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 func u(x1 , x2 ) = Ax1α x12−α for 0 < 0 < α < 1 . tion is of the Cobb-Douglas form, u(
u(x) = Ax1α x12−α ; α ∈ (0, (0, 1). By Hotelling Wold, Let u( pi (x) = We have
∂u (x) ∂x i 2 ∂u (x) j =1 j ∂x j
x
.
∂u( ∂u (x) = αAx1α−1 x12−α , ∂x 1 ∂u( ∂u (x) = (1 − α)Ax1α x2−α , ∂x 2 ∂u( ∂u (x) ∂u( ∂u (x) x1 + x2 = αAx1α x12−α + (1 − α)Ax1α x12−α ∂x 1 ∂x 2 = Ax1α x12−α . So,
αAx1α−1 x12−α α p1 (x) = = and x1 Ax1α x12−α (1 − α)Ax1α x2−α (1 − α) p2 (x) = = . x2 Ax1α x12−α
1
J&R Exercise 2.10 Hicks (1956) offered the following example to demonstrate how WARP can fail to result in transitive revealed revealed preferences when there are more than two t wo good. The consumer chooses bundle bundle i i x at prices p , i = 0, 1, 2, where:
1 p = 1 2 1 p = 1 11 p = 2
5 x = 19 9 12 x = 12 1227 x = 11
0
0
1
1
2
2
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.
p˜x˜ ≥ p¯ p˜x ¯ ⇒ p˜ p¯x ˜ > p¯ p¯x ¯. Note that WARP: p˜ 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 = p 0 x1 (WARP satisfied), p1 x1 = 36 ≥ 33 = p 1 x0 and p0 x1 = 48 > 48 > 42 42 = p 0 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 = p 1 x2 (WARP satisfied), p2 x2 = 50 ≥ 48 = p 2 x1 and p1 x2 = 39 > 39 > 36 36 = p 1 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 = p 2 x0 (WARP satisfied), 52 > 50 50 = p 2 x2 (WARP satisfied). p0 x0 = 42 ≥ 40 = p 0 x2 and p2 x0 = 52 > The second statement implies that x0 is revealed preferred to x2 , i.e., x0 R x2 . (b) We now have that x 0 R x2 and x 2 R x1 . Hence, transitivity would imply that x 0 R x1 .
However, when comparing bundles x1 and x0 we found x1 R x0 . Note Note tha thatt 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 U (w) = a + bw + bw + cw cw 2 . Consider the quadratic VNM utility function U ( (a) What restrictions if any must be placed on parameters a,b, and c for this function to display
risk aversion? aversion? (b) Over what domain of wealth can a quadratic VNM utility function be defined? (c) Given the gamble:
g = ((1/ ((1/2) ◦ (w + h + h)), (1/ (1/2) ◦ (w − h)), )), 0 . show that CE < E (g ) and that P > 0. (d) Show Show that this function, function, satisfying satisfying the restrictions restrictions in part (a), cannot represent represent preferences that display decreasing absolute risk aversion.
U (w) = a + bw + bw + cw cw 2 . First we need to assume that U (w) = b + b + 2cw 2cw > 0 . Let U ( (a) risk aversion ⇒ U (w ) = 2c < 0 ⇒ c < 0 . Thus, b > −2cw > 0 .
b + 2cw 2cw > 0 ⇒ w < − 2bc . (b) Since U (w) = b + g = ( 12 ◦ (w + h + h)), (c) Let g =
1 2
◦ (w − h)). Show that CE < E (g ) and P > 0 . u(C E ) = u( u (g ). By definition of the certainty equivalent, u(
u(g ) = = = = u(C E ) = = < =
1 1 [a + b( b(w + h + h)) + c + c((w + h + h))2 ] + [a + b( b(w − h) + c + c((w − h)2 ] 2 2 1 1 a + b[w + h + h + + w w − h] + c[(w [(w + h + h))2 + (w (w − h)2] 2 2 1 a + bw + bw + c[w2 + 2wh 2wh + + h h2 + w 2 − 2hw + hw + h h2 ] 2 a + bw + bw + c c((w2 + h2 ) u(g ) a + bw + bw + c c((w2 + h2 ) a + bw + bw + + cw cw 2 u(w).
Thus, U >0
u(C E ) < u(w) ⇔ CE < w. Then,
P = E ( E (g ) − C E = w − C E > 0 . (d) Decreasing absolute risk aversion means that
Ra (w) =
U (w) − U (w)
< 0. 0 .
b + 2cw 2 cw > 0 and 4c 4c2 > 0 we have that However, since b +
Ra (w) =
2c − b + 2cw 2 cw
3
4c2 = > 0. 0 . (b + 2cw 2 cw))2
J&R Exercise 2.27 0 , the VNM utility function u( u(w) = α + β ln( β ln(w w) displays decreasing absolute Show that for β > 0, risk aversion. aversion. u(w) = α + α + β β ln ln w. Let β > 0 and u( − wβ2 u (w) 1 Ra ( w ) = − =− β = . u (w ) w w Hence,
Ra (w) = −
1 < 0 w2
which means that u (w) displays decreasing absolute risk aversion.
J&R Exercise 2.28 u(x1 , x2 ) = ln(x ln(x1 ) + 2 l n (x (x2). If p p1 = p 2 = 1, will this person be risk loving, risk neutral, or Let u( risk averse when offered gambles over different amounts of income? u (x) = ln x1 + 2 ln x2 . Let p = (1, (1, 1). Note that the given Let u( given utility function function is defined defined over over bundles of commodities commodities and not over wealth/income levels. 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.
2 x∈R+
max ln x1 + 2 ln x2 s.t.
2 x∈R+
p1 x1 + p + p2 x2 = y = y x1 + x + x2 = y. y .
= 0 and x2 = 0. Note that at a maximum x1 (F OC )
L( p, λ) = ln x1 + 2 ln x2 + λ + λ((y − x1 − x2 ) ⇒
1 2 = λ , = λ , and x1 + x + x2 = y . x1 x2 0 and Hence, λ =
1
2
⇒ x2 = 2x1 . 2 x1 = y ⇒ x1 = y3 and x2 = 23y . Thus, x1 + 2x p = (1, (1, 1) equals So, the agents indirect utility given the prices p = x1
=
x2
y 2y 2 y v (y ) = ln + 2 ln 3 3 = ln y − ln 3 + 2[l 2[ln n 2 + ln ln y − ln3] = 3 ln ln y + 2 ln 2 − 3 l n 3. 3. Note that v (y ) =
3 y
and v (y ) = − y32 < 0 . Hence, the agent will be risk averse. 4
