Econ 201 Problem Set on Chapter 7
1. Complete the following table (round each answer to the nearest whole number): "utput # 1 % & ' $
Total Cost
Variable Cost
Fixed Cost
Marginal Cost
!erage Cost
!g. Var. Cost
!g. Fixed Cost
$ &# 1& 1#$
1# 11# $#
Solution: "utput # 1 % & ' $
Total Cost '# '$ # *+ 1#$ 1$# %##
Variable Cost # $ %# '# $ 11# 1#
Fixed Cost '# '# '# '# '# '# '#
Marginal Cost $ 1$ 1+ % '$ $#
!erage Cost '$ &# % % &# &&
!g. Var. Cost $ 1# 1& 1 %% %*
%.Trisha belie!es the production of a dress re-uires ' labor hours and % machine hours to produce. f Trisha decides to operate in the short run/ she must spend 0$## to lease her business space. lso/ a labor hour costs costs 01$ and a machine hours costs 0&$. hat is Trisha2s cost of production as a function of dresses produced3 Solution: 4ince the production of a dress re-uires spending 0# for labor and 0*# for
machine hours/ Trisha2s cost function is: C q
1&#q $##.
&. firm2s total total cost function is gi!en b5 the e-uation: TC 6 '### 7 $8 7 1#8 %. (1) rite an expression expression for each of the following following cost concepts: concepts: a. b. c. d. e. f.
Total Fixed Cost !erage Fixed Cost Total tal Var Varia iab ble Cost Cost !era !erage ge Varia ariabl blee Cos Costt !erag eragee Total tal Cos Costt Marginal Cost
(%) 9etermine the -uantit5 -uantit5 that minimies a!erage a!erage total cost. 9emonstrate that the predicted relationship between marginal cost and a!erage cost holds.
!g. Fixed Cost '# %# 1& 1# , *
Solution: PART (1) a. TFC '###
b. FC
'###
8
c. TVC TC TFC TVC $Q 1#Q %
d. VC
TVC 8
$8 1#8 % 8
e. TC
TC 8
'### $8 1#8 %
f. MC $ %#8
8
$ 1#Q
PART (2)
TC is minimied where MC is e-ual to TC. ;-uating MC to TC '### $8 1#8 % 8 '### $8 1# %
$ %#8
$8 %#8 %
'### 1#8 % 8%
'##
8 %#
TC is minimied at %# units of output.
'### $1# 1# 1#
%
1#
$#$
MC is indeed less than TC for -uantities smaller than %#. MC should exceed TC for an5 -uantit5 greater than %#. For example/ let 8 6 %$: MC 6 $ 7 %#(%$) 6 $#$ TC
'### $ %$ 1# %$ %$
%
'1$
MC is indeed greater than TC for -uantities greater than %#.
'.9a!5 Metal Compan5 produces brass fittings. 9a!52s engineers estimate the production function represented below as rele!ant for their long=run capital=labor decisions. 8 6 $##>#.? #.,/ where 8 6 annual output measured in pounds/
> 6 labor measured in person hours/ ? 6 capital measured in machine hours. The marginal products of labor and capital are: M@> 6 &##>=#.'? #.,
M@? 6 '##>#.? =#.%
9a!52s emplo5ees are relati!el5 highl5 sAilled and earn 01$ per hour. The firm estimates a rental charge of 0$# per hour on capital. 9a!5 forecasts annual costs of 0$##/### per 5ear/ measured in real dollars. a. b.
9etermine the firm2s optimal capital=labor ratio/ gi!en the information abo!e. Bow much capital and labor should the firm emplo5/ gi!en the 0$##/### budget3 Calculate the firm2s output
Solution: a.
#.'
? #.,
M@ >
&##>
M@ ?
'##>#.( ?
&## MDT4
'##
MDT4
;-uate to
#.%
? #.,
&##
'##
>#.' >#.( ? #.%
? #., >#.' >#.(
#.*$
? #., ? #.% >#.' >#.(
? #.% ? #.*$ > w 1$
. $# ? 1$ #.*$ > $# ? #.& #.*$ > r
? #.' ? 6 #.'> > b.
C 6 $##/### C 6 w> 7 r? $##/### 6 1$> 7 $#? ? 6 #.'> from optimal ratio $##/### 6 1$> 7 $#(#.'>) $##/### 6 1$> 7 %#> $##/### 6 &$>
> 6 1'/%,$.*1 or 1'/%, hours 4ubstitute to sol!e for ?. $##/### 6 1$(1'%,) 7 $#? $##/### 6 %1'/%+# 7 $#? %,$/*1# 6 $#? ? 6 $*1'.%# or ? 6 $*1' 8 6 $##(1'/%,) #.($/*1')#., 8 6 1$*/$,/1+1 $. paper compan5 dumps nondegradable waste into a ri!er that flows b5 the firm2s plant. The firm estimates its production function to be: 8 6 ?/ where 8 6 annual paper production measured in pounds/ ? 6 machine hours of capital/ and 6 gallons of polluted water dumped into the ri!er per 5ear. The marginal products of capital and labor are gi!en as follows: M@? 6
M@ 6 ?
The firm currentl5 faces no en!ironmental regulation in dumping waste into the ri!er. ithout regulation/ it costs the firm 0*.$# per gallon dumped. The firm estimates a 0&# per hour rental rate on capital. The operating budget for capital and waste water is 0&##/### per 5ear. a. b.
9etermine the firm2s optimal ratio of waste water to capital. Ei!en the firm2s 0&##/### budget/ how much capital and waste water should the firm emplo53 Bow much output will the firm produce3
Solution: a.
M@ 6 ? M@? 6 MDT4
?
?
Date of water charge to price of capital: @ @<
*.$ &#
.%$
;-uating MDT4 to ratio of input prices ?
#.%$/ ? 6 #.%$
b.
C 6 @ 7 @? ? &##/### 6 *.$# 7 &#? recall ? 6 #.%$ &##/### 6 *.$ 7 &#(#.%$) &##/### 6 *.$ 7 *.$ 6 %#/### gallons ? 6 #.%$ ? 6 #.%$(%#/###) ? 6 $### 8 6 ($###)(%#/###) 8 6 ##/###/###
