Ch14-16 - 180 RISKY ASSETS (Ch. 13) Chapter 14 NAME...

Info iconThis preview shows pages 1–19. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 16
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 18
Background image of page 19
This is the end of the preview. Sign up to access the rest of the document.

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—to-pay 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 “null-Ill “IIIIIII mull-Ill 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 well-off 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 well-off 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 revenue-maximizing 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 11-13($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 yo-yos 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 yo-yos? 1/3 . (b) What is the price elasticity of demand for yo-yos? *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 hundred-pound 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 upward-sloping 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 long-run 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 cents-a 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 ten-cent 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 quasi-linear 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 govt-unment 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 lOs-tfifimwt)——(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+zg-z 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 ...
View Full Document

Page1 / 19

Ch14-16 - 180 RISKY ASSETS (Ch. 13) Chapter 14 NAME...

This preview shows document pages 1 - 19. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online