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: 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 Quantity D

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 Quantity A to B: Increase in quantity demanded A 10 4 B 6 7 D0

Change in Demand Price Quantity D0 D0 to D1: Increase in Demand D1 6 7 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 for New homes? Washing machines in India Furniture in Nanaimo 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 $ Consumer Surplus = $24 - $8 = $16 Value of 4 units = $24 10 8 6 4 Expenditure on 4 units = $2 x 4 = $8 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 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 Qxs = 10 + 2Px Inverse Supply Function: 2Px = 10 + Qxs Px = 5 + 0.5Qxs

Change in Quantity Supplied Price Quantity S0 A to B: Increase in quantity supplied B 20 10 A 10 5

Change in Supply S0 to S1: Increase in supply Price S0 S1 8 6 5 7 Quantity S0 S1 8 5 7 6

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 D 7 6 5 6 12 Shortage 12 - 6 = 6 Quantity

If price is too high… Surplus 14 - 6 = 8 Price S D 9 6 14 8 7 8 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 PCs Quantity of PC’s S D S* P0 Q0 P* Q*

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 of Software S D* D P1 Q1 P0 Q0 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