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Economics 11 - Fall 2020

Final Exam, Version 1

PART 1: SHORT MULTIPLE CHOICE QUESTIONS (3.5 points each)

1. Maddalena loves cooking Spoons of Nutella (x) with Bread (y), so that her utility function is given by U (x, y) = 2 [min{2x, y}]1/3, then the expenditure function must be:

(a) E(px , py , u) = u + py u

(b) E(px , py , u) = u + py u

(d) E(px , py , u) = u + u (e) E(px , py , u) = u1/3 + u1/3

2. Francesca has an indirect utility function as V (px , py , I) = - , where I is her income, px represents the price of good x and py the price of good y . Unfortunately, the prices of good x and good y increased. The new prices are: p = 3px and p = 2py . Which should be her new income (I\ ) in order to guarantee the same utility as before the change in prices?

(a) I\ = I + 18py(3)px(2)

(b) I\ = I + 36

(d) I\ = I + 36p(p)

(e) I\ = 36I + 18pypx(3) - CORRECT ANSWER

3. Valeria has the following expenditure function: E(px , py , u) = 2U(pxpy ) , then his Marshallian demands for good x and y are:

(a) gx (px , py , I) = and gy (px , py , I) =

(b) gx (px , py , I) = and gy (px , py , I) =

(d) gx (px , py , I) = and gy (px , py , I) = - CORRECT ANSWER (e) gx (px , py , I) = and gy (px , py , I) =

4. Mario loves pizza (good x ) only. His utility for pizza is given by U(x) = 8x - x2 . A pizza costs $5 (px ) and Mario has $50 (income) in his wallet. How many pizzas will Mario eat? Which level of utility will Mario reach?

(a) Mario will consume 10 pizzas for a utility of 18

(b) Mario will consume 10 pizzas for a utility of 0

(d) Mario will consume 4 pizzas for a utility of 16 - CORRECT ANSWER (e) Mario will consume 4 pizzas for a utility of 20

5. Anna has preferences over good x and y: u(x, y) = 2x1/2y1/2 . She seeks to minimize her expenditure while reaching a utility level u. However she is not able to freely choose the amount of good y, that is she must consume y = y¯ . What is her compensated demand (hicksian demand) of the good x, x(px , py , u, y¯)?

(a) x(px , py , u, y¯) = u ╱ 、1/2

(b) x(px , py , u, y¯) = u ╱ 、1/2

(d) x(px , py , u, y¯) = u2 (e) x(px , py , u, y¯) = u ╱ 、1/2

6. Ciccio has preferences over goods x and y . We know that if his income increases by 1%, Ciccio will consume 0.4% more of good x and 1.6% more of good y. What share of his income does Ciccio spend in x (sx ) and in y (sy )?

(a) sx = 0.4 and sy = 0.6

(b) sx = 0.3 and sy = 0.7

(d) sx = 0.7 and sy = 0.3 (e) sx = 0.6 and sy = 0.4

7. Silvia has the following utility function U (x, y) = x2/3y4/3 + (min{x, y})2 . Her utility function is:

(a) homogeneous of degree 0

(b) homogeneous of degree 1

(d) homogeneous of degree 3 (e) not homogeneous

8. Andrea only consumes two goods, X and Y. We know that X is a Giﬀen good. Consider the following statements:

I. Y is a Normal good

II. Y is an Inferior good

III. Y is a gross substitute for X (the derivative of the marshallian for x with respect to the price of y is positive)

Which of the previous statements is TRUE?

(a) only III

(b) only II

(d) I and III - CORRECT ANSWER (e) II and III

9. It the utility function is given by U (x, y) = 2x + 3y and = 1, the indirect utility is

given by:

(a)

(b) - CORRECT ANSWER

(c)

(d)

10. Billo consumes only two goods X and Y. We know that X is not a Giﬀen good and it is also a inferior good. We also know that the Marshallian demand for X is homogeneous of degree 0. Consider the following statements:

I. as py increases, the Marshallian demand of X must increase

II. as py increases, the Marshallian demand of X must decrease

III. as px increases, the Hicksian demand of X must increase

Which of the previous statements is TRUE?

(a) only I - CORRECT ANSWER

(b) only II

(d) I and II (e) II and III

11. If the market demand function is QD = 20 - 3P and the market supply function is QS = 2 + 3P , the equilibrium price and quantity in the market are:

(a) P* = 4 and Q* = 8.

(b) P* = 1 and Q* = 17.

(d) P* = 3 and Q* = 11. - CORRECT ANSWER (e) P* = 2 and Q* = 14.

12. Consider the following production function F (K, L) = (min{L, 5K}) 3 . Given the prices of inputs K and L, which of the following statement is TRUE?

(a) Average cost equals marginal cost at all levels of production.

(b) Average cost exceeds marginal cost if q < 10

(d) Marginal cost is three times larger than average cost at all levels of production. - CORRECT ANSWER

(e) Average cost is three times larger than marginal cost at all levels of production.

13. Consider the production function F (K, L) = K L , which of the following statements is TRUE?

(a) The function exhibits increasing returns to scale and the marginal productivity of

capital is decreasing in capital. - CORRECT ANSWER

(b) The function exhibits increasing returns to scale and the marginal productivity of

capital is increasing in capital.

(d) The function exhibits constant returns to scale and the marginal productivity of capital is decreasing in capital.

14. Consider a cost function C(q, w, v) where v is the price of capital and w is the price of labor. Which of the following must be TRUE about C(q, w, v)

(a) The cost function is homogeneous is degree zero in the input prices w and v (b) The cost function is decreasing in the quantity produced q

(d) The cost function is convex in input prices w and v

(e) The cost function is homogeneous is degree one in input prices w and v - COR-

RECT ANSWER

15. Suppose the production function for good q is given by q = (3Kα + Lα )1/α where K and L are capital and labor inputs and 0 < α < 1. Consider the following statements about this function:

I. The function exhibits constant returns to scale.

II. The marginal rate of technological substitution, RTS = / 、1-α

III. The elasticity of substitution of the function is constant.

Which of these statements is TRUE?

(a) Only I

(b) I and II

(d) I, II, and III - CORRECT ANSWER (e) None of the options are correct

16. John wants to open a bakery shop. His cost function will be C (q) = 50 + q2 . He would be able to charge a constant price p for each unit of product. What is the lowest price such that he would earn non-negative proﬁts and enter the market?

(a) 25

(b) 20

(d) 10 - CORRECT ANSWER (e) 5

17. There is only one ﬁrm operating in the market for corn: Corn World. Corn World is

deciding how much corn to sell next year. It knows that the market demand for corn β

C(Q) = Q + 10. Compute the quantity that Corn World should produce to maximize proﬁts and the price at which it will be able to sell the produced corn:

(a) Q = and P =

(b) Q = and P = 2 - CORRECT ANSWER

(d) Q = and P =

18. Luke owns a price-taking ﬁrm and seeks to maximize proﬁts. He operates a production function that depends only on labor: f (L) = 10 ln(L). Which quantity q* does Luke produce, given the price of output p and the wage rate w?

(a) q* = 10 ln( )

(b) q* =

(d) q* = 10 ln( ) - CORRECT ANSWER

19. Suppose that a production function of a ﬁrm depends only on labor and it is given by q = f (l) = 2l . The price of labor w is equal to 2. The ﬁrm is a price taker and the price for the good it produces p is equal to 1. What is the proﬁt of the ﬁrm if the ﬁrm maximizes proﬁts?

(a) π = 2

(b) π = -1/3

(d) π = 1/2 - CORRECT ANSWER (e) π = 3

20. Eve runs a sandwich shop with the following cost function: C(q) = 49 + 8q + q2 , this cost function applies to Eve’s long run and short run decisions. Eve is a price-taker in the market for her sandwiches, and the market price is 20 dollars. What quantity q will her optimally choose to produce in the short-run and long-run?

(a) Short-run: q = 10 , long-run: q = 10

(b) Short-run: shutdown, long-run: exit

(d) Short-run: shutdown,long-run: q = 6

PART 2: ESSAY QUESTIONS

Essay Question 1 (30 Points)

Peter has utility over kettle corn (x) and diet Pepsi (y). Peter’s preferences are summarized

by the following expenditure function: E(px , py , u) = py ┌u + 1 - ln ╱ 、] . Where px is

the price of x and py is the price of y .

1. Find Peter’s Hicksian demands for goods x and y: (5 Points)

(a) hx (px , py , u) = and hy (px , py , u) =

(b) hx (px , py , u) = and hy (px , py , u) = u - ln ╱ 、- CORRECT ANSWER (c) hx (px , py , u) = u - ln ╱ 、and hy (px , py , u) =

(d) hx (px , py , u) = 3 and hy (px , py , u) = 3u - 3 ln ╱ 3、 (e) hx (px , py , u) = 3 and hy (px , py , u) = u - ln ╱ 、

Solution:

hx = =

hy = = ┌u + 1 - ln ╱ 、┐ + py [(-1) ] hy = u - ln ╱ 、

2. Find Peter’s indirect utility function: (5 Points)

(a) V (px , py , I) = 3 - 1 + ln ╱ 、- CORRECT ANSWER

(b) V (px , py , I) = 9 - 3 + 3 ln ╱ 、

(d) V (px , py , I) = I + 1 - ln ╱ 、 (e) V (px , py , I) = 3 - 1 + ln ╱ 、

Solution:

E(px , py , u) = py ┌u + 1 - ln ╱ 、┐

I = py ┌ V (px , py , I) + 1 - ln ╱ 、┐

= ┌ V (px , py , I) + 1 - ln ╱ 、┐

V (px , py , I) = - 1 + ln ╱ 、

3. Suppose that px and py double (increase in 100%). If Peter wants to achieve the same level of utility u, by what percentage will E(px , py , u) change? By what percentage will hx (px , py , u) and hy (px , py , u) change? (5 Points)

(a) E(px , py , u) stays the same, hx (px , py , u) and hy (px , py , u) double. (b) E(px , py , u) stays the same, hx (px , py , u) and hy (px , py , u) stay the same.

(d) E(px , py , u) doubles, hx (px , py , u) and hy (px , py , u) stay the same. - CORRECT ANSWER

(e) E(px , py , u) doubles, hx (px , py , u) and hy (px , py , u) less than double.

Solution:

E(px , py , u) is HD1 in prices. The hicksian demands are HD0 in prices.

4. Peter buys diet Pepsi (y) at Trader Joe’s, because the holidays diet Pepsi is on sale with a price discount of . The price of kettle corn (x) stays the same. If Peter wants to maintain the same level of utility he had before the discount, what is the percentage change in the consumption of x? (Think of the price change as a small change in price) (5 Points)

(a) -2/3%

(b) 2/3%

(d) 1/3% (e) 1%

Solution:

ln(hx ) = ln ╱ 、 = + ln(py ) - ln(px )

εx(c),py = 1

d ln q = 1 * (-1/3) = -1/3

5. Assume that px = 1, py = 3 and I = 101. The government decides to tax sugar food and beverages that contain a high concentration of sugar. The tax on the price of diet Pepsi is τy = 0.15 (p = py (1 + τy )) and the tax on the price of kettle corn is τx = 0.20 (p = px (1 + τx )). What is Peter’s consumption of kettle corn (x) after the government introduces the taxes on sugary food and beverages? (5 Points)

(a) gx = 0.96 - CORRECT ANSWER

(b) gx = 1.96

(d) gx = 8.4 (e) gx = 9.6

Solution:

Note that the hicksian demand for x does not depend on u. Thus:

1 py

gx = hx =

1 3 * 1.15

gx =

6. Consider the same case as in the previous part. That is px = 1, py = 3 and I = 101. The government decides to tax sugar food and beverages that contain a high concentration of sugar. The tax on the price of diet Pepsi is τy = 0.15 (p = py (1+τy )) and the tax on the price of kettle corn is τx = 0.20 (p = px (1 + τx )). What is Peter’s consumption of diet pepsi (y) after the government introduces the taxes on sugary food and beverages? (5 Points)

(a) gy = 28.94 - CORRECT ANSWER

(b) gy = 100

(d) gy = 7.33 (e) gy = 83.42

Solution:

I = pxgx + py gy

gy = - gx

gy = -

gy = - = -

Essay Question 2 (30 Points)

Mary always has a hot beverage early in the morning. She can consume either coﬀee (c) or tea (t). Her utility function is given by the following formula: u(c, t) = 2c + 3t. She currently has $10 of income and the prices of the two beverages are: pc = $1 and pt = $2.

1. Which of the following consumption bundles maximize Mary’s utility? (5 Points)

(a) 10 units of coﬀee and 0 units of tea. - CORRECT ANSWER

(b) 0 units of coﬀee and 5 units of tea.

(d) 2 units of tea and 6 units of coﬀee.

(e) 5 units of coﬀee and 0 units of tea.

Solution:

= > = ÷ c = = 10, t = 0

2. Which of the following statements is CORRECT at the current prices and income? (5 Points)

(b) εc,pc = -2/3 εc,pt = -1/3 εc,I = 1

(d) εc,pc = -1 εc,pt = 0 εc,I = 1 - CORRECT ANSWER (e) εc,pc = -1/3 εc,pt = 1/3 εc,I = 0

Solution:

At the current consumption bundle: c = I/pc :

ln(c) = ln(I) - ln(pc )

εc,I εc,pc εc,pt

= 1

= -1

= 0

3. The government decides to introduce a $0.5 tax on the price of coﬀee. The price of tea and Mary’s income stay the same. How much money will the government collect from taxes on coﬀee T? (5 Points)

(a) T = 5

(b) T = 5/3

(d) T = 0 - CORRECT ANSWER

(e) T = 15

Solution:

= < = = ÷ c = 0, t = = 10/2 = 5

The government does not collect any tax on Mary’s consumption.

Government revenue is:

T = 0.5 × gc (p, pt , I) = 0

4. As in the previous part, the government has decided to introduce a $0.5 tax on the price of coﬀee. The price of tea and Mary’s income stay the same. What is the total eﬀect, substitution eﬀect and income eﬀect of this change in the price of coﬀee on the consumption of tea. (Think of the change in the price of coﬀee as a large change) Note: compute TE, SE and IE without dividing by the price change. (5 Points)

(a) TEt = 5, SEt = 1.67, IEt = -6.67.

(b) TEt = 5, SEt = 6.67, IEt = -1.67. - CORRECT ANSWER

(d) TEt = 10, SEt = -5, IEt = 15.

(e) TEt = 0, SEt = -5, IEt = 5.

Solution:

u0 = V (pc , pt , I) = 2 × 10 = 20

TEt = gt (p, pt , I) - gt (pc , pt , I) = 5 - 0 = 5

SEt = ht (p, pt , u0 ) - ht (pc , pt , u0 ) = - 0 = 20/3

IEt = TEt - SEt = 5 - 20/3 = -5/3

5. Carol would like to help Mary to stay at the same level of utility as before the tax on the price of coﬀee. Carol is considering two alternatives. Giving Mary a monetary gift (Gm ), which increases Mary’s income. Or to buy some hot beverage for Mary (Gt for units of tea and Gc for units of coﬀee), after Mary has bought her optimal choice of beverages (Mary will therefore consume her optimal choice of coﬀee plus Gc and her optimal choice of tea plus Gt ). Which of the following gift combinations will help Mary to stay at the same level of utility as before the tax on coﬀee? (5 Points)

I. Gm = ; Gt = 0 ; Gc = 0

II. Gm = 0 ; Gt = ; Gc =

III. Gm = 0 ; Gt = 1 ; Gc = 1

(a) only I.

(b) I and II

(d) only III

(e) I , II and III

Solution:

u0 = V (pc , pt , I) = 2 × 10 = 20

Monetary gift:2 × = 2 × → I\ = I = 10 → Gm = I\ - I = 10/3

Non-monetary gifts:20 = 2 × (0 + Gc ) + 3 × (5 + Gt )

To check with monetary gift is the optimal one, plug in the options in the previous expression.

6. The government hires you to compute the sales tax on coﬀee that would allow the government to collect $1 in tax revenues from Mary. What is the sales tax on coﬀee τc that allows the government to achieve its tax revenue goal of $1? (After the sales tax, p = pc (1 + τc ).) (5 Points)