Quick lesson in some Mathematics used in Managerial Economics
description
Transcript of Quick lesson in some Mathematics used in Managerial Economics
![Page 1: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/1.jpg)
Quick lesson in some Mathematics used in
Managerial Economics
• Algebra• Derivatives (Marginal Analysis)
![Page 2: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/2.jpg)
Algebra
Translating from implicit functions to explicit functions:
X + 2y – 4 = 0Solve for x or y
Given Qd = 150 – 5P, determine the price function
![Page 3: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/3.jpg)
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
![Page 4: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/4.jpg)
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
![Page 5: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/5.jpg)
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
![Page 6: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/6.jpg)
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:
![Page 7: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/7.jpg)
• 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.
![Page 8: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/8.jpg)
Functions of several variables (Partial derivaties)
2221
2121 ),( cXXbXaXXXfy
Given the following function:
δy/δX1 = 2aX1 + bX2
δy/δX2 = bX1 + CX2
![Page 9: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/9.jpg)
Supply and Demand
![Page 10: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/10.jpg)
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.).
![Page 11: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/11.jpg)
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
![Page 12: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/12.jpg)
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?
![Page 13: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/13.jpg)
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
![Page 14: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/14.jpg)
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.
![Page 15: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/15.jpg)
• 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
![Page 16: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/16.jpg)
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
![Page 17: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/17.jpg)
What is Price?Could be absolute, relative, balance
or total
Absolute = Price of Product x (Px)
![Page 18: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/18.jpg)
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
![Page 19: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/19.jpg)
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
![Page 20: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/20.jpg)
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.
![Page 21: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/21.jpg)
Change in Quantity Demanded
Price
Quantity
D0
4 7
6
A to B: Increase in quantity demanded
B
10A
![Page 22: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/22.jpg)
Price
Quantity
D0
D1
6
7
D0 to D1: Increase in Demand
Change in Demand
13
![Page 23: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/23.jpg)
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
![Page 24: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/24.jpg)
Example
Determinants of demand for1.New homes?2.Washing machines in India3.Furniture in Nanaimo4.Pre-paid wireless telecom service
![Page 25: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/25.jpg)
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.
![Page 26: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/26.jpg)
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)
![Page 27: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/27.jpg)
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
![Page 28: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/28.jpg)
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
![Page 29: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/29.jpg)
Consumer Surplus:
• The value consumers get from a good but do not have to pay for.
![Page 30: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/30.jpg)
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
![Page 31: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/31.jpg)
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????
![Page 32: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/32.jpg)
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
![Page 33: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/33.jpg)
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
![Page 34: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/34.jpg)
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.
![Page 35: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/35.jpg)
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
![Page 36: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/36.jpg)
Change in Quantity Supplied
Price
Quantity
S0
20
10
B
A
5 10
A to B: Increase in quantity supplied
![Page 37: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/37.jpg)
Price
Quantity
S0
S1
8
75
S0 to S1: Increase in supply
Change in Supply
6
![Page 38: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/38.jpg)
Producer Surplus• The amount producers receive in excess of
the amount necessary to induce them to produce the good.
Price
Quantity
S0
Q*
P*
![Page 39: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/39.jpg)
Market Equilibrium
• Balancing supply and demand
– QxS = Qx
d
• Steady-state
![Page 40: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/40.jpg)
Price
Quantity
S
D
5
6 12
Shortage12 - 6 = 6
6
If price is too low…
7
![Page 41: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/41.jpg)
Price
Quantity
S
D
9
14
Surplus14 - 6 = 8
6
8
8
If price is too high…
7
![Page 42: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/42.jpg)
Comparative Static Analysis
• How do the equilibrium price and quantity change when a determinant of supply and/or demand change?
![Page 43: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/43.jpg)
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.
![Page 44: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/44.jpg)
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.
![Page 45: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/45.jpg)
Scenario 1: Implications for a Small PC Maker
• Step 1: Look for the “Big Picture.”• Step 2: Organize an action plan
(worry about details).
![Page 46: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/46.jpg)
Priceof
PCs
Quantity of PC’s
S
D
S*
P0
P*
Q0 Q*
Big Picture: Impact of decline in component prices on PC
market
![Page 47: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/47.jpg)
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?
![Page 48: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/48.jpg)
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?
![Page 49: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/49.jpg)
Priceof Software
Quantity ofSoftware
S
D
Q0
D*
P1
Q1
Big Picture: Impact of lower PC prices on the software market
P0
![Page 50: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/50.jpg)
Big Picture Analysis: Software Market
• Software prices are likely to rise, and more software will be sold.
• Use this to organize an action plan.
![Page 51: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/51.jpg)
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.
![Page 52: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/52.jpg)
Comparative Statics Analysis
• The rationing function of price is the change in market price to eliminate the imbalance between quantities supplied and demanded.
![Page 53: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/53.jpg)
Short-run Analysis• An increase in
demand causes equilibrium price and quantity to rise.
![Page 54: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/54.jpg)
Short-run Analysis
• A decrease in demand causes equilibrium price and quantity to fall.
![Page 55: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/55.jpg)
Short-run Analysis
• An increase in supply causes equilibrium price to fall and equilibrium quantity to rise.
![Page 56: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/56.jpg)
Short-run Analysis
• A decrease in supply causes equilibrium price to rise and equilibrium quantity to fall.
![Page 57: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/57.jpg)
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
![Page 58: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/58.jpg)
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.
![Page 59: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/59.jpg)
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
![Page 60: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/60.jpg)
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
![Page 61: Quick lesson in some Mathematics used in Managerial Economics](https://reader035.fdocuments.us/reader035/viewer/2022062409/56814e5b550346895dbbf67d/html5/thumbnails/61.jpg)
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