# Chapter 2 Economic Optimization by Mark Hirschey MANAGERIAL ECONOMICS 11th Edition

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```					   MANAGERIAL
ECONOMICS 11th Edition

By
Mark Hirschey
Economic Optimization
Chapter 2
OVERVIEW
   Economic Optimization Process
   Expressing Economic Relations
   Marginals as the Derivatives of Functions
   Marginal Analysis in Decision Making
   Incremental Concept in Economic Analysis
Chapter 2
KEY CONCEPTS
   optimal decision          derivative
   table
   inflection point
   graph                     second derivative
   equation                  profit maximization
   dependent variable        breakeven point
   independent variable
   marginal
   revenue maximization
   marginal revenue          average cost
   marginal cost              minimization
   marginal profit
Economic Optimization Process
   Optimal Decisions
   Best decision helps achieve objectives most
efficiently.
   Maximizing the Value of the Firm
   Value maximization requires serving
customers efficiently.
 What do customers want?
 How can customers best be served?
Expressing Economic Relations
   Tables and Equations
   Simple graphs and tables are useful.
   Complex relations require equations.
   TR= Price X Quantity
   If the Price function is \$ 2 then the Revenue
function is 2x
Expressing Economic Relations
   Total, Average, and Marginal Relations
   Marginal: Change in the dependent variable
caused by a 1-unit change in an independent
variable.
   Marginal Revenue: Change in total revenue
associated with a 1-unit change in output
   MC: Change in TC following a 1-unit change
in output
   MP=  in TP due to a 1-unit  in output.
Relation between Revenue and
Output; Revenue=\$1.5x
Revenue    Output
\$1.5       1
3         2
4.50      3
6         4
7.5       5
9         6
Revenue per time period (\$)
\$9 8 7 6 5 4
3 Total revenue = \$1.50 ´ output 2
1
0        1 2 3 4 5 6 7 8 9 Output
per time period (units)
Total, Marginal, and Average
Relations
Q     TP     MP    AP
0     \$0     \$0    -
1       19    19   \$ 19
2       52    33     26
3       93    41     31
4      136    43     34
5      175    39     35
6      210    35     35
7      217     7     31
8      208    -9     26
Maximization occurs when
marginal switches from positive to
negative.
   If marginal is above average, average is
rising.
   If marginal is below average, average is
falling.
   Profit per unit is rising when marginal
profit is greater than average profit per
unit.
Marginal Analysis. Evaluate the price (P) and the output (Q) data in the following table.

Q         P       TR       MR        AR
0       \$80
1        70
2        60
3        50
4        40
5        30
6        20
7        10
8         0

A. Compute the related total revenue (TR), marginal revenue (MR), and
average revenue (AR) figures:

B. At what output level is revenue maximized?
Profit Maximization. Fill in the missing data for price (P), total revenue (TR), marginal revenue (MR), total
cost (TC), marginal cost (MC), profit (), and marginal profit (M) in the following table.

Q         P       TR       MR        TC        MC                 M
0     \$160      \$ 0       \$ --     \$ 0        \$ --     \$ 0       \$ --
1      150       150       150       25        25       125       125
2      140                           55        30                 100
3                390                           35       300        75
4      120                  90       130                350
5      110       550                 175                           25
6                600        50                   55     370
7                630                 290         60                -30
8        80      640                 355                285
9                630                             75                -85
10                600                 525                  75

A. At what output (Q) level is profit maximized?

B. At what output (Q) level is revenue maximized?
Marginal Analysis: Tables. Susan Mayer is a sales representative for the Desperate
Insurance Company, and sells life insurance policies to individuals in the Phoenix area.
Mayer's goal is to maximize total monthly commission income, which is figured at 10%
of gross sales. In reviewing monthly experience over the past year, Mayer found the
following relations between days spent in each city and monthly sales generated.

Phoenix Scottsdale      Tempe
Days      Sales      Sales       Sales
0    \$ 5,000 \$ 7,500        \$ 2,500
1     15,000    15,000        6,500
2     23,000    21,500        9,500
3     29,000    27,000       11,500
4     33,000    31,500       12,500
5     35,000    35,000       12,500
6     35,000    37,500       12,500
7     35,000    39,000       12,500

A. Construct a table showing Mayer's marginal sales per day in
each city.

B. If administrative duties limit Mayer to only 10 selling days per
month, how should she spend them?

C. Calculate Mayer's maximum monthly commission income.
Graphing Total, Marginal, and
Average Relations
   Slope=Measure of the steepness of a line
   At any point along a total curve, the
corresponding average figure is given by
the slope of a straight line from the origin
to that point.
   At any point along a total curve, the
corresponding marginal figure is given by
the slope of a line drawn tangent to the
total curve at that point.
Important Relations
   The slope of the Profit curve is increasing
from the origin to point “C”. At point “C”
called an inflection point (or point of
diminishing returns), the slope of the
Profit curve is maximized. Marginal Profits
are maximized at that output.
   Between “C” and “E”, Profit continues to
increase and Marginal Profit is still positive
even though it is declining
Continued
   At point “E”, the Profit curve has slope 0
therefore Profit is neither rising nor failing.
MP is 0 and Profit is Maximized. Beyond
“E” the Profit curve has a negative slope
and MP is negative
Marginals as the Derivatives of
Functions
   Concept of a Derivative
   Derivative is a marginal relation.
   Derivatives and Slope
   Derivative of total revenue is marginal
revenue.
   Derivative of total cost is marginal cost.
   Derivative of total profit is marginal profit.
Marginal Analysis in Decision
Making
1. Finding Maximums or Minimums
    Maximum and minimum points occur where
marginal is zero.

  10,000  400Q  2Q   2
   At Q=100 we have maximum profit.
Beyond Q=100 MP is negative and profit is
decreasing.
Marginal Analysis in Decision
Making
2. Distinguishing Maximums from Minimums
   Maximum is where first derivative is zero,
second derivative is negative.
   Minimum is where first derivative is zero,
second derivative is positive.

  3,00  2400Q  350Q  8.33Q
2         3
   Marginal Profit is the first derivative of the
Profit Function

M  2400  700Q  25Q      2

Profit is either maximized or minimized at
the points where the first derivative or MP
is equal to 0.

M  25(Q  4)(Q  24)

Q=24 & Q=4
   The second derivative of Profit is:
d 2
2
 700  50Q
dQ

   Let’s evaluate the second derivative at Q=4
d 2
2
 700  50( 4)  500
dQ
   Because the second derivative is positive profit
is minimized at Q=4
   Let’s evaluate the second derivative at Q=24
d 2
2
 700  50(24)  500
dQ
   Because the second derivative is negative profit
is maximized at Q=24
Marginal Analysis in Decision
Making
   Maximizing the Difference Between Two
Functions
   Maximum profit requires MR = MC.
   Breakeven point. Profit is equal to 0
   Average Cost minimization occurs when
MR=AC
Profit Maximization: Equations. Austin Heating & Air Conditioning, Inc., offers
heating and air conditioning system inspections in the Austin, Texas, market. Prices
are stable at \$50 per unit. This means that P = MR = \$50 in this market. Total cost
(TC) and marginal cost (MC) relations are:

TC = \$1,000,000 + \$10Q + \$0.00025Q 2

MC = TC/Q = \$10 + \$0.0005Q

A. Calculate the output level that will maximize profit.

B. Calculate this maximum profit.
Average Cost Minimization. Better Buys, Inc., is a leading discount retailer of wide-
screen digital and cable-ready plasma HDTVs. Revenue and cost relations for a
popular 55-inch model are:

TR = \$4,500Q - \$0.1Q2

MR = TR/Q = \$4,500 - \$0.2Q

TC = \$2,000,000 + \$1,500Q + \$0.5Q2

MC = TC/Q = \$1,500 + \$1Q.

A. Calculate output, marginal cost, average cost, price, and profit
at the average cost-minimizing activity level.

B. Calculate these values at the profit-maximizing activity level.
Self Test Problem 1
   A. A table or spreadsheet for Presto output (Q), price (P), total
revenue (TR), marginal revenue (MR), total cost (TC), marginal cost
(MC), total profit (π), and marginal profit (Mπ) appears as follows:
Total      Marginal    Total     Marginal      Total       Marginal
Units    Price    Revenue     Revenue     Cost       Cost        Profit        Profit
0      \$60          \$0        \$60   \$100,000          \$5   (\$100,000)         \$55
1,000       55     55,000          50    105,500           6      (50,500)          44
2,000       50    100,000          40    112,000           7      (12,000)          33
3,000      45     135,000          30    119,500          8        15,500           22
4,000      40     160,000          20    128,000          9        32,000           11
5,000      35     175,000          10    137,500         10        37,500            0
6,000      30     180,000           0    148,000         11        32,000         (11)
7,000      25     175,000        (10)    159,500         12        15,500         (22)
8,000      20     160,000        (20)    172,000         13      (12,000)         (33)
9,000      15     135,000        (30)    185,500         14      (50,500)         (44)
10,000      10     100,000        (40)    200,000         15     (100,000)         (55)
   The price/output combination at which total profit is
maximized is P = \$35 and Q = 5,000 units. At that
point, MR = MC and total profit is maximized at \$37,500.
The price/output combination at which total revenue is
maximized is P = \$30 and Q = 6,000 units. At that
point, MR = 0 and total revenue is maximized at
\$180,000. Using the Presto table or spreadsheet, a
graph with TR, TC, and π as dependent variables, and
units of output (Q) as the independent variable appears
as follows
Self Test Problem 1
C.

To find the profit-maximizing output level analytically, set MR = MC, or set Mπ = 0,
and solve for Q. Because
MR = MC
Q = 5,000
At Q = 5,000,
P = \$35
π = -\$100,000 + \$55(5,000) - \$0.0055(5,000 2)
= \$37,500
(Note: 2π/Q2 < 0, This is a profit maximum because total profit is falling for Q > 5,000.)
To find the revenue-maximizing output level, set MR = 0, and solve for Q. Thus,
MR = \$60 - \$0.01Q = 0          Q =       6,000
At Q = 6,000,
P = \$30
π = TR - TC
= -\$100,000 + \$55(6,000) - \$0.0055(6,000 2) = \$32,000

(Note: 2TR/Q2 < 0, and this is a revenue maximum because total revenue is decreasing for output beyond Q > 6,000.)
   D.
   Given downward sloping demand and marginal revenue curves and
positive marginal costs, the profit-maximizing price/output
combination is always at a higher price and lower production level
than the revenue-maximizing price-output combination. This stems
from the fact that profit is maximized when MR = MC, whereas
revenue is maximized when MR = 0. It follows that profits and
revenue are only maximized at the same price/output combination
in the unlikely event that MC = 0.

In pursuing a short-run revenue rather than profit-maximizing
strategy, Presto can expect to gain a number of important
advantages, including     enhanced product awareness among
consumers, increased customer loyalty, potential economies of scale
in marketing and promotion, and possible limitations in competitor
entry and growth.       To be consistent with long-run profit
maximization, these advantages of short-run revenue maximization
must be at least worth Presto's short-run sacrifice of \$5,500 (=
\$37,500 - \$32,000) in monthly profits.
Self Test Problem 2

A.   A table or spreadsheet for Pharmed Caplets output (Q),
price (P), total revenue (TR), marginal revenue (MR),
total cost (TC), marginal cost (MC), average cost (AC),
total profit (π), and marginal profit (Mπ) appears as
follows:
Total      Marginal    Total     Marginal   Average     Total      Marginal
Units   Price   Revenue     Revenue     Cost       Cost       Cost       Profit      Profit
0    \$900          \$0       \$900    \$36,000       \$200        ---   (\$36,000)        \$700
100    \$890     89,000        \$880    \$60,000       \$280    600.00       29,000         600
200    \$880    176,000        \$860    \$92,000       \$360    460.00       84,000         500
300    \$870    261,000        \$840   \$132,000       \$440    440.00      129,000         400
400    \$860    344,000        \$820   \$180,000       \$520    450.00      164,000         300
500    \$850    425,000        \$800   \$236,000       \$600    472.00      189,000         200
600    \$840    504,000        \$780   \$300,000       \$680    500.00      204,000         100
700    \$830    581,000        \$760   \$372,000       \$760    531.43      209,000           0
800    \$820    656,000        \$740   \$452,000       \$840    565.00      204,000       (100)
900    \$810    729,000        \$720   \$540,000       \$920    600.00      189,000       (200)
1,000    \$800    800,000        \$700   \$636,000     \$1,000    636.00      164,000       (300)
 The price/output combination at which total profit is
maximized is P = \$830 and Q = 700 units. At that point,
MR = MC and total profit is maximized at \$209,000. The
price/output combination at which average cost is
minimized is P = \$870 and Q = 300 units. At that point,
MC = AC = \$440.
Using the Pharmed Caplets table or spreadsheet, a graph
with AC, and MC as dependent variables and units of
output (Q) as the independent variable appears as
follows:
To find the profit-maximizing output level analytically, set MR = MC, or
set Mπ = 0, and solve for Q.
Because          MR      =      MC
\$900 - \$0.2Q = \$200 + \$0.8Q
Q =      700
At Q = 700,
Price = TR/Q = (\$900Q - \$0.1Q2)/Q
=\$900 - \$0.1(700) = \$830
π= TR - TC
= \$900Q - \$0.1Q2 - \$36,000 - \$200Q - \$0.4Q2
= -\$36,000 + \$700(700) - \$0.5(7002)
=\$209,000
(Note: M2π/MQ2 < 0, and this is a profit maximum because profits are
falling for Q > 700.)
To find the average-cost minimizing output level, set MC = AC, and
solve for Q. Because
AC= TC/Q
= (\$36,000 + \$200Q + \$0.4Q2)/Q
= \$36,000Q-1 + \$200 + \$0.4Q,
it follows that:
MC= AC
\$200 + \$0.8Q =             \$36,000Q-1 + \$200 + \$0.4Q
0.4Q= 36,000Q-1
Q2 = 90,000      ; Q= 300
At Q = 300,       Price =\$900 - \$0.1(300)        =\$870
π =      -\$36,000 + \$700(300) - \$0.5(3002)
=        \$129,000
(Note: M2AC/MQ2 > 0, and this is an average-cost minimum because
average cost is rising for Q > 300.)
   D. Given downward sloping demand and marginal revenue curves
and a U-shaped, or quadratic, AC function, the profit-maximizing
price/output combination will often be at a different price and
production level than the average-cost minimizing price-output
combination. This stems from the fact that profit is maximized when
MR = MC, whereas average cost is minimized when MC = AC.
Profits are maximized at the same price/output combination as
where average costs are minimized in the unlikely event that MR =
MC and MC = AC and, therefore, MR = MC = AC.
   It is often true that the profit-maximizing output level differs from
the average cost-minimizing activity level.        In this instance,
expansion beyond Q = 300, the average cost-minimizing activity
level, can be justified because the added gain in revenue more than
compensates for the added costs. Note that total costs rise by
\$240,000, from \$132,000 to \$372,000 as output expands from Q =
300 to Q = 700, as average cost rises from \$440 to \$531.43.
Nevertheless, profits rise by \$80,000, from \$129,000 to \$209,000,
because total revenue rises by \$320,000, from \$261,000 to
\$581,000. The profit-maximizing activity level can be less than,
greater than, or equal to the average-cost minimizing activity level
depending on the shape of relevant demand and cost relations.


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