Economics 3030: Intermediate Microeconomics - Canvas

Economics 3030: Intermediate Microeconomics Mid-Term 1

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Economics 3030: Intermediate Microeconomics Mid-Term Examination 1 The examination contains 11 questions and is out of 107 points. Questions are divided into three sections: Section A (True / False) contains three questions at 4 points per question (Total 12 points). Section B (Multiple Choice) contains five questions at 7 points per question (Total 35 points). Section C (Short Answer Questions) contains three questions with points varying per question (Total 60 points).

NAME:

Section A: True / False (4 points per question, total 12 points)

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 and assume that

good 1 is Giffen. Define as the maximum quantity of good 1 that the consumer can consume at price given his budget income m. Any implies 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. Ben consumes only apples and bananas. His endowment is 5 units of apples and 10 units of bananas. Both goods are normal goods for Ben. At current prices, Ben is a net seller of apples. If the price of apples rises and the price of bananas stays the same, his demand for apples must decrease. (4 points)

A. TRUE

B. FALSE

Solution: If the consumer is a net seller (i.e. ) of a normal good, then the sign of the total effect is ambiguous since it will depend on the relative magnitudes of the (positive) combined income effect and the (negative) substitution effect:

x = x1,x2 ,...,xn( ) !x1 = m !p1

!p1 p1 > !p1 x1 < !x1

ω1 − x1 > 0

Δx1Δp1?!

= Δx1s

Δp1−!

+ Δx1m

Δm+!

ω1 − x1( )+

"#$ %$


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Thus, a good can be Giffen for an individual without being inferior. Conversely, an inferior good may be Giffen for a net demander, but cannot be Giffen for a net supplier

3. If an individual has standard convex preferences, then it is possible when there are only two goods, x and y, for y to be both a complement to x and an inferior good. (4 points)

A. TRUE

B. FALSE

Solution: See Figure 2 below in which we have modelled a rise in the price of good x, causing the budget line to pivot from AA to AB and moving the consumer from equilibrium point E0 to E1. Moving the new budget constraint out parallel until it is just tangent to the consumer’s original indifference curve allows us to decompose the movement from E0 to E1 into a substitution effect (E0 to E2) and an income effect (E2 to E1).

Figure 1

It is apparent that the substitution effect causes the consumer to move away from good x to good y such that point E2 must lay above and to the left of point E0. For good y to be a complement to good x, we require its consumption to decline with the increase in the price of x, implying that point E1 should lay below E0. But if good y were a inferior good, then E2 would have to lay below E1, good y must, in this case be a normal good. To summarize, when there are only two goods, x

y

x 0

E1

E2

E0

A

A B

I1

I2

y0

y1

y2


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and y, for y to be both a complement to x and an inferior good in terms of good y then it must be the case that:

• Substitution effect implies: • Complementary between good x and good y would imply: • Inferiority of good y would imply:

Which is impossible! Alternatively, you could prove this from the cross-price Slutsky equation:

Complementarity implies . The cross-price substitution effect is always positive such

that . If good y is inferior, then . Thus

That is, good y cannot be both inferior and a complement for good x. (Note: I would not have expected this as we did not cover the cross-price Slutsky equation in class).

Section B: Multiple Choice (7 points per question; total 28 points). N.B. Circle your answer to the question.

4. The following individuals have the following representations of their preferences.

Mick:

Keith:

Ronnie:

Charlie:

Bill:

Who has the same preferences as Ronnie?

a) Everyone

b) Everyone but Bill

c) Everyone but Keith

d) Everyone but Mick

e) Everyone but Keith, Mick, and Bill

f) Everyone but Keith and Bill

g) Everyone but Keith and Mick

h) Everyone but Mick and Bill

y2 > y0

y0 > y1

y1 > y2

ΔyΔpx

= Δys

Δpx

− Δym

Δmy

Δy Δpx < 0

Δys Δpx > 0 Δy Δm < 0

ΔyΔpx

+!

= Δys

Δpx

+!

− Δym

Δm−!

y

U (x, y) = x + y

U (x, y) = x + y( ) x + y( )2

U (x, y) = 10 x + y( ) −100

U (x, y) = ln10 x + y( ) U (x, y) = −10 x + y( ) +100


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(7 points)

Solution: Again, think of which are positive monotonic transformations (i.e. transformations which do not affect the optimal choices) of the individuals.

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

The supply function for the good is:

Which graph illustrates the inverse total supply and inverse demand functions for good x? (7 points)

A. Figure A

B. Figure B

C. Figure C

D. Figure D

E. None of the above

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

⎧⎨⎪

⎩⎪

qs p( ) = 5+10 p


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Figure A

Figure B

Figure C

Figure D

Solution: At any we have:

At any we have:

At any we have:

p

q 0

ps

pd

p

q 0

ps

pd

p

q 0

ps

pd

p

q 0

ps

pd

p > 40

qd p( ) ≡ qi

d

i=1

3

∑ p( ) = q1d p( ) = 10

20 ≤ p ≤ 40

qd p( ) ≡ qi

d

i=1

3

∑ p( ) = q1d p( ) + q2

d p( ) = 50− p

p < 20


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Note that:

It is apparent that and such that the equilibrium must lay on the lowest segment of the aggregate demand curve vis.

- see Figure 3:

Figure 2

(7 points)

6. 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.

qd p( ) = qi

d

i=1

q

∑ p( ) = q1d p( ) + q2

d p( ) + q3d p( ) = 230−10 p

qs p( ) = 5+10 p

⇒ps q( ) = −0.5+ 0.1q

ps 10( ) = −0.5+ 0.1⋅10 = 0.5< 40 p

s 30( ) = −0.5+ 0.1⋅30 = 2.5< 20

ps 117.5( ) = −0.5+ 0.1⋅117.5= 11.25

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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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 and

. 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:

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

Thus, her utility maximising bundle is where:

Such that:

See Figure 5:

x, y( ) = 20,45( )

px = 3py

y −105− 105

35⎛⎝⎜

⎞⎠⎟

x = 105− 3x

y = 4x

y∗ = 105− 3 14

y∗⎛⎝⎜

⎞⎠⎟

⇒

y∗ 1+ 34

⎛⎝⎜

⎞⎠⎟= 105

⇒

y∗ = 47⋅105

⇒y∗ = 60

x∗ = y∗

4= 60

4= 15


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Figure 3

7. Ben’s utility function over his income is given by . He drives to work every day and spends a lot of money in parking meters. Unlike his sister, Ben has always been a bit devious and on many days the thought of cheating and not paying for parking crosses his mind. He knows, however, that there is a one-in-two chance of being caught on any given day if he cheats, and that the cost of a parking fine is $36. If his daily income is $100, then what is the maximum amount that he will be willing to pay for one day of parking?

A. $36 B. $19 C. $18 D. $20 E. None of the above

Solution: Ben will be indifferent between paying x for parking or facing the ‘cheating lottery’ when the following equation holds:

y

x 0

y = 4x

45

11.5 15 20 35

80

105 y = 105 - 3x

I* 60

u y( ) = y


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8. Eileen and Bert have been married for nearly sixty years, but are always arguing.. Eileen’s inverse demand curve for cups of coffee is . Bert’s inverse demand curve for

cups of coffee is . What is their joint demand for coffee when the price per cup of coffee is $4 and $6 per cup respectively?

A. 5 cups and 6 cups B. 6 cups and 5 cups C. 5 cups and 6 cups D. 5 cups and 2 cups E. Some other amounts

Solution:

And:

Thus when only Eileen buys coffee and when both Eileen and Bert buy coffee. Thus, their joint (i.e. aggregate) normal and inverse demand functions are:

u 100− x( ) = 0.5u 100− 36( ) + 0.5u 100( )⇒

100− x( ) = 0.5 64 + 0.5 100

⇒

100− x( ) = 4+5

⇒x = 100− 92

⇒x = 19

pEd = 10− 2q

pBd = 5− 1

2 q

pEd = 10− 2q

⇒qE

d = 5− 12 p

pBd = 5− 1

2 q⇒qB

d = 10− 2 p

p ≥ 5 p < 5


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The results can be illustrated graphically though Eileen and Bert’s individual demand functions, see Figure 7, or though their joint (i.e. aggregate demand) function, see Figure 8:

Figure 4: Individual Demand Curves

qd =5− 1

2 p if p ≥ 5

15− 52 p otherwise

⎧⎨⎪

⎩⎪

⇔

pd =10− 2q if q ≤ 5

2

6− 25 q otherwise

⎧⎨⎪

⎩⎪

0 2 3 10/3 5 10 q

5

10

pEd = 10− 2q

⇔qE

d = 5− 12 p

pBd = 5− 1

2 q⇔qB

d = 10− 2 p

6

10/3

4

p


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Figure 5: Joint (i.e. Aggregate) Demand Curve

Thus:

And:

p

0 2 5/2 5 15 q

5

10

qd =5− 1

2 p if p ≥ 5

15− 52 p otherwise

⎧⎨⎪

⎩⎪

⇔

pd =10− 2q if q ≤ 5

2

6− 25 q otherwise

⎧⎨⎪

⎩⎪ 6

4

p = 4⇔qd 4 =( )15− 5

2 4( ) = 5

p = 6⇔qd 6( ) = 5− 1

2 6( ) = 2


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Section 3: Short Answer Questions (Points per question vary).

9. Let represent a utility maximising consumer’s preferences over goods and (with parameters , , , and ). The consumer has $100 of income that he can spend on and . A unit of costs $1 and unit of costs $12. What is the

consumer’s marginal rate of substitution of for in equilibrium, where:

(10 points)

Solution: Could you totally differentiate this utility function? Don’t answer that, because you don’t have to! We know that at the solution to the consumer’s problem is where MRS = ERS and the budget constraint is satisfied (i.e. tangency between indifference curve and budget line) such that in equilibrium the consumer’s MRS must be equal to the consumer’s ERS vis:

(10

u(x1,x2 ) =α x1βx2 + x1δ x2 x2

φ

x1 x2 α β δ φ

x1 x2 x1 x2

x2 x1 MRS21( )

MRS21 =dx2dx1 du=0

= −

∂u(x1,x2 )∂x1

∂u(x1,x2 )∂x2

= −ux1ux2

ERS21 =dx2dx1 dm=0

= −p1p2

= − 112


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10. President Pollack wants to reduce unruly behaviour amongst Cornell students – especially those on Econ 3030. Assume that student utility is defined over wealth, w, by

with and . All students have an initial

endowment of wealth and they face some probability, , of being fined an amount if they indulge in unruly behaviour. Two schemes are being considered: (i) increasing the amount of the fine by ten per cent; (ii) increasing campus security so as to increase the probability of detection by ten percent. (20 points)

A. Which scheme would you advise as being the most effective in reducing rowdy behaviour? (2 points)

B. Why? (18 points).

N.B. Ignore the cost of the schemes on Cornell’s finances.

Solution: Let . Student expected wealth is:

Increasing the fine by 10 per cent implies:

Increasing the probability of detection by 10 per cent implies:

Thus such that expected wealth is identical under the two schemes. But recall how we find expected utility; it is the point on the chord connecting the two individual wealth values – See Figure 4:

u = u w( )

∂u w( ) ∂w ≡ ′u w( ) > 0

∂2u w( ) ∂w2 ≡ ′′u w( ) < 0

w0 > 0 0p >0f >

wf = w0 − f

w = p w f( ) + 1− p( )w0

⇒

w = p w0 − f( ) + 1− p( )w0

⇒w = w0 − pf

w f = w0 − pf 1.1( ) = w0 − 1.1( ) pf

wp = w0 − p 1.1( ) f = w0 − 1.1( ) pf

w f = wp


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Figure 6

Thus, increasing the fine by 10 per cent reduces the student’s minimum possible level of wealth from to . This presents a new chord AD, the point on which above expected wealth from the increase in the fine (i.e. point F) is below the point above expected wealth from the increase in detection on the original chord AB (i.e.. point E). Thus expected wealth following the increase in the fine, , is less than expected wealth following the increase in detection,

. Thus, increasing the fine is the most effective policy in reducing rowdy behaviour.

w

u(w)

0 w

pu

u(w)

p fw w=

u

w ff

fu

A

B

C

D

E

F

w0 w f

w f w ff < w f

u f

u p


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11. Ben has preferences over consumption, c, and leisure, l, defined by the following utility function: . He is endowed with T units of time that he can divide between working, h, and leisure. When he works, he receives a wage per unit of time of w. He also receives Y dollars each period regardless of whether or not he works. Assume that the price of a unit of consumption is p. (30 points total).

Solution:

A. What is Ben’s total endowment income? (2 points)

B. What is Ben’s price of leisure? (2 points)

C. Find Ben’s demand functions for consumption and leisure. (15 points)

You could set up the Ben’s constrained maximization problem and solve. Alternatively, you might remember that with CD preferences, you spend a fraction of your income on each good that is equal to the weight of that good in your utility function. Either way:

Notice that we set the constraint up analogously to the consumer demand problem:

That is, the consumer faces prices of p and w for the two goods of c and l. Thus, his expenditure on these two goods must not exceed his endowment income of . Equivalently, it implies that Ben’s expenditure on the consumption good must equal the total of his earned and unearned income.:

The first-order utility maximising conditions are:

u = c0.1l0.9

wT +Y

w

L = c0.1l0.9 + λ wT +Y − pc − wl( )

pc − wl = wT + y⇔p1x1 + p2x2 = m

wT + y

pc = w T − l( )+ y

∂L∂c

= 0.1c−0.9l0.9 − λ p = 0

∂L∂l

= 0.9c0.1l−0.1 − λw = 0


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Thus:

So:

And:

D. What is Ben’s labor supply function? (1 point)

E. If Ben’s wage increases by 5%, by how much does his consumption of leisure change (N.B. your answer should be in percentage terms)? (10 points)

Ben’s elasticity of leisure demand with respect to the wage is:

∂L∂λ

= wT + y − pc − wl = 0

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lc= p

w⇒wl = 9 pc

wT + y − pc − wl = 0⇒wT + y − pc − 9 pc = 0⇒10 pc = wT + y⇒

c = 110

wT + yp

⎛⎝⎜

⎞⎠⎟

l = 9 pc

w= 9

10wT + y

w⎛⎝⎜

⎞⎠⎟

h = T − l = T − 910

wT +Yw

⎛⎝⎜

⎞⎠⎟= wT − 9Y

w

Elw =∂l∂wwl= − 910Yw2w109

wwT +Y

⎛⎝⎜

⎞⎠⎟

⇒

Elw = − YwT +Y


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Thus, his leisure drops by:

per cent

%Δl = − Y

wT +Y*5%

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Economics 3030: Intermediate Microeconomics - Canvas