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2 3. A g oo d x is inf e rior, but it is not a G i ↵ e n go o d. I f the s ubs titution e ↵ e c t due to an increase in p x is — 2, which of the following could possibly be the income e ↵ ect? (a) — 3 (b) — 1 (c) 0 (d) 1 (e) 3 4. Consider a firm with the production function F ( K, L ) = K 0 . 8 L 0 . 2 — 10. (Concretely, imagine that due to technological imperfections, the first 10 units of output are unus- able, so that if the firm produces 100 units of a good, only 90 are of su f cient quality). If the firm is currently using L = 20 units of labor and K = 20 units of capital, what is the elasticity of substitution? (a) 1 (b) 2 (c) 4 (d) 8 (e) 10 5. A consumer’s Marshallian demand for x is given by g x ( p x , p y , I ) = I . If the consumer’s indirect utility function is V ( p x , p y , I ) = 3 I , then we can conclude that (Hint: make sure that you derive the Hicksian demand for x ): (a) @ h x y (b) @ h x y (c) @ h x y (d) @ h x y (e) @ h x y > 0 and, hence, x and y are net complements > 0 and, hence, x and y are net substitutes < 0 and, hence, x and y are net complements < 0 and, hence, x and y are net substitutes = 0 and, hence, x and y are neither net complements nor net substitutes 6. A consumer’s Marshallian demand for x is g x ( p x , p y , I ) = 0 . 9 I . What is e c , the p x consumer’s compensated own price elasticity of demand for x ? p x +3 p y p x +3 p y @ @ @ @ @

9. If a firm has the the production function F ( K, L ) = min(3 K, 4 L ) What is its contingent demand for labor, L c ( w, v, q )? q 4 w +3 v q 3 w +4 v (c) q (d) q (e) q when w < 4 v , and 0 when w > 4 v w 3 3 10. Alice and Bob have identical preferences, u ( x, y ) = ln( x ) + y . However, they have di ↵ erent incomes – Alice’s income is I A > p y and Bob’s income is I B < p y . Suppose that p y increases by a very small amount, and consider the following statements: (i) At her current income, Alice views x as a gross substitute for y . (ii) At his current income, Bob views x as a gross substitute for y . (iii) At her current income, the increase in p y decreases Alice’s utility. (iv) At his current income, the increase in p y decreases Bob’s utility. Which of the above statements is correct? (a) (i) only (b) (ii) only (c) (i) and (iii) only (d) (ii) and (iv) only (e) All of the statements are correct. 3 4 (a) (b)

PART 2: ESSAY QUESTIONS Essay Question 1 (35 Points) Carlotta enjoys watching movies, x , while drinking sodas, y . Her preferences over movies and sodas can be represented by the utility function U ( x, y ) = ln( x ) + ln( y ) 11. Carlotta’s Marshallian demands for the two goods are given by which of the following expressions? (5 Points) (a) g x ( p x , p y , I ) = ln( I ) — ln( p x ) and g y ( p x , p y , I ) = ln( I ) — ln( p y ) (b) g x ( p x , p y , I ) = I and g y ( p x , p y , I ) = I . (c) g x ( p x , p y , I ) = p y I and g y ( p x , p y , I ) = p x I (d) g x ( p x , p y , I ) = p y (e) g x ( p x , p y , I ) = I and g y ( p x , p y , I ) = I — 1 and g y ( p x , p y , I ) = I 12. Find the expression for Carlotta’s indirect utility function. (5 Points) p x p y (b) V ( p x , p y , I ) = I — p x p y 2 p x (d) V ( p x , p y , I ) = I ( p x + p y ) 2 p y (e) V ( p ,p , I ) = 10 I + 1 x y 1 1 p 2 p 2 x y 13. Carlotta’s expenditure function is given by (5 Points) (a) E ( p x , p y , U ) = 4 p x p y exp( U ) (b) E ( p x , p y , U ) = ln ( p x + p y ) U (c) E ( p x , p y , U ) = p x p y U 1 1 (d) U 2 p 2 p p x ( p + p p y ( p + p p p p p p x p y p x + p y (a) V ( p x , p y , I ) = ln ⇣ p y ⌘ + I — 1 (c) V ( p x , p y , I ) = ln ⇣ I ⌘ + ln ⇣ I ⌘

2 U 2 ,p ,U ) = 2 p 2 p 2

14. Carlotta’s Hicksian demands are given by which of the the following expressions? (5 points) ⇣ ⌘ 1 ⇣ ⌘ 1 (a) x ( x , p y ,U ) = exp U 2 and h y ( p x , p y ,U ) = p x p exp U 2 ⇣ ⌘ 1 ⇣ ⌘ 1 ⇣ ⌘ 1 ⇣ ⌘ 1 (d) h x ( p x , p y , U ) = ⇣ p y ⌘ exp U and h y ( p x , p y , U ) = ⇣ p x ⌘ exp U ⇣ ⌘ 1 ⇣ ⌘ 1 (e) x ( x , p y ,U ) = U and h y ( p x , p y ,U ) = p x p U Carlotta’s income is 100 $ . Initially, the price of movies, p x , is 2 dollars and the price of sodas, p y , is 0.5 dollars. Governor Newsom decides to impose a sales tax on sodas for pu b l i c h eal t h r eason s t h at q u ad r up l es i t s p r i ce ( p 0 y = 2 ) . 15. T he t o t al e ↵ e c t , s u b s t i t u t i o n e ↵ e c t , a n d i n c o m e e ↵ e c t o n t h e de m a n d f or s o d a s p ro du c e d by the sales tax on sodas is (Hint: this is NOT a small change; also, compute the change in levels, do not divide by the price change generated by the sales tax): (5 points) The answers are in the order of TE, SE, and IE. (a) — 25, — 5, — 20 (b) — 150, — 75, — 75 (c) — 75, — 150, 75 (d) — 75, — 50, — 25 (e) — 50, 20, — 30 2 2 2 2 p y p U 2 p x p U 2 p y p U 2 p x p U 2 p 2 p 2 x x x x x x 2 2 2 2 p y p (b) ( , p y ,U ) = ln and h y ( p , p y ,U ) = ln (c) ( , p y ,U ) = 2 exp and h y ( p , p y ,U ) = 2 exp p y p

Essay Question 2 (30 Points) The firm Electrika produces electric cars using a technology that can be represented by the following production function 1 1 q = AK 4 L 4 where A > 0. 18. Which of the following statements is true? (6 points) (a) The production function exhibits constant returns to scale (b) The production function exhibits decreasing returns to scale (c) The production function exhibits increasing returns to scale (d) None of the above 19. Which of the following expressions gives the marginal rate of technical substitution (RTS)? (6 points) (a) A L (b) K 1 L (c) K 3 (d) K 4 L 4 20. Suppose that the wage rate is w and the rental rate is v . The firm’s contingent demands for factors of production are given by which of the following pairs of expressions? (6 Points) 1 2 2 w A 2 1 2 2 v A 2 (b) L = v 1 q and K = w 1 q w A v A (c) L = v q and K = w q w 2 A v 2 q 1 v 2 A w 2 q 1 v 2 w A 2 2 v A 2 K 4 L 1 v (a) L = w and K = 2 2 2 2 2 1 A and K 2 1 A (e) L = 1 v 1 and K = 1 w 1 q q (d) L w q 2 q 2

Electrika produces 100 cars a day. Currently, the price of labor, w , is 4, and the price of capital, v , is 64. Her technology is such that A = 100. Workers believe that their wages are too low. They therefore decide to strike. After weeks of bargaining, the firm decides to quadruple the price of labor to w = 16. 21. After the increase in the price of labor, the firm realizes the labor demand has to change. To produce the 100 cars a day, Electrika chooses to optimally reduce the number of workers by (the question asks about the reduction in number of workers) (6 Points) (a) 1 (b) 5 (c) 4 (d) 3 (e) 2 President Biden is not happy with Electrika’s decision to fire a large number of workers. He therefore decides to impose a tax on the price of capital so that Electrika employs the same number of workers as before the strike. 22. The firm continues to produce 100 cars. The tax on the price of capital under which it is optimal for Electrika to employ the same quantity of labor as before the strike is (the tax is in levels) (6 Points) (a) $271 (b) $239 (c) $84 (d) $192 (e) $129