Managerial Economics

Lecture: Optimization TechniqueDate: 08.06.2014

Maximizing profit

• Manager determine the quantity of output to be produced

• Quantity of sales to maximize profit

Optimization Techniques

• Methods for maximizing or minimizingan objective function• Examples– Consumers maximize utility by purchasingan optimal combination of goods– Firms maximize profit by producing andselling an optimal quantity of goods– Firms minimize their cost of production byusing an optimal combination of inputs

Optimization Techniques

• Concept of the DerivativeThe derivative of Y with respect to X isequal to the limit of the ratio ΔY/ΔX asΔX approaches zerody/dx=LimΔY/ΔX x→0

Geometric Relationships

• A marginal value is positive, zero, andnegative, respectively, when a totalcurve slopes upward, is horizontal, andslopes downward• A marginal value may be negative, butan average value can never be negative

Steps in Optimization

• Define an objective function of one ormore choice variables• Define the constraint on the values ofthe objective function• Determine the values of the choicevariables that maximize or minimize theobjective function while satisfying theconstraint

New Management Tools forOptimization

• Benchmarking (tool for improving productivity and quality)

• Total Quality Management (constantly improving the quality of products and the firm’s processes to deliver more value to customers; e.g. Six Sigma)

• Reengineering (radical redesign of all the firm’s processes to achieve major gains)

• Learning Organization (values continuing learning, both individual and collective)

Management Tools forOptimization

• Broad-banding (elimination of multiple salarygrades to foster movement among jobs withinthe firm and lower cost)• Direct Business Model (eliminating the time

and cost of third-party distribution)

Management Tools forOptimization

• Networking (forming of temporary strategicalliances among firms as per their core

competence)• Performance Management (holding executivesand their subordinates accountable for

delivering the desired results)

Other Management Tools forOptimization

• Pricing Power (ability of a firm to raise prices faster

than the rise in its costs and vice-versa)• Small-World Model (linking well-connectedindividuals from each level of the organization to

oneanother to improve flow of information and theoperational efficiency)

Other Management Tools forOptimization

• Strategic Development (continuous review ofstrategic decisions)• Virtual Integration (treating suppliers and

customersas if they were part of the company which reducesthe need for inventories)• Virtual Management (ability of a manager to

simulateconsumer behavior using computer models)

Univariate Optimization

• Given objective function Y = f(X)• Find X such that dY/dX = 0• Second derivative rules:• If d2Y/dX2 > 0, then X is a minimum• If d2Y/dX2 < 0, then X is a maximum

Example 1

• Given the following total revenue (TR) function, determine the quantity of output (Q) that will maximize total revenue:

• TR = 100Q – 10Q2• dTR/dQ = 100 – 20Q = 0• Q* = 5 and d2TR/dQ2 = -20 < 0

Example 2

• Given the following total revenue (TR) function, determine the quantity of output (Q) that will maximize total revenue:

• TR = 45Q – 0.5Q2• dTR/dQ = 45 – Q = 0• Q* = 45 and d2TR/dQ2 = -1 < 0

Example 3

• Given the following marginal cost function (MC), determine the quantity of output that will minimize MC:

• MC = 3Q2 – 16Q + 57• dMC/dQ = 6Q - 16 = 0• Q* = 2.67 and d2MC/dQ2 = 6 > 0

Example 4

• GivenTR = 45Q – 0.5Q2TC = Q3 – 8Q2 + 57Q + 2• Determine Q that maximizes profit (π):π = 45Q – 0.5Q2 – (Q3 – 8Q2 + 57Q + 2)

Example 4: Solution

• Method 1dπ/dQ = 45 – Q – 3Q2 + 16Q – 57 = 0– 12 + 15Q – 3Q2 = 0• Method 2MR = dTR/dQ = 45 – QMC = dTC/dQ = 3Q2 – 16Q + 57Set MR = MC: 45 – Q = 3Q2 – 16Q + 57• Use quadratic formula: Q* = 4

Multivariate Optimization

• Objective function Y = f(X1, X2, ...,Xk)• Find all Xi such that ∂Y/∂Xi = 0• Partial derivative:∂Y/∂Xi = dY/dXi while all Xj (where j ≠ i) areheld constant

Example 5

• Determine the values of X and Y thatmaximize the following profit function:π = 80X – 2X2 – XY – 3Y2 + 100Y• Solution∂π/∂X = 80 – 4X – Y = 0∂π/∂Y = – X – 6Y + 100 = 0Solve simultaneouslyX = 16.52 and Y = 13.91

Constrained Optimization

• Substitution Method– Substitute constraints into the objectivefunction and then maximize the objectivefunction• Lagrangian Method– Form the Lagrangian function by addingthe Lagrangian variable and constraint tothe objective function and then maximizethe Lagrangian function

Example 6

• Use the substitution method tomaximize the following profit function:π = 80X – 2X2 – XY – 3Y2 + 100Y• Subject to the following constraint:X + Y = 12

Example 6: Solution

• Substitute X = 12 – Y into profit:π = 80(12 – Y) – 2(12 – Y)2 – (12 – Y)Y – 3Y2 +

100Yπ = – 4Y2 + 56Y + 672• Solve as univariate function:dπ/dY = – 8Y + 56 = 0Y = 7 and X = 5

Example 7

• Use the Lagrangian method tomaximize the following profit function:π = 80X – 2X2 – XY – 3Y2 + 100Y• Subject to the following constraint:X + Y = 12

Example 7: Solution

• Form the Lagrangian functionL = 80X – 2X2 – XY – 3Y2 + 100Y + λ(X + Y – 12)• Find the partial derivatives and solvesimultaneouslydL/dX = 80 – 4X –Y + λ = 0dL/dY = – X – 6Y + 100 + λ = 0dL/dλ = X + Y – 12 = 0• Solution: X = 5, Y = 7, and λ = -53

Interpretation of theLagrangian Multiplier, λ

• Lambda, λ, is the derivative of theoptimal value of the objective functionwith respect to the constraint– In Example 7, λ = -53, so a one-unitincrease in the value of the constraint (from-12 to -11) will cause profit to decrease byapproximately 53 units