Advanced Microeconomic Analysis Solutions to Homework #2

Transcription

1 Advanced Microeconomic Analysis Solutions to Homework # Prove that Hicksian demands are homogeneous of degree 0 in prices. We use the relationship between Hicksian and Marshallian demands: x h i p, u) = x i p, ep, u)) where ep, u) is the expenditure function. Then we use the fact that ep, u) is homogeneous of degree in p, and Marshallian demand x i p, y) is homogeneous of degree 0 in p, y): x h i tp, u) = x i tp, etp, u)) = x i tp, etp, u)) = x i tp, tep, u)) = x i p, ep, u)) This is the same as the original value of x h i p, u), so it is homogeneous of degree We use the Slutsky equation: x i p, y) = xh i p, u ) x i p, u) x ip, y) p i p i The second term xh i p,u ) is always 0, and demand x i p, u) is always positive. Suppose x i is a normal good. By definition, x i 0. Therefore, the sign of x ip, u) x ip,y) is 0, so the sign of the left hand sign is 0. A decrease in own-price causes quantity demanded to increase. The converse of this statement is: if x ip,y) 0, then x i is a normal good. This depends on the relative magnitude of the two terms on the right-hand side; it may be possible for x i to be < 0 if the magnitude of xh i p,u ) is large. Therefore, the converse is not true. Suppose an own-price decrease causes a decrease in quantity demanded, i.e. x ip,y) p i > 0. Then the third term must be positive, so x ip,y) must be negative, therefore x i is an inferior good. The converse of this statement is: if x i is inferior therefore x ip,y) 0. This is not true if the magnitude of xh i p,u ) is not true. < 0), then x ip,y) p i > is large enough. Therefore, the converse

2 Plugging into the budget equation: Marshallian demand: Indirect utility: ux,..., x n ) = A i= x α i i, n α i = i= Lx,..., x n ) = Ax α...xαn n λp x x n y) = α iax α...xαn n λp i = 0 for i =...n x i x i ux α y ) = A λ = p x x n y = 0 α i x j = p i x j = p i α j x i α j x i p j p j α i p x + p 2 p p 2 α 2 α x p α n α x = y p ) α x i = y α i y j p α = j p i i α i αn y... ) α n = Ay α Expenditure function: use the relationship vp, ep, u)) = u Ay i= αi p i ) αi = u y = u A Hicksian demand: differentiate ep, u) with respect to p i. x h i p, u) = α iu Ap i j= p i= ) αj pj α j ) α... ) αi pi α i ) αn αn vp, p 2, p 3, y) = fy)p α pα 2 2 pα 3 3. In order for this to be a legitimate indirect utility, it must satisfy the following conditions all functions must be continuous): Homogeneous of degree 0 in p, y). Then fy) must be homogeneous of degree α + α 2 + α 3 ) Strictly increasing in y. Then fy) must be strictly increasing. Decreasing in p. Then α, α 2, α 3 must be 0. Quasiconvex in p, y). Then α + α 2 + α 3 ) and fy) must be convex. 2

3 vp, p 2, y) = wp, p 2 ) + zp,p 2 ) y. Homogeneous of degree 0 in p, y). Then zp, p 2 ) must be homogeneous of degree and wp, p 2 ) must be homogeneous of degree 0. Strictly increasing in y. This is always satisfied. Decreasing in p. wp, p 2 ) and zp, p 2 ) must be decreasing. Quasiconvex in p, y). wp, p 2 ) and zp, p 2 ) must be quasiconvex Given vp, y) = yp α pβ 2, α, β < 0. The direct utility is: ux) = min p vp, ) s.t. p x = = min p p α p β 2 s.t. p x + p 2 x 2 = Lp, p 2, λ) = p α p β 2 λp x + p 2 x 2 ) p = αp α p β 2 λx = 0 p 2 = βp α p β 2 λx 2 = 0 λ = p x + p 2 x 2 = 0 α β = p x p 2 = β p 2 x 2 α p x + p x β x 2 α x 2 = p = ux, x 2 ) = x + β α ) x x 2, p = α β x 2 x p 2 x + β α ), p 2 = x 2 + α β ) ) α ) β x 2 + α β ) We will show that any finite set of outcomes can be sorted into a sequence that is decreasing in preferability via induction on the number of elements. Suppose we have a sorted set of outcomes A = {a,..., a n }, such that a a 2... a n. We will show that if we add an additional element b to this set, it is possible to construct a n + -length sorted set containing a,...a n, b. Construct the sequence of true T) or false F) values by comparing b to each a i : b a ), b a 2 ),..., b a n ) By the completeness axiom, each element is well-defined and is either T or F. By the transitivity axiom, if b a i for some i, then b a i+, b a i+2,...b a n. Let j {,..., n} be the 3

4 index of the first element of A that is less preferred than b; that is, b a j is true and b a k is false for all k < j. We create a new sequence by inserting b at position j. Let the sequence {c,...c n+ } be defined as c = a, c 2 = a 2,...c j = b, c j+ = a j, c j+2 = a j+,...c n+ = a n. This sequence is sorted, therefore it has a best and worst element. Any sequence containing element is sorted, and has a best and worst element. By the proof above, this is also true for n = 2, 3, Suppose a a 2 a 3, and the gamble g = a 2 ). If g α a, α) a 3 ), then α must be strictly between 0 and. By the continuity axiom, we know α [0, ] must exist. Suppose α = 0. Then g = 0 a, a n ) a 2, which is a contradiction because a 2 a 3 by assumption. Therefore, α cannot be 0. Suppose α =. Then g = a, 0 a n ) a 2, which is a contradiction because a a 2 by assumption. Therefore, α cannot be. α must be 0, ) Suppose Uw) = a + bw + cw 2. This displays risk aversion if and only if it is concave, which is true iff c 0. A VNM utility function must be strictly increasing in wealth, so the region over which U ) is increasing is a valid domain. This is, b 2c ). Given the gamble g = 2 w + h), 2 w h)), then Eg) = w. We will show that CE < Eg). CE satisfies the condition UCE) = Ug) = 2 a + bw + h) + cw + h)2 ) + 2 a + bw h) + cw h)2 ) = a + bw + cw 2 + h 2 ) Since u ) is strictly increasing, if ux) > uy), then x > y. UEg)) = Uw) = a + bw + cw 2 ), which is strictly less than UCE) = a + bw + cw 2 + h 2 ) if h > 0. Therefore, CE < Eg). Since P = Eg) CE, then P > 0. For this utility function, R a w) = u w) u w) = 2c b+2cw, which is increasing in w. Therefore, this utility cannot represent preferences with decreasing absolute risk aversion Suppose a VNM utility function displays constant absolute risk aversion, so that R a w) = = α for all w. Then αu w) = u w) for all w, or d dw u w) = αu w) for all w. The u w) u w) only functional form for u w) that satisfies this is the exponential form, ux) = e αw. Then u w) = αe αw, u w) = α 2 e αw, and R a w) = α. 4

5 0.0 Q0 Suppose uw) = lnw). a) Find the Arrow-Pratt measure of absolute risk aversion. Is the utility function CARA, DARA, or IARA? R a w) is decreasing in w, so this is DARA. R a w) = u w) u w) = /w2 /w = w Suppose a consumer has an initial wealth of w 0 and is choosing a fraction x of his wealth, where 0 x, to invest in a risky asset. The risky asset has two outcomes: with probability p, it will give a return of 0 a total loss), and with probability p, it will give a return of r, so that if amount xw 0 is invested, the total return is rxw 0. The portion of wealth not invested in the risky asset is stored as cash, which has a certain return of 00%. Assume that the expected return is positive. b) In each of the two possible outcomes, what is the wealth of the consumer? In the bad outcome, total wealth is x)w 0. In the good outcome, total wealth is x)w 0 + rxw 0. c) Write down the expected wealth of the consumer, as a function of x. Expected wealth is p x)w 0 ) + p) x)w 0 + rxw 0 ) = w 0 + xr p) )). d) Write down the expected utility of the consumer, as a function of x. Expected utility is E[uw)] = p ln x)w 0 ) + p) ln x)w 0 + rxw 0 ) e) Find the value of x that maximizes the expected utility of the consumer. Taking the derivative with respect to x and setting it to 0, we get: E[uw)] x = p x + p)w 0r )) = 0 w 0 x + rx) Solving for x, we get x = r pr r. This is increasing in p and r. 5