EC611--Managerial Economics Optimization Techniques and New Management Tools Dr. Savvas C Savvides, European University Cyprus
Models and Data Model a framework based on simplifying assumptions it helps to organize our economic thinking based on a simplified picture of reality We focus on key elements Data the economist s link with the real world 1. time series 2. cross section Managerial Economics DR. SAVVAS C SAVVIDES 1
Real and Nominal Variables Many economic variables are measured in money terms Nominal values measured in current prices Real values adjusted for price changes compared with a base year measured in constant prices Managerial Economics DR. SAVVAS C SAVVIDES 2
Real & Nominal Values--Example 1960 1975 2003 Land Prices (Hilton Park Area, Nicosia) 2,500 27,000 125,000 Price Index (2000=100) 7.4 39.3 100.0 Real Land Price (in 2000 prices) 33,783 68,702 125,000 Real Land Price (in 1960 prices) 2,500 5,084 9,250 (2,500*100) / 7.4 = 33,783 (125,000*7.4) / 100 = 9,250 Managerial Economics DR. SAVVAS C SAVVIDES 3
Evidence in Economics Evidence collected and produced from empirical observation and testing may allow us to accumulate support for a theory, or to reject it, or indicate points for further research and investigation Scatter diagrams help us to test and validate economic theory with empirical reality Econometrics is a more sophisticated method that takes this task of empirically validating theory further using statistical techniques Managerial Economics DR. SAVVAS C SAVVIDES 4
Data & Scatter Diagrams Price Year Price Quantity 1 6.0 100 2 5.5 105 7.0 X (7.0, 80) 3 4 6.0 6.5 90 85 X X 5 6.0 87 6.0 X X X (6.0, 100) 6 7.0 80 X 7 6.5 88 80 100 Quantity Managerial Economics DR. SAVVAS C SAVVIDES 5
Economic Models: An Example Examples: 1. Quantity of CDs demanded depend on (or is a function of): f (Prices, income, preferences) 2. Revenues are a function of Sales: f (Q) Managerial Economics DR. SAVVAS C SAVVIDES 6
Expressing Economic Relationships Equations: TR = 100Q - 10Q 2 e.g. if Q=1 TR = 100(1) 10(31) 2 = 90 Tables: if Q=3 TR = 100(3) 10(3) 2 = 210 Q 0 1 2 3 4 5 6 TR 0 90 160 210 240 250 240 Graphs: 300 250 200 150 100 TR 50 0 0 1 2 3 4 5 6 7 Q Managerial Economics DR. SAVVAS C SAVVIDES 7
25 Managerial Economics DR. SAVVAS C SAVVIDES 8
Total, Average, & Marginal Cost AC = TC/Q e.g. for Q=3 AC = 180/3 =60 MC = TC/ Q For Q from 3 to 4: MC = (240-180)/(4-3) =60 / 1 = 60 AC 140 80 60 60 96 Managerial Economics DR. SAVVAS C SAVVIDES 9 Q 0 1 2 3 4 5 TC 20 140 160 180 240 480 - MC - 120 20 20 60 240
Total, Average, & Marginal Cost TC ($) 240 TC 180 120 60 0 0 1 2 3 4 Q AC, MC ($) MC 120 60 AC 0 0 1 2 3 4 Q Managerial Economics DR. SAVVAS C SAVVIDES 10
Profit Maximization Profit = TR - TC Q TR TC Profit 0 0 20-20 1 90 140-50 2 160 160 0 3 210 180 30 4 240 240 0 5 250 480-230 Managerial Economics DR. SAVVAS C SAVVIDES 11
Profit Maximization ($) 300 TC 240 TR 180 120 60 0 60 30 0-30 -60 0 1 2 3 4 5 Profit Q Managerial Economics DR. SAVVAS C SAVVIDES 12
Slope of a Line Slope between A & B P/ Q = -5 / +5 = - 1 Managerial Economics DR. SAVVAS C SAVVIDES 13
Slope of a Line Price Quantity Managerial Economics DR. SAVVAS C SAVVIDES 14
Slope of Non-Linear Relationships Total Revenue Slope of TR at A is positive: Slope of tangency at pt. A Slope of TR at B is negative Slope of tangency at pt. B A B TR Quantity Managerial Economics DR. SAVVAS C SAVVIDES 15
Concept of the Derivative (1) Optimization analysis can be conducted much more efficiently using differential calculus. This relies on the concept of the derivative, which resembles the concept of the margin. For example, if TR = Y and Q =X, the derivative of Y with respect to X is equal to the Y w.r.t. X, as the X approaches zero. dy dx = lim X 0 Y X Managerial Economics DR. SAVVAS C SAVVIDES 16
Concept of the Derivative (2) Let s expand on the right hand side. Since Y depends on X, Y = f ( X ) Y = X X = X (tautology). Add & subtract X on RHS. X = (X+ X) (X) Y = f(x+ X) f(x) Divide both sides by X Y/ X = [f(x+ X) f(x] / X Substituting the RHS of the last expression in the derivative expression, we get dy/ dx = [f(x+ X) f(x] / X Managerial Economics DR. SAVVAS C SAVVIDES 17
The Derivative An Example If Y = X 2 dy/ dx = [(X+ X) 2 X 2 ] / X dy/ dx = [ X 2 + 2X * X) + ( X) 2 -X 2 ] / X dy/ dx = [ (2X * X) + ( X) 2 ] / X dy/ dx = [ (2X * X)/ X ] + [( X) 2 / X] Cancelling the X terms dy/ dx = (2X + X) This says that at the limit, i.e., as X 0, the whole expression will approach 2X (since X=0) Managerial Economics DR. SAVVAS C SAVVIDES 18
Rules of Differentiation Constant Function Rule: The derivative of a constant, Y = f(x) = a, is zero for all values of a (the constant). Y = f( X) = a dy 0 dx = Y 10 Changes in X do not affect the value of Y. Horizontal lines have zero slope! Y = 10 Managerial Economics DR. SAVVAS C SAVVIDES 19 X
Rules of Differentiation Power Function Rule: The derivative of a power function, where a and b are constants, is defined as follows. b Y = f( X) = ax dy dx = bax b 1 Example: Y = 3X 2 Derivative: dy/dx = 2 * 3X 2-1 = 6X Managerial Economics DR. SAVVAS C SAVVIDES 20
Power Function --Example Equations: TR = 100Q - 10Q 2 Tables: Q 0 1 2 3 4 5 6 TR 0 90 160 210 240 250 240 Graphs: 300 250 200 TR TR 150 100 50 MR = dtr/dq = 100 20Q 0 0 1 2 3 4 5 6 7 MR Q Q 0 1 2 3 4 5 MR 100 80 60 40 20 0 Managerial Economics DR. SAVVAS C SAVVIDES 21
Rules of Differentiation Sum-and-Differences Rule: The derivative of the sum or difference of two functions U and V, is defined as follows. U = g( X) V = h( X) dy du dv = ± dx dx dx Y = U ± V Managerial Economics DR. SAVVAS C SAVVIDES 22
Rules of Differentiation Product Rule: The derivative of the product of two functions U and V, is defined as follows. U = g( X) V = h( X) Y = U V dy dv du = U + V dx dx dx Managerial Economics DR. SAVVAS C SAVVIDES 23
Rules of Differentiation Quotient Rule: The derivative of the ratio of two functions U and V, is defined as follows. U = g( X) V = h( X) Y ( du ) U( dv ) dy V = dx dx 2 dx V U = V Managerial Economics DR. SAVVAS C SAVVIDES 24
Rules of Differentiation Chain Rule: The derivative of a function that is a function of X is defined as follows. Y = f ( U ) U = g( X ) dy dy du = dx du dx Managerial Economics DR. SAVVAS C SAVVIDES 25
Optimization With Calculus (1) Optimization often requires finding the max. or the min. of a function (e.g. maxtr, mintc, or maxπ) Find X such that dy/dx = 0. This means that the curve of the function has zero slope Example: Given that TR = 100Q 10Q 2 d(tr) / dq = 100 20Q Setting dtr/dq =0, we get 0 =100 20Q 20Q = 100 Q* = 5 Therefore, Total Revenues are maximized at Q* = 5 To find the optimum Price, we go to the demand equation from which the TR function derived: P = 100 10Q P* = 100 10 (5) = 50 Managerial Economics DR. SAVVAS C SAVVIDES 26
Optimization With Calculus (2) Equation: TR = 100Q - 10Q 2 300 TR 250 200 TR 150 100 50 0 0 1 2 3 4 5 6 7 MR = dtr/dq = 100 20Q = 0 20Q = 100 Q MR Q = 5 Managerial Economics DR. SAVVAS C SAVVIDES 27
Optimization With Calculus (2) To distinguish between a max and a min, we use the second derivative. Second derivative rules: If d 2 Y/dX 2 > 0 (positive), then X is a minimum. If d 2 Y/dX 2 < 0 (negative), then X is a maximum. In the example, we found d(tr) / dq = 100 20Q d 2 (TR)/dQ 2 = - 20 (negative) Therefore, we know that the TR function is at a maximum ( top of the hill ) at Q = 5 Managerial Economics DR. SAVVAS C SAVVIDES 28
Multivariate Optimization Multivariate functions: TR = f (Sales, Advertising, prices, ) TC = f ( wages, interest, raw materials, ) Demand = f (price, income, P of substitutes, ) To optimize a function that has more than one independent variables, we use the partial derivative. Managerial Economics DR. SAVVAS C SAVVIDES 29
Multivariate Optimization (2) The Partial Derivative: The partial derivative (indicated by ) is used in order to isolate the marginal effect of each one of the independent variables. The same rules of differentiation apply, except that when we differentiate the dependent variable w.r.t. one variable, we hold all other variables constant. Managerial Economics DR. SAVVAS C SAVVIDES 30
Partial Derivative--Example Suppose that Profits (π) are a function of the sales of products X and Y as follows: π = f (X, Y) = 80X 2X 2 XY 3Y 2 + 100Y To find the partial derivative of Π w.r.t X, we hold Y constant (i.e. Y =0) to get: π / X = 80 4X Y To find the partial derivative of Π w.r.t Y, we hold X constant (i.e. X =0) to get: π / Y = 100 X 6Y Managerial Economics DR. SAVVAS C SAVVIDES 31
Max or Min Multivariate Functions Example (cont) To max or min a multivariate function, we set each partial derivative equal to zero and solve the resulting simultaneous equations: π / X = 80 4X Y = 0 π / Y = 100 X 6Y = 0 To solve these simultaneous equations, we multiply the 1 st by (-6) and the 2 nd by (-1) to get: - 480 + 24X +6Y = 0 100 X 6Y = 0-380 + 23X = 0 Therefore, X = 380 / 23 = 16.52 Managerial Economics DR. SAVVAS C SAVVIDES 32
Max or Min Multivariate Functions Example (cont) Substituting X = 16.52 into the first equation, we find the value of Y: 80 4 (16.52) Y = 0 80 66.08 Y = 0 Y = 13.92 Thus, the firm maximize Profits when it sells 13.92 unit of Y and 16.52 units of X. Thus: π = 80X 2X 2 XY 3Y 2 + 100Y π = 80(16.52) 2(16.52) 2 16.52 * 13.92 3(13.92) 2 + 100(13.92) π = 1,356.52 Managerial Economics DR. SAVVAS C SAVVIDES 33
Constrained Optimization So far, we dealt with unconstrained optimization However, in most real life situations, firms are faced with a series of constraints (budget, capacity, lack of raw materials, etc). In these cases, we need to optimize (max or min) the objective function (profits, revenues, costs, market share, etc) subject to the constraints faced by the firm. We have two methods to solve constrained optimization problems: 1. Substitution Method (used for simple functions) 2. Lagrangian Method (used for complex functions) Managerial Economics DR. SAVVAS C SAVVIDES 34
New Management Tools Benchmarking: finding out what processes or techniques excellent firms use and adopt & adapt Total Quality Management: the constant improvements in product quality and processes to deliver consistently superior service and value to customers Reengineering: seeks to completely reorganize the firm (processes, departments, entire firm). Radically redesigning processes to achieve significant gains in speed, quality, service, profitability The Learning Organization: continuous learning both on the individual level as well as on the collective level. It is based on five ingredients: a new mental model - achieve personal mastery develop system thinking develop shared vision strive for team learning Managerial Economics DR. SAVVAS C SAVVIDES 35
Other Management Tools Broad Banding: eliminating multiple layers of salary levels, and increasing labor flexibility Direct Business Model: dealing directly with the consumer, eliminating distributors and saving on time and costs (e.g, Dell ) Networking: the formation of strategic alliances to increase the synergies and capitalize on individual competences Pricing Power: being able to increase prices faster than costs thus increasing profits Small-World Model: large firms may gain efficiency by simulating the operation of small firms by breaking up the process in smaller scale and linking the units or individuals through organizational systems Virtual Integration: the blurring of traditional boundaries between manufacturer and its suppliers and manufacturer and customer supply chain management Virtual Management: the simulation of the production process and consumer behavior using computer models Managerial Economics DR. SAVVAS C SAVVIDES 36