Chapter 2 Marginal Analysis and Optimization Techniques Calculus 1

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 ANALYSIS A MARGINAL is the amount added to the total by an additional unit of an activity. Marginal revenue is (DTotal Revenue)/(DOutput) 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 average Cost 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 averages Economic decision rules are mostly defined in terms of marginals and elasticities 4

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. 5

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 6

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 -30 Average profits = total profits/number of ads marginal 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. 7

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. 8

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 9

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 Newspaper 40 /300=.133 15 /100=.15 20 /200=.10 30 /300=.100 13 /100=.13 15 /200=.075 22 /300=.073 10 /100=.10 12 /200=.06 18 /300=.060 9 /100= .09 10 /200=.05 14 /300=.037 6 /100= .06 8 /200=.04 10

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. 11

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 (p) depends upon the number of desktops (X) and notebooks (Y) produced daily: p = (32X – X2) + (48Y – 2Y2) How many units of X and Y should the firm produce daily? Ans: Marginal Profit of X = p/X = 32 – 2 X Marginal Profit of Y = p/Y = 48 – 4 Y Since 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 12

a. If the marginal is above the average, the average rises 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 13

Total, Average, and Marginal Relationships Graphical Form $ Total Profits Average v. Total curves - given any point on a total curve, A, the corresponding average is the slope of the straight line, OA, connecting point A with the origin, 0. Marginal A 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. Average Ads

From an Average or Marginal to the Total 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 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. 17

Average and Marginal Curves Straight Line Marginal and Average: given a linear average curve, the marginal is also linear with the same vertical intercept and twice the slope. Average and Marginal Curves Price Average Revenue = Price if the firm charges a single price For a linear demand curve (Average Revenue curve), Marginal Revenue has same intercept and twice the slope Quantity

Why Marginal Revenue is below Average Revenue (Price)

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 dQ Total Costs = 5 Q Marginal Cost = d Total Cost= 5 Total 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 30

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 31

Totals TC TR Total Profits Quantity Marginals MC MR Quantity 34