Chapter 2Marginal Analysis and Optimization

Techniques

Marginal Analysis

Calculus

Total and Activities

Total: profits, revenues, costs, production, output, demand, …

Activity: factors affecting total such as outputs, advertising, price, …Total Revenue = f(output, advertising)

Demand = f(price, advertising, income, …)

Total Output = f(labor hours, machine hours)

Total Cost = f(output)

MARGINAL ANALYSISA MARGINAL is the amount added to the total by an

additional unit of an activity. Marginal revenue is (Total Revenue)/(utput)

An AVERAGE is simply the total divided by the activity level.Average revenue is (Total Revenue)/(Output)

An ELASTICITY is a marginal divided by an averageCost elasticity is (Marginal Cost)/(Average Cost)

If cost elasticity = 0.5, a 10% increase in output leads to a 5% increase in cost

Data is often in the form of totals and averagesEconomic decision rules are mostly defined in terms of

marginals and elasticities

Two Rules of Marginal Analysis

Assume our goal is to maximize profit

RULE 1 - Optimal Activity Level (Unconstrained)

An Activity should be expanded so long as its marginal profitability is positive.

Example:

If an added $1 million of advertising increases profits by $100,000, increase advertising.

The marginal profitability of advertising is $0.10 per dollar of advertising

Stop advertising if (a) the marginal profit of advertising is zero or negative(b) Advertising budget could be better used on other activities

Example – relationship between the marginal, average and total profitability of advertising

Number of Total Average Marginal

ads Profits Profits Profits

0 0 - -

1 80 80 80

2 180 90 100

3 270 90 90

4 280 70 10

5 250 50 -30Average profits = total profits/number of adsmarginal profits= change in profits /change in no. of ads

RULE 2 - Relative Activity Levels

Expand all activities to levels at which they yield the same marginal profits PER UNIT OF SCARCE RESOURCE.

Example of Relative Activity Levels

• The manager of a small retail firm wants to maximize the effectiveness of her firm's weekly advertising budget of $1,100.

• The manager has the option of advertising – on the local television station, – on the local AM radio station, or – in the local newspaper.

• As a class project, a managerial economics class at a nearby college estimated the impact on the retailer's sales of varying levels of advertising in the three media.

The estimates of the increases in weekly sales

Increase in Units Sold

Number of Ads TV Radio Newspaper

From 0 to 1 40 15 20

1 2 30 13 15

2 3 22 10 12

3 4 18 9 10

4 5 14 6 8

The prices of ads in the three media are $300 for each TV ad, $100 for each radio ad, and $200 for each newspaper ad.

The marginal benefit per dollar's worth of expenditures for the first radio ad is greater than for either of the other two media:

TV Radio Newspaper40 /300=.133 15 /100=.15 20 /200=.1030 /300=.100 13 /100=.13 15 /200=.07522 /300=.073 10 /100=.10 12 /200=.0618 /300=.060 9 /100= .09 10 /200=.0514 /300=.037 6 /100= .06 8 /200=.04

Example Continued

The first ad selected by the manager will be a radio ad- the activity with the largest marginal benefit per dollar expenditure. ...

By selecting three radio ads, two television ads, and one newspaper ad, the manager of the firm maximized sales subject to the expenditure constraint.

Note that for this combination the marginal benefits per dollar expenditure for the three media are equal to 0.10.

Calculus Example of Marginal Analysis

• A firm manufactures desktop (X) and notebook (Y) computers receives a daily allotment of only 7 CPUs. The firm’s profit function () depends upon the number of desktops (X) and notebooks (Y) produced daily:

= (32X – X2) + (48Y – 2Y2)

• How many units of X and Y should the firm produce daily?

Ans: Marginal Profit of X = X = 32 – 2 X

Marginal Profit of Y = Y = 48 – 4 YSince each computer uses one CPU, equating Marginal Profit per CPU for X and Y:

(32 – 2X)/1=(48 – 4Y)/1 or 2Y – X = 8

and combining it with the resource allotment constraint of 7 CPUs, X + Y = 7,

we get 2Y – 8 = X and X = 7 – Y, 2Y – 8 = 7 - Y

X = 2 and Y = 5

Total, Average, and Marginal Relationships Tabular Form

Example - the effect of advertising on profits

Number of Total Average Marginal

ads Profits Profits Profits

0 0 - -

1 80 80 80

2 180 90 100

3 270 90 90

4 280 70 10

5 250 50 -30

1. First units - total, average, and marginal figures are equal (if total profits are $0 at 0 ads)

2. Total v. Marginal - the total is the sum of the preceding and current marginals

3. Average v. Marginal -

a. If the marginal is above the average, the average rises

b. If the average and the marginal are equal, the average is unchanged

c. If the marginal is below the average, the average falls

Total, Average, and Marginal Relationships Graphical Form

Total Profits

Ads

$

Marginal

Average

Average v. Total curves - given any point on a total curve, A, the corresponding average is theslope of the straight line, OA, connecting point A with the origin, 0.

Marginal v. Total Curves – given any point on the total curve (A), the marginal is equal to the slope of a line tangent to the total at that point.

A

From an Average or Marginal to the Total

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9

Average Cost

Marginal Cost

Total v. Average - the total (profits, costs, or revenues) of any output 0Q is the area of the rectangle inscribe under the average curve which has 0Q as its base.

Total v. Marginal - the total (profits, costs, or revenues) of any output is the portion of the area under the marginal (profit, cost, or revenue) curve which lies above 0Q.

Average and Marginal Curves

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8

Average

Marginal

Price

Quantity

Average Revenue = Priceif the firm charges a single price

For a linear demand curve (Average Revenue curve),Marginal Revenue has same intercept and twice the slope

Straight Line Marginal and Average: given a linear average curve, the marginal is also linear with the same vertical intercept and twice the slope.

Why Marginal Revenue is below Average Revenue (Price)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10

Average

Marginal

XY (Scatter) 5

Profit Maximizing Price and Quantity Example

Price = 10 - Q (Average Revenue)Total Revenues = (Price)(Quantity) = 10 Q - Q2 Marginal Revenue = dTotal Revenue = 10 - 2 Q

dQTotal Costs = 5 QMarginal Cost = d Total Cost= 5

dQTotal Profits = Total Revenues - Total Costs

= 10 Q - Q2 - 5 Q = 5 Q - Q2 Marginal profits = d Total Profit/dQ = 5 - 2 Q

= Marginal Revenue - Marginal Cost = 10 - 2Q - 5

Marginal Profit = Marginal revenues - Marginal costs

M = MR - MC = (10 - 2 Q) - 5 = 5 - 2 Q

For profit maximization, M = 0 or MR = MC

In the above example, M = 5 - 2 Q = 0

Q = 5/2 = 2.5 units

Price = 10 - (5/2) = $7.50 per unit

= 25/4= $6.25

-4

-2

0

2

4

6

8

10

12

0 2 4 6 8 10 12

-10

-5

0

5

10

15

20

25

30

35

0 2 4 6 8

Quantity

Totals

TR

TC

Quantity

Marginals

MC

MR

Total Profits