University of Bath DEPARTMENT OF ECONOMICS COURSEWORK TEST 1: SUGGESSTED SOLUTIONS

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1 University of Bath DEPARTMENT OF ECONOMICS COURSEWORK TEST 1: SUGGESSTED SOLUTIONS First Year INTRODUCTORY MICROECONOMICS (ES11) 4 TH NOVEMBER 216, 17: (75 minutes) ANSWER ALL QUESTIONS The coursework test aer comrises four ages and is divided into two sections: Section A (True / False) contains five questions at 6 marks er question (total 3 marks); Section B (Multile Choice) comrises seven questions at 1 marks er question (total 7 marks). The examination aer thus comrises 1 marks. Note: An incorrect answer will result in a ten er cent reduction in the mark available for the question; thus an incorrect answer to a true / false question will result in a reduction of.6 mark and an incorrect answer to a multile choice question will result in a reduction of 1 mark. CANDIDATES MUST NOT TURN OVER THE PAGE AND READ THE TEST PAPER UNTIL INSTRUCTED TO BY THE CHIEF INVIGILATOR PLEASE WRITE BOTH YOUR CANDIDATE NUMBER AND THE LETTER SET OUT IN THE BOTTOM RIGHT HAND CORNER OF THIS PAGE ON YOUR ANSWER SHEET ANSWER PAPERS WITHOUT EITHER WILL NOT BE MARKED THIS QUESTION PAPER MUST BE HANDED IN WITH THE SCRIPT AT THE END OF THE TEST A 1

2 Section A: True / False (6 marks er question) 1. Grace consumes two goods, x and y, and always devotes 9 er cent of her income to the former. If the rice of good y falls by 9 er cent and her income falls by 9 er cent, then she will be neither better off nor worse off as a result: A B TRUE FALSE Solution: Write Grace s original budget constraint as: x x + y y = m where: x x =.9m y y =.1m And her new budget constraint as: x x 1 + y 1 y 1 = m 1 x x y y 1 =.91m since m 1 =.91m and y 1 =.1 y. Grace s original consumtion of good y, y, would now cost: 1 y y =.1 y y =.1(.1m ) =.1m which would leave Grace with: m 1 y 1 y =.91m.1m =.9m to send on the consumtion of good x. Thus, it is aarent than Grace could afford her original bundle x, y ( ). Given, however, the fall in the rice of good y, Grace will increase her utility by increasing (decreasing) her consumtion of good y (x) until equality between MRS and ERS is restored at E 1 see Figure 1: 2

3 B y E 1 A y E I 1 I x B A x Figure 1 2. If in a two good world, goods x and y are comlements for one another at all ossible ( ), then the own rice elasticity of demand, Ε = q rices x, y each good is greater than one at all ossible rices: A. TRUE B. FALSE Solution: Consider good x and define the own rice elasticity of good x as: Ε x = x x x x Further, define the budget constraint as: x x + y y = m ( )( q) >, for Now, hold y and m constant and consider a change in x : ( ) m = x + y x y = x x 1 x + x x x + y y x = 3

4 Thus: y = 1 x + x x y x x y = x 1+ x x x y x x y x = x y ( Ε x 1) If the two goods are comlements for one another then ( y x ) <, which imlies that Ε x < 1. Intuitively, if Ε x < 1 then a one er cent increase in the rice of good x will lead to a less than one er cent decrease in the demand for good x such that exenditure on good x increases. If the rice of good y and money income remain constant, then sending on good y must fall, which imlies that the demand for good y falls eole live in the village of Barton-under-Wold. In 217 the village will celebrate its 1 th anniversary. As art of its celebrations the Parish Council is considering whether to stage a fireworks dislay, which would cost 19,. The Council has surveyed of the residents, asking them the maximum they would be willing to ay to attend the dislay. Their resonses were all different, varying uniformly from lus 2 to minus 5 - some residents (e.g. et owners) resonded with a negative rice since they would ay for the dislay not to take lace. From their maximum willingness to ay resonses the Council has constructed the following demand function: q d ( ) = 2 1 where q d ( ) the number of residents who would be willing to ay at least rice to attend the dislay. The Council decides that it will run the dislay and recou the cost by increasing local taxation if and only if this increases the net well-being of its citizens. The Council is non-corrut and is seeking only to maximise the well-being of the residents, and the residents answered the survey honestly. The Council decides to run the fireworks dislay. A. TRUE B. FALSE Solution: See Figure 2: 4

5 2 A B C D 2 25 q -5 E The normal demand function imlies: q d d ( ) = 2 1 ( q) = 2 q 1 Figure 2: Villagers Demand for Firework Dislay The residents would therefore obtain a total consumer surlus from the dislay of: Area ABC Area CDE CS = ( ) ( 5 5) Thus, the Council should not go ahead with the dislay. = 1875 < Ben sends all of his income on green eggs and ham. Green eggs are an inferior good for him. If his income increased by 25 er cent and rices did not change, then his exenditure on ham must have increased by more than 25 er cent of his original income: A. TRUE B. FALSE Solution: Write Ben s original budget constraint as: x x + y y = m 5

6 where x denotes green eggs and y denotes ham. His new budget constraint may be written as: x x + y y = m where x < x by inferiority. Thus: ( ) = ( m m) x ( x x) y y y y Δy = Δm x Δx y Δy =.25m x Δx y Δy >.25m since Δx = x x <. Intuitively, since green eggs are an inferior good then it must be the case that Ben reduces his consumtion, and thus exenditure, on green eggs as a result of the 25 er cent increase in money income. To maintain his budget constraint, Ben must have increased his exenditure on ham by more than 25 er cent. 5. Ben has standard convex references and buys from two shos, each selling a different tye of good. Visiting one of these shos requires a bus journey. If the bus fare rises, then he must increase his urchases from the other sho rovided that both goods are normal and both continue to be bought. A. TRUE B. FALSE Solution: The bus fare constitutes a fixed cost in obtaining goods from one of the shos, say y - see Figure 3. The initial budget constraint is thus ABC, where AB is the bus fare and the sloe of BC embodies the relative rices set by the two shos. When the bus fare is increased, the budget constraint becomes ADE. If both goods are normal and both continue to be bought, then Ben will reduce his consumtion of both goods. 6

7 A y B D E E 1 I 1 I C x Figure 3 Section B: Multile Choice (1 marks er question) 6. The weekly demand and suly functions for wine urchased in UK suermarkets and other retail outlets are: q d q s ( ) = 5 5 ( ) = 1 +1 where denotes rice er bottle and q denotes millions of bottles er week. The government imoses a sales tax of 3 a bottle. Ignoring costs (e.g. health) arising from alcohol consumtion, the tax reduces society s net welfare by: A. 75 million a week B. 5 million a week C. 4 million a week D. 3 million a week E. 15 million a week Solution: The demand and suly functions imly: 7

8 q d d ( ) = 5 5 ( q) = 1 q 5 And: q s s ( ) = 1 +1 ( q) = 1+ q 1 Original equilibrium: ( d q ) = 1 q 3 1 q = 9 q = 3 5 q = 1+ 1 = ( s q ) And: d ( 3) = = 4 3 = 4 = 1+ 1 = ( s 3) After the tax: d ( q t ) = 1 q t 3 1 q = 6 t q t = 2 ( ) + q t 5 = = t s ( q t ) And: 8

9 d ( 2) = 1 2 t = 6 See Figure 4: 1 5 = 6 = 1+ 3 ( ) = s 2 t ( ) t s A C B t = 3 s 1 d q Figure 4 Net welfare is defined as CS + PS + T, where T = tq. It is aarent that this is reduced by the triangle ABC where: Area ABC = = Ben sends his entire budget and consumes 19 units of x and 18 units of y. The rice of x is three times the rice of y. His income doubles and the rice of y triles, but the rice of x stays the same. If he continues to buy 18 units of y, what is the largest number of units of x that he can afford? A. 42 B. 32 C. 22 D. 12 E. None of the otions offered Solution: Consider first Ben s original budget constraint: 9

10 19 x +18 y = m 57 y +18 y = m m = 75 y since x = 3 y. Now, let θ denote the largest number of units of x that Ben can afford if he continues to buy 18 units of good y when his income doubles and the rice of good y triles: θ x + 54 y = 2m 3θ y + 54 y = 15 y 3θ + 54 = 15 3θ = 96 θ = Ben and Grace have the same, standard convex references over income and leisure and are free to chose how many hours to work each day. Ben is aid 1 an hour and chooses to work 9 hours a day. Grace is aid 9 an hour for the first 8 hours she works and 18 an hour for any time she works beyond 8 hours a day. Which of the following statements is comletely true? A. Since she has the same tastes as Ben and can earn the same income by working 9 hours a day, Grace chooses to work 9 hours a day B. Ben would be better off facing the same ay schedule as Grace C. Grace would refer Ben s ay schedule to her own D. Grace will work less than 9 hours a day E. None of the above Solution: Ben s (linear) ay schedule imlies a maximum income of 1*24 = 24. Grace s (kinked) ay schedule imlies a maximum income of 9*8 + 18(24-8) = 36. It is aarent that the two schedules intersect at T-9 hours since 1*9 = 9 = 9*8 + 18*1. Since Ben and Grace have identical references and thus identical indifference curves, then Ben would be better off facing the same ay schedule as Grace See Figure 3: 1

11 36 m E 1 24 E 9 I 1 I h. 1..h. 9..h. T-9 T-8 T=24 L Figure 3 ( ) = ( ) 3 and the suly function is ( ) = ( ) 2, where is the rice in and q is bottles of wine. The government 9. The demand function for wine is q d q s imoses a rice ceiling of 32 er bottle and, to avoid shortages, agrees to ay wine roducers a subsidy for each bottle of wine sufficient to ensure suly equals demand. How much would the subsidy er bottle have to be? A. 6 B. 7 C. 8 D. 9 E. None of the above Solution: q d ( ) = ( ) 3 q d ( ) = d ( q) = q And: 11

12 q s ( ) = ( ) 2 q s ( ) = 5+ 7 s ( q) = q Thus, original equilibrium: q ( d ) = = 5+ 7 = q s 1 = 376 = 37.6 which imlies: q ( d ) = ( ) ( ) ( ) = = 5+ 7( 37.6) = q s q = At a rice ceiling of = 32 we have: q d and: q s ( ) = ( ) = ( ) = 285 ( ) = 229 Thus, buyers demand 285 bottles at the rice celling of 32. The suly rice er bottle at 285 bottles is: s ( 285) = ( 285) = 4 Thus, wine roducers would need to be aid a subsidy of 8 er bottle see Figure 4 12

13 127 s ( q) Subsidy = 8 s sub ( q) d ( q) q 5 7 Figure 4 1. The town of Dunnyness in Scotland has 15 residents and a distillery. The only whisky the residents drink is 2 year-old whisky sulied by the distillery. The total demand from 1 of the 15 residents is given by the artial demand function: q 1 d ( ) = 25 1 d where q 1 ( ) = number of tots consumed er day by these 1, residents, and is the rice er tot in. The artial demand function for other 5 is given by: q 2 d ( ) = 5 if 5 otherwise The distillery releases tots from its store of 2 year old whisky according to the daily suly function: q s ( ) = 5 +1 Which grah below reresents the inverse total demand and inverse suly functions for the 2 year old whisky? A. Figure A B. Figure B C. Figure C D. Figure D 13

14 E. None of the above s s d q d q Figure A Figure B s s d q d q Figure C Figure D Solution: The normal demand curve for Grou 1 imlies: q 1 d ( ) = 25 1 d 1 ( q) = 25.1q This is the inverse demand curve for Grou 1. When rice falls to = 5, this grou ( ) = 2 and Grou 2 enter the market, demanding q d 2 ( 5) = 25 and ( ) = q d 1 ( 5) + q d 2 ( 5) = 45. Since ( ) = 25, it must be the case that as rice falls from = 5 to =, an additional demand q 1 d 5 rendering the inverse demand curve horizontal until q d 5 q 1 d 5 tots are demanded by Grou 1, rendering a total demand of ( ) = q d 1 ( ) + q d 2 ( ) = 5. q d 14

15 The normal suly curve imlies the inverse suly curve: q s ( ) = 5 +1 s ( ) = 5+.1q 25 s 5 d q Figure Grace consumes goods x and y. Her demand for x is given by x( x,m) =.5m 5 x. Now her income is 42, the rice of x is 3, and the rice of y is 1. If the rice of x rises to 4 and if we denote the income effect on her demand for x by DI and the substitution effect on her demand for x by DS, then: A. DI = -.2 and DS = -4.8 B. DI =.3 and DS = 4.7 C. DI = -.3 and DS = -4.7 D. DI = -4.7 and DS = -.3 E. None of the above Solution: Grace s income is originally 42 and the rice er unit of good x is 3 and the rice er unit of good y is 1. Thus, her demand for good x will be: 15

16 x( x,m) =.5m 5 x x ( 3, 42) =.5( 42) 5( 3) ( ) = = 6 x 3, 42 If the rice of good x rises to 4 er unit, then her demand at the new rice will be: ( ) =.5m 5 x x,m x 4, 42 x ( ) =.5( 42) 5( 4) ( ) = 21 2 = 1 x 4, 42 The total change in demand is thus: Δx = x x,m ( ) x( x,m) ( ) x ( 3, 42) Δx = x 4, 42 Δx = 1 6 = 5 We decomose this total change as follows: Substitution Effect Recall that the substitution effect is defined as: ( ) x( x, m) Δx s = x x, m Thus we need to calculate how much we would have to change Grace s income by to make the original consumtion of 5 units of good x a week just affordable when the rice of good x is 4 a unit. We therefore aly the formula set out in equation (3) of the Slutsky Equation handout vis: Δm = xδ x = 6 ( 4 3) = 6 Thus, we would have to increase Grace s income by 6 to hold her urchasing ower constant following the rise in the rice of good x. Thus: m = m + Δm = = 426 Note that if her money income were 426, then Grace could urchase her original bundle of goods. Her original bundle was 6 units of good x, which originally cost her 3*6 = 18, imlying that she sent = 42 on good y. With an income of 426 and a new (higher) rice of good x of 4 er unit, Grace can still urchase 6 units of good x, thereby 16

17 sending 24 on good x, and have = 42 to urchase her original quantity of good y. Grace s demand for good x when she faces the new rice of 4 er unit and has 426 money income is, however: ( ) =.5 x x, m x 4, 426 m 5 x ( ) =.5( 426) 5( 4) ( ) = = 1.3 x 4, 426 That is, even though Grace can afford the original bundle when m = $426, the increase in the rice of good x has induced a substitution effect away from good x. This substitution effect is: ( ) x 1 ( x,m) Δx s = x x, m ( ) x ( 3, 42) Δx s = x 4, 426 Δx s = = 4.7 Thus, holding her urchasing ower constant, the rise in the rice of good x has induced Grace to substitute away from good x by 4.7 units. Income Effect Recall from equation (5) in the Slutsky equation handout that the income effect is defined as: Δx n = x x, m ( ) x ( x, m ) Thus in terms of our examle, we have: Δx n = x x,m ( ) x ( x, m ) ( ) x ( 4, 426) Δx n = x 4, 42 Δx n = =.3 Thus good x is a normal good for this consumer the rise in the rice of good x has induced a negative income effect of.3 units. 12. Grace s inverse demand function for coffee is d ( q) = 5 1 2q, where q denotes the number of cus of coffee er day. Her own-rice elasticity of demand for coffee in equilibrium is Ε = q er day? ( )( q) = 2 3. How many cus of coffee does Grace consume 17

18 A. 8 B. 1/3 C. 4 D. 6 E. None of the above Solution: d ( q) = q q d ( ) = 1 2 Thus: Ε = q q = = 2 3 ( ) 2 = = 1 2 = 2 Thus: q d ( ) = 1 2 = ( ) = 6 18