# Optimal Decisions Using Marginal Analysis

Chapter two

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Optimal Decisions Using Marginal Analysis

Managerial Economics, 8e

William F. Samuelson ● Stephen G. Marks

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

RECAP

Chapter 2: Optimal Decisions Using Marginal Analysis

U = number of actual users

M = number of potential users

a = % of actual users lost in each period

b = % of potential users gained in each period

The sustainable user base is:

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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RECAP

Chapter 2: Optimal Decisions Using Marginal Analysis

= cost of reducing a by 1%

= cost of increasing b by 1%

The optimal allocation of the marketing budget is:

Calculate the derivatives and find the optimal ratio

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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RECAP

Chapter 2: Optimal Decisions Using Marginal Analysis

The derivatives are:

T

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EXAMPLE 1: MICROCHIP MANUFACTURER

Demand:

Revenue:

Chapter 2: Optimal Decisions Using Marginal Analysis

Other factors than price that may impact demand are held fixed. They would affect the parameters.

Sales are lots (of 100 chips) per week.

Price per lot is in thousands of dollars.

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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Figure 2.1 The demand curve for microchips

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Figure 2.2 revenue from microchips

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

EXAMPLE 1:

Cost:

Profit:

Chapter 2: Optimal Decisions Using Marginal Analysis

Quadratic function that increases, then decreases.

Cost is in thousands of dollars.

Fixed cost: \$100K per week. Variable cost: \$38K per lot.

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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Figure 2.3 The cost of microchips

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Figure 2.4 Profit from microchips

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Figure 2.5 A close-up view of profit

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

EXAMPLE 1:

Profit:

Chapter 2: Optimal Decisions Using Marginal Analysis

Derivative of Q) is the slope of the profit function at Q. Slope is zero at that maximizes profit.

Recall the profit function.

First-order condition:

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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Figure 2.6 Total profit and marginal profit

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

DIFFERENTIATION

Chapter 2: Optimal Decisions Using Marginal Analysis

Suppose we have a function of the form:

The derivative is:

Special cases:

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DIFFERENTIATION

Chapter 2: Optimal Decisions Using Marginal Analysis

For instance, for the function:

Derivatives of the additive terms are:

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DIFFERENTIATION

Chapter 2: Optimal Decisions Using Marginal Analysis

The derivative of a sum of terms is the sum of the derivatives of the additive terms:

So:

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EXAMPLE 2:

Profit:

Chapter 2: Optimal Decisions Using Marginal Analysis

Derivative of Q) is the slope of the profit function at Q. Slope is zero at that maximizes profit.

Suppose we have a quadratic profit function.

First-order condition:

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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Figure 2A.1 The Firm’s Profit Function

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

EXAMPLE 3:

Chapter 2: Optimal Decisions Using Marginal Analysis

Consider hypothetically a cubic profit function:

It has two flat points, where its derivative is zero. One is a minimum, one is a maximum.

Profit:

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Figure 2A.2 A Second Profit Function

Chapter 2: Optimal Decisions Using Marginal Analysis

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

EXAMPLE 3:

Chapter 2: Optimal Decisions Using Marginal Analysis

Recall the cubic profit function:

Solutions of the first-order condition are Q = 2 and Q = 10. For a maximum, we also need > 6:

First-order condition:

Second-order condition:

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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MAXIMA AND MINIMA

Chapter 2: Optimal Decisions Using Marginal Analysis

Near a maximum, a function rises, then falls. Its derivative changes from positive to negative and keeps declining. The derivative of its derivative is negative.

Near a minimum, a function falls, then rises. Its derivative changes from negative to positive and keeps increasing. The derivative of its derivative is positive.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

22

EXAMPLE 4: MULTIVARIATE OPTIMIZATION

Profit:

Chapter 2: Optimal Decisions Using Marginal Analysis

Maximize separately with respect to P and A.

Suppose the firm simultaneously decides on pricing and an advertising budget.

First-order conditions:

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

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EXAMPLE 4:

Chapter 2: Optimal Decisions Using Marginal Analysis

Solve jointly by substituting one into the other.

Recall the dual first-order conditions.

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Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.

24

EXAMPLE 5: CONSTRAINED OPTIMIZATION

Profit:

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