Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics in a Global Economy Chapter 2 Optimization Techniques and New Management Tools

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University OPTIMIZATION Managerial economics is concerned with the ways in which managers should make decisions in order to maximize the effectiveness or performance of the organizations they manage. To understand how this can be done we must understand the basic optimization techniques. Managerial economics is concerned with the ways in which managers should make decisions in order to maximize the effectiveness or performance of the organizations they manage. To understand how this can be done we must understand the basic optimization techniques. Functional relationships: relationships can be expressed by graphs: relationships can be expressed by graphs:

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University This form can be expressed in an equation:  Q = f ( P )(1)  Though useful, it does not tell us how Q responds to P, but this equation do.  Q = p (2) Marginal Analysis  The marginal value of a dependent variable is defined as the change in this dependent variable associated with a 1-unit change in a particular independent variable. e.g.

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Note: Total profit is maximized when marginal profit shifts from positive to negative Note: Total profit is maximized when marginal profit shifts from positive to negative.

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Average Profit = Profit / Q  Slope of ray from the origin:  Rise / Run  Profit / Q = average profit  Maximizing average profit doesn’t maximize total profit MAX C B profits Q PROFITS quantity

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Marginal Profits =  /  Q  Q 1 is breakeven (zero profit)  maximum marginal profits occur at the inflection point (Q 2 )  Max average profit at Q 3  Max total profit at Q 4 where marginal profit is zero  So the best place to produce is where marginal profits = 0. profits max Q2Q2 marginal profits Q Q average profits average profits Q3Q3 Q4Q4 (Figure 2.1) Q1Q1

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University  2005 South-Western Publishing Differential Calculus in Management  A function with one decision variable, X, can be written as: Y = f(X)  The marginal value of Y, with a small increase of X, is M y =  Y/  X  For a very small change in X, the derivative is written: dY/dX = limit  Y/  X  X  B

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Marginal = Slope = Derivative  The slope of line C-D is  Y/  X  The marginal at point C is  Y/  X  The slope at point C is the rise (  Y) over the run (  X)  The derivative at point C is also this slope X C D Y YY XX

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Optimum Can Be Highest or Lowest Optimum Can Be Highest or Lowest  Finding the maximum flying range for the Stealth Bomber is an optimization problem.  Calculus teaches that when the first derivative is zero, the solution is at an optimum.  The original Stealth Bomber study showed that a controversial flying V-wing design optimized the bomber's range, but the original researchers failed to find that their solution in fact minimized the range.  It is critical that managers make decision that maximize, not minimize, profit potential!

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Quick Differentiation Review  Constant Y = cdY/dX = 0Y = 5 FunctionsdY/dX = 0  A Line Y = c XdY/dX = cY = 5X dY/dX = 5  Power Y = cX b dY/dX = bcX b-1 Y = 5X 2 Functions dY/dX = 10X Name Function Derivative Example

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University  Sum of Y = G(X) + H(X) dY/dX = dG/dX + dH/dX Functions Functions exampleY = 5X + 5X 2 dY/dX = X  Product of Y = G(X) H(X)  Two Functions dY/dX = (dG/dX)H + (dH/dX)G example Y = (5X)(5X 2 ) example Y = (5X)(5X 2 ) dY/dX = 5(5X 2 ) + (10X)(5X) = 75X 2 Quick Differentiation Review

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University  Quotient of Two Y = G(X) / H(X) Functions dY/dX = (dG/dX)H - (dH/dX)G H 2 Y = (5X) / (5X 2 ) dY/dX = 5(5X 2 ) -(10X)(5X) (5X 2 ) 2 Y = (5X) / (5X 2 ) dY/dX = 5(5X 2 ) -(10X)(5X) (5X 2 ) 2 = -25X 2 / 25X 4 = - X -2 = -25X 2 / 25X 4 = - X -2  Chain RuleY = G [ H(X) ] dY/dX = (dG/dH)(dH/dX) Y = (5 + 5X) 2 dY/dX = 2(5 + 5X) 1 (5) = X dY/dX = 2(5 + 5X) 1 (5) = X Quick Differentiation Review

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University  USING DERIVATIVES TO SOLVE MAXIMIZATION AND MINIMIZATION PROBLEMS  Maximum or minimum points occur only if the slope of the curve equals zero.  Look at the following graph

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University y dy/dx Max of x Slope = 0 value of x Value of dy/dx which Is the slope of y curve Value of Dy/dx when y is max x x Note that this is not sufficient for maximization or minimization problems.

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Min value of y Max value of y d2y/dx2 > 0 d2y/dx2 < 0 value of dy/dx y Dy/dx x x

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University  Graphs of an original third-order function and its first and second derivatives. (what if the second order = 0)

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Optimization Rules

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Applications of Calculus in Managerial Economics  maximization problem:  A profit function might look like an arch, rising to a peak and then declining at even larger outputs. A firm might sell huge amounts at very low prices, but discover that profits are low or negative.  At the maximum, the slope of the profit function is zero. The first order condition for a maximum is that the derivative at that point is zero. If  = 50Q - Q 2, then d  /dQ = ·Q, using the rules of differentiation. Hence, Q = 25 will maximize profits where Q = 0.

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University More Applications of Calculus  minimization problem: Cost minimization  supposes that there is a least cost point to produce. An average cost curve might have a U-shape. At the least cost point, the slope of the cost function is zero. The first order condition for a minimum is that the derivative at that point is zero. If TC = 5Q 2 – 60Q, then dC/dQ = 10Q Hence, Q = 6 will minimize cost Where: 10Q - 60 = 0.

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University More Examples  Competitive Firm: Maximize Profits  where  = TR - TC = P Q - TC(Q)  Use our first order condition: d  /dQ = P - dTC/dQ = 0.  Decision Rule: P = MC. a function of Q Max  = 100Q - Q 2 First order = Q = 0 implies Q = 50 and;  = 2,500

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Second Order Condition: one variable  If the second derivative is negative, then it’s a maximum  If the second derivative is positive, then it’s a minimum Max  = 100Q - Q 2 First derivative Q = 0 second derivative is: -2 implies Q =50 is a MAX Max  = X 2 First derivative 10X = 0 second derivative is: 10 implies Q = 10 is a MIN Problem 1Problem 2

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Extra examples

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Partial Differentiation  Economic relationships usually involve several independent variables.  A partial derivative is like a controlled experiment- it holds the “other” variables constant  Suppose price is increased, holding the disposable income of the economy constant as in Q = f (P, I )  then  Q/  P holds income constant.

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Problem:  Sales are a function of advertising in newspapers and magazines ( X, Y) Max S = 200X + 100Y -10X 2 -20Y 2 +20XY  Differentiate with respect to X and Y and set equal to zero.  S/  X = X + 20Y= 0  S/  Y = Y + 20X = 0  solve for X & Y and Sales

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Solution: 2 equations & 2 unknowns  X + 20Y= 0  Y + 20X = 0  Adding them, the -20X and +20X cancel, so we get Y = 0, or Y =15  Plug into one of them:  X = 0, hence X = 25  To find Sales, plug into equation:  S = 200X + 100Y -10X 2 -20Y 2 +20XY = 3,250

Managerial Economics Prof. M. El-Sakka CBA. Kuwait University

Key Terms  Marginal profit  Average profit  Marginal cost  Marginal revenue  Marginal analysis  Optimization  Derivative of Y with respect to X (dy/dx)  Differentiation  Second derivative  Partial derivative  Constrained optimization  Lagrangian multiplier method  Lagrangian function