PART 1: SHORT MULTIPLE CHOICE QUESTIONS (3.5 points each)
1. Gonzalo’s preferences over coffee C and tarts T can be represented by the utility function U(C, T) = min{C, 3T}. Which pair of expressions gives the corresponding Hicksian demands? (a) hC(pC, pT , u)=0 and hT (pC, pT , u) = u 3 (b) hC(pC, pT , u) = u and hT (pC, pT , u)=3u (c) hC(pC, pT , u) = u and hT (pC, pT , u) = u 3 (d) hC(pC, pT , u) = ⇣pT pC ⌘ 1 2 u and hT (pC, pT , u) = ⇣pC pT ⌘ 1 2 u
2. Suppose a consumer’s preferences over goods x and y can be represented by the utility function U(x, y) = p2x + y. Suppose the consumer currently possesses the bundle (x, y) = (5, 10). This consumer is willing to trade away this bundle for which of the following bundles? (a) (x, y) = (8, 3) (b) (x, y) = (4, 8) (c) (x, y) = (2, 15) (d) (x, y) = (7, 7) 1
3. Consider the following two utility functions: U1(x, y) = x 3 4 y 1 4 and U2(x, y) = 6 ln(x) + 2 ln(y). Of the following statements: (i) The two utility functions imply the same MRS (ii) The two utility functions lead to the same indirect utility function (iii) The two utility functions lead to the same Marshallian demands Which one(s) are true? (a) Only (i) (b) Only (i) and (ii) (c) Only (i) and (iii) (d) Only (ii) and (iii) (e) All statements are true
4. Suppose Phoebe’s preferences over books (B) and food F can be represented by the utility function U(B,F) = B2F3. Phoebe spends what percentage of her income on food? (a) 30 percent (b) 20 percent (c) 60 percent (d) 40 percent (e) 70 percent
5. Suppose that, for a certain consumer, whenever the price of x is twice the price of y (ie, px = 2py), the consumer optimally consumes twice as much y as x (ie, y⇤ = 2x⇤). Which of the following utility functions represent preferences consistent with this behavior? (a) U(x, y) = 2 ln(x) + y (b) U(x, y)=3x + y (c) U(x, y) = min{4x, 5y} (d) U(x, y) = ln(x) + 2 ln(y) (e) U(x, y) = xy 2
6. Suppose that Bob’s preferences over pizza x and burgers y can be represented by the utility function U(x, y) = x + y. Suppose that Bob minimizes expenditure in an effort to attain a utility level of 100. Suppose that initially, px = 2 and py = 1. But, then, due to a change in economic conditions, the price of burgers changes to p0 y = 3. By how much must Bob’s expenditure INCREASE in order to maintain the same utility level of 100? (WE ARE ASKING FOR THE INCREASE IN INCOME) (a) 300 (b) 100 (c) 50 (d) 200
7. Isabel is folding origami paper cranes for her etsy business. For each paper crane, she needs one piece of paper K and two hours L (the cranes are very elaborate). Which of the following describes her production function? (a) F(K, L) = min{2K, L} (b) F(K, L) = min{K, 2L} (c) F(K, L)=2K + L (d) F(K, L) = L + 2K
8. Suppose a utility function is given by u(x, y) = x↵y1↵ for 0 < ↵ < 1. Goods x and y are: (a) Net substitutes and neither gross complements nor gross substitutes (b) Net substitutes and gross substitutes (c) Net complements and gross complements (d) Gross substitutes and neither net complements nor net substitutes 3
9. Suppose we have a production funciton F(K, L) such that a small change in RTS generates a small change in the capital-labor ratio K/L. Which if the following statements is true (a) Elasticity of substitution is high and the isoquant is relatively linear (b) Elasticity of substitution is high and the isoquant is sharply curved (c) Elasticity of substitution is low and the isoquant is relatively linear (d) Elasticity of substitution is low and the isoquant is sharply curved
10. Consider the following statements: (i) If when the price of the good X rises and everything else stays the same, the income effect IEX and substitution effect SEX move in opposite directions and IEX dominates SEX if X is a Giffen good (ii) If when the price of the good A rises and everything else stays the same, the substitution effect SEB and the income effect IEB for the good B reinforce each other if B is an inferior good. (iii) If when the price of the good S falls and everything else stays the same, the income effect IES and substitution effect SES move in opposite directions if S is a normal good. Which of the previous statements is true? (a) (i) (b) (i) and (ii) (c) (ii) and (iii) (d) (i) and (iii) 4
11. Consider the following production functions: (i) F(K, L)=(K + L) 3 2 (ii) F(K, L) = K 1 2L1 3 (iii) F(K, L) = K 1 3L2 3 (iv) F(K, L) = min{3K, 2L} Which of the above production functions exhibits decreasing returns to scale? (a) (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (ii) and (iv)
12. Which of the following elasticities can be generated by a Marshallian demand function that is homogeneous of degree 0 in prices and income if good x is normal, and good x is a gross complement for y? (a) ex,px = 1.5, ex,py = 0.5, and ex,I = 1 (b) ex,px = 1, ex,py = 0.25, and ex,I = 0.75 (c) ex,px = 1, ex,py = 0.25, and ex,I = 1.25 (d) ex,px = 0.5, ex,py = 0.5, and ex,I = 0.1
13. Patrick has utility function U(x, y) = x 2 5 y 3 5 . What’s his compensated own price elasticity of demand for good y? (a) 0 (b) -0.2 (c) -0.4 (d) 0.2 (e) 0.4 5
14. If the production function is given by q(K, L) = min{2K2, L2} and the firm wants to produce quantity q0, what are the contingent demands functions for capital and labor? (a) Kc = q0 2 1 2 , Lc = q 1 2 0 (b) Kc = q 1 2 0 , Lc = q 1 2 0 (c) Kc = q 1 2 0 , Lc = ⇣q0 2 ⌘ 1 2 (d) Kc = q0 2 1 2 , Lc = q0 2 1 2 (e) None of the above
15. There is only one Subway on campus, whose total cost function is C(Q) = 25+4Q+Q2. Subway has market power. Thus, the price depends on the quantity produced by it. The demand function is Q = 140 P. What is the optimal quantity and price that maximizes Subway profit? (a) P = 140, Q = 0 (b) P = 122, Q = 18 (c) P = 114, Q = 26 (d) P = 106, Q = 34 (e) P = 100, Q = 40
16. Subway’s cost function did not change, i.e. C(Q) = 25 + 4Q + Q2. However, ASUCLA worries about students’ welfare and decides to fix the price to P = 10. Choose all correct statements from below: I Subway produces Q = 3 in the short run. II Subway incurs a loss in the short run. III Subway shuts down in the short run. (a) I, III (b) I, II (c) II, III (d) I, II, III 6
17. A factory has production function F(K, L) = (5L1 2 + 2K 1 2 )2, what’s the elasticity of substitution ? (a) 0 (b) 2 5 (c) 1 (d) 2 (e) 5 2
18. Lightning, a firm producing electric cars, has a constant returns to scale production function. To produce any quantity it always minimizes total costs. In 2022, Lightning optimally used 20 units of capital and 50 units of labor to produce 1000 units of output. In 2023, the firm decides to produce 800 units of output. In both years, the price of capital is $100 and the price of labor is $20. The total cost of producing the 800 units of output in 2023 is (a) 3000 (b) 4500 (c) 1500 (d) 2400 (e) 917
19. If a price-taking firm’s production function is given by q = 5p l, then its profit function is given by (a) 25p2 2w (b) 5p2 2w (c) 5p2 4w (d) 25p 2w (e) 25p2 4w 7
20. A factory has production function F(K, L) = K 3 4L1 4 . In the short-run, the capital is fixed at the level K¯ and the firm minimizes cost, the wage is w and the capital rental price is v. What is the marginal cost function? (a) q4 K¯ 3 (b) 4 q3 K¯ 3 (c) w q4 K¯ 3 + vK¯ (d) 4w q3 K¯ 3 (e) 4w q3 K¯ 3 + vK¯ 8 PART 2: ESSAY QUESTIONS Essay Question 1 (30 Points) Suppose that Maurice’s preferences over fish x and pasta y can be represented by the utility function U(x, y) = ⇣ x 1 2 + y 1 2 ⌘2
21. Maurice’s Marhsallian demands for the two goods are given by which of the following pairs of expressions? (5 points) (a) gx(px, py, I) = I 2px and gy(px, py, I) = I 2py (b) gx(px, py, I) = I px and gy(px, py, I)=0 (c) gx(px, py, I) = I px+py and gy(px, py, I) = I px+py (d) gx(px, py, I) = pyI px(px+py) and gy(px, py, I) = pxI py(px+py) (e) gx(px, py, I) = p2 yI p2 x+p2 y and gy(px, py, I) = p2 xI p2 x+p2 y
22. Which of the following statements is NOT true? (5 Points) (a) Maurice’s preferences over fish x and pasta y are homothetic (b) Maurice’s Marshallian demands satisfy the budget constraint (c) The share of income that Maurice spends on x does not vary with the price of y, py (d) If Maurice’s income quadrupled (and nothing else changed), his purchases of x and y would quadruple (e) If the prices of both goods tripled and Maurice’s income also tripled, he would not alter his purchases of the two goods 9
23. Maurice’s indirect utility function that corresponds to the utility function U(x, y) = ⇣ x 1 2 + y 1 2 ⌘2 is (5 points) (a) V (px, py, I) = p2 xpyI px+py (b) V (px, py, I) = q(px+py)I pxpy (c) V (px, py, I) = ln px ln py + ln I (d) V (px, py, I) = (px+py)I pxpy (e) V (px, py, I) = I px+py
24. Suppose that Maurice’s income is 120 and that the prices are px = py = 1. Given the utility function representing Maurice’s preferences U(x, y) = ⇣ x 1 2 + y 1 2 ⌘2 , what is the value of Maurice’s utility at his optimal consumption bundle? (5 points) (a) 180 (b) 240 (c) 60 (d) 4 p15 (e) 200
25. A new president is elected, who believes that carbohydrates are unhealthy. He therefore enacts a tax of 1 dollars per unit of pasta. Maurice’s optimal consumption of fish and pasta becomes (5 points) (a) x⇤ = 100 and y⇤ = 10 (b) x⇤ = 62 and y⇤ = 36 (c) x⇤ = 80 and y⇤ = 20 (d) x⇤ = 54 and y⇤ = 24 (e) x⇤ = 20 and y⇤ = 60 10
26. Maurice begs his sister to transfer him enough resources so that he can enjoy the same level of utility he enjoyed before the introduction of the tax. Katrina, the sister, replies that she can only transfer enough resources so that Maurice can regain half of the utility he has lost due to the introduction of the sales tax. Which amount S does Katrina transfer to Maurice? (5 points) (a) S = 16 (b) S = 10 (c) S = 30 (d) S = 18 (e) S = 20 11 Essay Question 2 (35 Points) Daniel produces bikes with the following production function: F(K, L) = ↵KL + L2 where ↵ > 0 and > 0.
27. Does this production function exhibit increasing, constant, or decreasing returns to scale? (4 points) (a) Increasing returns to scale (b) Constant returns to scale (c) Decreasing returns to scale (d) Increasing returns to scale, then decreasing returns to scale
28. Find the marginal product of labor and marginal product of capital. (4 points) (a) MPL = ↵K, MPK = ↵L (b) MPL = ↵K + 2L, MPK = ↵L (c) MPL = ↵K + L, MPK = ↵L (d) MPL = ↵K, MPK = L
29. How does the marginal rate of technical substitution depend on ? (PUT L ON THE HORIZONTAL AXIS AND K ON THE VERTICAL AXIS WHEN COMPUTING RTS) (4 points) (a) RT S is increasing in (b) RT S is constant in (c) RT S is decreasing in (d) None of the above 12
30. Find the elasticity of substitution: (5 points) (a) =1+ 2 ↵ · L K (b) =1+ ↵ 2 · L K (c) = ↵ 2 · L K (d) = ↵ · K L
31. Let w be the price of labor, v the price of capital, and q is the quantity produced. Find the contingent demand for capital (you can use the first order conditions, don’t worry about being at a corner). (5 points) (a) Kc = q · ⇣ v ↵wv ⌘ (b) Kc = q · ⇣ v ↵wv ⌘ · ↵w2v ↵v (c) Kc = q 1 2 ⇣ v ↵wv ⌘ 1 2 · ↵w2v ↵v (d) Kc = q 1 2 ⇣ v ↵wv ⌘ 1 2 Daniel has a contract with Metro Bike Share to produce 100 bikes. The price of labor is w = 4 and the price of capital is v = 5. ↵ and have values such that his contingent labor demand is Lc = q 1 2 ✓ v 5w 3v ◆1 2
32. How many units of labor will Daniel demand? (4 points) (a) Lc = 5 (b) Lc = 10 (c) Lc = 20 (d) Lc = p 5 2 13
33. Daniel’s employees feel they are undervalued and go on strike. After weeks of negotiations, they settle on a new wage of w = 7. How many employees will Daniel fire to optimally produce the 100 units in his contract with Metro Bike Share? (4 points) (a) 15 employees (b) 5 employees (c) 2.5 employees (d) 8 employees
34. The California Labor Commissioner catches wind of Daniel unjustly firing workers and decides to punish him by imposing a tax of ⌧ dollars per unit of capital hired by the firm. The California Labor Commissioner wants the tax ⌧ to be such that Daniel will hire the same number of workers as before the strike. Daniel still has the same contract with Metro Bike Share to produce 100 units. What should the tax ⌧ be? (5 points) (a) ⌧ = 75 4 = 18.75 (b) ⌧ = p 15 2 = 10.61 (c) ⌧ = 15 4 = 3.75 (d) ⌧ = 45 4 = 11.25 14
Essay Question 3 (35 Points)
While at the beach, Charlotte wants to buy souvenirs, which can be either seashells or pearls. Her preference over seashells (x) and pearls (y) is represented by the following utility function: u(x, y) = x 1 2 + y 1 2 35. Which of the following utility functions could NOT also represent Charlotte’s preference? (5 Points) (a) u(x, y)=(x 1 2 + y 1 2 )2 (b) u(x, y) = 1 x 1 2 + y 1 2 (c) u(x, y) = ln(x 1 2 + y 1 2 ) (d) u(x, y) = x + y (e) u(x, y) = x 1 2 + y 1 2 10
36. What is Charlotte’s Marshallian demand for seashells, x⇤(px, py, I), and her Marshallian demand for pearls, y⇤(px, py, I)? (5 Points) (a) x⇤(px, py, I) = I 2px , y⇤(px, py, I) = I 2py (b) x⇤(px, py, I) = Ipy px+py , y⇤(px, py, I) = Ipx px+py (c) x⇤(px, py, I) = Ipy pxpy+2p2 x , y⇤(px, py, I) = Ipx pxpy+(p2 y/2) (d) x⇤(px, py, I) = I 2px py , y⇤(px, py, I) = I 2py + py (e) x⇤(px, py, I) = I px , y⇤(px, py, I) = I+ py
37. What happens to Charlotte’s optimal consumption of seashells, x⇤, when the parameter increases (while prices and income remain constant), and why? (5 Points) (a) x⇤ increases because the marginal utility of x increases. (b) x⇤ increases because the marginal utility of y decreases. (c) x⇤ decreases because the marginal utility of x decreases. (d) x⇤ decreases because the marginal utility of y increases. (e) x⇤ remains constant because has no effect on the marginal utility of x. 15 For the rest of the problem, use = 1.
38. If the price of seashells is px = 2, the price of pearls is py = 8, and Charlotte has income I = 40, how much utility does she get from consuming her optimal consumption bundle? (5 Points) (a) 3 (b) 5 (c) 8 (d) 6 (e) 13
39. The opening of an oyster farm nearby has caused a positive supply shock for pearls, so that the price of pearls drops from py = 8 to p0 y = 2. If I and px remain unchanged, how much utility does Charlotte get from her new optimal consumption bundle? (5 Points) (a) 2 p10 ⇡ 6.32 (b) 10 (c) 15 (d) 12p2 ⇡ 16.97 (e) 3 p5 ⇡ 6.71 16
40. The government take advantage of the lower prices of pearls to raise revenue by imposing a lump-sum tax of ⌧ on Charlotte. Thus, her new income after the lump-sum tax is I0 = 40 ⌧ and the government earns ⌧ tax revenue from Charlotte. However, the government does not want to make Charlotte worse off than she was before the decline in the price of pearls. Under the new prices (px = 2 and p0 y = 2), what lump-sum tax ⌧ should the government set so that Charlotte’s maximized utility is equal to what it was in the original scenario (i.e. the utility from Question 38)? (5 Points) (a) 28 (b) 8 (c) 24 (d) 12 (e) 15
41. Instead of imposing a lump-sum tax, the government considers imposing a per-unit tax (a sales tax) on the consumption of pearls. Charlotte’s income is once again I = 40, the price of seashells is still px = 2, and the price of pearls becomes p00 y =2+ ⌧ . The tax revenue generated by the per-unit tax is ⌧ · y⇤, where y⇤ is the optimal quantity of pearls Charlotte consumes. Additionally, the government is still concerned about Charlotte’s welfare and wants to ensure that she keeps the same utility as the original scenario (i.e. the utility from Question 38). How much revenue does the government raise through the per-unit tax on pearls? (Hint: before starting with calculations, think carefully about the change in the price of pearls and on the effect of the sales tax on it) (5 Points) (a) 2 (b) 6 (c) 8 (d) 12 (e) 16 17 Essay Question 4 (30 Points) George operates a small car wash business in LA. He has to pay a fixed cost of $100 to rent the store, and the variable cost of washing q cars is 2q + q2. As there are many car washes around, Chris is a price taker and the price for each wash is fixed at p.
42. What is the total cost function for George’s car wash? (5 points) (a) C = 2q + q2 (b) C =2+ q (c) C =2+2q (d) C = 100 + 2q + q2 (e) C = 102 + q
43. In the short run, if the price for each wash is p = 10 and George is a price taker, how many car should George wash to maximize profit? (5 points) (a) 1 (b) 4 (c) 3 (d) 2 (e) 0
44. At p = 10, when George maximizes profit, which of the following statements is correct (use the cost function you derived in Question 42 for both the short and long run)? (5 points) (a) In the long run, he will produce 4 car wash. (b) In the long run, he decides whether to exit by comparing price with average variable cost. (c) In the short run, he earns a negative profit. (d) In the short run, he decides to shut down the business. (e) All of the above are incorrect. 18 There is a sudden economic shock and George becomes the only car wash business in LA. Now he has market power and faces the demand curve Q = 124 2p.
45. At what price will he maximize profit? (5 points) (a) 133 (b) 35 (c) 157 (d) 52 (e) 42
46. Continue from the previous question. Suppose the George is required to pay a fixed license fee before continuing this business, what is the amount of license fee that makes him indifferent between operating and exiting? (5 points) (a) 650 (b) 143 (c) 197 (d) 500 (e) 400 The economy starts booming and everyone works full time. People have therefore little time to wash their cars on their own. Consequently, the market demand for car washes shifts outward to Q = 1044 2p. Because of the shift, businesses outside the market evaluate whether to enter the car wash market. Each business has the same cost function as George’s.
47. In the long run perfect competitive market equilibrium, how many firms will there be in the market? (5 points) (a) 40 (b) 100 (c) 120 (d) 24 (e) 96
