NATIONAL UNIVERSITY OF SINGAPORE

EC2101: MICROECONOMIC ANALYSIS I

Semester 1, AY 2014/2015

MIDTERM EXAMINATION SOLUTION

2 October 2014

Time allowed: 70 minutes

INSTRUCTIONS TO CANDIDATES 1. Write down your name here: . 2. Write down your name, matric number, and tutorial number on the answer book. 3. Write down your answers for ALL questions (including MCQs) in the answer book. 4. Write down your answers for ALL MCQs in CAPITAL letters on the SAME page of

the answer book. 5. You are NOT allowed to write in pencils. 6. This examination paper comprises four (4) printed pages, including this page. 7. This examination comprises two (2) parts. Part I contains eight (8) MCQs. Part II

contains two (2) structured questions. 8. This is a CLOSED book exam. 9. You may use a non-programmable calculator, but no cell phone or laptop is allowed. 10. The total mark for this paper is 100.

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I. Multiple Choice Questions (5 marks each) Choose the BEST answer

1. The graph below shows two budget lines. Which of the following scenarios is consistent with the change in budget line from BL1 to BL2 ?

A. Price of x and price of y both drop while income remains the same. B. Price of x and income both drop while price of y remains the same. C. Price of y rises while income and price of x remain the same. D. Price of x drops while income and price of y remain the same.

2. A firm’s production function is Q =min(L3, K2) . Which of the following statements is

true? A. If the firm uses 3 units of labor and 2 units of capital, it produces 1 unit of

output. B. If the firm uses 2 units of labor and 3 units of capital, it produces 1 unit of output. C. Labor and capital are perfect substitutes. D. All of the above.

3. Suppose a firm’s initial production function is Q = KL+ K where L ≥1 and K ≥1 .

Which production below does NOT represent a technological progress when compared with the initial production function?

A. Q = 3(KL+ K ) . B. Q = (KL+ K )2 . C. Q = KL+ L . D. Q = KL+ L+ K .

4. A consumer’s indifference curves over two goods, tea and ice cream, are upward sloping. Then it is possible that

A. The consumer likes both tea and ice cream. B. The consumer likes ice cream but hates tea. C. The consumer likes ice cream but neither likes nor dislikes tea. D. The consumer hates both tea and ice cream.

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5. A consumer buys two goods, rice and housing. At the initial optimal basket, the

consumer buys both goods. When the price of rice decreases while the price of housing and the consumer’s income remain constant, the consumption of rice increases by 4 units. If rice is an inferior good, regarding the substitution effect (SE) and income effect (IE) with respect to rice, which of the following is true?

A. 0 ≤ SE ≤ 4 and 0 ≤ IE ≤ 4 . B. −4 ≤ SE ≤ 0 and −4 ≤ IE ≤ 0 . C. SE > 4 and IE < 0 . D. SE < 0 and IE > 4 .

6. Suppose a consumer’s preference satisfies the three assumptions -- completeness,

transitivity and more is better. The consumer has an income of $18. When the price of x is $2 and the price of y is $1, the consumer’s optimal choice is 6 units of x and 6 units of y. When the price of x becomes $1 and the price of y becomes $2, assuming income does not change, the optimal choice CANNOT be A. 10 units of x and 4 units of y. B. 8 units of x and 5 units of y. C. 6 units of x and 6 units of y. D. 4 units of x and 7 units of y. All the 4 baskets are on the budget line given the new prices. Basket D is under the budget line given the initial prices. Therefore, the initial optimal basket, 6 units of x and 6 units of y, is strictly preferred to basket D, and the initial optimal basket is still affordable under the new prices. Thus basket D cannot be the optimal choice given the new prices as the consumer can get higher utility by consuming the initial optimal basket.

7. A consumer likes both food (F) and clothing (C). Assume MRSF,C is diminishing as F increases and C decreases along the same indifference curve, thus her indifference curves are smooth and convex. Further assume that her indifference curves do not intersect the axes. The price of food is PF > 0 , the price of clothing is PC > 2 and her income is I > 0 . Initially her optimal basket is 4 units of food and 2 units of clothing. When the price of food increases by $1 per unit and the price of clothing decreases by $2 per unit, assuming her income remains constant, compared to the initial optimal basket, which of the following statements is true regarding her new optimal basket?

A. The consumer buys more clothing and her utility is higher. B. The consumer buys more clothing but it is unclear how her utility will change. C. The consumer still buys the initial optimal basket and her utility is unchanged. D. It is unclear how her optimal basket and her utility will change.

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8. A consumer buys two goods, x and y. When the price of x changes while the price of y and the consumer’s income remain constant, regarding compensating variation (CV) and equivalent variation (EV), which of the following statements is true?

A. If the substitution effect (with respect to x) is 0, CV must be 0. B. If the substitution effect (with respect to x) is 0, EV must be 0. C. CV and EV cannot both be 0. D. None of the above.

II. Structured Questions

1. (28 marks) A consumer has utility function U(x, y) = x +3y . The price of x is $2, the price of y is $4, and the consumer’s income is $24. (x and y do not have to be integers.) a) (6 marks) What is the optimal basket?

Since MUx

Px=12<MUy

Py=34

, the consumer only buys y and his optimal basket is x=0,

y=24/4=6.

b) (8 marks) Suppose the consumer receives a cash subsidy of $8, what is the new

optimal basket? At the new optimal basket, what is the level of utility?

The consumer still only buys y, and his optimal basket is x=0, y=(24+8)/4=8. At this optimal basket, the utility for the consumer is 3*8=24.

c) (7 marks) Suppose instead of the cash subsidy, the consumer receives a per unit

discount on every unit of x purchased (effectively, the price of x is lower). To be as well off as with the cash subsidy in part b), how much should the discount be? Briefly explain. With the discount, how much y does the consumer buy? Why?

Since there is no change in income and the price of y, if the discount is such that only consuming y is still an optimal basket, he can buy 6 units of y and his utility is only 18. Thus, to get a utility of 24, the discount should be such that the consumer only buy x and x=24. To afford 24 units of x, the price of x must be 24/24=$1. Thus the discount should be 2-1=$1. With the discount, effectively, the price of x is $1, the consumer

only buys x since MUx

Px=11>MUy

Py=34

, and the optimal basket is x=24 and y=0.

The graph below shows the optimal basket with the discount. The two dotted green lines are the budget lines in part a) and part b). The red line is the budget line with $1 discount on x. With the discount, the optimal basket is on the horizontal axis, at x=24 and y=0.

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d) (7 marks) Suppose the cash subsidy in part b) is not available. In addition, instead of receiving a discount on every unit of x, the price of the first four units of x is still $2 and the consumer receives a $1.2 per unit discount on each additional unit of x purchased beyond the first four units. (For example, if the consumer buys 5 units of x, the price for the first four units is $2 and the consumer receives a $1.2 discount for the fifth unit.) What is the optimal basket now? Briefly explain. Compared to part b), is the consumer better off, as well off, or worse off?

With the $1.2 discount, when x<=4, the budget line is the same as the budget line in part a), with a slope of -0.5. When x>4, the budget line becomes flatter as the price of x is effectively $2-$1.2=$0.8. The slope of this part of the budget line is -0.2. If the consumer spends all the money on x, the consumer can buy 24 units of x since the first 4 units costs 4*$2=$8, with the remaining $24-$8=$16, the consumer can buy $16/$0.8=20 units of x. Thus the horizontal intercept of the budget line is x=24. See graph below. The red line is the budget line with the discount. It is clear from the graph that x=24, y=0 is the optimal basket. The consumer is as well off as in part b) since his utility is still 24. More rigorously, since the budget line has two parts, we need to know which part of the budget line does the optimal basket lie on. If the optimal basket lies on the part of

the budget line where x<=4, the consumer will only buy y as MUx

Px=12<MUy

Py=34

and the optimal basket is x=0 and y=6. The utility is 18. If the optimal basket lies on the part of the budget line where x>4, the consumer should only buy x as MUx

Px=10.8

>MUy

Py=34

and the optimal basket is x=24 and y=0. The utility is 24.

Since 24>18, the optimal basket is x=24 and y=0.

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2. (32 marks) A consumer has utility function U(x, y) = (x + 5)y . (x and y do not have to be integers.)

a) (11 marks) Suppose the price of x is $1 and the consumer’s income is $35.

Calculate the income and substitution effects (with respect to y) when the price of y decreases from $4 to $2. (Round all your calculations to two decimal places.)

The initial optimal basket

MUx = y and MUy = x +5 . The tangency condition is yx +5

=PxPy=14

, the budget line

is x+4y=35. Solving the two equations together, we have x=15 and y=5.

The new optimal basket

The tangency condition is yx +5

=PxPy=12

, the budget line is x+2y=35. Solving the two

equations together, we have y=10.

The intermediate basket

At the initial optimal basket, the consumer gets a utility of (15+5)*5=100. Thus the new basket must satisfy (x + 5)y =100 , and is tangent to a budget line with the same

slope as the new budget line. Thus the tangency condition is yx +5

=PxPy=12

. Solving

these two equations, we have y=7.07. The substitution effect is 7.07-5=2.07. The income effect is 10-7.07=2.93.

b) (14 marks) Let the price of x be Px > 0 , the price of y be Py > 0 , and the

consumer’s income be I > 0 . Derive the demand functions for x and y respectively. When Px increases while Py and I remain constant, will the consumer buy less x? Briefly explain.

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If there is an interior solution, then the tangency condition is yx +5

=PxPy

, the budget

line is Pxx + Py y = I , solving for the two equations together, the demand function for x

is x =I −5Px2Px

, and the demand function for y is y =I +5Px2Py

.

Note that when 5Px>I, there is no interior solution because the quantity of x will be negative. In this case, we have a corner solution where the consumer only buys y.

Therefore, the demand function for x is x =

I −5Px2Px

, Px ≤I5

0, Px >I5

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The demand function for y is y =

I +5Px2Py

, Px ≤I5

IPy, Px >

I5

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$$

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$$

If 5Px<=I, we have interior solution and the demand of x is x =I −5Px2Px

.

Differentiating the demand of x with respect to the price of x, we have ∂x∂Px

= −I2Px

2< 0 . Alternatively, the demand for x can be written as

x =I −5Px2Px

=I2Px

− 2.5 . Thus as the price of x increases, the consumer indeed buys

less x. If 5Px>I, the consumer does not buy any x. Thus as the price of x increases, the consumption of x is unchanged. If initially the price of x satisfies 5Px<=I, but after the price increase, the price of x satisfies 5Px>I, the consumer also buy less x (from x>0 to x=0).

c) (7 marks) Draw the Engel curves of x and y respectively. Label your graph clearly.

When I<5Px, the demand for x is 0. When I>=5Px, the demand for x is x =I −5Px2Px

.

Rewriting the demand, we have I = 2Pxx +5Px . Thus the Engel curve, with x on the horizontal axis and I on the vertical axis, is vertical when I<5Px. And when I>=5Px, the Engel curve for x is a straight line with a slope of 2Px.

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When I<5Px, the demand for y is y = IPy

. Or, I = Py y . When I>=5Px, the demand for

y is y =I +5Px2Py

. Or I = 2Py y −5Px . Therefore, the Engel curve for y has a slope of Py

when I<5Px, and a slope of 2Py when I>=5Px.