on X creates efficiency and
distribution effects
Price
Quantity
S
D
5
6 12
Shortage12 - 6 = 6
6
Price control…
7
Price
Quantity
S
D
9
146
8
8
If price is too high… (your turn..)
7
Comparative Static Analysis
• How do the equilibrium price and quantity change (for a specific product) 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.
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.
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
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
Demand and Supply Interdependencies
• Event: Vegetarians switch to meat• Reaction:• In meat market• In a complementary product
market• In a substitute product market
Demand and Supply in the SR and LR
Event: An unexpected flood destroys lettuce crops
1.In SR, quality drops, lettuce heads are smaller. Increase in SS reduces price despite a lower stock of lettuce
2.In MR, less lettuce arrive in markets. Price increase due to shortage
3.In LR, SS may return to its original level4.Don’t forget the effects on substitutes
and complements. Remember, expectations play a role too
Now your turn
• Lean meat, fatty meat and medical services
• Demand for lean meat increases
Case: Burger King
• Event: Price of lower margin- hamburger falls
• What happens to demand for hamburgers and the demand for high-margin fries?