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Introductory Microeconomics (ES10001) Coursework 1

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University of Bath

DEPARTMENT OF ECONOMICS COURSEWORK TEST 1: SUGGESSTED SOLUTIONS

First Year

INTRODUCTORY MICROECONOMICS (ES10001)

_____________________________________

3RD NOVEMBER 2017, 17:30 – 18.45 (75 minutes) _____________________________________

ANSWER ALL QUESTIONS

The coursework test paper comprises four pages and is divided into two sections: Section A (True / False) contains five questions at 6 marks per question (total 30 marks); Section B (Multiple Choice) comprises seven questions at 10 marks per question (total 70 marks). The examination paper thus comprises 100 marks. Note: An incorrect answer will result in a ten per cent reduction in the mark available for the question; thus an incorrect answer to a true / false question will result in a reduction of 0.6 mark and an incorrect answer to a multiple choice question will result in a reduction of 1 mark.

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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


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Section A: True / False (6 marks per question)

1. If there is no endowment effect, then the demand curve for a Giffen good is always upward sloping:

A. TRUE B. FALSE

Solution: Assume the consumer consumes a bundle of goods x = x1,x2 ,...,xn( ) and assume

that good 1 is Giffen. Define !x1 = m !p1 as the maximum quantity of good 1 that the consumer can consume at price !p1 given his budget income m. Any p1 > !p1 implies x1 < !x1 in order for the budget constraint to be satisfied. Intuitively, if demand increases with price, and if the price rises to such an extent that the consumer is devoting all of his budget income to the consumption of that good, then any further increases in price must result in a decrease in consumption of good 1 such that the demand curve for good 1 becomes downward sloping (i.e. backward bending).

2. In the country Janlandia the whole population eat one meal a day, choosing either chocolate pie or vegetable curry. Everyone has identical income and tastes. Chocolate pie is the cheaper meal and is an inferior good for everyone. The Janlandian Government is considering imposing an ‘unhealthy food’ tax on chocolate to reduce chocolate consumption. But an expert adviser argues that chocolate consumption will definitely increase if its price goes up. The expert is correct:

A. TRUE B. FALSE

Solution: Chocolate pie is an inferior good so an increase in its price will lead to a fall in the demand for chocolate pie via the substitution effect but an increase in the demand for chocolate pie via the income effect. Thus the overall effect of the demand for chocolate pie is ambiguous. If the income effect dominates the substitution effect, then the overall demand for chocolate pie will increase. But this is not certain and so the ‘expert’ is incorrect.

3. If in a two good world, goods x and y are complements for one another at all possible prices

px , py( ) , then the own price elasticity of demand, Ε = − ∂q ∂p( ) p q( ) > 0 , for

each good is greater than one at all possible prices:

A. TRUE B. FALSE

Solution: Consider good x and define the own price elasticity of good x as:

Ε x = − ∂x

∂px

⋅px

x

Further, define the budget constraint as:


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pxx + py y = m

Now, hold py and m constant and consider a change in px :

∂m∂px

=∂ pxx + py y( )

∂px

= 0

⇒

1⋅ x + px ⋅∂x∂px

+ py ⋅∂y∂px

= 0

Thus:

∂y∂px

= − 1py

x + ∂x∂px

⋅ px

⎛

⎝⎜⎞

⎠⎟

⇒

∂y∂px

= − xpy

1+ ∂x∂px

⋅px

x⎛

⎝⎜⎞

⎠⎟

⇒∂y∂px

= xpy

Ε x −1( )

If the two goods are complements for one another then ∂y ∂px( ) < 0 which (assuming

x > 0 and py > 0 ) implies that Ε x <1 . Intuitively, if Ε x <1 then a one per cent increase in

the price of good x will lead to a less than one per cent decrease in the demand for good x such that expenditure on good x increases. If the price of good y and money income remain constant, then spending on good y must fall, which implies that the demand for good y falls.

4. Ben spends all of his income on green eggs and ham. Green eggs are an inferior good for him. If his income increased by 25 per cent and prices did not change, then his expenditure on ham must have increased by more than 25 per cent of his original income:

A. TRUE B. FALSE

Solution: Write Ben’s original budget constraint as:

pxx + pyy = m

where x denotes green eggs and y denotes ham. His new budget constraint may be written as:

px ′x + py ′y = ′m


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where ′x < x by inferiority. Thus:

py ′y − y( ) = ′m −m( )− px ′x − x( )⇒pyΔy = Δm − pxΔx⇒pyΔy = 0.25m − pxΔx⇒pyΔy > 0.25m

since Δx = ′x − x < 0 . Intuitively, since green eggs are an inferior good then it must be the case that Ben reduces his consumption, and thus expenditure, on green eggs as a result of the 25 per cent increase in money income. To maintain his budget constraint, Ben must have increased his expenditure on ham by more than 25 per cent.

5. 2500 people live in the village of Barton-under-Wold. In 2017 the village will celebrate its 1000th anniversary. As part of its celebrations the Parish Council is considering whether to stage a fireworks display, which would cost £19,000. The Council has surveyed of the residents, asking them the maximum they would be willing to pay to attend the display. Their responses were all different, varying uniformly from plus £20 to minus £5 - some residents (e.g. pet owners) responded with a negative price since they would pay for the display not to take place.

From their ‘maximum willingness to pay’ responses the Council has constructed the following demand function:

qd p( ) = 2000−100 p

where qd p( ) denotes the number of residents who would be willing to pay at least price

p𝑝! to attend the display. The Council decides that it will run the display and recoup the cost by increasing local taxation if and only if this increases the net well-being of its citizens. The Council is non-corrupt 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 display.

A. TRUE B. FALSE

Solution: See Figure 1:


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Figure 1: Villagers’ Demand for Firework Display

The normal demand function implies:

qd p( ) = 2000−100 p

⇒

pd q( ) = 20− q100

The residents would therefore obtain a total consumer surplus from the display of:

Area ABC – Area CDE ≡ CS = 1

220× 2000( )− 5×500( )⎡⎣ ⎤⎦ = £18750 < £19000

Thus, the Council should not go ahead with the display.

Section B: Multiple Choice (10 marks per question)

6. The weekly demand and supply functions for wine purchased in UK supermarkets and other retail outlets are:

qd p( ) = 50−5p

qs p( ) = −10+10 p

p

0 2000 2500 q

20

-5

A

E

B C D


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where p denotes price per bottle and q denotes millions of bottles per week. The government imposes a sales tax of £3 a bottle. Ignoring costs (e.g. health) arising from alcohol consumption, the tax reduces society’s net welfare by:

A. £75 million a week B. £50 million a week C. £40 million a week D. £15 million a week E. None of the above

Solution: The demand and supply functions imply:

qd p( ) = 50−5p

⇒

pd q( ) = 10− q5

And:

qs p( ) = −10+10 p

⇒

ps q( ) = 1+ q10

Original equilibrium:

pd q∗( ) = 10− q∗

5= 1+ q∗

10= ps q∗( )

⇒3

10q∗ = 9

⇒q∗ = 30

And:

pd 30( ) = 10− 305

= 4 = 1+ 3010

= ps 30( )⇒p∗ = 4

After the tax:


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pd qt∗( ) = 10−

qt∗

5= 1+ 3( ) + qt

∗

10= pt

s qt∗( )

⇒3

10qt∗ = 6

⇒qt∗ = 20

And:

pd 20( ) = 10− 205

= 6 = 1+ 3( ) + 2010

= pts 20( )

⇒pt∗ = 6

See Figure 2:

Figure 2

Net welfare is defined as CS + PS + T, where T = tq. It is apparent that this is reduced by the triangle ABC where:

Area ABC =

12⋅3⋅10 = 15

p

q 0

6

1

4

ps

pd

pts

10

20 30 50

3

A

B

C

t = £3


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7. There are 3 consumers (1, 2, 3) of good x. Each consumer has a unique demand function:

q1d p( ) = 10

q2d p( ) =

20 for p < 2040− p for 20 ≤ p ≤ 40

0 for p > 40

⎧

⎨⎪⎪

⎩⎪⎪

q3

d p( ) = 200−10 p for p < 200 for p ≥ 20

⎧⎨⎪

⎩⎪

The supply function for the good is:

qs p( ) = 5+10 p

Which graph illustrates the inverse total supply and inverse demand functions for good x?

A. Figure A B. Figure B C. Figure C D. Figure D E. None of the above

Figure A

Figure B

p

q 0

ps

pd

p

q 0

ps

pd


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Figure C

Figure D

Solution: At any p > 40 we have:

qd p( ) ≡ qi

d

i=1

3

∑ p( ) = q1d p( ) = 10

At any 20 ≤ p ≤ 40 we have:

qd p( ) ≡ qi

d

i=1

3

∑ p( ) = q1d p( ) + q2

d p( ) = 50− p

At any p < 20 we have:

qd p( ) = qi

d

i=1

q

∑ p( ) = q1d p( ) + q2

d p( ) + q3d p( ) = 230−10 p

Note that:

qs p( ) = 5+10 p

⇒ps q( ) = −0.5+ 0.1q

It is apparent that ps 10( ) = −0.5+ 0.1⋅10 = 0.5< 40 and p

s 30( ) = −0.5+ 0.1⋅30 = 2.5< 20 such that the equilibrium must lay on the lowest segment of the aggregate demand curve vis.

ps 117.5( ) = −0.5+ 0.1⋅117.5= 11.25 - see Figure 3:

p

q 0

ps

pd

p

q 0

ps

pd


10

Figure 3

8. Ben spends his entire budget and consumes 19 units of x and 18 units of y. The price of x is three times the price of y. His income doubles and the price of y triples, but the price 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 above

Solution: Consider first Ben’s original budget constraint:

19px +18py = m⇒57py +18py = m⇒m = 75py

since px = 3py . 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 price of good y triples:

p

q 5 10 30 117.5 230

ps

pd

40

20

0

11.25

qd p( ) = q1

d p( ) = 10

qd = q1

d + q1d = 50− p

qd = q1

d + q2d + q3

d = 230−10 p


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θ px + 54 py = 2m⇒3θ py + 54 py = 150py⇒3θ + 54 = 150⇒3θ = 96⇒θ = 32

9. Ben and Grace have the same, standard convex preferences over income and leisure and are free to chose how many hours to work each day. Ben is paid £10 an hour and chooses to work 9 hours a day. Grace is paid £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 completely 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 pay schedule as Grace C. Grace would prefer Ben’s pay schedule to her own D. Grace will work less than 9 hours a day E. None of the above

Solution: Ben’s (linear) pay schedule implies a maximum income of £10*24 = £240. Grace’s (kinked) pay schedule implies a maximum income of £9*8 + £18(24-8) = £360. It is apparent that the two schedules intersect at T-9 hours since £10*9 = £90 = £9*8 + £18*1. Since Ben and Grace have identical preferences and thus identical indifference curves, then Ben would be better off facing the same pay schedule as Grace – See Figure 4:


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Figure 4

10. The demand for wheat in Tuckland is ppqd −= 24)( . There are no imports or exports of wheat. The supply of wheat in Tuckland depends on the weather in the growing season. In a good year the supply is q

s( p) = 6+ 2 p and the Government imposes a sales tax of £1.50 per unit of wheat. In a bad year the supply is q

s( p) = −6+ 2 p and the Government provides a subsidy to suppliers of £1.50 per unit of wheat. The total tax collected by the Government in a good year exceeds the total subsidy paid by the Government in a bad year by:

A. £3 B. £5 C. £7 D. £9 E. None of the above

Solution: First, note the inverse demand curve:

qd ( p) = 24− p ⇔⇒pd (q) = 24− q

Now, consider the inverse supply curve in a good year. Without the tax, we have:

m

L 0

E1

E0

240

T-9 T-8 T=24

I0

90

I1

360

£10.p.h.

£18.p.h.

£9.p.h.

72


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qs( p) = 6+ 2 p⇒ps(q) = −3+ 0.5q

With the tax, we have:

qs( p) = 6+ 2 p⇒ps(q) = −3+ 0.5q

pts(q) = −3+ t( ) + 0.5q

⇒pt

s(q) = −1.5+ 0.5q

Thus, in equilibrium:

pd (qt∗) = 24− qt

∗ = −1.5+ 0.5qt∗ = pt

s(qt∗)

⇒48− 2qt

∗ = −3+ qt∗

⇒3qt

∗ = 51⇒qt∗ = 17.5

Thus tax revenue collected is T = tqt∗ = £1.50×17 = £25.50 . Now, in a bad year, we have:

qs( p) = −6+ 2 p⇒ps(q) = 3+ 0.5q

With the subsidy, we have:

pss(q) = 3− s( ) + 0.5q

⇒ps

s(q) = 1.5+ 0.5q

Thus, in equilibrium:


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pd (qs∗) = 24− qs

∗ = 1.5+ 0.5qs∗ = ps

s(qs∗)

⇒48− 2qt

∗ = 3+ qt∗

⇒3qs

∗ = 45⇒qs∗ = 15

Thus subsidy provided is S = sqs∗ = £1.50×15= £22.50 .

11. There are only two goods, good 1 and good 2. When prices are 21 =p and 22 =p , Grace spends her entire income of £12 on quantities x1 = 3 and x2 = 3 . Her own price arc elasticity of demand for good 1 is

13 over the range of prices p1 = 2 to 31 =p . The

price of good 1 increases from 21 =p to ′p1 = 3. In the new equilibrium Grace purchases what quantity of good 2?

A 6 7 B. 21 8 C. 24 7 D. 33 2 E. None of the above

Solution: Initial situation:

p1x1 + p2x2 = M

⇒2× 3( ) + 2× 3( ) = 12

Now, for good 1:


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Ε11 = −Δx1

Δp1

⋅p 1

x1

= −′x1 − x1( )′p1 − p1( ) ⋅

p1+ ′p1( ) 2x1 + ′x1( ) 2

= 13

⇒

Ε11 = −′x1 − 3( )

3− 2( ) ⋅2+ 3( ) 23+ ′x1( ) 2

= 13

⇒

Ε11 =3− ′x1

1⋅ 2.5

3+ ′x1( ) 2= 1

3

⇒15 3− ′x1( ) = 3+ ′x

⇒45−15 ′x1 = 3+ ′x

⇒

′x1 =4216

= 218

Thus, in the new equilibrium:

′p1 ′x1 + p2 ′x2 = M

⇒3× 21

8( ) + 2× ′x2( ) = 12

⇒63+16 ′x = 96⇒′x = 33 16

12. Grace insists on consuming 4 times as much of y as she consumes of x (so she always has y = 4x). She will consume these goods in no other ratio. The price of x is 3 times the price of y. Grace has an endowment of 20 x’s and 45 y’s which she can trade at the going prices. She has no other source of income. What is Grace’s gross demand for x? (7 points)

A. 105 B. 65 C. 15 D. 12 E. None of the above

Solution: First, ascertain Grace’s budget line. Her initial endowment is

x, y( ) = 20,45( ) and

px = 3py . Thus, Grace could sell all of her 20 units of good x and acquire an additional 60

units of good y, or alternatively, sell all of her 45 units of good y and acquire an additional 15 units of good x. Thus, her budget line is defined by the equation:


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y −105− 105

35⎛⎝⎜

⎞⎠⎟

x = 105− 3x

And we know that her preferences imply that she will only consume where:

y = 4x

Thus, her utility maximising bundle is where:

y∗ = 105− 3 14

y∗⎛⎝⎜

⎞⎠⎟

⇒

y∗ 1+ 34

⎛⎝⎜

⎞⎠⎟= 105

⇒

y∗ = 47⋅105

⇒y∗ = 60

Such that:

x∗ = y∗

4= 60

4= 15

See Figure 5:


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Figure 5

y

x 0

y = 4x

45

11.5 15 20 35

80

105 y = 105 - 3x

I* 60

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