economic optimization

8/12/2019 Economic Optimization

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Chapter 2

ECONOMIC OPTIMIZATION

The purpose of managerial economics is to provide a systematic framework for problem analysisand solution. The pluses and minuses of various decision alternatives must be carefullymeasured and weighed. Costs and benefits must be reliably measured; time differences must beaccurately reflected. The collection and characterization of relevant information is the mostimportant step of this process. After all relevant information has been gathered, managers mustaccurately state the goal or goals that they seek to achieve. Without a clear understanding ofmanagerial obectives, effective decision making is impossible. !nce all relevant informationhas been gathered, and managerial obectives have been clearly stated, the managerial decisionmaking process can proceed to the consideration of decision alternatives. "ffective managerialdecision making is the process of efficiently arriving at the best possible solution to a given

problem. #f only one solution is possible, then no decision problem e$ists. When alternativecourses of action are available, the decision that produces a result most consistent withmanagerial obectives is the optimal decision. The process of arriving at the best managerialdecision, or best problem resolution, is the focus of managerial economics.

This chapter introduces fundamental principles of economic analysis, which are essentialto all aspects of managerial economics and form the basis for describing demand, cost, and profit relations. !nce basic economic relations are understood, the tools and techni%ues ofoptimization can be applied to find the best course of action.

CHAPTER OUTLINE

MMMMMMMMMMMMMMMCDIV. ECONOMIC OPTIMIZATION PROCESS

A. Optimal Decisi!s" The best decision produces the result most consistent withmanagerial objectives.

1. Economic concepts and methodology are used to select the optimalcourse of action in light of available options and objectives.

B. Ma#imi$i!% the Val&e ' the (irm" In managerial economics, the primaryobjective of management is maimi!ation of the value of the firm. Influencesthat must be considered include"

1#$%$. prices and the &uantity sold.

1#$%1. cost relations.

1#$%'. the appropriate discount rate.

MMMMMMMMMMMMMMMCDV. REVENUE RELATIONS



(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Price a!) Ttal Re*e!&e" Total revenue is the amount sold in dollars.

1. )i*e all financial data, revenue figures are often studied using computerspreadsheets that show functional relations using formulas and e&uations.

'. If price is constant regardless of the &uantity sold, the relation between&uantity sold and total revenue is simply T& ' ( ) *.

a. +enerally spea*ing, T& f -/.

b. Total revenue -T&/, the variable to the left of the e&ual sign, iscalled the dependent variable. Its value depends on the si!e of thevariable variables to the right of the e&ual sign.

c. 0ariables on the righthand side of the e&ual sign are called

independent variables. Their values are determined independently ofthe functional relation epressed by the e&uation.

Mar%i!al Re*e!&e" 2arginal revenue is the change in total revenue caused by a

oneunit change in the number of units sold -*/. In calculus terminology, +& 'T&-*.

3$3'.+enerally spea*ing, a marginal relation is the change in the dependentvariable caused by a oneunit change in an independent variable.

3$34.5hen marginal revenue is positive, total revenue is increasing. If marginalrevenue is negative, total revenue is decreasing.

3$3%.5hen a linear relation eists between price and the number of units sold, both price and marginal revenue relations begin at the same point on the ais, but marginal revenue falls twice as fast as price with respect tooutput.

666666666666666666666666666666666666666666666666666666666Re*e!&e Ma#imi$ati! E#ample" 6evenue maimi!ation occurs at the point of

greatest total revenues.

'7%8'. 6evenue maimi!ation involves consideration of revenue relationsonly. 9ost relations are not considered.

'7%84. To find the revenuemaimi!ing output level, set +& $, and solvefor *.



'7%8%. To be consistent with longrun profit maimi!ation, the advantagesof shortrun revenue maimi!ation must be at least sufficient tocompensate for the corresponding loss in shortrun profitability.

MMMMMMMMMMMMMMMCDVI. COST RELATIONS

A. Ttal Cst" Total costs are comprised of fied costs and variable epenses thatfluctuate with output.

1. Because all costs are variable in the long run, longrun fied costs alwayse&ual !ero. In the long run, the firm has complete fleibility with respectto input use.

'. The short run is the operating period during which the availability of atleast one input is fied.

B. Mar%i!al a!) A*era%e Cst" 2arginal cost is the basis for shortrun operating

decisions: average cost is the basis for longrun planning decisions.

1. 2arginal cost is the change in cost associated with a 1unit change inoutput. In calculus terminology, +C ' TC-*. 2arginal costs must becovered by added revenues to justify production in the short run.

'. Average cost is total cost divided by output, or AC ' TC-*. In the longrun, average costs must be covered by revenues or the company will goout of business.

4. 5hen marginal cost is greater -less/ than average cost, average cost is

increasing -decreasing/.

9. A*era%e Cst Mi!imi$ati! E#ample" The averagecost minimi!ing activitylevel is the optimal production level for a target level of output.

1. Average cost is minimi!ed when +C ' AC.

'. Average cost maimi!ation involves consideration of cost relations only.6evenue relations are not considered.

MMMMMMMMMMMMMMMCDVII. PRO(IT RELATIONS

A. Ttal a!) Mar%i!al Pr'it" Total profit is maimi!ed when the optimal level ofoutput is produced efficiently.

1. Total profit e&uals total revenue minus total cost. +raphically, it ise&uivalent to the vertical distance between the total revenue and total costcurves at any output level.



'. 2arginal profit is the change in total profit associated with a 1unit changein output. In calculus terminology, +/ ' /-*. -Economists use the6oman letter ;(< for price and the +ree* letter pi for profit./.

B. Pr'it Ma#imi$ati! E#ample" The profitmaimi!ing activity level is theoptimal production level for an optimal level of output.

4'##$. Total profit is maimi!ed when +/ ' +& 0 +C $. E&uivalently, profits are maimi!ed when +& ' +C.

4'##1. The slopes of the total revenue and total cost curves measuremarginal revenues - +&/ and marginal costs - +C /. 5here these slopes aree&ual, +& +C and profit is maimi!ed.

4'##'. 5hereas revenue maimi!ation involves consideration of revenuerelations only and average cost maimi!ation involves consideration ofcost relations only, profit maimi!ation involves consideration of both

revenue and cost relations.

MMMMMMMMMMMMMMMCDVIII. THE INCREMENTAL CONCEPT INECONOMIC ANAL+SIS

. Mar%i!al Vers&s I!creme!tal C!cept" Incremental analysis involveseamining the impact of alternative managerial decisions on revenues, costs, and profits. It focuses on changes or differences between available alternatives.

. The incremental change is the difference resulting from a given decision.

. 2arginal relations measure only the effect associated with unitarychanges in output.

. I!creme!tal Pr'its" Incremental profit is the profit gain or loss associatedwith a given managerial decision.

1. 5hen incremental profit is negative, total profit declines.'. Incremental profit is positive -and total profit increases/ if the incremental

revenue associated with a decision eceeds incremental cost.

. I!creme!tal C!cept E#ample" The incremental concept is important formanagerial decision ma*ing because it focuses attention on the differencesamong available alternatives.

1. 6evenues and costs that are unaffected by a decision are irrelevant andshould be ecluded from analysis.

MMMMMMMMMMMMMMMCDI,. SUMMAR+



PROBLEMS & SOLUTIONS

P2.1 Marginal Analysis. Characterize each of the following statements as true or false,and e$plain your answer.

A. #f marginal revenue is greater than average revenue, the demand curve isdownward sloping.

B. (rofit is minimized when total revenue e%uals total cost.

C. 1iven a downward2sloping demand curve and positive marginal costs, profit2ma$imizing firms always sell more output at lower prices than revenue2ma$imizing firms.

D. +arginal cost must be less than average cost for average cost to decline asoutput e$pands.

E. +arginal profit is the difference between marginal revenue and marginal cost,and always e$ceeds zero at the profit2ma$imizing activity level.

P2.1 SOLUTION

A. =alse. Because average revenue is falling along a downward sloping demandcurve, marginal revenue must be less than average revenue for the demand curveto slope downward.

-. =alse. (rofits are maimi!ed when marginal revenue e&uals marginal cost.

(rofits e&ual !ero at the brea*even point where total revenue e&uals total cost.(rofits are minimi!ed when the difference between total revenue and total costis at a maimum.

C. =alse. (rofit maimi!ation involves setting marginal revenue e&ual to marginalcost. 6evenue maimi!ation involves setting marginal revenue e&ual to !ero.+iven a downward sloping demand curve and positive marginal costs, revenuemaimi!ing firms charge lower prices and offer greater &uantities of output thanfirms that maimi!e profits.

D. True. Average cost falls as output epands so long as marginal cost is less thanaverage cost. If this condition is met, average costs decline whether marginalcosts are falling, rising or constant.

E. =alse. 2arginal profit e&uals marginal revenue minus marginal cost, and e&uals!ero at the profit maimi!ing activity level.

P2.2 Revenue Mai!i"a#i$n% Tales. +arketing director (eter (etrelli has derived the following price-demand information from a market e$periment for a new closet2 space organizer product called +a$ 3eadroom4



Pri'e Pr$(u'# De!an(

5677 7

896 :

867

86 <

877 8

<96 6

<67 =

<6 9

<77 >

96 ?

67 :7

6 ::

77 :

:96 :<

:67 :8

:6 :6

A. @se a spreadsheet to calculate total and marginal revenue at each level of

product demand.

B. At what price level is total revenue ma$imized Why

P2.2 SOLUTION

A. Total and marginal revenue at each level of product demand are as follows"

PriceP

Pr)&ct

Dema!)

Ttal

Re*e!&eTR/P

Mar%i!al

Re*e!&eMR/ TR >

?@$$ $

%3@ 1 ?%3@ ?%3@

%@$ ' 8$$ %'@

%'@ 4 1,'3@ 43@

%$$ % 1,#$$ 4'@



PriceP

Pr)&ct

Dema!)

Ttal

Re*e!&eTR/P

Mar%i!al

Re*e!&eMR/ TR >

43@ @ 1,73@ '3@

4@$ # ',1$$ ''@

4'@ 3 ','3@ 13@

4$$ 7 ',%$$ 1'@

'3@ 8 ',%3@ 3@

'@$ 1$ ',@$$ '@

''@ 11 ',%3@ -'@/

'$$ 1' ',%$$ -3@/

13@ 14 ','3@ -1'@/

1@$ 1% ',1$$ -13@/1'@ 1@ 1,73@ -''@/

-. At a price level of ?'@$ and product demand of 1$ units, total revenue ismaimi!ed at a level of ?',@$$. (rior to that point, the added sales from adecrease in price more than compensate for the loss in revenue from chargingcurrent customers a lower price. At prices lower than ?'@$, the loss in revenuefrom charging current customers a lesser price is greater than the gain inrevenues from new customers, and total revenue declines. As seen in themarginal revenue column, total revenue increases so long as marginal revenue is positive, but declines when marginal revenue is negative.

P2.0 A*era%e Cst Mi!imi$ati!" Ta1les. 2ende! 2achine Tools, )td., has been as*edto submit a bid on the projected cost of sophisticated production machinery. To helpin the bid development process, Isaac 2ende!, head of product &uality control, has prepared the following schedule of projected volume and production costs"



Ou#)u#

T$#al

C$s#

7 567,777

: 6:,967 68,777

< 6=,967

8 =7,777

6 =<,967

= =>,777

9 9<,67

> 9?,677

? >=,67

:7 ?<,677

:: :7,67

: :::,677

:< :,67

:8 :<8,677

:6 :67,777

A. @se a spreadsheet to calculate the marginal and average cost at each level of production.

B. At what level of production is average cost minimized Why

P2.* SOLUTION

A. 2arginal and average cost at each level of production appears as follows"



O&tp&t

Ttal

Cst

Mar%i!al

Cst

A*era%e

Cst

$ ?@$,$$$

1 @1,3@$ ?1,3@$ ?@1,3@$.$$' @%,$$$ ','@$ '3,$$$.$$

4 @#,3@$ ',3@$ 17,81#.#3

% #$,$$$ 4,'@$ 1@,$$$.$$

@ #4,3@$ 4,3@$ 1',3@$.$$

# #7,$$$ %,'@$ 11,444.44

3 34,'@$ @,'@$ 1$,%#%.'8

7 38,@$$ #,'@$ 8,843.@$

8 7#,'@$ #,3@$ 8,@74.44

1$ 84,@$$ 3,'@$ 8,4@$.$$

11 1$','@$ 7,3@$ 8,'8@.%@

1' 111,@$$ 8,'@$ 8,'81.#3

14 1'','@$ 1$,3@$ 8,%$4.7@

1% 14%,@$$ 1','@$ 8,#$3.1%

1@ 1@$,$$$ 1@,@$$ 1$,$$$.$$

-. 2inimum average costs of ?8,'81.#3 are reali!ed at an activity level of 1' unitsof output. otice from the marginal cost column that average cost falls so long

as marginal cost is less than average cost. Average cost rises so long asmarginal cost is greater than average cost. Average cost reaches a minimumwhen marginal cost switches from being lower than average cost to beinggreater than average cost.

P2. Pr'it Ma#imi$ati!" Sprea)sheet A!al3sis. )incoln Burrows, a writing instrumentdesign specialist at =o 6iver tate (en, Inc., has been as*ed to project the profitmaimi!ing activity level for a given design during the coming period. 6elevantdemand and cost information are as follows"



+ P TC

7 5:,777

5677

: ?=7 987 ?7 ?>7

< >>7 :,7

8 >87 :,8=7

6 >77 :,977

= 9=7 :,?87

9 97 ,:>7

> =>7 ,87

? =87 ,==7

:7 =77 ,?77

:: 6=7 <,:87

: 67 <,<>7

:< 8>7 <,=7

:8 887 <,>=7

:6 877 8,:77

A. Construct a table Bor spreadsheet showing total revenue, marginal revenue,marginal cost, average cost, total profit, marginal profit, and average profit

for this product at the various activity levels indicated previously.

B. #dentify the profit2ma$imizing activity level.

C. #s the profit2ma$imizing activity level the same activity level as that at whichminimum average costs are e$perienced Why or Why not

P2., SOLUTION

A. A table -or spreadsheet/ showing total revenue, marginal revenue, marginal cost, average cost,total profit, marginal profit, and average profit for this product at the various activity levels isas follows"



PTR /P

MR /

TR4 TCMC /

TC4 AC /TC4

5 /TR6TC

M5 /

54 A5 /54

$ ?1,$$$ ?$ ?@$$ -?@$$/

1 8#$ 8#$ ?8#$ 3%$ ?'%$ ?3%$.$$ ''$ ?3'$ ?''$.$$' 8'$ 1,7%$ 77$ 87$ '%$ %8$.$$ 7#$ #%$ %4$.$$

4 77$ ',#%$ 7$$ 1,''$ '%$ %$#.#3 1,%'$ @#$ %34.44

% 7%$ 4,4#$ 3'$ 1,%#$ '%$ 4#@.$$ 1,8$$ %7$ %3@.$$

@ 7$$ %,$$$ #%$ 1,3$$ '%$ 4%$.$$ ',4$$ %$$ %#$.$$

# 3#$ %,@#$ @#$ 1,8%$ '%$ 4'4.44 ',#'$ 4'$ %4#.#3

3 3'$ @,$%$ %7$ ',17$ '%$ 411.%4 ',7#$ '%$ %$7.@3

7 #7$ @,%%$ %$$ ',%'$ '%$ 4$'.@$ 4,$'$ 1#$ 433.@$

8 #%$ @,3#$ 4'$ ',##$ '%$ '8@.@# 4,1$$ 7$ 4%%.%%

1$ #$$ #,$$$ '%$ ',8$$ '%$ '8$.$$ 4,1$$ $ 41$.$$

11 @#$ #,1#$ 1#$ 4,1%$ '%$ '7@.%@ 4,$'$ -7$/ '3%.@@

1' @'$ #,'%$ 7$ 4,47$ '%$ '71.#3 ',7#$ -1#$/ '47.44

14 %7$ #,'%$ $ 4,#'$ '%$ '37.%# ',#'$ -'%$/ '$1.@%

1% %%$ #,1#$ -7$/ 4,7#$ '%$ '[email protected] ',4$$ -4'$/ 1#%.'8

1@ %$$ #,$$$ -1#$/ %,1$$ '%$ '34.44 1,8$$ -%$$/ 1'#.#3

-. (roduction and sale of 1$ units at a price of ?#$$ -or 8 units and a price of?#%$/ is the profitmaimi!ing activity level. (rofits are maimi!ed at a level of?4,1$$. (rior to this point, the marginal revenue associated with additionalsales eceeds their associated marginal cost, marginal profit is positive, and total

profit rises with an epansion in output. ubse&uent to this point, the marginalrevenue associated with additional sales is less than their associated marginalcost, marginal profit is negative, and total profit falls with an epansion inoutput.

C. o, the profitmaimi!ing activity level is not the same activity level as that atwhich minimum average costs are eperienced. In this problem, marginal costsare a constant ?'%$ per unit. 5ith fied costs of ?@$$, average costs continueto fall as output epands: they approach ?'%$ as a lower limit. Althoughaverage costs continue to diminish as output epands, ever lower prices must beoffered to generate these added sales. Total profits fall as output epands beyond 1$ units because marginal revenues are less than marginal costs beyondthat point. It is important to recogni!e that profit maimi!ation re&uires acomparison of marginal revenues and marginal costs: average cost minimi!ationinvolves a comparison of marginal cost and average cost relations only -revenueeffects are not considered/.

P2.- Marginal Analysis% Tales. +ichael Dcofield, a student at +innesota Dtate@niversity, is preparing for final e$ams and has decided to devote five hours to the study of managerial economics and finance. DcofieldEs goal is to ma$imize the



average grade earned in the two courses, and must decide how much time to spendon each e$am. Dcofield realizes that ma$imizing the average grade in the twocourses is e%uivalent to ma$imizing the sum of the grades. According to DcofieldFsbest estimates, the grades achieved will vary according to the schedule shown below.

Managerial E'$n$!i's

inan'e

/$urs $0 S#u(y ra(e

/$urs $0 S#u(y ra(e

7 6 7 67

: 86 : =

=6 9

< 96 < >:

8 >< 8 >>6 ?7 6 ?<

A. Gescribe the manner in which Dcofield could make use of the marginal conceptin managerial economics to assist in determining the optimal allocation of fivehours between the two courses.

B. 3ow much time should Dcofield spend studying each subect

C. #n addition to managerial economics and finance, Dcofield is also taking amarketing course. Dcofield estimates that each hour spent studying marketing

will result in an eight point increase on the marketing e$amination score.Dcofield has tentatively decided to spend three hours preparing for themarketing e$am. #s DcofieldFs attempt to ma$imize the average grade in allthree courses with three hours devoted to marketing and five hours devoted tomanagerial economics and finance Ballocated as in part H an optimaldecision Why or Why not

P2.- SOLUTION

A. An optimal allocation of study time is one that will permit cofield to maimi!ethe average grade earned in the managerial economics and finance courses. This

maimi!ation will occur when cofield allocates each hour of study time to thatcourse where the marginal grade value of study time is greatest.

-. To determine how much time cofield should spend studying each subject atable illustrating the marginal grade value of each hour of study time must beconstructed. This table reads as follows"



Ma!a%erial Ec!mics (i!a!ce

H&rs ' St&)3 7ra)e

Mar%i!al7ra)e Val&e

H&rs ' St&)3 7ra)e

Mar%i!al7ra)e Val&e

$ '@ $ @$ 1 %@ '$ 1 #' 1'

' #@ '$ ' 3' 1$

4 3@ 1$ 4 71 8

% 74 7 % 77 3

@ 8$ 3 @ 84 @

5ith only five hours to study, cofield should spend three hours on managerialeconomics and two hours on finance.

C. o, cofieldCs decision to spend three hours studying for the mar*eting eam isincorrect if the objective is to maimi!e the average grade received inmanagerial economics, finance and mar*eting. Dnly two hours should beallocated to studying mar*eting because an additional hour spent on financewould increase the total grade achieved by nine points: one point more than theeight point gain associated with the third hour spent preparing for the mar*etingeam, and will lead to a maimum average grade.

P2.8 Mar%i!al A!al3sis" Ta1les. 2onica +eller is a regional media consultant for=riendly Images, Inc., a ew or*based media consultant. +eller has gathered thefollowing data on wee*ly advertising media ependitures and gross sales for a major

client, +reenwich 0illage=s 9entral (er* 9offeehouse.



A(ver#ising

E)en(i#ure

r$ss Sales $ll$ing Pr$!$#i$n in #3e $ll$ing Me(ia%

Nes)a)er Ra(i$ Televisi$n

57 5:7,777 5:7,777 5:7,777

:77 :,777 :8,777 :<,777

77 :<,>77 :9,=77 :6,=77

<77 :6,877 7,77 :>,777

877 :=,=77 ,777 :>,=77

677 :9,77 ,877 :>,>77

A. Construct a table Bor spreadsheet showing marginal sales following promotion

in each media. BAssume here and throughout the problem that there are no synergistic effects across different media.

B. #f the Central (erk Coffeehouse has an advertising budget of 5677 per week,how should it be spent Why

C. Calculate the profit ma$imizing advertising budget and media allocationassuming the Central (erk Coffeehouse enoys an average profit contributionbefore media e$penditures of =I on store2wide sales. 3ow much arema$imum weekly profits Bbefore ta$es

P2.4 SOLUTION

A. A table showing marginal sales generated by promotion in each media appearsas follows"



A)*ertisi!%

E#pe!)it&re

Mar%i!al Sales (ll9i!% Prmti! i! the

(ll9i!% Me)ia"

Ne9spaper Ra)i Tele*isi!

?$

1$$ ?',$$$ ?%,$$$ ?4,$$$

'$$ 1,7$$ 4,#$$ ',#$$

4$$ 1,#$$ ',#$$ ',%$$

%$$ 1,'$$ 1,7$$ #$$

@$$ #$$ %$$ '$$

-. Fsing the data in part A, and given a ?@$$ advertising budget, gross sales and

profit contribution are maimi!ed by allocating ?4$$ to radio and ?'$$ totelevision advertising. Irrespective of whether the 9entral (er* 9offeehousesee*s to maimi!e revenues or profit, this ependiture allocation is optimal.

C. +iven an average profit contribution before media ependitures of #G onstorewide sales, an additional dollar of advertising will be profitable so long asit returns more than ?1#.#3 in additional revenues. That is, the profitcontribution on additional revenues of ?1#.#3 is just sufficient to cover mediacosts of ?1 - ?1#.#3 $.$#/. The profit maimi!ing advertising budget is ?8$$ per wee* allocated as" ?'$$ on newspaper, ?%$$ on radio, and ?4$$ ontelevision.

2aimum wee*ly profits are"

Base sales ?1$,$$$H ewspaper sales 4,7$$H 6adio sales 1',$$$H Television sales 7,$$$

?44,7$$ +ross margin $.$#

+ross profit ',$'7 2edia costs 8$$

et profit ? 1,1'7

-before ta/



P2.5 Marginal Analysis% E6ua#i$ns. American #dol (roducts, #nc., has developed a new pocket2sized wireless communications device. This productFs reliability level,measured by its failure rate, is a function of the amount of on2site preventivemaintenance and warranty work4

Jailure &ate ' 7.: 2 7.7:t K 7.77:t ,

+arginal Jailure &ate ' J- t ' 27.7: K 7.77t,

where t represents the amount of time devoted to on2site preventive maintenance andwarranty work4

A. Det the marginal failure rate e%ual to zero to determine the failure rate2minimizing level of on2site preventive maintenance and warranty work.

B. What is the optimal failure rate

P2.5 SOLUTION

A. et 2= $ to find the failure rateminimi!ing level of onsite preventivemaintenance and warranty wor*"

2= $.$1 H $.$$'t

$ $.$1 H $.$$'t

$.$$'t $.$1

t @ hours

=or this to be a failurerate minimi!ing, rather than maimi!ing, level of onsite preventive maintenance and warranty wor*, the failure rate must be increasing beyond this point. This is indeed the case, as can be illustrated with a simplenumerical eample -see part B/.

-. At t @, note that"

= $.1 $.$1t H $.$$1t'

$.1 $.$1-@/ H $.$$1-@'/

$.$3@ -or 3.@G/To confirm that this is failurerate minimi!ing level of t, notice that = isincreasing for t @, e.g., = $.$3# or 3.#G when t #.

P2.7 Pr$0i# Mai!i"a#i$n% E6ua#i$ns. 1regory 3ouses, #nc. operates with the followingrevenue and cost functions4



T& ' 5:77* 2 57.6* BTotal &evenue

+& ' T&- * ' 5:77 2 5:* B+arginal &evenue

TC ' 5:,677 2 5:7* K 57.6* BTotal Cost

+C ' TC- * ' 25:7 K 5:* B+arginal Cost

where * represents the %uantity of output produced and sold as measured bythousands of hours of temporary accounting services B777.

A. Det +/ ' +& 2 +C ' 7 to determine the profit2ma$imizing price-outputcombination for AT#.

B. Dhow that marginal revenue e%uals marginal cost at this profit2ma$imizing

output level.

P2.7 SOLUTION

A. et 2J 26 29 $ to maimi!e profits, where"

2J 26 29

$ ?1$$ ?1 H ?1$ ?1

' 11$

@@ -$$$/

And,

( T6>

-?1$$ ?$.@'/>

?1$$ ?$.@ ?1$$ ?$.$@-@@/

?3'.@$

-. At @@ -$$$/, note that"

26 ?1$$ ?1 ?1$$ ?@@ ?%@

29 ?1$ H ?1 ?1$ H ?@@ ?%@



This numerical finding illustrates the general result that if 2J 26 29 $,then 26 29 will always be true.

P2.8 Pr$0i# 9ersus Revenue Mai!i"a#i$n. West Wing (roducts, #nc., based in Gurham, Lew 3ampshire, produces and sells a wide variety of replacement parts ande%uipment for light aircraft. Meo +c1arry, a product line specialist for thecompany, is reviewing the company= s #nternet marketing strategy for a popular flight manual. Gemand and cost relations for the guidebook are given by thee%uations4

( ' 5:66 2 57.76* BGemand

+& ' T&- * ' 5:66 2 57.:* B+arginal &evenue

TC ' 596,777 K 56* K 57.7:6* BTotal Cost

+C ' TC- * ' 56 K 57.76* B+arginal Cost

where * is the %uantity produced and sold per week.

A. Calculate the revenue2ma$imizing price-output combination.

B. Calculate the profit2ma$imizing price-output combination.

C. Are the differences in your answers to parts A and H typical or atypical "$plain.

P2.8 SOLUTION

A. et 26 $ to find the revenuemaimi!ing output level"

26 ?1@@ ?$.1

$ ?1@@ ?$.1$.1 1@@

1,@@$

Because total revenue is declining beyond this point, 1,@@$ is a point ofmaimum revenues. And,

( ?1@@ ?$.$@-1,@@$/

?33.@$

-. et 2J 26 29 $ to find the profitmaimi!ing output level"



2J 26 29

$ ?1@@ ?$.1 ?@ ?$.$'@

$.1'@ 1@$

1,'$$

Because total profit is declining for 1,'$$, 1,'$$ is a point of maimum profits. And,

( ?1@@ ?$.$@-1,'$$/

?8@

C. This is a typical result: so long as the product demand curve slopes downward

and marginal cost eceeds !ero, revenue maimi!ation results in greater output-here, 1,@@$ versus 1,'$$/ and lower prices -here, ?33.@$ versus ?8@/ than istrue with profit maimi!ation.

P2.1: Pr$0i# Mai!i"a#i$n 9ersus Average C$s# Mini!i"a#i$n. #mmensely popularmystery writer Dtan +arsh has ust published a new book titled NDouth (ark,Colorado.@ &elevant monthly demand and cost relations for this hard cover titleare4

( ' 5<6 2 57.77796* BGemand

+& ' T&- * ' 5<6 2 57.77:6* B+arginal &evenue

TC ' 567,777 K 56* K 57.7776* BTotal Cost +C ' TC- * ' 56 K 57.77:* B+arginal Cost

where * is the number of books produced and sold per month.

A. Calculate the profit2ma$imizing price-output combination and profit level.

B. Calculate the average cost2minimizing price-output combination and profitlevel.

C. Contrast your answers to parts A and H.

P2.1: SOLUTION

A. To find the profitmaimi!ing activity level, set 2J 26 29 $"

2J 26 29



$ ?4@ ?$.$$1@ ?@ ?$.$$1

$.$$'@ 4$

1',$$$.

( ?4@ ?$.$$$3@-1',$$$/

?'#.

J T6T9

?4@ ?$.$$$3@' ?@$,$$$ ?@ ?$.$$$@'

?@$,$$$ H ?4$ ?$.$$1'@'

?@$,$$$ H ?4$-1',$$$/ ?$.$$1'@-1',$$$'/

?14$,$$$

Because total profit is declining for 1',$$$, 1',$$$ is a point ofmaimum profits per month. - Lote" A publisherCs price of ?'# is consistent witha boo*store price of roughly ?4@./

-. To find the average costminimi!ing activity level, set 29 A9"

29 A9 T9>

?@ H ?$.$$1 -?@$,$$$ H ?@ H ?$.$$$@'/>

?@ H ?$.$$1 ?@$,$$$> H ?@ H ?$.$$$@

$.$$$@ @$,$$$>

' 1$$,$$$,$$$

1$,$$$

( ?4@ ?$.$$$3@-1$,$$$/

?'3.@$

J ?@$,$$$ H ?4$-1$,$$$/ ?$.$$1'@-1$,$$$'/

?1'@,$$$



Because average cost is rising for 1$,$$$, 1$,$$$ is a point ofminimum average costs per month.

C. (rofit maimi!ation involves a comparison of the marginal revenues andmarginal costs of production. Averagecost minimi!ation involves aconsideration of cost relations only. Therefore, it is not surprising that thesetwo approaches often yield different price>output combinations. In this instance,the added revenues associated with an additional ',$$$ units beyond the averagecostminimi!ing level of 1$,$$$ is so great as to overcome the disadvantage ofsomewhat higher average costs at the 1',$$$ units versus 1$,$$$ units activitylevel.

economic optimization

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