Quick lesson in some Mathematics used in Managerial Economics

1
Quick lesson in some Mathematics used in
Managerial Economics

• Algebra
• Derivatives (Marginal Analysis)

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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

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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

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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

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Second order condition

• The second derivative (the derivative of what is
  already is a derivative) should be negative
• d2f/dQ2 lt 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

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Example

• Manager wants to maximize profit (?)
• ? 4Q Q2
• df/dQ 4 -2Q
• df/dQ 0 when Q Q 2
• ? 4
• But how do you know that ?4 is the maximum?
  Check 2nd order condition

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• d2f/dQ2 -2 lt0 ? maximum
• Note that second derivative is negative at every
  point, not just at Q. This means Q2 is a
  global maximum for this function.
• For every other value of Q, profits are smaller.

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Functions of several variables (Partial
derivaties)
Given the following function
dy/dX1 2aX1 bX2 dy/dX2 bX1 CX2
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Supply and Demand
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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.).

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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

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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?

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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

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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.

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• 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

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Market Demand Curve

• Market demand is the sum of all the individual
  demands.
• Law of Demand
• The demand curve is downward sloping.

Price
D
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What is Price?

• Could be absolute, relative, balance or total
• Absolute Price of Product x (Px)

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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)

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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

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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.

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Change in Quantity Demanded
A
B
D0
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Change in Demand
D0 to D1 Increase in Demand
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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

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Example

• Determinants of demand for
• New homes?
• Washing machines in India
• Furniture in Nanaimo
• Pre-paid wireless telecom service

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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.

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• 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 (firms advertising budget)

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• ? 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

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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

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Consumer Surplus

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

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Consumer SurplusThe Continuous Case
Price
10
8
6
4
Expenditure on 4 units 2 x 4 8
2
D
1 2 3 4 5
Quantity
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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????

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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.

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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

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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.

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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

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Change in Quantity Supplied
A to B Increase in quantity supplied
B
A
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Change in Supply
S0 to S1 Increase in supply
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Producer Surplus

• The amount producers receive in excess of the
  amount necessary to induce them to produce the
  good.

Price
S0
P
Q
Quantity
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Market Equilibrium

• Balancing supply and demand
• QxS Qxd
• Steady-state

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If price is too low
Price
Quantity
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If price is too high
Price
Quantity
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Comparative Static Analysis

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

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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.

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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.

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Scenario 1 Implications for a Small PC Maker

• Step 1 Look for the Big Picture.
• Step 2 Organize an action plan (worry about
  details).

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Big Picture Impact of decline in component
prices on PC market
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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?

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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?

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Big Picture Impact of lower PC prices on the
software market
Price of Software
P0
Quantity of Software
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Big Picture Analysis Software Market

• Software prices are likely to rise, and more
  software will be sold.
• Use this to organize an action plan.

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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.

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Comparative Statics Analysis

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

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Short-run Analysis

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

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Short-run Analysis

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

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Short-run Analysis

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

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Short-run Analysis

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

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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

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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.
  

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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

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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

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Supply, Demand, and PriceThe 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