ch08solution manual

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Besanko & Braeutigam – Microeconomics, 5 th editionSolutions Manual

Chapter 8

Cost Curves

Solutions to Review Questions

1. What is the relationship between the solution to the firm’s long-run cost-minimizationproblem and the long-run total cost curve?

The long-run total cost curve plots the minimized total cost for each level of output holding input prices fixed !n other "ords, for a given set of input prices, the long-run total cost curverepresents the total cost associated "ith the solution to the long-run cost minimization pro#lemfor each level of output

2. !plain wh" an increase in the price of an input t"picall" causes an increase in the long-run total cost of producing an" particular level of output.

$hen the price of one input increases, the isocost line for a particular level of total cost "illrotate in to"ard the origin %ssuming the isocost line "as tangent to the isouant for the firm'sselected level of output, "hen the isocost line rotates it "ill no longer touch the original isouant!n order for an isocost line to reach a tangenc( "ith the original isouant, the firm "ould need tomove to an isocost line associated "ith a higher level of cost, i.e. an isocost line further to thenortheast

#. $f the price of labor increases b" 2% percent& but all other input prices remain the same&would the long-run total cost at a particular output level go up b" more than 2% percent&less than 2% percent& or e!actl" 2% percent? $f the prices of all inputs went up b" 2% percent&would long-run total cost go up b" more than 2% percent& less than 2% percent& or e!actl" 2%percent?

!f the price of a single input goes up leaving all other input prices the same and the level ofoutput constant, total cost "ill rise #ut #( a smaller percentage than the increase in the input price This occurs #ecause the firm "ill su#stitute a"a( from the no" relativel( more expensivela#or to the no" relativel( less expensive other inputs So, if the price of la#or rises #( )*+holding all other input prices constant, total cost "ill rise #( less than )*+!f the prices of all inputs go up #( the same percentage, total cost "ill rise #( exactl( that same percentage So, if input prices rise #( )*+, total cost "ill also rise #( )*+

'. (ow would an increase in the price of labor shift the long-run average cost curve?

%n increase in the price of la#or "ould result in a long-run total cost curve that lies a#ove theinitial long-run total cost curve at ever( uantit( except *Q = Since AC TC Q= , increasing

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total cost "ill raise average cost at ever( uantit( except *Q = Therefore, the long-run averagecost curve "ill shift up

). a* $f the average cost curve is increasing& must the marginal cost curve lie above theaverage cost curve? Wh" or wh" not?

b* $f the marginal cost curve is increasing& must the marginal cost curve lie above theaverage cost curve? Wh" or wh" not?

a3 $hen MC AC > , average cost is increasing, and "hen MC AC < , average cost isdecreasing So, if the average cost curve is increasing it must lie below the marginal cost curve #3 !f the marginal cost curve is increasing, it ma( lie a#ove or #elo" the average cost curveThe onl( determining factor here is "hether or not marginal cost lies a#ove or #elo" averagecost !f it lies a#ove, average cost "ill #e increasing and if it lies #elo", average cost "ill #edecreasing 4no"ing that marginal cost is increasing or decreasing tells us nothing a#outaverage cost

+. ,etch the long-run marginal cost curve for the flat-bottomed/ long-run average costcurve shown in 0igure 8.11.

%-

M-

T-

$hen average cost is falling, marginal cost "ill lie #elo" average cost, and "hen average cost isincreasing, marginal cost "ill lie a#ove average cost 6ver the flat-#ottomed portion "hereaverage cost is neither increasing nor decreasing, marginal cost and average cost "ill #e eual

. Could the output elasticit" of total cost ever be negative?

The output elasticit( of total cost, "hen simplified can #e "ritten as

,TC Q MC AC

ε =

Since AC TC Q= , and since TC and Q must al"a(s #e positive, AC "ill al"a(s #e positiveMarginal cost, MC , represents the change in total cost associated "ith an increase in output$hen output increases, total cost must al"a(s rise for a given set of input prices, impl(ing that MC is also al"a(s positive Therefore, the output elasticit( of total cost must al"a(s #e positive

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8. !plain wh" the short-run marginal cost curve must intersect the average variable costcurve at the minimum point of the average variable cost curve.

Because fixed cost does not change, marginal costs reflect the change in varia#le costs Thus, as

"ith the relationship #et"een an( average and marginal, if average varia#le cost is decreasing,marginal cost must #e #elo" average varia#le cost, and if average varia#le cost is increasing,marginal cost must lie a#ove average varia#le cost This implies marginal cost "ill intersectaverage varia#le cost at the minimum of average varia#le cost

. ,uppose the graph of the average variable cost curve is flat. What shape would theshort-run marginal cost curve be? What shape would the short-run average cost curve be?

!f the average varia#le cost curve is flat, average varia#le cost is neither increasing nordecreasing Marginal cost "ill therefore #e eual to average varia#le cost and the marginal costcurve "ill therefore also #e flat Since average fixed cost is al"a(s declining, and since average

total cost is the vertical sum of average varia#le and average fixed costs, average total cost mustalso #e declining at all levels of Q if average varia#le cost is constant 7raphicall(, averagetotal cost "ill #e declining and as(mptotic to the average varia#le cost curve

1%. ,uppose that the minimum level of short-run average cost was the same for ever"possible plant size. What would that tell "ou about the shapes of the long-run average andlong-run marginal cost curves?

The long-run average cost curve is the envelope to the short-run average cost curves associated"ith each level of output !f each of these short-run average cost curves has the same minimum point, the long-run average cost curve "ill #e a horizontal line tangent to all of these minimum

points Because the long-run average cost curve "ill #e flat, long-run average cost is neitherincreasing nor decreasing, and the long-run marginal cost curve "ill also #e flat and eual tolong-run average cost

11. What is the difference between economies of scope and economies of scale? $s itpossible for a two-product firm to en3o" economies of scope but not economies of scale? $sit possible for a firm to have economies of scale but not economies of scope?

8conomies of scale refer to a situation "hen average total cost for a single product declines asthe level of output for that product increases These economies of scale might occur, forexample, #ecause "orkers can specialize in tasks as the level of output increases and the

"orkers' productivit( ma( increase 8conomies of scope refer to efficiencies that arise "hen afirm produces more than one product !n particular, economies of scope exist if one firm producing N products does so at a lo"er total cost than N separate firms producing the sameuantities of each product individuall(The notion of economies of scale can actuall( #e applied to a multi-product firm as "ell $e canuse this extension to further refine the distinction #et"een economies of scale and scopeSuppose a firm is producing N products, "ith output levels measured #( / ), , , N Q Q QK !f itoperates "ith economies of scale, the total cost of production "ill rise #( less than /+ "hen

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production of all outputs increases #( /+ !f it operates "ith diseconomies of scale, the totalcost of production "ill rise #( more than /+ "hen production of all outputs increases #( /+B( contrast, economies of scope exist if it is less costl( to have the outputs produced #( one firminstead of #( N firms, each specializing in the production of one of the outputs :ote that information a#out economies of scope does not tell us "hether the firm has economies

of scale !f a production process has economies of scope, there ma( not #e economies of scale;urther, information a#out economies of scale does not tell us "hether the firm has economies of scope !f a production process has economies of scale, there ma( not #e economies of scope

12. What is an e!perience curve? What is the difference between economies of e!perienceand economies of scale?

The experience curve represents the relationship #et"een average varia#le cost and cumulative production volume over time 6ne "ould expect that as cumulative production volumeincreased, average varia#le cost "ould fall 8conomies of scale refer to a situation "hen averagecost declines as the level of output for that product increases "ithin a given time frame

!n general, economies of scale "ould occur if the average cost curve declined as the level ofoutput increased 8conomies of experience "ould occur if, as cumulative production volumeincreased, the average cost curve shifted do"n"ard for all levels of output So, economies ofscale refer to lo"er average costs that occur as output increases and economies of experiencerefer to lo"er average costs for all levels of output as cumulative production volume increases

Solutions to Problems

8.1. 4he following incomplete table shows a firm’s various costs of producing up to + units

of output. 0ill in as much of the table as possible. $f "ou cannot determine the number in abo!& e!plain wh" it is not possible to do so.

The ta#le is reproduced #elo" ;irst, since fixed costs are independent of uantit(, the entire T;column can #e easil( filled in <roceeding through the ta#le ro" #( ro", for = / it is eas( tosee that T> = T – T; = 2*, and the rest of the ro" is similarl( straightfor"ard ;or = ),T = T> ? T; = /2*, and the rest of the follo"s easil( ;or = 9, all "e have is T; = )*@thus, "e cannot infer an(thing else ;or = 0, T = A% = 92* !t's then possi#le to get T>and %>@ ho"ever, "e cannot find M since "e don't kno" T or T> for = 9 ;or = 5,





the important step is to use M53 = T53 – T03 to find T53 = 55* ;or = C, theimportant step is T> = %>A = D)*

T T> T; % M %>/ 1%% 2* )* /** 2* 2*

) /2* 1+% )* E* 2* 2*9 - - 2% - - -0 92* 9C* )* ) - E*5 55* 59* )* //* 1% /*CC D0* D)* )* /)99 /E* 12%

8.2. 4he following incomplete table shows a firm’s various costs of producing up to + unitsof output. 0ill in as much of the table as possible. $f "ou cannot determine the number in abo!& e!plain wh" it is not possible to do so.

!t helps to re"rite this ta#le adding an extra column for Total ;ixed osts at each level of outputThe T; for = ) is Fust )A9* = C*, and this is also the T; value for ever( other output level

Then for = /, "e kno" T = %A = /**, T> = T – T; = 0* and the rest isstraightfor"ard Similarl( "e can fill in the ro"s for = ), 9, 0, and C ;or = 5, "e need touse the fact that MC3 = TC3 – T53 to infer T53 = )5* The rest is straightfor"ard

T T> %; % M %> T;/ /** 0* C* 1%% 0* 0* C*) //* )% #% 55 /* )5 C*9 /)* C* )* 0* 1% )* C*0 /2* /)* /5 05 C* #% C*5 )5* /E* /) 5* D* 92 C*C ##% )D* /* 55 8% 05 C*

8.#. 4he following incomplete table shows a firm’s various costs of producing up to + unitsof output. 0ill in as much of the table as possible. $f "ou cannot determine the number in abo!& e!plain wh" it is not possible to do so.





Q TC TVC TFC AC MC AVC

/ /2 2 /* /2 2 2) 9* )* /* /5 /) /*9 0C 9C /* 0C9 /C /)0 CC 5C /* CC0 )* /C5 E* 2* /* /2 )0 /CC //2 /*2 /* //2C )2 /2

8.'. 4he following incomplete table shows a firm’s various costs of producing up to + unitsof output. 0ill in as much of the table as possible. $f "ou cannot determine the number in abo!& e!plain wh" it is not possible to do so.

Q TC TVC TFC AC MC AVC

/ )* /* /* )* /* /*) 9C )C /* /2 /C /9

9 55 05 /* /299 /E /50 2) D) /* )*5 )2 /25 //) /*) /* ))0 9* )*0C /00 /90 /* )0 9) ))99

8.). 5 firm produces a product with labor and capital& and its production function isdescribed b" Q 6 LK . 4he marginal products associated with this production function are

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MP L 6 K and MP K 6 L. ,uppose that the price of labor e7uals 2 and the price of capitale7uals 1. erive the e7uations for the long-run total cost curve and the long-run averagecost curve.

Starting "ith the tangenc( condition, "e have

)

/)

L

K

MP w MP r

K

L

K L

=

=

=

Su#stituting into the production function (ields

) 3

)

Q LK

Q L L

Q L

==

=

<lugging this into the expression for K a#ove gives

))

Q K =

;inall(, su#stituting these into the total cost euation results in

) )) )

0)

2

Q QTC

QTC

TC Q

= +

=

= and average cost is given #(

2

2

QTC AC Q Q

AC Q

= =

=

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8.+ 5 firm’s long-run total cost curve is . erive the e7uation for the

corresponding long-run average cost curve& 9iven the e7uation of the long-run

average cost curve& which of the following statements is true:

a. 4he long-run marginal cost curve lies below for all positive 7uantities

b. 4he long-run marginal cost curve is the same as the for all positive

7uantities

c. 4he long-run marginal cost curve lies above the for all positive 7uantities

d. 4he long-run marginal cost curve lies below for some positive 7uantities

and above the for some positive 7uantities

%:S$8GH cThe euation of the AC curve is AC Q3 = /***Q !t is increasing in Q 7iven the relationship #et"een AC and MC curves, the fact that the AC curve is increasing means that the MC curvemust lie a#ove the AC curve

8. 5 firm’s long-run total cost curve is . erive the e7uation for the

corresponding long-run average cost curve& 9iven the e7uation of the long-run

average cost curve& which of the following statements is true:

a. 4he long-run marginal cost curve lies below for all positive 7uantitiesb. 4he long-run marginal cost curve is the same as the for all positive

7uantities

c. 4he long-run marginal cost curve lies above the for all positive 7uantities

d. 4he long-run marginal cost curve lies below for some positive 7uantities

and above the for some positive 7uantities

%:S$8GH aThe euation of the AC curve is AC Q3 = TC Q3 = /***/) = /***Q-(1/2) This is adecreasing function of 7iven the relationship #et"een AC and MC curves, the fact that the AC curve is decreasing means that the MC curve must lie #elo" the AC curve





8.8. 5 firm’s long-run total cost curve is TC ;Q* 6 1%%%Q < #%Q2 = Q# . erive the e!pressionfor the corresponding long-run average cost curve and then setch it. 5t what 7uantit" isminimum efficient scale?

) 9

)

/*** 9*

/*** 9*

TC Q Q Q AC

Q Q

AC Q Q

− += =

= − +

7raphicall(, average cost is

*

)**

0**C**

2**

/***

/)**

* 5 /* /5 )* )5 9*

>

5 v e r a g

e C o s t

Minimum efficient scale occurs "here the average cost curve reaches a minimum, /5Q = for

this cost function

8.. 5 firm’s long-run total cost curve is TC ;Q* 6 '%Q < 1%Q2 = Q#& and its long-runmarginal cost curve is MC ;Q* 6 '% < 2%Q = #Q2 . ver what range of output does theproduction function e!hibit economies of scale& and over what range does it e!hibitdiseconomies of scale?

;rom the total cost curve, "e can derive the average cost curve, )/*0*3B QQQ AC +−= The

minimum point of the % curve "ill #e the point at "hich it intersects the marginal cost curve,ie )) 9)*0*/*0* QQQQ +−=+− This implies that AC is minimized "hen Q = 5 B(definition, there are economies of scale "hen the AC curve is decreasing ie Q I 53 anddiseconomies "hen it is rising Q J 53

8.1%. 0or each of the total cost functions& write the e!pressions for the total fi!ed cost&average variable cost& and marginal cost ;if not given*& and draw the average total cost andmarginal cost curves.a* TC ;Q* 6 1%Q

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b* TC ;Q* 6 1+% = 1%Q

c* TC ;Q* 6 1%Q2& where MC ;Q* 6 2%Q

d* TC ;Q* 6 1%@Q& where MC ;Q* 6 ) / @Q

e* TC ;Q* 6 1+% = 1%Q2& where MC ;Q* 6 2%Q

a3 TFC *, AVC /*, MC /*

#3 TFC /C*, AVC /*, MC /*

c3 TFC *, AVC /*Q

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MC AC = /*

MC = /*

AC




d3 TFC *, AVC Q/*

e3 TFC /C*, AVC /*Q

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MC

AC

MC

AC

AC

MC




8.11. 5 firm produces a product with labor and capital as inputs. 4he production functionis described b" Q 6 LK . 4he marginal products associated with this production function are MP L 6 K and MP K 6 L. Aet w 6 1 and r 6 1 be the prices of labor and capital& respectivel".a* 0ind the e7uation for the firm’s long-run total cost curve as a function of 7uantit" Q.

b* ,olve the firm’s short-run cost-minimization problem when capital is fi!ed at a 7uantit"of ) units ;i.e.& K 6 )*. erive the e7uation for the firm’s short-run total cost curve as afunction of 7uantit" Q and graph it together with the long-run total cost curve.c* (ow do the graphs of the long-run and short-run total cost curves change when w 6 1and r 6 '?d* (ow do the graphs of the long-run and short-run total cost curves change when w 6 'and r 6 1?

a3 ost-minimizing uantities of inputs are eual to L = KQ Kr w3 and K = KQ Kr w3Lence, in the long-run the total cost of producing Q units of output is eual to TC Q3 = /* ?)KQrw3 ;or w = / and r = / "e have TC Q3 = )KQ

#3 $hen capital is fixed at a uantit( of 5 units ie, K = 53 "e have Q = K ! L = 5 L Lence,in the short-run the total cost of producing Q units of output is eual to STC Q3 = 5 ? Q5

c3 $e have L = KQ Kr w3 and K = KQ Kr w3 Lence, TC Q3 = )KQrw3 and STC Q3 = 5r ? wQ5 $hen w = / and r = 0 "e have TC Q3 = 0KQ and STC Q3 = )* ? Q5

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - /)

TC

Q)5

5

"TC Q3

TC Q3




d3 $hen w = 0 and r = / "e have TC Q3 = 0KQ and STC Q3 = 0Q5

8.12. 5 firm produces a product with labor and capital. $ts production function isdescribed b" Q 6 min; L& K *. Aet w and r be the prices of labor and capital& respectivel".

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - /9

TC

Q)5

5

"TC Q3

TC Q3

/**

)*

"TC Q3, w =/, r = 0

TC Q3, w =/, r = 0

TC

Q)5

/*

"TC Q3

TC Q3

)50

"TC Q3, w = 0, r = /

TC Q3, w = 0, r = /




a* 0ind the e7uation for the firm’s long-run total cost curve as a function of 7uantit" Q andinput prices& w and r .b* 0ind the solution to the firm’s short-run cost minimization problem when capital is fi!edat a 7uantit" of ) units ;i.e.& K 6 )*. erive the e7uation for the firm’s short-run total costcurve as a function of 7uantit" Q. 9raph this curve together with the long-run total cost

curve for w 6 1 and r 6 1.c* (ow do the graphs of the long-run and short-run total cost curves change when w 6 1and r 6 2?d* (ow do the graphs of the long-run and short-run total cost curves change when w 6 2and r 6 1?

a3 The inputs are complementar( and the cost-minimizing firm uses them in proportions/H/ Lence, "e have TC Q3 = Qw ? r 3

#3 !f w = r = /, then TC Q3 = )Q !n the short run, it is impossi#le to produce more than 5units This is #ecause min L,53 cannot #e an( greater than 5 To produce Q ≤ 5 units, "e set L =

Q $ith w = r = /, this implies "TC Q3 = /5 ? Q /* is the fixed cost of the indivisi#le input, 5is the fixed cost of la#or, and Q is the varia#le cost of la#or3 The diagram #elo" sho"s TC Q3and "TC Q3 for Q ≤ 5 "hen w = r = /

c3 ;or w = / and r = ) "e have TC Q3 = 9Q and "TC Q3 = Q ? /* for Q not larger than 5


TC

Q5

5

"TC Q3

TC Q3




d3 ;or w = ) and r = / "e have TC Q3 = 9Q and "TC Q3 = )Q ? 5 for Q not larger than 5

8.1#. 5 firm produces a product with labor and capital. $ts production function isdescribed b" Q 6 L = K . 4he marginal products associated with this production function are MP L 6 1 and MP K 6 1. Aet w 6 1 and r 6 1 be the prices of labor and capital& respectivel".a* 0ind the e7uation for the firm’s long-run total cost curve as a function of 7uantit" Q

when the prices labor and capital are w 6 1 and r 6 1.


TC

Q5

/*

5

"TC Q3

TC Q3

"TC Q3, w = /, r = )

TC

Q

5

/*

/5

"TC Q3

TC Q3

"TC Q3, w = ), r = /

TC Q3, w = /, r = )

TC Q3, w = ), r = /




b* 0ind the solution to the firm’s short-run cost minimization problem when capital is fi!edat a 7uantit" of ) units ;i.e.& K 6 )*& and w 6 1 and r 6 1. erive the e7uation for the firm’sshort-run total cost curve as a function of 7uantit" Q and graph it together with the long-run total cost curve.c* (ow do the graphs of the short-run and long-run total cost curves change when w 6 1

and r 6 2?d* (ow do the graphs of the short-run and long-run total cost curves change when w 6 2and r 6 1?

a3 $ith a linear production function, the firm operates at a corner point depending on"hether w I r or w J r !f w I r , the firm uses onl( la#or and thus sets L = Q !n this case, thetotal cost including the fixed cost3 is wQ !f w J r , the firm uses onl( capital and thus sets K = Qin this case, the total cost is rQ $hen w = r = /, the firm is indifferent among com#ination of L and K that make L ? K = /* Thus, "e have TC Q3 = Q

#3 $hen capital is fixed at 5 units, the firm's output "ould #e given #( Q = 5 ? L !f the

firm "ants to produce Q I 5 units of output, it must produce 5 units and thro" a"a( 5 – Q ofthem The total cost of producing fe"er than 5 units is constant and eual to 5, the cost of thefixed capital ;or Q J 5 units, the firm increases its output #( increasing its use of la#or !n particular, to produce Q units of output, the firm uses Q – 5 units of la#or, for a cost of Q – 5, and5 units of capital, for a cost of 5 Thus, "TC Q3 = Q – 5 ? 5 = Q

c3 !n the long run, since w I r , the firm produces its output entirel( "ith la#or Thus, TC Q3

= Q, Fust as in part #3 !n the short-run, "ith capital fixed at 5 units, the firm's output "ould #egiven #( Q = 5 ? L !f the firm "ants to produce Q I 5 units of output, it must produce 5 units ofoutput and thro" a"a( 5 – Q of them !t can produce this output using its fixed stock of 5 unitsof capital and no la#or The total cost of producing Q I 5 units of output "hen the price ofcapital is ) per unit is /*

;or Q J 5 units, the firm increases its output #( increasing its use of la#or !n particular, to produce Q units of output, the firm uses Q – 5 units of la#or, for a cost of Q – 5, and 5 units of

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - /C

TC

Q5

"TC Q3

TC Q3

"TC Q3 &TC Q3

5




capital, for a cost of /* Thus, "TC Q3 = Q – 53 ? /* = Q ? 5

:otice that "hen K = 5, w = /, and r = ), the "TC curve strictl( lies a#ove the TC curve This is #ecause K = 5 is never an optimal capital choice for the firm "hen w = / and r = ) %s a resultthe firm's total costs are al"a(s higher in the short run than the( are in the long run

d3 The total cost curve is the same as in part #3, ie TC Q3 = Q This is #ecause the cheaperinput in this case capital3 continues to have a price of / per unit !n the short run, "ith capital

#eing fixed at 5 units, the cost of producing Q I 5 is 5 To produce more than Q units, the firmuses Q – 5 units of la#or at a total cost of )Q – 53 = )Q – /* !t also uses 5 units of capital at atotal cost of 5 Thus, for Q J 5, "TC Q3 = )Q – /* ? 5 = )Q – 5

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - /D

TC

Q5

/*"TC Q3 w = /, r = )

"TC Q3

"TC Q3 &TC Q3

5 TC Q3 w = /, r = )




8.1'. Consider a production function of two inputs& labor and capital& given b" Q 6 ;@ L =@ K *2. 4he marginal products associated with this production function are as follows:

Aet w 6 2 and r 6 1.a* ,uppose the firm is re7uired to produce Q units of output. ,how how the cost-minimizing 7uantit" of labor depends on the 7uantit" Q. ,how how the cost-minimizing7uantit" of capital depends on the 7uantit" Q.

b* 0ind the e7uation of the firm’s long-run total cost curve.c* 0ind the e7uation of the firm’s long-run average cost curve.d* 0ind the solution to the firm’s short-run cost minimization problem when capital is fi!edat a 7uantit" of units ;i.e.& K 6 *.e* 0ind the short-run total cost curve& and graph it along with the long-run total cost curve.f * 0ind the associated short-run average cost curve.

a3 Starting "ith the tangenc( condition "e have

/ ) / ) / )

/ ) / ) / ) )/

0

0

L

K

MP w

MP r

L K L L K K

K

L

K L

−

−

=

+ = +

=

=<lugging this into the total cost function (ields


TC

Q5

5"TC Q3 = "TC Q3, w = ), r =/

TC Q3 = TC Q3, w = ), r = /

/*

"TC Q3 &TC Q3

"TC Q3, w = ), r =/




)/ ) / )

)/ )

0 3

9

E

E

Q L L

Q L

Q L

Q L

= +

= =

=

!nserting this #ack into the solution for K a#ove gives

0

E

Q K =

#30

)E E

)

9

Q QTC

QTC

= +

=

c3)9

)9

TC Q AC Q

Q

AC

= =

=

d3 $hen EQ ≤ the firm needs no la#or !f EQ > the firm must hire la#or, setting E K = and plugging in for capital in the production function (ields)

/ ) / )

/ ) / )

/ ) / )

)/ )

E

9

9

9

Q L

Q L

L Q

L Q

= + = +

= −

= − Thus,

/)

)

9 if E

* if E

Q Q L

Q

− > = ≤

e3

( ) )

/ )) 9 E "hen E

E "hen E

Q QTC

Q

− + >= ≤

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7raphicall(, short-run and long-run total costs are sho"n in the follo"ing figure

***

5**

/***

/5**

)***

)5**

* 5 /* /5 )* )5 9*

>

4 o t a l C o s t

SGT-

NGT-

E

f3

( ) )

/ )) 9 Eif E

Eif E

QQ

TC Q AC Q

QQ

− + >= = ≤

8.1). 4ric"cles must be produced with # wheels and 1 frame for each tric"cle. Aet Q be thenumber of tric"cles& W be the number of wheels& and F be the number of frames. 4he price

of a wheel is P W and the price of a frame is P F .a* What is the long-run total cost function for producing tric"cles& TC ;Q,P W , P F *?b* What is the production function for tric"cles& Q;F,W *?

a3 8ach tric(cle reuires the purchase of three "heels at price P # and one frame at price P F Thus, TC Q, P # , P F 3 = Q9 P # ? P F 3 #3 Three "heels and one frame are perfect complements in production Thus the productionfunction is Q F$ # 3 = minO F , /93# P :otice that F$ # 3 = /, 93 (ields Q /, (F$ # 3 = ), C3(ields Q ), etc

8.1+. 5 hat manufacturing firm has the following production function with capital andlabor being the inputs: Q 6 min;' L& K *Bthat is it has a fi!ed-proportions productionfunction. $f w is the cost of a unit of labor and r is the cost of a unit of capital& derive thefirm’s long-run total cost curve and average cost curve in terms of the input prices and Q.

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - )*




The fixed proportions production function implies that for the firm to #e at a cost minimizingoptimum, K L D0 = and #oth of these eual Q Therefore, L = Q0 and K = QD So the firm's

total cost is Qr w

rQwQrK wL QD0

RD0 +=+=+

The average cost curve is

D0

r w

QTC L%AC +== :ote that this average cost curve is

independent of and is simpl( a straight line

8.1. 5 pacaging firm relies on the production function Q 6 KL = K & with MP L 6 K and MP K 6 L = 1. 5ssume that the firm’s optimal input combination is interior ;it uses positiveamounts of both inputs*. erive its long-run total cost curve in terms of the input prices& w

and r . erif" that if the input prices double& then total cost doubles as well.

Since "e can assume an interior solution, the tangenc( condition must hold Therefore the

optimal #undle must #e such that

/ r

w

L

K =

+

This means /

w

rK L =+ Su#stituting this #ack into

the production function, "e see thatr

Qw K

w

rK Q == so,

)

This implies that /−=

w

Qr L The total cost curve is then rK wLTC += = ) wwrQ − !f "e

su#stitute )w and )r in the place of w and r respectivel(, "e get TC ) =TC wwrQwQr w A))03)B3)3B)B) =−=− , so total cost does indeed dou#le "hen input

prices dou#le

8.18. 5 firm has the linear production function Q 6 # L = ) K & with MP L 6 # and MP K 6 ).

erive the e!pression for the 1ong-run total cost that the firm incurs& as a function of Qand the factor prices& w and r.

%s "e sa" in hapter D, linear production functions usuall( have corner solutions !n this case,the firm "ill use onl( la#or if

59or,,

r w

r

w M%T" K L <>

Similarl(, it "ill use onl( capital if59

r w>

!f the firm does use la#or, then it "ill use9

Q L = "ith a total cost of wQ9 Similarl( if it uses

capital it "ill use5

Q K = "ith a total cost of rQ5

Therefore, the firm's total cost curve can #e expressed as P5,

9minO Q

r wTC =

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8.1 5 firm uses two inputs: labor and capital. 4he price of labor is and the price of

capital is . 4he firm’s long-run total cost is given b" the e7uation Dased

on this e7uation& which change would cause the greater upward rotation in the long-runtotal cost curve: a 1% percent increase in or a 1% percent increase in ? Dased on "our

answer& is the firm’s production operation more capital intensive or labor intensive?!plain "our answer.

% /* percent increase in r "ould cause the TC curve to rotate up"ard more than a /* increase inw !n fact, a /* percent increase in r "ould cause a 053A/* = 2 percent increase in TC for an( positive level of output , "hile a /* percent increase in w "ould cause onl( a /53A/* = ) percent increase in TC for an( positive level of output Q The fact that total costs are moreresponsive to a change in the price of capital than to a change in the price of la#or, suggests thatthe firm's production operation is more capital intensive than la#or intensive

8.2%. When a firm uses K units of capital and L units of labor& it can produce Q units ofoutput with the production function Q 6 K @ L. ach unit of capital costs 2%& and each unit of labor costs 2). 4he level of K is fi!ed at ) units.a* 0ind the e7uation of the firm’s short-run total cost curve.b* n a graph& draw the firm’s short-run average cost.

a3 ;rom the production function "e see that 5Q L= , so the amount of la#or reuired to

produce Q is given #()

)5

Q L = The short run total cost function is

))

)5 )* )5 )*B53 /** )5

Q

C L K Q

= + = + = + #3

8.21. When a firm uses K units of capital and L units of labor& it can produce Q units ofoutput with the production function Q 6 @ L = @ K . ach unit of capital costs 2& and eachunit of labor costs 1.

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - ))




a* 4he level of K is fi!ed at 1+ units. ,uppose Q E '. What will the firm’s short-run totalcost be? ;(int: (ow much labor will the firm need?*b* 4he level of K is fi!ed at 1+ units. ,uppose Q > '. 0ind the e7uation of the firm’s short-run total cost curve.

a3 8ven if the firm hires zero units of la#or, "ith K fixed at /C it can still produce up to Q /C* + = 0 units of output So for 0≤Q , L * is the cost-minimizing choice of la#or and the

short-run total cost function is Fust the cost of capitalH C rK & wL )/C3 ? /*3 = 9)

#3 ;or Q ' 0, the firm needs to hire positive amounts of la#or, according to /C+= LQ or L Q 03) So for Q ' 0, the short-run total cost function is C Q3 = rK & wL )/C3 ? /Q 03) = 9) ? Q 03)

8.22. Consider a production function of three inputs& labor& capital& and materials& given b"Q 6 LKM . 4he marginal products associated with this production function are as follows:

MP L 6 KM & MP K 6 LM & and MP M 6 LK . Aet w 6 )& r 6 1& and m 6 2& where m is the price perunit of materials.a* ,uppose that the firm is re7uired to produce Q units of output. ,how how the cost-minimizing 7uantit" of labor depends on the 7uantit" Q. ,how how the cost minimizing7uantit" of capital depends on the 7uantit" Q. ,how how the cost-minimizing 7uantit" ofmaterials depends on the 7uantit" Q.b* 0ind the e7uation of the firm’s long-run total cost curve.c* 0ind the e7uation of the firm’s long-run average cost curve.d* ,uppose that the firm is re7uired to produce Q units of output& but that its capital isfi!ed at a 7uantit" of )% units ;i.e.& K 6 )%*. ,how how the cost-minimizing 7uantit" of labordepends on the 7uantit" Q. ,how how the cost-minimizing 7uantit" of materials depends

on the 7uantit" Q.e* 0ind the e7uation of the short-run total cost curve when capital is fi!ed at a 7uantit" of)% units ;i.e.& K 6 )%* and graph it along with the long-run total cost curve.f* 0ind the e7uation of the associated short-run average cost curve.

a3 8uating the #ang for the #uck #et"een la#or and capital implies

5

/5

L

K

MP w

MP r

KM

LM

K L

=

=

=

8uating the #ang for the #uck #et"een la#or and materials implies

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - )9




5

)

5)

L

M

MP w

MP

KM

KL

L M

=

=

=

<lugging these into the production function (ields

9

9

/ 9

55 3

)

)5

))

)5))5

LQ L L

LQ

Q L

Q L

=

=

=

=

Su#stituting into the tangenc( condition results a#ove implies

/ 9)

5)5Q

K =

and

/ 95 )

) )5

Q M

=

#3/9 / 9 /9

/ 9

) ) 5 )5 5 )

)5 )5 ) )5

)/5

)5

Q Q QTC

QTC

= + +

=

c3/ 9

/5 )

)5

TC Q AC

Q Q

= =

d3 Beginning "ith the tangenc( condition





5

)

5)

L

M

MP w

MP

KM

KL

L M

=

=

=

Setting 5* K = and su#stituting into the production function (ields

)

55*3

)

/)5

/)5

LQ L

Q L

Q L

=

=

=

Su#stituting this result into the tangenc( condition result a#ove implies

5/)5)

)*

Q

M

Q M

=

=

e3 !n the short run,

5 5* )/)5 )*

) 5*5

Q QTC

QTC

= + +

= +

7raphicall(, short-run and long-run total cost curves are sho"n in the follo"ing figure





***

)***

0***C***

2***

/****

/)***

* /*** )*** 9*** 0*** 5***

>

4 o t a l C o s t

SGT-

NGT-

f3 Short run average cost is given #(

) 5*

5

Q

TC AC Q Q

+

= =

8.2#. 4he production function Q 6 KL = M has marginal products MP K 6 L& MP L 6 K & and MP M 6 1. 4he input prices of K & L& and M are '& 1+& and 1& respectivel". 4he firm isoperating in the long run. What is the long-run total cost of producing '%% units of output?

;irst, notice that if the firm uses L it must necessaril( use K and vice versa@ there is no pointusing a positive amount of one of these inputs and zero of the other Thus, there are three possi#le solutions to the long-run cost minimization pro#lemH i3 interior, using positive amounts

of K$ L$ and M @ ii3 a corner "ith K L * and M ' *@ or iii3 a corner "ith M * #ut positiveamounts of both K and Lsing approach i3, "e euate MP K /r = MP M s and MP L /w = MP M s to get K /C and L 0sing the production constraint then (ields M = Q – C0 Total cost using this approach "ill #eC KLM Q3 = /C03 ? 0/C3 ? /Q C03 = Q & C0sing approach ii3, "e have K L = * and the input demand for M comes from the productionconstraintH M Q Total cost "ill #e C M Q3 = Qsing approach iii3, M * and the tangenc( condition #et"een K and L (ields MP Lw = MP K r ,or K 0 L om#ined "ith the production function, "e get the input demand functions

Q K )= and Q L)/

= Total cost "ill #e

C KLQ3 = ( ) ( ) ( ) QQQ /C*/)0/C)

/ =++

omparing the three approaches, it is eas( to see that C M Q3 I C KLM Q3 for all values of QLence, a cost-minimizing firm "ill never use K , L, and M simultaneousl(@ it could produce thesame output at less cost #( Fust using M ;urthermore, C M Q3 I C KLQ3 onl( for Q * )5C So for Q 0**, the firm should set M * and, follo"ing approach ii3, set K 0* and L = /* Totalcost "ill #e C KL 9)*

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8.2'. 4he production function Q 6 KL = M has marginal products MP K 6 L& MP L 6 K & and MP M 6 1. 4he input prices of K & L& and M are '& 1+& and 1& respectivel". 4he firm isoperating in the short run& with K fi!ed at 2% units. What is the short-run total cost ofproducing '%% units of output?

$ith K fixed at )* units, the production function #ecomes Q )* L & M Thus, L and M are perfect su#stitutes Since MP Lw = /)5 J MP M s = /, the marginal product per dollar spent onla#or is al"a(s higher than that on materials So the cost-minimizing input com#ination is M *"ith L solving 0** )* L & *, or L )* The short run total cost is C /C)*3 ? 0)*3 ? /*3 =0**

8.2). 4he production function Q 6 KL = M has marginal products MP K 6 L& MP L 6 K & and MP M 6 1. 4he input prices of K & L& and M are '& 1+& and 1& respectivel". 4he firm isoperating in the short run& with K fi!ed at 2% units and F fi!ed at '%. What is the short-run

total cost of producing '%% units of output?$ith K fixed at )* and M fixed at 0*, the production function #ecomes Q )* L & 0* To produce 0** units, the firm needs to hire la#or until 0** = )* L & 0*, or L /2 The short-runtotal cost is C /C/23 ? 0)*3 ? /0*3 = 0*2

8.2+. 5 short-run total cost curve is given b" the e7uation STC ;Q* 6 1%%% = )%Q2. erivee!pressions for& and then setch& the corresponding short-run average cost& averagevariable cost& and average fi!ed cost curves.

)

3 /*** 5* 3 /*** 3 5*

3 5*

/*** 3

"TC Q Q"TC Q

"AC Q QQ Q

AVC Q Q

AFC QQ

= += = +

=

=

7raphing B 3"AC Q , 3 AVC Q , and 3 AFC Q (ields

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - )D




***

)****

0****C****

2****

/*****

/)****

*** 5** /*** /5** )***

>

C o s t

S%-

%>-%;-

8.2. 5 producer of hard dis drives has a short-run total cost curve given b" STC ;Q* 6 K

= Q2 /K . Within the same set of a!es& setch a graph of the short-run average cost curves forthree different plant sizes: K 6 1%& K 6 2%& and K 6 #%. Dased on this graph& what is theshape of the long-run average cost curve?

*

/

/

)

)

9

9

* /5 9* 05 C*

>

C o s t

S%-/* S%-)* S%-9*

Since each of these short-run average cost curves reaches a minimum at an average cost of )*,the long-run average cost curve associated "ith these short-run curves "ill #e a horizontal line,tangent to the #ottom of each of these curves, at a long-run average cost of )*

8.28. 0igure 8.18 shows that the short-run marginal cost curve ma" lie above the long-runmarginal cost curve. Get& in the long run& the 7uantities of all inputs are variable& whereasin the short run& the 7uantities of 3ust some of the inputs are variable. 9iven that& wh" isn’tshort-run marginal cost less than long-run marginal cost for all output levels?





$ith some inputs fixed, it is likel( that the fixed level is not optimal given the firm's sizeTherefore, it ma( #e more expensive to produce additional units in the short run than in the longrun "hen the firm can emplo( the optimal, i.e., cost minimizing, uantit( of the fixed input

8.2. 4he following diagram shows the long-run average and marginal cost curves for afirm. $t also shows the short-run marginal cost curve for two levels of fi!ed capital: K 6 1)%and K 6 #%%. 0or each plant size& draw the corresponding short-run average cost curve ande!plain briefl" wh" that curve should be where "ou drew it and how it is consistent with theother curves.

The SG% curves are sho"n #elo" 8ach curve must satisf( t"o reuirements- i3 the SG%curve must #e tangent to the NG% curve at the output level at "hich the SGM curve for that particular plant size intersects the NGM curve@ and ii3 the SG% curve must reach a minimumat the output level at "hich it intersects its o"n SGM curve;or the plant size of 9** this is easil( achieved #( Fust dra"ing a curve tangent to the NG%curve at its minimum point, since this is also the point at "hich the NGM and the correspondingSGM curves intersect at the output level Q 53;or a plant size of /5*, these t"o points must #e kept in mind and the curve must #e dra"ncarefull( to compl( "ith #oth ;irst, SG% is tangent to NG% at Q ), "here NGMintersects SGM for K /5* Second, SG% reaches its minimum "here it intersects SGM,near Q )0

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - )E




8.#%. ,uppose that the total cost of providing satellite television services is as follows:

where Q1 and Q2 are the number of households that subscribe to a sports and movie

channel& respectivel". oes the provision of satellite television services e!hibit economies ofscope?

8conomies of scope exist if

/ ) / ) , 3 ,*3 *, 3 *,*3TC Q Q TC Q TC Q TC − < −

!n this case

/ ) / )

/ /

) )

, 3 /*** ) 9

,*3 /*** )

*, 3 /*** 9

*,*3 *

TC Q Q Q Q

TC Q Q

TC Q Q

TC

= + += += +=

So, economies of scope exist if

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - 9*

SG%, 4=/5*SG%, 4=9**




/ ) / )

) )

/*** ) 9 3 /*** ) 3 /*** 9

9 /*** 9

* /***

Q Q Q Q

Q Q

+ + − + < +< +<

"hich is certainl( true Thus, in this case the cost of adding a movie channel "hen the firm isalread( providing a sports channel is less costl( #( /***3 than a ne" firm suppl(ing a moviechannel from scratch 8conomies of scope exist for this satellite T> compan(

8.28. 5 railroad provides passenger and freight service. 4he table shows the long-run totalannual costs TC ;F, P *& where P measures the volume of passenger traffic and F the volumeof freight traffic. 0or e!ample& TC ;1%&#%%* 6 1&%%%. etermine whether there are economiesof scope for a railroad producing F 6 1% and P 6 #%%. Driefl" e!plain.

TC /*, 9**3 = /*** "hile TC /*, *3 ? TC *, 9**3 = 5** ? 0** = E** Thus TC /*, 9**3 JTC /*, *3 ? TC *, 9**3 so economies of scope do not exist at this output level

8.2. 5 researcher has claimed to have estimated a long-run total cost function for theproduction of automobiles. (is estimate is that TC ;Q& w& r * 6 1%%w<Hr HQ#& where w and r arethe prices of labor and capital. $s this a valid cost functionBthat is& is it consistent withlong-run cost minimization b" the firm? Wh" or wh" not?

9/**Q r TC

w=

This TC function implies that for a fixed Q and r , increasing w "ould lo"er long-run totalcost !f the firm "ere minimizing cost in the long run, #( using the optimal com#ination of K and L , it "ould not #e possi#le to reduce total cost "hen w is increased

%s ;igures 29 and 20 in the text illustrate, "hen one input price increases, the total long-runcost "ill increase Therefore, this long-run total cost function is not consistent "ith long-run costminimization #( the firm

8.#%. 5 firm owns two production plants that mae widgets. 4he plants produce identicalproducts and each plant ;i * has a production function given b" Qi 6 @ K i Li & for i 6 1& 2. 4heplants differ& however& in the amount of capital e7uipment in place in the short run. $n

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particular& plant 1 has K 1 6 2)& whereas plant 2 has K 2 6 1%%. $nput prices for K and L are w

6 r 6 1.a* ,uppose the production manager is told to minimize the short-run total cost ofproducing Q units of output. While total output Q is e!ogenous& the manager can choosehow much to produce at plant 1;Q1* and at plant 2;Q2*& as long as Q1 = Q2 6 Q. What

percentage of its output should be produced at each plant?b* When output is optimall" allocated between the two plants& calculate the firm’s short-run total& average& and marginal cost curves. What is the marginal cost of the 1%%thwidget? f the 12)th widget? 4he 2%%th widget?c* (ow should the entrepreneur allocate widget production between the two plants in thelong run? 0ind the firm’s long-run total& average& and marginal cost curves.

a3 8ssentiall(, the firm's production function is

)/

)/

)/

/*5

/**)5

L L

L L

QQQ

+=

+=

+=

That is, the firm has t"o varia#le inputs, L/ and L) The marginal products are MP L/ = )5 L/3 –*5 and MP L) = 5 L)3 –*5 sing the tangenc( condition, "e see that L) = 0 L/ sing the productionfunctions at each plant, "e see

//)) 0)*/* Q L LQ ===

Since total output Q Q/ ? Q), "e have Q) = 0Q – Q)3 or Q) = 2Q Similarl(, Q/ = )Q So thefirm should produce 2* percent of output at plant ) and )* percent at plant / #3 om#ining the a#ove tangenc( condition and the production constraint, "e find the inputdemands are L/ = Q)C)5 and L) = 0Q)C)5 !ncluding the cost of capital, total cost is then C

/)5 ? Q)/)53 %verage cost is AC /)5Q3 & Q/)53 Marginal cost is MC = )Q/)5 MC /**3 = /C, MC /)53 = ), and MC )**3 = 9)

c3 !n the long run, the plants are identical so the entrepreneur should split productioneuall( #et"een the t"o plants ie Q/ = Q)3 Thus the total production function can #e "ritten as

//

/

)/

)

)

L K

Q

QQQ

=

=

+=

%gain, "e can vie" total output as depending on the choice of onl( t"o inputs Sincethe plants are identical the entrepreneur "ill hire eual amounts of capital at each plant@ and

similarl( for la#or3 Minimizing cost implies r w M%T" K L // , = or K / = L/ !nput demands arethen L/ = K / = *5Q so that total cost is C = w L/ ? L)3 ? r K / ? K )3 = )Q Nong-run average costis AC = ) and long-run marginal cost is MC = )

8.#1 5 railroad has two t"pes of services: freight service and passenger service. 4he stand-alone cost for freight service is where e7uals the number of ton-miles of

op(right . )*/0 1ohn $ile( & Sons, !nc hapter 2 - 9)




freight hauled each da" and is the total cost in thousands of dollars per da". 4he stand-

alone cost for passenger service is where e7uals the number of

passenger-miles per da" and is the total cost in thousands of dollars per da". When a

railroad offers both services 3ointl" its total is o the

provision of passenger and freight service e!hibit economies of scope?

The provision of passenger and freight service do not exhi#it economies of scope8conomies of scope "ould #e present if, for all positive values of Q/ and Q), the total cost ofoffering #oth services together is less than the sum of the stand-alone costs of offering eachserviceLo"ever, in this case, the reverse is trueH the sum of the stand-alone costs of freight and passenger services is /,5** & Q1 & 2Q2 "hich is less than the total cost if #oth services areoffered together, "hich is ),***& Q1 & 2Q2

8.#2 ,uppose that the e!perience curve for the production of a certain t"pe of semi-conductor has a slope of 8% percent. ,uppose over a five-"ear period& cumulativeproduction e!perience increases b" a factor of 8. $nput prices over this period did notchange. 5t the beginning of the period& average variable cost was I1% per unit. 4his cost isindependent of the level of output at an" particular point in time. What is "our bestestimate of average variable cost at the end of this five-"ear period?

The pro#lem tells us that %> is independent of output at an( point in time and that factor pricesdid not change !t is plausi#le to conclude that the level of %> over the five-(ear period isaffected #( changes in cumulative experience!f cumulative experience has increased #( a factor of 2, this means that cumulative experiencehas dou#led three times from N to ) N , then from ) N to 0 N , and then again from 0 N to 2 N 3 $itha slope of 2* percentH• The first dou#ling reduced %> to 2* percent of "hat the( had #een, ie, from /* to 2• The second dou#ling reduced %> to 2* percent of the ne" level, ie, from 2 to *23A2 =

C0• The third dou#ling reduced %> to 2* percent of this ne" level, ie, from C0 to *23AC0

= 5/)

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