In Class Homework Ch 2 f704y

Microeconomics MScE: In Class Homework Ch2 Bettina Klaus

J&R Exercise 2.7 Derive the consumer’s inverse demand functions, p1 (x1 , x2 ) and p2 (x1 , x2 ), when the utility function is of the Cobb-Douglas form, u(x1 , x2 ) = Axα1 x1−α for 0 < α < 1. 2 ; α ∈ (0, 1). By Hotelling Wold, Let u(x) = Axα1 x1−α 2 pi (x) =

∂u(x) ∂xi . P2 ∂u(x) j=1 xj ∂xj

We have ∂u(x) = αAxα−1 x1−α , 1 2 ∂x1 ∂u(x) = (1 − α)Axα1 x−α 2 , ∂x2 ∂u(x) ∂u(x) x1 + x2 = αAxα1 x21−α + (1 − α)Axα1 x1−α 2 ∂x1 ∂x2 = Axα1 x21−α . So, αAxα−1 α x21−α 1 and = α 1−α x1 Ax1 x2 (1 − α)Axα1 x−α (1 − α) 2 . p2 (x) = = 1−α x2 Axα1 x2

p1 (x) =

1

J&R Exercise 2.10 Hicks (1956) offered the following example to demonstrate how WARP can fail to result in transitive revealed preferences when there are more than two good. The consumer chooses bundle xi at prices pi , i = 0, 1, 2, where:     1 5 p0 = 1 x0 = 19 2 9     1 12 1 1    p = 1 x = 12 1 12     1 27 p2 = 2 x2 = 11 1 1 (a) Show that these data satisfy WARP. Do it by considering all possible pairwise comparisons of the bundles and showing that in each case, one bundle in the pair is revealed preferred to the other. (b) Find the intransitivity in the revealed preferences. WARP: p˜x˜ ≥ p˜x¯ ⇒ p¯x˜ > p¯x¯. Note that p0 x0 = 42, p1 x1 = 36, and p2 x2 = 50. (a) Compare x0 , x1 : Note that p0 x1 = 48 and p1 x0 = 33. Thus, p0 x0 = 42 48 = p0 x1 (WARP satisfied), p1 x1 = 36 ≥ 33 = p1 x0 and p0 x1 = 48 > 42 = p0 x0 (WARP satisfied). The second statement implies that x1 is revealed preferred to x0 , i.e., x1 R x0 . Compare x1 , x2 : Note that p1 x2 = 39 and p2 x1 = 48. Thus, p1 x1 = 36 39 = p1 x2 (WARP satisfied), p2 x2 = 50 ≥ 48 = p2 x1 and p1 x2 = 39 > 36 = p1 x1 (WARP satisfied). The second statement implies that x2 is revealed preferred to x1 , i.e., x2 R x1 . Compare x2 , x0 : Note that p2 x0 = 52 and p0 x2 = 40. Thus, p2 x2 = 50 52 = p2 x0 (WARP satisfied), p0 x0 = 42 ≥ 40 = p0 x2 and p2 x0 = 52 > 50 = p2 x2 (WARP satisfied). The second statement implies that x0 is revealed preferred to x2 , i.e., x0 R x2 . (b) We now have that x0 R x2 and x2 R x1 . Hence, transitivity would imply that x0 R x1 . However, when comparing bundles x1 and x0 we found x1 R x0 . Note that x0 R x1 and x1 R x0 cannot be true at the same time. Therefore, transitivity is not satisfied. 2

J&R Exercise 2.25 Consider the quadratic VNM utility function U (w) = a + bw + cw2 . (a) What restrictions if any must be placed on parameters a, b, and c for this function to display risk aversion? (b) Over what domain of wealth can a quadratic VNM utility function be defined? (c) Given the gamble: g = ((1/2) ◦ (w + h), (1/2) ◦ (w − h)), show that CE < E(g) and that P > 0. (d) Show that this function, satisfying the restrictions in part (a), cannot represent preferences that display decreasing absolute risk aversion. Let U (w) = a + bw + cw2 . First we need to assume that U 0 (w) = b + 2cw > 0. (a) risk aversion ⇒ U 00 (w) = 2c < 0 ⇒ c < 0. Thus, b > −2cw > 0. (b) Since U 0 (w) = b + 2cw > 0 ⇒ w < − 2cb . (c) Let g = ( 12 ◦ (w + h), 12 ◦ (w − h)). Show that CE < E(g) and P > 0. By definition of the certainty equivalent, u(CE) = u(g). 1 1 [a + b(w + h) + c(w + h)2 ] + [a + b(w − h) + c(w − h)2 ] u(g) = 2 2 1 1 = a + b[w + h + w − h] + c[(w + h)2 + (w − h)2 ] 2 2 1 = a + bw + c[w2 + 2wh + h2 + w2 − 2hw + h2 ] 2 = a + bw + c(w2 + h2 ) u(CE) = u(g) = a + bw + c(w2 + h2 ) < a + bw + cw2 = u(w). Thus, U 0 >0

u(CE) < u(w) ⇔ CE < w. Then, P = E(g) − CE = w − CE > 0. (d) Decreasing absolute risk aversion means that 0 U 00 (w) 0 < 0. Ra (w) = − 0 U (w) However, since b + 2cw > 0 and 4c2 > 0 we have that 0 2c 4c2 0 Ra (w) = − = > 0. b + 2cw (b + 2cw)2 3

J&R Exercise 2.27 Show that for β > 0, the VNM utility function u(w) = α + β ln(w) displays decreasing absolute risk aversion. Let β > 0 and u(w) = α + β ln w. Ra (w) = −

− wβ2 u00 (w) 1 = − . = β u0 (w) w w

Hence,

1 <0 w2 which means that u (w) displays decreasing absolute risk aversion. Ra0 (w) = −

J&R Exercise 2.28 Let u(x1 , x2 ) = ln(x1 ) + 2 ln(x2 ). If p1 = p2 = 1, will this person be risk loving, risk neutral, or risk averse when offered gambles over different amounts of income? Let u(x) = ln x1 + 2 ln x2 . Let p = (1, 1). Note that the given utility function is defined over bundles of commodities and not over wealth/income levels. In order to determine the risk attitude of the agent when offered different gambles over amounts of income, we first have to find the indirect utility function associated with p and y. max u(x)

s.t.

p 1 x1 + p 2 x2 = y

max ln x1 + 2 ln x2

s.t.

x1 + x2 = y.

x∈R2+ x∈R2+

Note that at a maximum x1 6= 0 and x2 6= 0. (F OC)

L(p, λ) = ln x1 + 2 ln x2 + λ(y − x1 − x2 ) ⇒

2 1 = λ, = λ, and x1 + x2 = y. x1 x2 Hence, λ 6= 0 and x11 = x22 ⇒ x2 = 2x1 . Thus, x1 + 2x1 = y ⇒ x1 = y3 and x2 = 2y . 3 So, the agents indirect utility given the prices p = (1, 1) equals y 2y + 2 ln 3 3 = ln y − ln 3 + 2[ln 2 + ln y − ln 3] = 3 ln y + 2 ln 2 − 3 ln 3.

v(y) = ln

Note that v 0 (y) =

3 y

and v 00 (y) = − y32 < 0. Hence, the agent will be risk averse. 4