Econ 301 Juliano

8/17/2019 Econ 301 Juliano

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Multiple Choice Problems

1. eThe correct answer is e because elasticity is -0.5Assume that the price of one hot tub is P, then an individual must have an income of at

least 100P in order to purchase a hot tub. Therefore, the quantity demanded at this price is

1000000100P!10000P. "upposin# further that the price of a hot tub doubles, the price

now becomes $P, therefore ma%in# the income required to be $00P. &uantity demanded

at this point decreases to 100000$00P!5000P. 'ecall that the formula for price elasticity

of demand is

$. b(. a

At a price of )100, the quantity of tic%ets demanded is $*+$(+!50

At the price of )110, the total quantity demanded is 1$+1$+!$5


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. bConsumers problems

1.a. es. 'ay prefers more to less of each #ood. This can be confirmed by ta%in# first

order derivatives of the utility function to show that the level of utility is

positively related to the amount of each #ood consumed.

/This value is positive

/this value is positive

b. hen utility is 2, the function can be written as

'ewritin# it in the form of y!m3c

The intercepts for this line will be

X Y

0 50100 0

At a utility level of 1


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'ewritin# in the form y!m3c

4ntercept coordinates are

X Y

0 100

100 0

hen !$, we can write the utility function as

'ewritin# this function in the form y!m3c

The intercept coordinates for this line will be

X Y

0 200

100 0

Plottin# the three curves in one #raphs

c. 6or any bundle, the indifference curve is a line passin# throu#h point /100, 0.

The resultant indifference map is a star shaped ray of indifference curves all


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passin# throu#h /100, 0. es, the utility is homothetic because each utility

function can be e3pressed as a function of the other.d. 4f 'ay7s income is less than )100, he will consume only #ood and nothin# of

#ood 8. 9ut if his income is above )100, he will consume only #ood 8 and

nothin# of #ood . :iven the utility function and the prices of #ood 8 and #ood

/both )1, we can derive his demand for #ood 8 as a function of his income. 6or

all income levels above )100, 'ay will consume 1 unit of #ood 8 for every

additional )1 of income. Therefore, 8!4-100, where 8 is the quantity of 8

consumed and 4 is 'ay7s income

$.

a The #eneral form of a consumption leisure bud#et line is

here ; is consumption levels, w is the wa#e rate, < is hours of leisure, T is total

time available and = is the part of income that is independent of wor%in# hours.:iven total time T, wa#e rate w, and 31 hours of leisure, the a#ent faces a bud#et

constraint #iven by the equation.


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b to solve for utility ma3imi>in# levels of 31 and 3$, we proceed as follows

=a3

s.t

6rom this problem, we can derive the lan#ra#e as

?ifferentiatin# this function with respect to , , and

@@@@@@@@@@@@@@@@@1

@@@@@@@@@@@@@@@@@$

!0@@@@@@@@@@@@@@@@@@@@@(

6rom 1 and $, we can solve for the e3pansion paths of 31 and 3$

6rom 1

6rom $


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quatin# the two

"ince is common on both sides it cancels out, leavin#

Bbtainin# reciprocals on both sides and then square roots results in

e can plu# equivalents of , and into the bud#et constraint to obtain their optimal values.

6actorin# out

?ividin# both sides by


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b At first, increasin# wa#e will cause an individual to wor% more and ta%e less

hours of leisure. This is the case because substitution effect is hi#her than income

effect hence wor% is relatively more profitable than leisure. Cowever, as wa#es

%eep increasin#, the substitution effect %eeps declinin# while income effect

increases until a point where income effect is bi##er than substitution effect. At

this point, an individual consumes more of leisure hours and less of wor%

True or False

1. 6alse. Dot all the sections of the AT; symboli>e ?'". Bnly the upward slopin# part of

the curve is associated with decreasin# returns to scale. The downward slopin# part is

associated with increasin# returns to scale.$. True. At =P1w1E=P$w$, the slope of the isoquant e3ceeds that of the isocost, implyin#

that the firm can maintain same level of output by increasin# the use of input 1 and

decreasin# the use of input $ until equilibrium is reached.(. True. henever the =; curve is below the AT; curve, it implies that an additional unit

of input will cost less than the previous unit, which means the avera#e cost is also

declinin#. Bn the other hand, if =; the A;, then the A; is increasin# because every

additional unit is costin# more, which implies hi#her costs on avera#eF Therefore, the

avera#e cannot increase when =; is declinin#.. 6alse. Avera#e cost is minimi>ed when y!10

'ecall that

At minimum, the derivative of the A; function is equal to >ero


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=ultiplyin# each term by

Ta%in# the square root on both sides

y!10.

5. 6alse. :iven that the inputs in this production function are perfect substitutes, the optimal

solution is a corner solution, implyin# that the firm ma3imi>es profit by usin# only one

input. Then, if the price of 31 doubles and that of 3$ triples, then the firm is better off

producin# usin# 31 only because it is cheaper. Therefore, the firm7s cost will only double.*. 6alse. "ince the firm shows decreasin# returns to scale, the effect of the ta3 and subsidy

do not cancel out. ssentially, the increase in inputs used due to the subsidy will result in

less than proportionate increase in output. This means that the overall impact is ne#ativeG

hence the optimal quantity has to chan#e.+. 6alse. 4f the firm is in a competitive mar%et, an increase in its output price must cause its

quantity to fall because buyers will shift to other firms.Production problems

1./a 4f the firm is e3periencin# a fi3ed proportions production function, the total

cost is equal to the sum of costs of both inputs.

/b :iven that the firm7s inputs are perfect substitutes, then the firm optimal

bundle is a corner solution, which means it can only produce with one input.


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As such, it chooses the input which has a lower cost. Cowever, the input costs

may be equal, in which case the firm is indifferent. Therefore, there are (

possible cost functions.

4f w1Ew$, T;!w1y.4f w$Ew1, T;!w$y4f w1!w$!w, T;!wy

$. :iven the production function y!min/$31, (3$ then the conditional demand for 31

isH

hen , the production function becomes y!min/$31, 1$

"ince the proportions are fi3ed, $31!1$

31!*T;!*w1w$

(. The total cost function is

Production function

This problem is a cost minimi>ation problem which is solved as follows.

=in

s.t


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6ormulatin# Bbtainin# 6irst order conditions

@@@@@@@@@@i

@@@@@@@@@@.ii

@@@@@iii

e can solve e3pansion paths of 31 and 3$ usin# equations 1 and $

6rom 1,

@@@@@@@@@@@@..iv

6rom $,


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@@@@@@@@@@@..v

quatin# iv and v

"ubstitutin# these values into equation ( one at a time

4ntroducin# the e3ponent on both sides


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=ultiplyin# throu#h by

;ollectin# li%e terms

The cost function will be


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.

:iven that H

a. Avera#e variable cost

b. =ar#inal cost

c. Avera#e cost is minimi>ed when the derivative of AI; is put to >ero


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d. se, the supply curve can be derived from the mar#inal cost curve by ta%in# the

positively slopin# part of the curve. To do that we need to determine the turnin# point of

=; to %now where the supply curve be#ins by differentiatin# =; with respect to y and

puttin# it to >ero.

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Econ 301 Juliano