Quick lesson in some Mathematics used in Managerial Economics

1. Quick lesson in some Mathematics used in Managerial Economics • Algebra • Derivatives (Marginal Analysis)

2. 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

3. 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

4. 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

5. 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

6. 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:

7. 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.

8. Functions of several variables (Partial derivaties) Given the following function: δy/δX1 = 2aX1 + bX2 δy/δX2 = bX1 + CX2

9. Supply and Demand

10. 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.).

11. 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

12. 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?

13. 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

14. 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.

15. 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

16. Market demand is the sum of all the individual demands. Law of Demand The demand curve is downward sloping. Quantity Market Demand Curve Price D

17. 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)

18. What is Price? Could be absolute, relative, balance or total Absolute = Price of Product x (Px)

19. 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)

20. Balance & Total • Balance = PPC/PRP PPC = price paid by customers PRP = price received by producers Quite useful also to express Balance as PPC-PRP Focus 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

21. Total Price = Px + TC TC = transaction costs

22. 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.

23. Price A to B: Increase in quantity demanded 10 6 4 7 Quantity Change in Quantity Demanded A B D0

24. Change in Demand Price 6 D1 D0 Quantity 7 13 D0 to D1: Increase in Demand

25. 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

26. Example Determinants of demand for • Parking at VIU? • Washing machines in India • Furniture in Nanaimo • Pre-paid wireless telecom service

27. 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.

28. 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)

29.  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

30. 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

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

32. Value of 4 units = $24 Consumer Surplus = $24 - $8 = $16 Consumer Surplus:The Continuous Case Price $ 10 8 6 4 Expenditure on 4 units = $2 x 4 = $8 2 D 1 2 3 4 5 Quantity

33. 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????

34. 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

35. Price S0 Quantity 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.

36. 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

37. 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.

38. 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

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

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

41. Producer Surplus • The amount producers receive in excess of the amount necessary to induce them to produce the good. Price S0 P* Q* Quantity

42. Market Equilibrium • Balancing supply and demand • QxS = Qxd • Steady-state

43. Price controls (ceilings) and its effect in different markets Price control reduces incentive to produce Scarcity of x creates excess demand for x and increases the effective price paid The higher price leads to increased demand for substitutes Increased demand for substitute y increases the price Old consumers of substitute are hurt Production of substitute increases Price control on X creates efficiency and distribution effects

44. Price control… S 7 6 5 D Shortage 12 - 6 = 6 6 12 Price Quantity

45. If price is too high… (your turn..) S 7 8 9 D 6 14 8 Price Quantity

46. Comparative Static Analysis • How do the equilibrium price and quantity change (for a specific product) when a determinant of supply and/or demand change?

47. 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.

48. Scenario 1: Implications for a Small PC Maker • Step 1: Look for the “Big Picture.” • Step 2: Organize an action plan (worry about details).

49. Big Picture: Impact of decline in component prices on PC market Price of PCs S S* P0 P* D Quantity of PC’s Q* Q0

50. 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?