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Unformatted text preview: 180 RISKY ASSETS (Ch. 13) Chapter 14 NAME
Consumer’s Surplus Introduction. In this chapter you wiil study ways to measure a con—
sumer’s valuation of a. good given the consumer’s demand curve for it.
The basic logic is as follows: The height of the demand curve measures
how much the consumer is wiliing to pay for the last unit of the. good
purchasedwthe willingness to pay for the marginal unit. Tl‘ierefore the
sum of the willingnesses—topay for each unit gives us the totai willingness
to pay for the consumption of the good. In geometric terms, the total willingness to pay to consume some
amount of the good is just the area. under the demand curve up to that
amount. This area is called gross consumer’s surplus or total beneﬁt
of the consumption of the good. If the consumer has to pay some amount
in order to purchase the good, then we must subtract this expenditure in
order to calculate the (net) consumer’s surplus. When the utility function takes the quasilinear form, aha) + m, the
area under the demand curve measures u(:i:), and the area under the
demand curve minus the expenditure on the other good measures U(LL‘) ~1—
771. Thus in this case, consiu'ner’s surplus an exact measure of
utility, and the change in consun'ier’s surplus is a. n‘ionetary measure of a
change in“ utility. If the utility function has a. different form, consumer’s surplus will not
he an exact measure of utility, but it will often he a good approximation.
However, if we want more exact measures, we can use the ideas of the
compensating variation and the equivalent variation. Recali that the compensating variation is the amount of extra income
that the consumer would need at the new prices to he as well off as she
was facing the old prices; the equivalent variation is the amount of money
that it would he 1‘1ecessary to take away from the consumer at the old
prices to make her as weil off as she would be, facing the new prices.
Although different in general, the change in eonsun‘ier’s sur1.)lus and the
compensating and equivalent variations wiil be the same if preferences are
quasilinear. In this chapter you wili practice: 0 Calcuiatin ‘ consumer’s sur 3111s and the chance in consumer’s sur ilus
l b o Calcuiating compensating and equivalent variations Example: Suppose that the inverse demand curve is given by P(q) m
100 — 10g and that the consumer currently has 5 units of the good. How
much money wouid you have to pay him to compensate him for reducing
his consumption of the good to zero? Answer: The inverse demand curve has a height of 100 when q = 0
and a height of 50 when (,7 = 5. The area under the demand curve is a. trapezoid with a base of 5 and heights of 100 and 50. We can calculate i
g.
"a
3‘.
if;
I;
t
i; mum.“ m....,..~y..., mmsmmmwmm whenmamm«A.»Amy»,w».uxrramwmwmqowwam 182 CONSUMER’SSURPLUS (Ch. 34) NAMF 183 the area 0f thls trapemld by applymg the formula (a) If the equilibrium rent for an apartment turns out to be $20, which (heightl + heightg). consumers will get apartments? A , B , C , D . NEH Area of a trapezoid = hase X In this 0839 we have A m 5 X l (100 + 50) _ $375 (3)) If the equiiibriun'l rent for an apartn'lent turns out to be $20, what
. . , , , 2 m . Example: Suppose new that the consumer is purchasmg the 5 umts at a 18 the consumer s (not) s11; plus genelated 111 this market for person A. price of $50 per unit. If you require him to reduce his purchases to zero,
how much money would be necessary to compensate him? In this case, we saw above that his gross beneﬁts decline by $375.
On the other hand, he has to spend 5 X 50 2 $250 less. The decline in
net surplus is therefore $125. 20 . For person 13? 5 . (c) If the equilibrium rent is $20, what is the total net consumers’ surplus generated in the market? 50 . Example: Suppose that a consumer has a utiiity function u(2:1,w2) :
$1 + 3:2. Initially the consumer faces prices (1,2) and has income 10.
If the prices change to (4, 2), calculate the compensating and equivalent
variations. Answer: Since the two goods are perfect substitutes, the consumer
will initially consume the bundle (10,0) and get a utility of 10. After the
prices change, she will consume the bundle (0,5) and get a utiiity of 5.
After the price change she would need $20 to get a utility of 10; therefore
the compensating variation is 20 ~ 10 m 10. Before the price change, she
would need an income of 5 to get a utility of 5. Therefore the equivalent
variation is 10 —— 5 x 5. 3 (d) If the equilibrium rent is $20, what is the total gross consumers surplus in the market? 130 . (e) If the rent declines to $19, how much does the gross surpius increase? 0. ( f ) If the rent declines to $19, how much does the net surplus increase? 4. Calculus 14.3 Quasimedo consumes earplugs and other things. His utiiity
function for earplugs (r: and money to spend on other goods y is given by 14.1 (0) Sir Pius consumes mead, and his demand function for tankards
of mead is given by 19(3)) 2: 100 ~— p, where p is the price of mead in shillings. 2
u(rt,y) = 100:1: — £— + y. (a) If the price of mead is 50 shillings per tankard, how many tankards of 2 mead Will he (70118111118? 50  (a) What kind of utility function does Quasimodo have? Quasilinear . (b) How much gross consumer’s surplus does he get from this censump~ tion? 3 , 750 . (b) What is his inverse demand curve for earl'ﬂugs? p H — SC . (c) If the price of earplugs is $50, how many earplugs will he consume? l
50. i
i (c) How much money does he spend on mead? 2 , 500 . (d) “that is his net consun'ler’s surplus from mead consumption? 1,250. {d} If the price of earplugs is $80, how many earplugs will he consume? 20. 14.2 (0) Here is the table of reservation prices for apartments taken from Chapter 1: (8) Suppose that Quasimodo has $4,000 in total to spend a month. What i
i
l
is his total utility for earplugs and money to spend on other things if the A. B C D E F G H i 40 25 30 35101835 5 Person Price price of earplugs is $50? $5 , 250 . 184 CONSUMER’S SURPLUS (Ch. 14) (f) What is his total utility for earplugs and other things if the price of earplugs is $80? $4 , 200 . (9) Utility decreases by 1 , 050 when the price changes from $50 to
$80. (it) What is the change in (net) consumer’s surplus when the price changes from $50 to $80? 1 , 050 . 14.4 (2) In the graph below, you see a representation of Sarah Gamp’s
indifference curves between cucumbers and other goods. Suppose that
the reference price of cucumbers and the reference price of “other goods”
are both 1. Other goods “nullIll
“IIIIIII
mullIll Cucumbers (a) W’hat is the minimum amount of money that Sarah would need in order to purcl'iase a bundle that is indifferent to A? 20 .
(b) What is the minimum amount of money that Sarah would need in
order to purchase a bundle that is indifferent to B? 30 . (c) Suppose that the reference price for cucumbers is 2 and the reference
price for other goods is 3. How much money does she need in order to purchase a bundle that is indifferent to bundle A? 30 . (d) What is the minimum amount of money that Sarah would need to purchase a bundle that is indifferent to B using these new prices? 40 . NAME W 185 (e) No matter what prices Sarah faces, the amount of money she needs
to purchase a bundle indifferent to A must be (higher, lower) than the amount she needs to purchase a bundle indifferent. to B. lower . 14.5 (2) Bernicc’s preferences can be reln'esented by vim, y) m min {512, y},
where (I: is pairs of earrings and y is dollars to spend on other things. She
faces prices (pm, pg) = (2,1) and her income is 12. ((1,) Draw in pencil on the graph beiow some of Bernice’s indifference
curves and her budget constraint. Her optimai hundie is 4 pairs of earrings and 4 dollars to spend on other things. Dollars for other things i6 1
~ éencd
8 ,. Blue 4
0 4 8 12 i6 Pairs of earrings (3)) The price of a pair of earrings rises to $3 and Bernice’s income stays
the same. Using blue ink, draw her new budget constraint on the graph above. Her new optimal bundle is 3 pairs of earrings and
3 dollars to spend on other things. (6) What bundle wouid Bernice choose if she faced the original prices and I . a l l ‘1 had Just enough income to reach the new irrdlfier‘eirce curve? (3, .
Draw with red ink the budget line that passes through this bundle at
the original prices. How much income would Bernice need at the original prices to have this (red) budget line? $9 . \ebr‘s‘mo: Nnxavzrb’n/rw»; I)is!\KIr/dxmvmﬂrmcmeemwtgwmxww>mmgwavxmwrmxwwmyrrrmrgmyo:«3x sewn«mammogram Calculus 186 CONSUMER’SSURPLUS (Ch. 14) (cl) The maximum amount that Bernice would pay to avoid the price increase is $3 . This is the (compensating, equivalent) variation in income. Equival ent . (e) W‘ hat bundle would Bernice choose if she faced the new prices and had just enough income to reach her original indifference curve? (4, 4) .
Draw with black ink the budget line that passes through this bundle at . g ' 1 v I r
the new prices. How much income would Bernice have With this budget? $16. (f) In order to be as well—off as she was with her originai bundle, Bernice’s original income would have to rise by $4 . This is the (compensating, equivalent) variation in income. Compensating . 14.6 (0) Ulrich likes video games and sausages. In fact, his preferences
can be represented by u(ac,y) = 111(33 + 1) + y where a: is the number of
video games he plays and y is the number of doliars that he spends on
sausages. Let pm be the price of a video game and m be his income. (:1) Write an expression that says that Ulrich’s marginal rate of substiw
tution equals the price ratio. ( Hint: Remember Donald Fi‘ibble from Chapter 6'?) 1/(CC + 1) 2 p1,; . (1)) Since Ulrich has quasil inear preferences, you can soive this
equation aione to get his demand function for video games, which is a: 3 1/193; " 1 . His demand function for the dollars to spend on sausages is y 2 m — 1 +19. (c) Video games cost $.25 and Ulrich’s income is $10. Then Ulrich den mands 3 video games and 9 . 25 dollars’ worth of sausages. His utility from this bundle is 10 . 64: . (Round off to two decimal
places.) (of) If we took away all of Ulrich’s video games, how much money wouid he need to have to spend on sausages to be just as welloff as before? $10.64. Calculus NAME _.. __ __ 187 ((3) Now an amusement tax of $.25 is put on video games and is passed on in full to consumers. With the tax in place, Ulrich demands 1
video game and 9 . 5 dollars’ worth of sausages. His utility from this bundle is 10 . 19 . (Round off to two decimal places.) ( f ) Now if we took away ail of Ulrich’s video games, how much money
would he have to have to spend on sausages to be just as well—off as with the bundle he purchased after the tax was in place? $1 0 . 19 . (Q) What is the change in Ulrich’s consumer surplus due to the tax? How much money did the government collect from Ulrich by means of the tax? $ . 25 . 14.7 (1) Lolita, an intelligent and charming Holstein cow, consumes
only two goods, cow feed (made of ground corn and oats) and hay. Her
preferences are represented by the utility function U (.13, y) m a? — {1.2/2 +y,
where a: is her consumption of cow feed and y is her consumption of hay.
Lolita has been instructed in the mysteries of budgets and optimization
and always maximizes her utility subject to her budget constraint. Lolita
has an income of $721. that she is allowed to spend as she wishes on cow
feed and hay. The price of hay is always iii, and the price of cow feed will
be denoted by p, where 0 < p g 1. (it) Write Lolita’s inverse demand function for cow feed. (Hint: Loiita’s
utility function is quasilinear. When 3,: is the numeraire and the price of
:r: is p, the inverse demand function for someone with quasilinear utility f(:i:) i— y is found by simply setting p = f’(:i:).) p = 1 W 33' . (b) If the price of cow feed is p and her income is m, how much hay does
Lolita choose? (Hint: The money that she doesn’t spend on feed is used to bay bay.) m — p(1  p) . ((2) Plug these numbers into her utility function to find out the utility level that she enjoys at this price and this income. ’21, = m—l— mp)2/2 . (d) Suppose that Lolita’s daily income is $3 and that the price of feed is
$.50. What bundle does she buy? /2, 1 / . What bundle would she buy if the price of cow feed rose to $1? (0, . 188 CONSUMER’SSURPLUS (Ch. 14) (c) How much money would Lolita. be willing to pay to avoid having the price of cow feed rise to $1? 1/8 . This amount is known as the equ ival €111: variation. (f) Suppose that the price of cow feed rose to $1. How much extra money
would you have to pay Lolita to make her as welloff as she was at the old prices? 1/8 . This amount is known as the compensating
variation. Which is bigger, the compensating or the equivalent variation, or are they the same? Same . (9) At the price $.50 and income $3, how much (not) consumer’s surplus
is Lolita getting? 1/8 . 14.8 (2) F. Flintstone has quasiiinear preferences and his inverse demand
function for Brontosaurus Burgers is P05) = 30 — 2!). Mr. Fiintstone is
currently consuming 10 burgers at a price of 10 dollars. (a) How much money would he be willing to pay to have this amount
rather than no burgers at all? $200 . What is his level of (net) consumer’s surplus? $ 100 . (b) The town of Bedrock, the only supplier of Brontosaurus Burgers,
decides to raise the price from $10 a burger to $14 a burger. What is Mr. Flintstone’s change in consumer’s surplus? At price
$10, consumer’s surplus is $100. At $14,
he demands 8 burgers, for net consumer’s
surplus of %Uﬁix 8):: 54. The change in consumer’s surplus is —$36. 14.9 (1) Karl Kapitalist is willing to produce p/ 2 — 20 chairs at every
price, p > 40. At prices below 40, he will produce nothing. if the price of chairs is $100, Karl will produce 30 chairs. At this price, how much is his producer’s surplus? %(60 X 30) = . 14.10 (2) Ms. Q. Mote loves to ring the church bells for up to 10
hours a day. Where m is expenditure on other goods, and a: is hours of
bell ringing, her utility is u(m,:1:) = m + 32: for m S 10. If :6 > 10, she
develops painful blisters and is worse off than if she didn’t ring the beils. NAME m 189 Her income is equal to $100 and the sexton allows her to ring the boil for
10 hours. (a) Due to complaints from the villagers, the sexton has decided to restrict
Ms. Mote to 5 hours of beli ringing per day. This is bad news for Ms. Mote. in fact she regards it as just as had as iosing $15 dollars of
income. (1)) The sexton reients and offers to let her ring the bells as much as she
likes so long as she pays $2 per hour for the privilege. How much ringing does she do now? 10 hours . This tax on her activities is as had as a loss of how much income? $20 . (c) The villagers continue to compiain. The sexton raises the price of
hell ringing to $4 an hour. How much ringing does she do now? 0 hours . This tax, as compared to the situation in which she couid ring the bells for free, is as bad as a loss of how much income? $30 . 190 CONSUMER’S SURPLUS (Ch. l4) Chapter 15 NAME Market Demand Introduction. Some problems in this chapter will ask you to construct
the market demand curve from individual demand curves. The market
demand at any given price is simply the sum of the individual demands at
that price. The key thing to remember in going from individual demands
to the market demand is to add quantities. Graphically, you sum the
individual demands horizontally to get the market demand. rThe market
demand curve will have a kink in it whenever the market price is high
enough that some individual demand becomes zero. Sometin'ies you will need to ﬁnd a consumer’s reservation price for
a good. Recall that the reservation price is the price that makes the
consumer indifferent between having the good at that price and not hav
ing the good at all. Mathematically, the reservation price 32* satisfies
u(0,m) x 15(1, m 13*), where m is income and the quantity of the other
good is measured in dollars. Finally, some of the problems ask you to calculate price and/or in
come elasticities of demand. These problems are especially easy if you
know a little calculus. If the demand function is D(p), and you want to
calculate the price elasticity of demand when the price is p, you only need
to calculate dD(p)/dp and multiply it by p/q. 15.0 Warm Up Exercise. (Calculating elasticities.) Here are
some drills on price elasticities. For each demand function, ﬁnd an ex
pression for the price elasticity of demand. rThe answer will typically be
a function of the price, p. As an example, consider the linear demand
curve, 13(3)) = 30 — 63). Then dD{p)/dp : —6 and p/q : 13/(30 — 63)}, so
the price elasticity of demand is —6p/(30 — tip). ((1) mp) = crisp. —p/(60 — p).
(6) De) = cw bp —bp/(a  5P)
(c) De) = 40p”. —2.
(93} Du?) = fin—b. —b (a) De) = (22+ 3)“? ~2p/(p + 3). 192 MARKETDEMAND (Ch. 15) (f) De) = to Mr". “1929/ (p + a) 15.1 (0) In Gas Pump, South Dakota, there are two kinds of consnn'iers,
Buick owners and Dodge owners. Every Buick owner has a demand funcu
tion for gasoline DB (p) = 20 w 5}) for p g 4 and 1913(3)) m 0 if p > 4.
Every Dodge owner has a demand function D 13(3)) m 15 ~ 3p for p S 5
and 409(3)) = U for p > 5. (Quantities are measured in gallons per week
and price is measured in dollars.) Suppose that Gas Pump has 150 con—
sumers, 100 Buick owners, and 50 Dodge owners. (a) If the price is $3, what is the total amount demanded by each indi— vidual Buick Owner? 5 . And by each individual Dodge owner? 6. (b) What is the total amount demanded by all Buick owners? 500 . r W hat is the total amount demanded by all Dodge owners? 300 . (a) What is the total amount demanded by all consumers in Gas Pump at a price of 3? 800 . (d) On the graph below, use blue ink to draw the demand curve repre—
senting the total demand by Buick owners. Use black ink to draw the
demand curve representing total demand by Dodge owners. Use red ink
to draw the market demand curve for the whole town. ((3) At what prices does the market demand curve have kinks? At p24 andp25. (f) When the price of gasoline is $1 per gallon, how much does weekly demand fall when price rises by 10 cents? 65 gallons . (9} When the price of gasoline is $4.50 per galion, how much does weekly demand fall when price rises by 10 cents? 15 gallons . (It) When the price of gasoline is $10 per gallon, how much does weekly demand fall when price rises by 10 cents? Remains at zero . NAME _..__. m 193 Dollars per gallon i s r : r a = E :
i i E i ' 5 l l l g \. a
r J E z s a .
g l ,2 2 : 5 g :
i s j i i E
’ E : l E
.l i .E .5
I 3. i i l i 3 s l :
4 . _l 2 5 l l E l 2 3 l i g l 1 2
E. i .E. E .z .E .E
. i g l g 2 l i; 5
3 E i .1. i. i s 2 J a i l i 5
s i j l i : l 4 : aluehne l E l 3 I i i 2' i ' l g g
g _ l 3 E E ‘ l
2 l l l. l
: s ' i ' g 2 5 a
5 s 3 ‘ é ' l
a ..... .. s = i .l
i ' 2 l l
" l : E i ' i
A. l e j ’ , f , Black 3
.. f ﬁne: : % >
0 500  000 ISOO 2000 2500 3000 Gallons per week 15.2 (0) For each of the following demand curves, compute the inverse
demand curve. ((1)1392) minax{10—2p,0}. I — if q < (b) De) =100W 29(q) I 10, 000/ C12  (c)!nD(p) =10 ~41). p(q) = (10 m in (d) 111 13(3)) 2 11120— 21113). p(q) = ‘V 20/6]. 15.3 (0) The demand function of dog breeders for electric dog polishers
is qb : max{200—p, O}, and the demand function of pet owners for electric
dog polishers is ((0 m max{90 — 413,0}. (a) At price 19, what is the price elasticity of dog breeders’ demand for
electric dog polishers? ~“10/ — p) . What is the price elasticity of pet owners’ demand? —4p/ ( — . 194 MARKETDEMAND (Ch. 15) (b) At what price is the dog breeders’ elasticity equal to M1? $100 . At what price is the pet owners’ elasticity equal to ~11? $1 1 . 25 . (c) On the graph below, draw the dog breeders’ demand curve in blue
ink, the pet owners’ demand curve in red ink, and the market demand
curve in pencil. (d) Find a nonzero price at which there is positive total demand for dog polishers and at which there is a kink in the demand curve. $22 . 5O .
What is the market demand function for prices below the kink? — 5p . What is the market demand function for prices above the kink? 200—10. ((2) Where on the market demand curve is the price elasticity equal to —i? $100 . At what price will the revenue from the sale of electric dog poiishers be maximized? $100 . If the goal of the sellers is to
maximize revenue, will electric dog polishers be sold to breeders only, to pet owners only, or to both? Breeders only . NAME M 195 Rte 300 2 e E i 3 ..F .3 ' a _ 2 = : g;  E " i' :3 ‘ 3 ; i i i l i i i i i E i : : z : a 3 ' i 200 _ 3 3 g : 3 s " i g E i i \‘ é i g g i I50 r 2 _‘ 5 '. t Bl I. 5 >‘ 3 E E r l E I00 ,ue r i ..... "i t. a i W.§ u r s E E 50 in ,é “.3 i .Hg 3 .m3 Pencil line i ' 2 f z \ i : ' E a a? i i" '  ._ i S ‘ 2 g ~+_: 290 0 50 200 250 300 Quantity Calculus 15.4 (0) The demand for kitty litter, in pounds, is lnD(p) = 1,000 —— p + 111m, where p is the price of kitty litter and m is income. (a) What is the price elasticity of demand for kitty litter when p = 2 and m = 500? “*2 . When p = i and m. m 500? —3 . When p = 4 and m :— 1, 500? *4 . (b) What is the income elasticity of demand for kitty litter when p = 2 and m = 500‘? 1 . When p x 2 and m = 1,000? 1 . When 33 m 3 and m = 1,500? 1 . 196 MARKETDEMAND (Ch. 15) (a) What is the price elasticity of demand when price is p and income is I m? —p . The income elasticity of demand? 1 . Calcuius 15.5 (0) The demand function for drangles is q(p) m (p + 1)“? (a) What is the price elasticity of demand at price p? —2p/ i . (b) At What price is the price elasticity of demand for drangles equal to ~i? When the price equals 1. (c) Write an expression for total revenue from the sale of drangles as a function of their price I pg 2 p/(p i 1)2 . Use calculus to ﬁnd the revenuemaximizing price. Don’t forget to check the mamamumcmmamn. Differentiating and solving gives p:: 1. Q ((1!) Suppose that the demand function for drangles takes the more general
form (1(3)) 2 (19+ a)""b where e > 0 and b > 1. Calculate an expression for the price elasticity of demand at price 3). “bp/ (19 "h a) . At what price is the price elasticity of demand equal to #1? p = 03/ ‘ . 15.6 (0) Ken’s utility function is 1LK(:1:1,:122) 2 (E1 + 332 and Barbie’s
utility function is 1113($1,332) = (23] +1)(m2 + l). A person can buy 1
unit of good 1 or 0 units of good I. It is impossibie for anybody to buy
fractional units or to buy more than 1 unit. Either person can buy any
quantity of good 2 that he or she can afford at a price of $1 per unit. (a) Where m is Barbie’s wealth and 331 is the price of good 1, write an
equation that can be solved to ﬁnd Barbie’s reservation price for good 1. (m — p1 + = m — 1 . “that is Barbie’s reservation price
for good 1? p I (m I— 1)/2 . What is Ken’s reservation price for good 1'? $1 . (b) If Ken and Barbie each have a wealth of 3, plot the market demand
curve for good 1. NAME .—__...,mm 197 Price 0  2 3 4
Quantity 15.7 (0) The demand function for yoyos is D(p, M’) x 4 — 2p + “130 114,
where p is the price of yowyos and M’ is income. If M‘ is 100 and p is l, (a) What is the income elasticity of demand for yoyos? 1/3 . (b) What is the price elasticity of demand for yoyos? *2/3 . 15.8 (0) If the demand function for zarfs is P = 10 —— Q, (a) At what price will total revenue realized from their sale he at a max— imum? P 3 (b) How many zarfs will be sold at that price? Q = 5 . 15.9 (0) The demand function for football tickets for a typicai game at a
large midwestern university is 13(3)) = 200, 000  10, 000p. The university
has a clever and avaricious athletic director who sets his ticket prices so
as to maximize revenue. The university’s football stadium hoids 100,000
spectators. (a) Write down the inverse demand function. Z 
q/10,000. 198 MARKETDEMAND (Ch. 15) (5) Write expressions for total revenue ) Z m (12/ 10, and marginal revenue MR 2 20 — (1/5, 000 as a function of the
nun‘iber of tickets sold. (c) On the graph below, use blue ink to draw the inverse demand function
and use red ink to draw the marginal revenue function. On your graph,
also draw a vertical blue line representing the capacity of the stadium. Price
w a __ .,Hhmm m.m
: g ' r z 1 =
s g z i g i i f 3 i i i
: g E i l i i E
25 i l E i l l 5 i L .E ._ .5. .E .E ,3 .5
3 ; i E i i i i
g E , ,,,.., i ,.,
20 g : 3 7 Stadium; capacity g g ,I g Blatklmje ; g i i g 2 3 i ' 9 . E ‘ i i
s z 1 . i 2 3 . 1 E 5 ‘s é a f ? l i i i l i i i
E 2 ‘ i i a ' i g g i z r a V; 3 2 i i i
i E i Blueiine , "i ' 'E 'i E m 3 ' §.ﬁ ‘ g ’ §.r i i i.i
i 5 i ‘ l i z i i i
g g i .i
g l i i
5 g g ii . i ‘ ‘ E i E i l 0 20 40 60 80 100 i 20 I40 I 60 Quantity x 000 (d) What price will generate the n‘iaximum revenueP $10 . What quantity will be sold at this price? 100 , 000 . (6) At this quantity, what is marginal revenue? 0 . At this quantity,
what is the price elasticity of demand? —1 . Will the stadium be full?
Yes. (f) A series of winning seasons caused the demand curve for football
tickets to shift upward. The new demand function is q(p) = 300,000 — 10,0001). What is the new inverse demand function? Z '—
q/10,000. NAME W 199 (9) Write an expression for marginal rovcnue as a function of output. M’R(q) : "m (1/5, . Use red ink to draw the new demand function and use black ink to draw the new marginal revenue function. (h) Ignoring stadium capacity, what price would generate maximum revenue? $15 . What quantity would be sold at this price? 150,000. (2') As you noticed above, the quantity that would maximize total revenue
given the new higher demand curve is greater than the capacity of the
stadium. Clever though the athletic director is, he cannot sell seats he
hasn’t got. He notices that his marginal revenue is positive for any number
of seats that he sells up to the capacity of the stadium. Therefore, in order to maximize his revenue, he should sell 100 , 000 tickets at a price
of $20 . (31') When he does this, his marginal revenue from soiling an extra seat is 10 . The elasticity of demand for tickets at this price quantity
combination is E = —2 . 15.10 (0) The athletic director discussed in the last problem is consid—
ering the extra revenue he would gain from three proposals to expand the
size of the football stadium. Recall that the demand function he is now
facing is given by q{p) = 300, 000 —— 10, 00039. (a) How much could the athletic director increase the total revenue per
game from ticket sales if he added 1,000 new seats to the stadium’s capac— ity and adjusted the ticket price to maximize his revenue? 9 , 900 . (b) How much could he increase the revenue per game by adding 50,000 new seats? $250 , 000 . 00,000 new seats? (Hint: The athletic director still wants to maximize revenue.) $250 , 000 . (c) A zealous alumnus offers to build large a stadium as the athletic
director wouk‘l like and donate it to the university. There is only one hitch.
The athletic director must price his tickets so as to keep the stadium full.
If the athletic director wants to maximize his revenue from ticket sales, how large a stadium should he choose? 150 , 000 seats . 200 MARKET DEMAND (Ch. 1 5 ) Chapter 16 NAME Equilibrium Introduction. Supply and demand problems are bread and butter for
economists. In the problems below, you will typically want to solve for
equilibrium prices and quantities by writing an equation that sets supply
equal to demand. Where the price received by suppliers is the same as the
price paid by demanders, one writes supply and demand as functions of
the same price variable, p, and solves for the price that equalizes supply
and demand. But if, as happens with taxes and subsidies, suppliers face
different prices from demanders, it is a good idea to denote these two
prices by separate variables, ps and pd, Then one can solve for equiiibrium
by solving a system of two equations in the two unknowns p5 and pd. The
two equations are the equation that sets supply equal to demand and
the equation that relates the price paid by demanders to the not price
received by suppliers. Example: The demand function for commodity a: is g = 1,000 — 10m,
where pd is the price paid by consumers. The supply function for :12 is
(1 = 1.00 + 20103, where p, is the price received by suppliers. For each unit
sold, the government collects a tax equal to half of the price paid by com
sumers. Let us ﬁnd the equilibrium prices and quantities. In equilibrium,
supply must equal demand, so that 1, 000 —~ 10m : 100 + 2031,. Since the
government collects a tax equal to half of the price paid by consumers,
it must be that the sellers only get half of the price paid by consumers,
so it must be that 335,. = pig/2. Now we have two equations in the two
unknowns, p3 and pd. Substitute the expression pd/Q for p, in the first
equation, and you have 1,000 — mm = 100 + 10m. Solve this equation
to ﬁnd pd Z 45. Then 1),, = 22.5 and q x 550. 16.1 (0) The demand for yak butter is given by 120 M 4m and the
supply is 2333 m— 30, where pd is the price paid by demanders and gas is
the price received by suppiiers, measured in dollars per hundred pounds.
Quantities demanded and supplied are measured in hundredpound units. (a) On the axes below, draw the demand curve (with blue ink) and the
supply curve (with red ink) for yak butter. q 202 EQUlLIBRiUM (Ch. 16) Price 80 4o p 92E 2 Res “nei. _,
Pi 5 i i . ‘ ..
4‘2" ql a I I
o 20 4o 60 so 100 I20 Yak butter (1)) Write down the equation that you wouid solve to ﬁnd the equilibrium price. SOlVG _ 4p 2 2p — . (c) What is the equilibrium price of yak butter? $25 . What is the equilibrium quantity? 20 . Locate the equilibrium price and quantity
on the graph, and label them p1 and q1. (d) A terrible drought strikes the central Ohio steppes, traditional home
land of the yaks. The supply scheduie shifts to 2393‘  60. The demand
schedule remains as before. Draw the new supply schedule. Write down
the equation that you would solve to ﬁnd the new equilibrium price of yak butter. 120 — 4p 2 2p — . (c) The new equililniun’l price is 30 and the quantity is 0 .
Locate the new equilibrium price and quantity on the graph and label
them pg and Q‘g. (f) The government decides to relieve stricken yak butter consumers and
producers by paying a subsidy of $5 per hundred pounds of yak butter
to producers. If pd, is the price paid by demanders for yak butter, what is the total amount received by producers for each unit they produce?
pd  5 . When the price paid by consumers is pd, how much yak butter is produced? 2190; — 50 . NAM E __..__._. . W 20 3 (9) Write down an equation that can be solved for the equiiibriun'i price paid by consumers, given the subsidy program. 2230; — I — 419d . What are the equilibrium price paid by consumers and the equiiibrium quantity of yak butter now? pd : /6 ,
q 3170/3 m 50 2 20/3. (h) Suppose the government had paid the subsidy to consumers rather
than prochicers. What would be the equilibrium net price paid by con suniers? /6 . The equilibrium quantity would be /3 . 16.2 (0) Here are the supply and demand equations for throstles, where
p is the price in dollars:
Dip) = 40 — 29 8(19) 2 10 + 19. On the axes beiow, draw the demand and supply curves for throstles,
using biue ink. 0 IO 20 3O 40
Throstles
(a) The equilibrium price of throstles is 15 and the equilibrium quantity is 25 .
(1)) Suppose that the government decides to restrict the industry to selling
only 20 throstles. At what price would 20 throsties he demanded? 20 . r How many throstles wouid suppliers supply at that price? 30 . At what price would the suppliers suppiy only 20 units? 33 10 . 204 EQUILIBRIUM (Ch. 16) (c) The government wants to make sure that only 20 throstles are bought,
but it doesn’t want the ﬁrms in the industry to receive more than the
minimum price that it would take to have them supply 20 throsties. One
way to do this is for the government to issue 20 ration coupons. Then
in order to buy a throstle, a consumer would need to present a ration
coupon along with the necessary amount of money to pay for the good.
If the ration coupons were freely bought and sold on the open market, what would be the equilibrium price of these coupons? 35 10 .
(a!) On the graph above, shade in the area that represents the deadweight loss from restricting the supply of throstles to 20. How much is this exm
pressed in dollars? (Hint: What is the formula for the area of a triangle?) $25. 16.3 (0) The demand curve for ski lessons is given by D(pD) = 100—223;;
and the supply curve is given by S = 3313. (a) What is the equilibrium price? $20 . What is the equilibrium
quantity? 60 . (b) A tax of $10 per ski lesson is imposed on consumers. Write an equation
that relates the price paid by demanders to the price received by suppliers. pp 2 195 l . Write an equation that states that supply equals demand.  2191) = BPS . (c) Solve these two equations for the two unknowns PS and pp. With the $10 tax, the equilibrium price pp paid by consumers would be $26 per lesson. The total number of lessons given would be 4:8 . (d) A senator from a mountainous state suggests that although ski lesson
consumers are rich and deserve to be taxed, ski instructors are poor and
deserve a subsidy. He proposes a $6 subsidy on production while main—
taining the $10 tax on consumption of ski lessons. Would this policy have
any different effects for suppliers or for den‘lanciiers than a tax of $4 per lesson? NO . 16.4 (0) The demand curve for salted codﬁsh is D(P) : 200 — SP and
the supply curve 5 = 0?. NAME M 205 (a) 011 the graph below, use blue ink to draw the demand curve and the supply curve. The equilibrium market price is $20 and the equilibrium quantity sold is 100 . Price
40 “nonsuwwmmw .M.u...“...tWWW................._........an.“uwwuqu.umMmummwwmvwwmmmwmwwwwﬁé 30 ., IO a
i 0 50 '00 l50 200 Quantity of codfish (b). A quantity tax of $2 per unit sold is placed on salted codﬁsh. Use red 3
ink to draw the new supply curve, where the price 011 the vertical axis remains the price per unit paid by demaudcrs. The new equilibrium price i paid by the demanders will be $2 1 and the new price received by the i suppliers will be $ 19 . The equilibrium quantity sold will be 95 . (c) The deadweight loss due to this tax will be 5 = 2 X . On
your graph, shade in the area that represents the deadweight loss. 16.5 (G) The demand function for merino ewes is D{P) = 100/1"i and
the supply function is 8(1)) = P. (a) What is the equilibrium price? $ 10 . strawmpvm wmrwnvaswmmw Ammu iywwwnweu {
i‘
i
t
l;
L
y
>.
t 206 EQUILEBRIUM (Ch. 16) (b) What is the equilibrium quantity? 10 . (c) An ad valorem tax of 300% is imposed on merino ewes so that the
price paid by demanders is four times the price received by suppliers.
What is the equilibrium price paid by the demanders for merino ewes new? $20 . W’ hat is the equilibrium price received by the suppliers for merino ewes? $5 . W hat is the equilibrium quantity? 5 . 16.6 (0) Schrecklich and LaMerde are twojustiﬁably obscure nineteenth
century in'ipressionist painters. The world’s total stock of paintings by
Schrecklich is 100, and the world’s stock of paintings by LaMerde is 150.
T he two painters are regarded by connoisseurs as being very similar in
style. Therefore the demand for either painter’s work depends both on its
own price and the price of the other painter’s work. The demand function
for Schrecklichs is 133(1)) = 200—4133 WQPL, and the demand function for
Lahrlerdes is DEM?) = 200 m 3131, — P3, where PS and PL are respectively
the price in dollars of a Schrecldich painting and a LaMerde painting. ((1) Write down two simultaneous equations that state the equilibrium
condition that the demand for each painter’s work equals suppiy. The equations are 200 — 4P3 w— ZPL = 100 and
200 m 3PL — Pg : 150. ( b ) Solving these two equations, one ﬁnds that the equiiibrium price of Schrecklichs is and the equilibrium price of lilaMerdes is 10 . {c} On the diagram below, draw a line that represents all combinations of
prices for Schrecklichs and LaMerdes such that the supply of Schrecklichs
equals the demand for Schrecldichs. Draw a second line that represents
those price combinations at which the demand for LaMerdes equals the
supply of LaMerdes. Label the unique price combination at which both
markets clear with the letter E. NAME % 207 PI
40 30 20 I0 (d) A fire in a bowling alley in Haintramck, Michigan, destroyed one of
the world’s largest collections of works by Schrecklich. The fire destroyed
a total oi 10 Schrecklichs. After the fire, the equilibrium price of Schredrw lichs was 23 and the equilibrium price of LaMerdes was 9 . (c) On the diagram you drew above, use red ink to thaw a line that shows
the locus of price combinations at which the demand for Sclirecidichs
equals the supply of Schrecidichs after the fire. On your diagram, label
the new equilibrium combination of prices E" . 16.7 (0) The price elasticity of demand for oatmeal is constant and
equal to —1. When the price of oatmeal is $10 per unit, the total amount
demanded is 6,000 units. (a) \Nrite an equation for the demand function. (I = 60, /p .
Graph this demand function below with blue ink. (Hint: If the demand
curve has a constant price elasticity equai to c, then 13(3)) 2 up“ for some
constant a. You have to use the data of the problem to solve for the
constants c and c that apply in this particular case.) *7/ .. I ”' .. ‘i 208 EQUILIBRIUM (Ch. lo) Price 20 I5 i0 0 2 4 6 8 I0 l2 Quantity (thousands) (b) If the supply is perfectly inelastic at 5,000 units, what is the equilib rium price? $ 12 . Show the supply curve on your graph and label the
equilibrium with an E. (c) Suppose that the demand curve shifts outward by 10%. Write down the new equation for the demand function. q E 66, / p . Sup
pose that the suppiy curve remains vertical but shifts to the right by 5%. Solve for the new equilibrium price and quantity 5, . d By what percentage a)proxirnately did the equilibrium )rice rise?
1 . 1 It rose by about 5 percent . Use red ink to draw the
new demand curve and the new supply curve on your graph. {6) Suppose that in the above problem the demand curve shifts outward
by 23% and the supply curve shifts right by 31%. By approximately what percentage will the equilibrium price rise? By about (33 *~ 3/) percent. 16.8 (0) An economic historian* reports that econometric studies in
dicate for the pro—Civil War period, 1820~1860, the price elasticity of
demand for cotton from the American South was approximately ——1. Due
to the rapid expansion of the British textile industry, the demand curve
for Ameri ran cotton is estimated to have shifted outward by about 5%
per year during this entire period. * Gavin Wright, The Political Economy of the Cotton South, W. W.
Norton, 1978. NAME 209 (a) If during this period, cotton production in the United States grew by
3% per year, what (approxin'rately) must be the rate of change of the price ofcottou during this period? It would rise by about 2% a year. (3)) Assuming a constant price elasticity of W1, and assuming that when
the price is $20, the quantity is aiso 20, graph the demand curve for cotton. What is the total revenue when the price is $20? . What is the total revenue when the price is $10? . Price of cotton
40 30 m rmmwi I0 0 I 0 20 30 40
Quantity of cotton (6) If the change in the quantity of cotton supplied by the United States is
to be interpreted as a movement along an upwardsloping long—run supply
curve, what would the elasticity of supply have to be? (Hint: From 1820
to 1860 quantity rose by about 3% per year and price rose by 2 %
per year. [See your earlier answer] If the quantity change is a nicwmncnt
along the longrun supply curve, then the long—run price elasticity 1111th be what?) 1 . 5 0/0 . (d) The American Civil: War, beginning in 180], had a devastating oil'ect
on cotton production in the South. Production fell by about 50% and
remained at that level throughout the war. What would you predict would be the effect on the price of cotton? It would double if demand didn’t change. 210 EQUItiBRiUM (Ch. 16) (e) What wouid be the oiled; on total revenue of cotton farmers in the Smdh? Since the demand has elasticity of M1, the revenue would stay the same. ( f ) The expansion of the British textile industry ended in the 1860s,
and for the remainder of the nineteenth century, the demand curve for
American cotton remained approximately unchanged. By about 1900,
the South approximately regained its prewar output level. What do you think happened to cotton prices then? They would recover to their old levels. 16.9 (0) The number of bottles of chardonnay demanded per year is
$1,000,000  60,0001”, where P is the price per bottle (in US. dollars).
The number of bottles supplied is 40,0001”. (0;) What is the equilibrium price? $10 . What is the equilibrium
quantity? . (6) Suppose that the government introduces a new tax such that the wine
maker must pay a tax of $5 per bottle for every bottle that he produces. What is the new equilibrium price paid by consumers? $ 12 . What is
the new price received by suppliers? $7 . What is the new equilibrium
quantity? 280 , 000 . 16.10 (0) The inverse demand function for bananas is Pd, 2 18 * 3Qd
and the inverse supply function is R, 3 6+ Q3, wi‘iere prices are measured
in cents. (a) If there are no taxes or subsidies, what is the equilibrium quantity?
3 . What is the equilibrium market price? 9 cents . (b) If a subsidy of 2 cents per pound is paid to banana growers, then
in equilibrium it still must be that the quantity demanded equais the
quantity supplied, but now the price received by sellers is 2 cents higher
than the price paid by consumers. \Vhat is the new equilibrium quantity? 3 . 5 . What is the new equilibrium price received by suppliers? 9 . 5
cents . What is the new equiiibriuin price paid by demanders? 7 . 5 cents. NAME m 211 (c) Express the change in price as a percentage of the original price. 0 . .
‘16 . 66/, . If the CI‘OSSwGldStECity of demand between bananas and
apples is +5, what will happen to the quantity of apples demanded as a
consequence of the banana subsidy, if the price of apples stays constant? (State your answer in terms of percentage change.) ‘8 . 3300 . 16.11 (1) King Kanuta rules a small tropical isiand, Nutting Atoll,
whose primary crop is coconuts. If the price of coconuts is P, then King
Kanuta’s subjects will den'iand EXP) :2 1, 200 — iOOP coconuts per week for their own use. The number of coconuts that will be supplied per week
by the island’s coconut growers is 8(1)) = 100P. (a) rthe equilibrium price of coconuts will be 6 and the equilib rium quantity suppiied will be 600 . ((1) One day, King Kanuta decided to tax his subjects in order to collect
coconuts for the Royal Larder. rThe king required that every subject
who consumed a coconut would have to pay a coconut to the king as a
tax. Thus, if a subject wanted 5 coconuts for himseif, he would have
to purchase 10 coconuts and give 5 to the king. When the price that
is received by the sellers is pg, how much does it cost one of the king’s subjects to get an extra coconut for himseif? 2195 . (c) When the price paid to suppiiers is 333, how many coconuts wili the
king‘s subjects demand for their own consun'iption? (Hint: Express p13 in terms of pg and substitute into the demand function.) Since pL)x:2pS, they consume 1,200——200pg. (d) Since the king consumes a coconut for every coconut consumed by
the subjects, the total amount demanded by the king and his subjects is
twice the amount demanded by the subjects. Therefore, when the price
received by suppliers is pg, the total number of coconuts demanded per week by Kanuta and his subjects is 2, w" . (e) Solve for the equilibrium value of 103 24/ 5 , the equilibrium total
number of coconuts produced 480 , and the equilibrium total number of coconuts consumed by Kanuta’s subjects. 240 . 212 EQUlLlBRlUM (Ch. 16) ( f ) King Kanuta’s subjects resented paying the extra coconuts to the
king, and whispers of revolution spread through the palace. Worried by
the hostile atmosphere, the king changed the coconut tax. Now, the
shopkeepers who sold the coconuts would be responsible for paying the
tax. For every coconut sold to a consumer, the shopkeeper would have to pay one coconut to the king. This plan resulted in /2 = coconuts being sold to the consumers. The shopkeepers got / 5 per
coconut after paying their tax to the king, and the consumers paid a price of /5 per coconut.g 16.12 (1) On August 29, 2005, Hurricane Katrina caused severe damage
to oil installations in the Gulf of Mexico. Although this damage could
eventually be repaired, it resulted in a substantial reduction in the short
run supply of gasoline in the United States. In many areas, retail gasoline
prices quickly rose by about 30% to an average of $3.06 per gallon. Georgia governor Sonny Perdue suspended his state’s 7.5 centsa
gallon gas tax and 4% sales tax on gasoline purchases until Oct. 1. Gov—
ernor Perdue explained that, “I believe it is absolutely wrong for the state
to reap a tax windfall in this time of urgency and tragedy.” Lawmakers
in several other states were considering similar actions. Let us apply supply and demand analysis to this problem. Before
the hurricane, the United States consumed about 180 million gallons of
gasoline per day, of which about 30 million gallons came from the Gulf
of Mexico. In the short run, the supply of gasoline is extremely inelastic
and is limited by reﬁnery and transport capacity. Let us assume that the
daily short run supply of gasoline was perfectly inelastic at 180 million
gallons before the storm and perfectly inelastic at 150 million gallons after
the storm. Suppose that the demand function, measured in millions of
gallons per day, is given by Q = 240 — 301’ where P is the dollar price,
including tax, that consumers pay for gasoline. (at) W hat was the market equilibrium price for gasoline before the hurri cane? $2 . 00 After the hurricane? $3 . 00 (b) Suppose that both before and after the hurricane, a government tax of
10 cents is charged for every gallon of gasoline sold in the United States.
How much money would suppliers receive per gallon of gasoline before the hurricane? $ 1 . 90 After the hurricane? $2 . 90 (c) Suppose that after the hurricane, the federal government removed the
gas tax. What would then be the equilibrium price paid by consumers?
$3 . 00 How much money would suppliers receive per gallon of gasoline? $3 . 00 How much revenue would the government lose per day by 213 NAMEW removing the tax? $ 1 5 mi 11 ion What is the net effect of removing
the tax on gasoline prices? There iS no ef f ect . Who are the
gainers and who are the losers from removing the tax? Gasoline
suppliers gain and government loses $15
million per day. (all Suppose that after the hurricane, the tencent tax removed in some
states but not in others. The states where the tax is removed constitute
just half of the demand in the United States. Thus the demand schedule
in each half of the country is Q = 120 —— 1513 where P is the price paid by
consumers in that part of the country. Let P* be the equilibrium price
for consumers in the part of the country where the tax is removed. In
equilibrium, suppliers must receive the same price per gallon in all parts
of the country. Therefore the equilibrium price for consumers in states
that keep the tax must be $10" + $0.10. In equilibrium it must be that the
total amount of gasoline demanded in the two parts of the country equals
the total supply. Write an equation for total demand as a function of P [3:120 — 1513* + 120 _ + . Set demand equal to supply and solve for the price paid by consumers in the states
that remove the tax $2 . 95 and for the price paid by consumers
in states that do not remove the tax. 13* = How much
money do suppliers receive per gallon of gasoline sold in every state?
$2 . 95 How does the tax removal affect daily gasoline consumption in
each group of states? Increases consumption by“ . 75
million gallons in states that remove tax
and decreases consumption by .75 million
gallons in states that do not. ( e ) If half of the states remove the gasoline tax, as described above, some
groups will be better off and some worse off than they would be if the tax were left in place. Describe the gains or losses for each of the following
groups. Consumers in the states that remove the tax Gain from 5 cent price reduction. Consumers in other states Lose from 5 cent price 214 EQUlLlBRlUM (Ch. 16) increase. Gmmmemmwmm Gain from 5 cent increase in
revenue per gallon. Governments of the states that remove the tax LOSS $15
million per day in revenue. Governments of states that do not remove the tax Because
consumption falls by 0.75 million gallons
per day, they lose $75,000 per day. 16.13 (2) Citizens of Zephyronia consume only two goods and do not
trade with the outside world. There is a ﬁxed supply of labor in Zephy—
ronia, which can be applied to production of either good. Good 1 is
produced at constant returns to scale, using one unit of labor for each
unit of output. Good 1 generates no externalities. Good 2 is also pro—
duced at constant returns to scale, with one unit of labor needed for each
unit of output, but every unit of Good 2 that is produced also generates
one unit of poilution. Both industries are perfectly competitive. Zepher—
nia has 1,000 citizens, and each citizen of this economy supplies 10 units
of labor and each has a quasilinear utility function
i 1
(1021.362, s) = 231+ 4552 ‘ 5333 _ 10003‘ where 231 and 2:2 are her consumptions of goods 1 and 2, and where s is
the total amount of pollution generated by the production of good 2. The
currency of Zephyronia is known as the Pull. We will treat labor as the
numeratrc so that each consumer has an income of 10 Puffs. (a) If there are no controls on pollution, then in competitive equilibrium,
the price of each good must be equal to the marginal cost of producing the good. The marginal cost of good 1 is 1 Puff and the marginal cost of good 2 is 1 Puff . Suppose that individuals assume that
the effect of their own purchases on pollution levels is negiigihle. Then
each citizen chooses :01 and 9:2 to maximize utility subject to the budget constraint £131 + 332 = . Each citizen will choose to consume
3 units of :02. Total production of good 2 in Zephyronia will
be 3 , 000 units, and the total amount of pollution generated will he 3 , 000 units. Total utility of each citizen of Zephyronia will be 7,1.(7, 3, 3000) = 11.5. NAME WWW—w.— 215 (1)) Suppose that the govermnent imposes a pollution tax of 1 Puff for
each unit of pollution produced by any ﬁrm and divides the revenue col~
looted from the tax equally among all citizens. Then the marginal cost of producing a unit of good 2 will rise to 2 Puffs and since the inn dustry is competitive, the price of good 2 will be 2 Puff S . Suppose
that individuals ignore the tiny effect of their own purchases on the level
of pollution and on the amount of their own rebate. Then each citizen wili choose to consume 2 units of good 2 and the total amount
of pollution will be 2, . The total revenue of the governn'iem'. from
the pollution tax will be 2 , 000 Puffs . Each citizen will there
fore receive a rebate of 2 Puffs and hence will have a total income
of 12 including wages and rebate. Therefore each consumer will
consume 8 units of good 1, 2 units of good 2, and will experience 2 , 000 units of pollution. The utility of each consumer will now be U(8, 2, = (c) If you did the previous section correctly, you will have found that
with a pollution tax of 1 Puff per unit of pollution produced, the total
level of poliution produced will be 2,000 units. Suppose that instead of
taxing pollution, the government required that ﬁrms must have a pollution
permit for each unit of pollution produced and the govtunment issued
exactly 2,000 pollution permits. Suppose that these permits are purchased
and used by 1:)1'oducers of good 2. How many units of good 2 will be produced? 2 , 000 At what price will the total demand for good 2 equai 2,000? 2 (Hint: Each individual’s demand for good 2 is
given by the equation 0:1 = 4—p2. Market demand is the total demand of
1,000 individuals like this, so market demand is D(p2) = 4, 000 —— 1,000132.
In equilibrium, the total amount of good 2 demandcxl must equal the
supply.) (0!) (Extra Credit) Show that a tax of t = 1 Puffs per unit of pollution
is the tax rate that results in the highest total utility for consumers.
With a tax of t, the price of good 2 is
1—tt and each citizen consumes 3——t units of good 2. Total pollution is 3:21000(3 216 EQUlLIBRIUM (Ch. 16) t), government revenue is lOOOtC3——t), each
citizen receives a rebate of t(3——t). Each
consumer consumes lOstﬁﬁmwt)——(1+—ﬂ(3——ﬂ
units of good 1, 3——t units of good 2, and experiences 1000(3——t) units of smoke.
Calculating utility and simplifying, we find each has utility‘(3/2)Gi—t)ﬁt+t) which is maximized when t==1. 16.14 (2) Suppose that in Zephyronia all is as in the previous problem
except that there are alternative ways of producing good 2, some of which
produce less pollution than others. A method of production that emits 2
units of pollution per unit of output requires C(z} units of labor per unit
of output, where dz) = 2 + 22 — 2z.
(a) If there is no tax on pollution, ﬁrms will choose the technology that
minimizes labor cost per unit. To do this, they choose 2: == 1 and then the marginal cost of a unit of good 2 will be 1 Puff . In
competitive equiiibrium, the price of good 1 will be 1 Pufl and the price of good 2 will be 1 Puff . The total amount of good 2 demanded in Zephyronia wiil he 3 , 000 units and the amount of polintion generated will be 3 , 000 units . (b) Suppose that producers must pay a tax of 1 Puﬁ per unit of pollution
generated. Again they will choose a technology that minimizes their per
unit cost. When they take the tax into account, their production costs
per unit will be c(z)—tzm2+22w2z+zx2+zgz
where z is the amount of poilution generated per unit of output. What
level of z will they choose? z = 1 /2 With this choice, what is the unit
cost for producers of good 2 including the tax? 7/4 Puffs What is the erpiililn'iurn price of good 2? Puffs At this price, how NAME—W 217 much of good 2 will be produced? 2 , 250 units What is the total amount of pollution that will be generated? 2, X 1/2 = 1, (0) Suppose alternatively that the governi'nent requires pollution permits
and issues 1,125 of these permits. With 1,125 permits, how many units of good 2 can be produced? 2 , 250 Find the price of good 2 at which
supply equals demand when this number of units is produced? 7/4: Puffs ﬁf’f’é'kﬁ‘ ‘u ...
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