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Unformatted text preview: 101A Midterm Professor Steve Goldman and GSI Ellen Myerson March 107 1998 Answer all questions: Point values are shown in parentheses. In addition,
the deﬁnitions for the terms in italics (questions 1 and 4) are worth 2 each. 1. (12) True, False. Please deﬁne all terms in italics and explain your answers
brieﬂy but clearly. (a) The Hicksian Compensated Income is always larger than the Slutsky Compensated Income regardless of whether price has increased or
decreased. (b) The Independence Axiom is not satisﬁed by an individual whose util
ity function satisfies the empected utility hypothesis. (c) The expenditure function is concave in prices only if the indiﬁerence
curves exhibit a decreasing Marginal Rate of Substitution. (d) The use of present value to compare mutually exclusive investment opportunities requires that individuals can borrow or lend at a con—
stant rate of interest. 2. (5) Is the following consistent with utility maximization in a two commod—
ity world: 0 An individual when confronted with prices of (4, 8) chooses (1, 5). o The same individual facing prices of (6, 9) chooses (5,3). 3. Joe’s Utility function depends upon his wealth as
U(W) = W’ where W is initially $10. He is offered the following gamble: pay $1 and
receive $6 with probability 1/4 and lose another $1 with probability 3/4.
(a) ) 3) What is his expected level of wealth if he gambles?
(b 5 (
( ) If he is an expected utility maximizer, should he accept the gam—
ble? 4. (Remember to deﬁne the terms in italics.) Consider the utility function given by:
U(a:m,$ ) = 3.73mi? . and prices are pm, p and income is given by I. (a)
(b)
(C) 10) Find the Marshallian Demands for x and a: . U] 10) Find the Hicksian Compensated demands for mm and ac . ( ( (10) Verify the Slutsky Equation for the demand for good 1 with
respect to a price change in good 1. (
( (d) 3) What bundle is demanded When 15:3,? '2'2'anch‘ : 12° 1 (e) 7) What is the price elasticity of demand for Jim at the values in d? Econ 101A, Spring 1998
Prof. Goldman TA: Ellen Myerson
Midterm Solutions Deﬁnitions:
(1a) Hieksian Compensated Income: The minimum amount of income needed to be able to reach a given utility level under given prices.
(la) Slutsky Compensated Income: The amount of income needed to be able to purchase a given commodity bundle (X 0) under the given prices. (lb) Independence Axiom: It says that if two lotteries differ only in one of their prizes,
then the lotteries must be ordered in the same way as those prizes, i.e. L >— L'<:> aL + (l — a)L">— aL'+(1— a)L",wherea 6 (0,1]. (lb) Expected utility hypothesis: An individual faced with risk will make choices in
order to maximize his/her expected utility. (1c) Expenditure function: e( p, u) = min{ p  xl U (x) _>_ u} . In words, the expenditure function tells you the minimum amount of money needed given a price vector p to reach
utility level u. (10) Marginal rate of substitution: The amount of good Y you would need to be
given in order to keep your utility unchanged when one unit of X is taken away. (Or the
amount of Y you would give up in order tokeepyonrutilityunchanged when given
another unit of X). It is the negative of the slope of an indiﬁerence curve. (ld) Present value: The current value of a sum of money that would just be sufﬁcient
to produce a given ﬁJture income stream when invested at the market rate of interest.
(43) Marshallian Demand: The Marshallian demand for a commodity is the quantity
demanded as a function of the prices and the consumer’s income. x( p, I) = arg max{U (x)! p  x S I} , where x is a vector of commodities and p is the price X vector. (40) Slutsky equation: —~’ = ﬂ — X ,. . A IS the total effect of a prlce change on
a: (9P, 61
“‘1? “E C the quantity demanded, B is the substitution effect, and C is the income effect. The Slutsky equation decomposes the change in quantity demanded into the income and
substitution effects. (4e) Price elasticity of demand: The price elasticity of demand measures the
responsiveness or sensitivity of the quantity demanded to changes in prices. i%
X l 0°19l
absolute value.) 77 = . (It doesn’t matter if you leave it as a negative ’nUmber without taking the (1) True, False, Uncertain:
(a) False. Since the Slutsky Compensated Income is suﬁicient to purchase thr original
bundle of goods, it must be at least as great as the minimum income necessary to provide the utility from that bundle. In the case of perfect complements, there is no
substitution effect and the Hicksian and Slutsky compensated incomes would be equal. (1b) False. The independence axiom IS satisﬁed by an individual whose utility ﬁmction
satisﬁes the expected utility hypothesis. It can be proved that the expected utility
hypothesis implies the independence axiom. In class we saw that the independence
axiom was used to prove the expected utility hypothesis. (1c) False. The expenditure ﬁinction is concave in prices whether or not the
indifference curves exhibit a decreasing MRS. The proof of concavity did not depend on
the MRS (see the expenditure function lecture notes for the proof of concavity). (1d) False. The present value can be calculated when the interest rate is a function of
time. An individual can therefore compare the present value of mutually exclusive
investment opportunities whether or not the interest rate is constant. Suppose an asset
has an expected stream of payouts given by q(t) and the instantaneous rate of interest at
any moment in time I is p( r) . Then the present value of the stream is: .0 —lp(r)dr
P(O) = J; q(t)e 0 dt . This can be calculated for any two investment opportunities. The rationale for present value does require that, at any instant, borrowing and lending
may be carried out at the same interest rate. (2) Let P0 = (4,8), X0 = (1,5), P'= (6,9), and X'= (5,3). Then P°X° = 4+40= 44, and POX'z 20 + 24 = 44. This means that. if she is maximizing utility, since the
consumer chose X0 when he/she could have bought X’, it must be because she prefers X0 to X’. Also, P'X'= 30+ 27 = 57 and P'X0 2 6+ 45 = 51. This means
that under prices P’ the consumer chose bundle X’ when X0 was affordable. This is
not consistent with utility maximizing behavior because if the consumer is
maximizing utility, she will choose the same bundle under both the original and new
prices since both bundles are affordable under the new and old prices. Therefore this
consumption behavior is not consistent with utility maximizing behavior. (3) (a) If he gambles and wins, his wealth is 10 — l + 6 = 15. This happens with
probability %. If he gambles and loses, his wealth is 10 — l — l = 8 . This happens
with probability %. Therefore, his expected level of wealth if he gambles is 39 1 3 —15 +— =—= .7. 4( ) 4(8) 4 95 (b) His expected utility from gambling is: 1 3 l 3
—U15+—U8=—12 —82=1.25
4 < ) 4 () 4(5>+4() 04 His utility if he does not gamble is U(lO) = 100. So if he is an expected utility
maximizer, then he should accept the gamble because his expected utility from
gambling is greater than if he does not gamble. (4)(a) Maximize utility subject to the budget constraint in’order'to ﬁnd the Marshallian
demands. The Lagrangian for this problem is: L(x1,x2 ,1) = 3x1x2 + AU — pix1 — pzxz) . Because utility is increasing in x1 and x2
(both partial derivatives are positive for positive x1 and xz), the budget constraint will be
binding at the optimal solution. Also, there will be an interior solution because if either
x1 or X2 were zero, then utility would be zero. However, if both are positive then utility is
positive. Therefore, at the optimal solution, L1 = L2 = L, = 0. (1)111: 3x2 _ﬂp1 : 0 (2)L2 = 3xi_2~102 = 0 (3)101 : 1—p1x1_p2x2 = 0
p1 x
(l) and (2) => x—2 = p— . (Note that this is the condition that the MRS equals the price
1 2 ratio at an interior solution.) => x2 = ﬁx, . Put this into the budget constraint (3):
p2 , I
:>x, =32?
1
,, I
:>x2 =5.
2 x; and x; are the Marshallian demands. They are ﬁinctions of I and p; and p 2. (4b) There are two ways to ﬁnd the Hicksian compensated demand functions. One
would be to solve for the expenditure ﬁinction and use this to get the compensated
demands. The other way is to plug the Marshallian demands into the utility ﬁinction and
then solve for I in terms of the prices and u. Then put this expression into the
Marshallian demand functions. I will do this second method below. 1 I 3 12
U(x1,x2)=3[——J[ j:—
2pl 2102 4 plpz 4 1 :12 =§up1p2 31:2 371171172
2 1
Tu
1 , 2?; V317: These are the Hicksian compensated demands. The quantity demanded is a ﬁinction of
the prices and utility level. (40) I will denote the Marshallian demand for good 1 d1 and the Hicksian compensated demand for good 1 as h]. From the answer to part (a), d 1 ad] 1 m d dd] 1 (H)
= — , —— = — , an ~— = —.
’ 2pl Q11 21212 61 2p1 Using the answer from (b), ﬂ=_l lp_23u (***)
471 2 V3191 The Slutsky equation for the demand for good 1 with respect to a price change in good w] ﬁll ml
is — = — — a’x — .
5111 ﬁn 0"!
I
From (*), the leﬁ hand side is —— 2 2 . From (**) and (***), the right hand side is
[71
1/11)2 [11(1) .312 ..
— ~ ——u — — ~— . Now re lace u w1th — (found at the beglnning of
2 3 pi 2p] 2p. p 4 plpz (4b)). Therefore, the right hand side becomes 1 112 I 1 j 1 ( I ] 1 I
—— ——~ — — :— — =——. Thisisequaltotheleﬁhand
2 V4 pf [2171 2P1 41012 4p? 2 pf side found above. Therefore the Slutsky equation has been veriﬁed. (4d) Use the Marshallian demands here because you are given the prices and income. . . 12 12
Thlsglvesyou x1=ﬁ=2 and x2 =ﬁ=3
ﬁr, p1 1 p1 12 3
4e =———=—— —. Atthevalues ind, =———=—l. It’sﬁne if ou
( ) ’7 a), x1 2p]2 x1 ’7 2(9) 2 ( y take the absolute value and say that the elasticity is equal to l.) ...
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