Instructor' s Manual to accompany
\
The Structure of Economics A Mathematical Analysis Third Edition
Eugene Silberberg University ofWashington Wing Suen University ofHang Kong
glrwin D McGraw-Hill Boston ßnrr Ridge, IL Dnbnqne, lA Madison, WI New York San Francisco St. Lonis ßangkok Bogobi Caracas Lisbon London Madrid Mexico City Milan New Delhi Seonl Singapore Sydney Taipei Toronto
The third edition of The Structure of Economics contains two brand new chapters. Chapter 15, Contracis and Incentives, and Chapter 16, Markets with Imperfect Information, cover the exciting recent developments in information economics. Wing Suen ofthe University ofHong Kong wrote these chapters. We discarded the old Chapter 19 on stability of equilibrium in order to accommodate this new material. Also discarded is most ofthe old Chapter 2, the calculus review, which we feel is no Ionger important. Wehave maintained, however, the discussion ofTaylor's series, and some topics that are not typically covered in elementary calculus courses, such as continuous compounding.
The traditional chapters of the text contain many emendations and clarifications, which we hope will prove useful. We discarded the very general primal-dual analysis bf c~mparative statics presented as Sec. 7.5 in the second edition, these results being of little practical use, in favor of greater analysis of the more useful models where the parameters enter eilher the objective function or the constraint. We have striven to keep the informal tone of the text in both the old and new chapters, and to focus on deriving interestiflg and useful results, usmg the most elementary math that is required to do the job.
There is a tendency to treat comparative statics in a very terse manner, as though the whole subject could be summarized by the solution to the matrix equation (H)(&!oa) = (-fxa). We, however, see comparative statics as the core methodology of economic science. As long as economists are not able to measure tastes and other important functions which determine behavior, the only scientific (i.e., refutable) propositions we can derive will be Statements about how decision variables change when parameters change, under the assumption of stability of those underlying functions. Thus the third edition to this text remains fiercely devoted to the goal of deriving refutable propositions from maximization hypotheses, and understanding the mathematical structures !hat yield these results.
CHAPTER 1
2. Assertions are those
b~havioral
postulates which we believe to be universally true, such
as "more is preferred to less,t' diminishing MRS, etc. Assumptions, on the other hand,
are the test conditions of a given expedment; they are true only at a given moment, e.g., the price of gasoline rose 20 cents a gallon at a certain time, etc. 4. This question is answered in the introductory paragraph. Of course, economists would Iove to be able to predict total quantities. This will be impossible as long as some variables are not measured.
2
11. We will once again derive dx' Jdt < 0. This can be seen by setting R( x)
=0 in Example
1. Thus this theory yields the same refutable propositions as those in Examples 1 through 3. 13. The side of the can uses 1r Dh; the "waste," made up of .the· eight·corner pieces when a
circle is cut from two end squares, is k(2D 2 a
=2 -
~,
-
~1r( J?-J')
= lcD (2- ~) :: kD2 a, where 2
and where 0 ::; k :5 1 indicat~s the ability to recycle this material. The
objective function is thus minimize
rrhD + 2D 2
subject to
rrD h=l. 4
-
kD 2 a
2
Using the constraint to eliminate h,
minimize
~ + D 2 (2 ;_ ka) "D
41r
=0 rrh + 2D(2- ka) = 0.
- rrD• +2D(2- ka) -
or
Therefore
h
v= When k = 0 (no recycling),
CHAPTER4 Section 4.2 3. For 0 < a,ß< 1,a+ß< 1,
4-k(4-rr)
4-2ka
-jj =
"
=
"
t; when k = 1 (no waste), -jj = 1.
3 YLL
= <>(<>- l)L"-2 KP < 0
YKK
= ß(ß -l)L"J(P-2 < 0
YLLYKK- YLK
lf <> + ß
4. YLL
= aß(l- a- ß)L2a-2K2P-2 > 0.
=1, the function is weakly concave. =-a/L 2 < 0;
YKK
=-ß/K 2 < 0·,
YLK
= UKL =0;
YLLYkK
= aß/L 2K 2 > 0.
Section 4.5
5. "t
= M Rt -
M C - ty = 0
1r2=MR2-MC=O
where M R; = T R;'. Differentiating with respect to t,
8yj
8y2
"117ft+ "127ft = 1
where the
"'i 's are given as in Problem 6, Section 4.2.
This problern has the same
structure as the profit maximization problern with respect to a change in factor prices. We find, similarly,
(a)
( b) The denominator is positive here, but
-1r
12 =
C"(y) = MG'
rising, then 8yif8t > 0. (c) This follows the profit-maximizing firm example exactly.
J
0. If marginal cost is
4
Additional Comment: It is even the ease that {}y• f8t
= 8yjf8t + 8y2f8t does not have a determinate sign:
where D = 1<111<22- .-~ 2 > 0. Hence, only if the marginal revenue curve is downward sloping in market 2 (a condition not implied by profit maximization) will total output sold by this monopalist decrease in rcsponse to a tax increase. This result is known as the Edgeworth taxation paradox.
6. Consider ditch-diggers and shovels. If the wage of ditch-diggers fall, holding the number of shovels fixed will tend to reduce the firm's demand for ditch-diggers.
7. The model is now
maximize.-
=TR
1
+ TR 2 -
C(y)- ty1 - ty 2
producing the first-order relations, .-,=MRt-MC-t=O .-2
=MR2-MC-t=O.
Since the parameter t enters 6oth first-order relations, it will not be possible to sign either 8yif8t or 8y2f8t. Letting D
=1<111<22- .-~2 ,
(a) {}y•fat= (.-11 +1<2z-2.-,2)/D
< 0.
8. Unless the revenue and cost functions can be measured, these two models are observationally equivalent. The parameter t enters both models identically; oyj I 8t < 0 is implied in both models and no other results are forthcoming. 9. The objective function is
5
The first and second-order conditions are
.., =pft -
w, =0,
1r2
= pfa- (1 + t)w2 =0
",u .. -pf" u < 0'
D
=p{!u/22 -1[2) > 0.
(c) Differentiating the first-order identities with respect to w2 and t, ~
= -w1 7f
=
8=' W2f.?.:'·
10. The objective function is
The model is essentially the sru:ne as the text problem; note, however, that the factor demands are not homogeneaus of degree 1 in factor priees. There are no comparative statics relations available for output price, since p is endogenous, being embedded in
11. This is an examination problem. It follows the earlier monopolistic discrimination models. Note, however, in part (e), output price is endogenous. There is an implied profit-maximizing output price. It makes no sense to ask about any other, non-implied price. 12. The fundamental identity is
Differentiating with respect to p and then
W2,
[)y' - [)y'
[)y' 8:t2
-[)p - op+ o:t~ op [)y' - [)y' 8:t2
aw2
a.,g aw2 · Using the second equation to eliminate oy' I a.,g =
from the first, and remembering that
oy' j8w 2 == -8:t2f8p.
[) •
[) '
.J!_ := .J!_ -
ap - ilp Since 8:t2jow2
< 0, oy' fop > oy' f8p.
(~)2
_P_
(~) ·
6 CHAPTER5
Text 2. Expanding by the first column: lAI = auAu. Since Aa is also the determinant of an upper-triangular matrix, the result follows by induction.
APPENDIX
2.
Multiply (AB)- 1AB =I by
s- 1A- 1 on the right.
4. Taking the transpese of AA- 1
= I,
(A- 1 )' A' =I'= I. However, (A')- 1 A' =I by
definition. Since inverses are unique, (A-l)' = (A')- 1 • 5. Apply matrix multiplication. 8. This follows because AA' = I implies that A' A = I (taking the transpese of both sides).
CHAPTER 6
1. and 2. These problems are straightforward generalizations of the text material. In each case, if c>; enters only the ;
=
- L;.; times the ratio of some border-preserving principal minor of H to the whole
Hessian determinant, H. These two determinants must have opposite sign, and thus the result follows. 5. The equivalence of 4( c) and 4( d) is shown in Chapter 10 in the text. 6. The right-hand matrix for"' is (-1,0,0)'; for ß it is (0,-A,-x.)'. Thus, ßzjjßa -H11 f H
=
> 0. Since ß enters two first-order equations, no refutable propositions are
possible: ßzj ßß
(b) Note that ßz';./ßa
= -A H21 _ x H
= -Ht./H = -H.tfH,
2
Hat
H.
ßA• jßa = -Hta/H
= -H.tfH.
The result follows by substitution. When Chapter 7 has been covered, do this problern again, using the duality results derived there.
7
8. (b) Follows from second partials, AC;; = -AC/;;/f. Note that 8(-AC' f;)f8x; =
-;-AC'/;;- f;8AC' f8x; =-AC'/;; sii)ce 8AC' f8x; = 0. (c) 8x,f8w, = (-/22 (/21
+ x,ff(h2f,-
!12!2))/AC(f"h2-
+ x,ff(!Hh- f2,J,))/AC(!ld22- ff2) ~
ff2)
~
0;8x,j8w, =
0; reverse .the ones and twos to
get ax,j8w2, 8x2/8w2. (d) From first-order conditions, f; = w;/AC' = wd' /(w,xi+w2xi), multiply through by x; and sum the two equations. This is not Euler's Theorem. This result, an identify in
w,
and w2 (not
x,
and x 2) holds only lft the point of miilimum AC,
not for all x,, x2.
CHAPTER 7 l(a) and (b). Fora, this follows the text exactly. Note that if x 1 and x 2 are held fixed, k cannot be varied without violating the constraint. (c) From the geometry in (a), rPa(a) = fa, and rPaa- !aa > 0. However,
rPa(a)
E
/a(xi(o:), xi(a),o:).
Differentiating both sides with respect to a,
from which the result fo!lows. (d) The primal-dual problern is
subject to
g(x 1,x2)
=k
8
producing the Lagrangian
The first-order conditions are
c, = " c2
-
)..g,
=0
= 12- )..u2 = o
Ca= Ja-
c, = -<~>· +).. =0 C>.
= k- g(z 1,z2) = 0.
The third and fourth equations are the envelope theorem for <> and k. The second-order matrix ts
c" c12 c•• C2a H=
J.a
0
-g,
ha
0
-g·
f1a
faa
faa-
aa
-
0
0
0
-ka
1
-g,
-ga
0
1
0
The border-preserving principal minor formed by eliminating the first two rows and columns,
faa- aa H11.22
-
u
=
1
1 = -Uaa- aa)
0
yields the only implied curvature property for , aa (e) From ka, we have
* = I!J:.
> !aa·
Using d(1) and d(3),
>0
9
from whieh the result follows by the implied differentiation. (f) Of eourse, 8>.' f8k does not have an implied sign, unless f is eoneave and g is convex in"'' and
"'2·
(g) lf /(:z:,, :z:2, 01)
= h(:z:,, :z:2) +
OI:Z:l,
!aa E 0, thus from (a) or (d), tf>aa- 0
> 0.
2. This problern is a special case of problern 1, for the parameter 01. The reciprocity
condition in (c) follows from
using the results of part b.
3. This problern is worked out in Cha.pter 8. It is a. useful exercise for students to do on their own, however. Campare with the tr11-ditiona.l methodology of Cha.pter 6. 4. Long-Run
Short-Run:
s.t. w,:z:,
+ w2:z:2 = k
(a) 1t'•
=7r' = L 81r' 81r' 8L 81r' 81r' --=--=-----=-aw, aw, aw, aw, aw,
="''- >.:z:, = -:z:,(l + >.).
10
In long-run, .\ = 0 from first-order eonditions. Therefore, ~ =
,\ = l?h. -·I Wt
= ill W2
-xi.
In short-run,
=
W, therefore,
I. Therefore,
ehr' -llw1 =-xj(! + .\') 1)2~·
1)2~•
IJ2L
llxi
-->-->----+-llw( llw( . llw, llw1 >
[ll(xf(l + .\'))] > 0 llw1
_ [ll(xl(l + .\'))] = -xj ( /).\') . llw, llw1 _(I+.\') (llxf) llw 1
"'
(~~:) + (~::) < 0
This doesn't imply ~ (b) Show~
< 0 sinee ~
0.
••' From the enveIope theorem,
;:.::J..
8w, ·
Therefore, the elasticities cannot be compared.
(c) However, if it is assumed that ~ < 0, since ~
•~· < 0; t here.ore, ' ow, more elastic.
8
8
"' "' ~~
" ' ' ) < 0, or < "' (.....-
= '1rkw
•·· ~
1
==
'1t':U 1 k
••' 't.e., t he L - Rdeman d'1s < ~·
11
5. The first-<>rder conditions are
Multiplying through by
xt and summing,
I: /;x; a r a A' I: g;k; a A'rk
= 84> k k, 8
84>
Thus
8k
-r=T· Integration yields log
=log k +log a,
or
= ak.
CHAPTER 8
1. The technology available to the firm is just one constraint facing the firm; it says nothing about how a firm uses that technology. In order to be well-defined, i.e., to be useful in deriving testable implications, cost functions must consider additional behavioral assumptions and constraints, such as wealth ma.ximization. With differing behavioral postulates, different costs will be associated with a given output Ievel. 2. Empirical reasons suggest that the .cost-minimization model should yield an interior solution, which in turn implies that the isoquants be postulated tobe convex to the origin. Our observations refute the implications derived from the assumption of concave-tothe-<>rigin isoquants. Such concave isoquants imply the firms will hire only one input. This behavior is not observed. It is for this reason only that convex (to the origin)
isoquants are postulated.
12
5. (a) 1r•
= max{maxfpyw,x, y .Z:t ,::2
w 2x 2 + >.(f(x 1 , x2)- y)]}
= max{py+ max[-w,x 1 - w 2 x 2 + >.(f(x 1 ,x 2 ) - y)]} y
:l:l ,.t:.
(b) The inner minimum is precisely the cost minimization prob lern; the outer maximum specifies that this occurs at the ;>rofit maximizing Ievel of output. (c) The first order conditions of the Lagrangian are:
L,, =
-w, + >.f
1
=0
The matrix of the second derivatives of L is
0
(L;;) =
0
o Vu -1
,,
0
-1
>.!12
!t
!2
0
Its determinant, H, must be negative and its second-order border-preserving principle minors, H;;, must be positive. Expanding H by the first column and then by the remairring first row yields H = ->. 2(/u/22- !(2 )
< 0. The comparative statics are
found using (L;;) and Cramer's Rule. In particular, the reciprocity condition falls out directly:
13
The supply slope follows from the signs given by the second order conditions, ßy•
=_H
ßp
11 _ _ ( +)
H-
(-)
>
0'
These results also follow immediately from the theorems of Chapter 7. 6. Salve the first order conditions of the Lagrangian
From the first two conditions
or
Substitute this into the constraint,
then
14
And
x;
can be found directly, by summetry. C' is found by substituting
xi, x;
into
= kt yll/( a, +a,)}W~a,f(a, +a,)}W~[a,/(a, +a>)]+l
+ k2y[l/( a, +a,)] w\"• /( a, +a,)Jw;[a, /(a, +a, )]+ 1 = (kt + k2 )yll/( a, +a,)lw\"•/( a, +<>>)]W~a,f( a, +a,)}
where k;
= (a;/ai)la;/(a;+a;ll.
Now, verify that
Now, differentiating C' with respect to, say, w1 yields
15
( : : ) -1
Therefore, (8C' jßw!)
= (::).
= xj. Similarly, (8C' jßw2) =: x;.
7. (b) Solve the first equation in part (a) for FLK· Substitute into the second equation for !KL (since hK
= !KL)·
(c) In part (a), !LL < 0 implies hK
> 0.
(e) From equation (7-13), factor out,\ from each of the first two rows; then factor out 1/.\ from the last column and -1 from the last row.
CHAPTER.9
3. In each part, the cost function can be checked first tobe sure it makes sense, i.e., that it is homogeneous of degree one in w,, w2 • Also, the production function which is found can be verified by using it in the cost minimization problern and rederiving the original cost function.
l
16
(a)
(W2) t
Ct = x ,1 = -1 2 W!
C2
Rearrange
x2
>
e>
= x,2 = -21 (w•)-! -W! e! .
and substitute it into xj:
(b)
Substitute xj into x2: x2
= y +log(.\)
or
y
= X2 + logx,.
(c)
c, = xj = yw, (wi + wn-!- = y
Rearrange
x2
and substitute it into xi:
(1 + ( ::)
2
• ).
17
4. By definition,
f(tx1, ... , txn-1• Xn)
S
t' f(x1, ... , Xn).
By Euler's theorem:
n
I:;
/;x;
=rj(x1, ... ,xn)
n-1
I:;
/;x; = sf(x,, ... ,xn)·
i=l
Subtracting the second summation from the first leaves,
fnxn = f(xt, ... , Xn)(r- s),
(:;J
Xn
= y(r- s).
Rearrange this to solve the differential equation:
or
18
oy = (r _ s) (ox.) Y
Xn
J oy = (r-s)J ox. Y
Xn
log y = (r- s) log Xn +log (g(x,, ... , Xn_ 1 )),. or
y=xnr-• g (Xt, ...
,Xn-1
)
is the most general production function. If all x,'s have the same properties, the function 18
5. If y
= f(x,, ... , :tn) = F(h(x,, ... , x.)) is homothetic, h can be homogeneous of degree
one in
Zt, ...
,xn. If f is homothetic in
x1, ... ,Xn-1
also, then h must be homogeneaus
of some degree s in x 1 , ... ,Xn-t· Thus, using Eulers theorem as in question 9-4, h must have the form hf'x2' ... x:;•. Therefore, f(x 1 , ... ,xn) must have the form F(kxf 1 x~'
... x!:").
7. If f = F(h(:c,,x2)), then fi = F'h, and fi; = F'h•; + F"h,h;. Substitutethese into equation (!t-28):
1
19
"= x,x,(f? fu- 2fthft2 + fl !22) (F' h,)(F' h,)[F' h 1 x 1+
= x,x,[(F'h,)2(F'hu + F"hD- 2(F'ht)(F'h,)(F'h F'~x,]
. . . +(F'h 2 ) 2 (F'h 22 +
Then,
1,
.
+ F"h 1 ~ 2 )
F"h~)J
"= h,h,jh,,h when f
is homothetic, since h is then homogeneaus of degreeone.
Intuitively, since for homothetic functions each isoquant is a radial blow-up of the others, the properties of any one isoquant reflect the properties of all isoquants.
CHAPTER 10 2. Upward-sloping or flat indifference curves imply that the consumer may be "" better off when both goods are increased. 4. >.M
= (Uj fpt) = (U2fp,).
Since x; arehomogeneaus of degree zero, Ut
is also homogeneaus of degree zero in PI, P2 and M. Then,
>.M(tp tp tM)-. U;'(tp, 'tp,, tM) 1' 21 t Pi
= c' Ut(p,, p,, M) . P1
= t-1>,M(Pt,P2,M) 5. From the first-order conditions of the utility maximization model:
= U;(xj'f, xr)
20
or ·
Substitute in
Z2P2
= M- "'tPt from the constraint
then
And, by symmetry,
Now Substitute zf' il i .
xr into M)a' (
= (<>t ~ <>2 Pt and into
<>t
M)a'
~ "'' p,
Y1 I
21
Differentiate
u•
with respect to M,
6. By Euler's Theorem for xf":
t (&;fl) J=l
I;. (:!tr) X; J
PJ
Pi+ .
(~t) M =0,
i
=l, ... ,n
+ ( ";~) (&xfl) =0 &M
(&xfl) &p1
L:eff+efL=O,
(10- 53)
i=l, ... ,n.
j
&"1)1
n
LPi
;=t
&!J = 1
"L.JPi (x;-Mx;-M) --= &xtt 1 . &M
(10- 54)
J
Differentiate the budget constraint with respect to p;:
&xl)l
n
M
LPiT+x; ::0, i=l
~Pi(~)(:;)
c:r)
= -xfl (~)
LKie.J1=-Ki j
By Euler's Theorem for
xY:
i=l, ... ,n
p,
1
i=l, ... ,n.
(10-
59)
22
M) = 0, 2:P; ;'. n=l p '2:::. (Pi) (oxf) =0 x, op n
({)
1
(10- 60)
1
1
'2::: ef; = 0,
i
= 1, ... , n.
j
Differentiate the constraint with respect to p;:
'tu;( 8op,"'Y)=o,
i=1, ... ,n.
i=l
Multiplylng by A,
LPi (axf) =o {)p, j
2:Pi (P') ("i) . M x 1
1
(axf) {)p,
=0
:L: tc;ef. = 0,
(10-61) . i
= 1, ... , n.
j
9. LPiSik
= l:PjSkj = 0
i
j
LPiSik + Pk'kk = 2:PjSkj + PkSH = 0 i#
j#
LPiBik = LPiSkj =·-p.csu. i;tlr:
j#k
Then, since su < 0,
2:PiSik i#
=2:PjBkj > 0. #k
(10- 62)
:ll I
23
11. This is just eq. (10-22). 12. This is an application of Roy's identity and eqs. (10-74) and (10-75) for the demand curves with endowments.
13. L = U(o:1,o:2) + >.(1- Pl"l- p2o:2); The first-order conditions are: L1
= U1 -
>.p1
=0
L2
=u2 -
>.p2
=o
(a) Using the first-order conditions,
" 1 "1 (0oU)
M
oU) "2
+ ( 0" 2
M
=>.
+ >. M P2"2M = ,\M (PI"r + P2o:r) =
M
M
P1"1
,\M.
(b) By the envelope theorem,
(c):
(d) lf U(o: 1,o:2) is homogeneaus of degree r, then by part (a),
However, using part (c), and then part (a),
24
(au·) (au·) - - Pt+
8p,
- - P2
8p,
'= ->.M
=-rU*.
Thus U*(Pt 1 P2) is homogeneaus of degree -r by the converse of Euler's tbeorem.
=
or
U(zf!,zr)
= r'U*(Pt.P2)· 14.
Tbe first-order conditions are:
L,
= U( -
L2 ;", u~
Lp
>.p,
=0
- >.p2 =o
=M- PlZl -p2Z2 =o.
The second--order condition is
Uf'
\~
'
Ii
D=
0
U!j
-p2 > 0.
I.,
or,
v =-pw:· - PW~ > o. Therefore,
u:' < 0, (b) If U;' < 0, i=l,2,
or
u~
< 0, or both.
25
(+)(-) > 0 (+) . 15. In the two-good model, the goods must be net substitutes, i.e., 8xlf f8pl > 0. From the budget constraint (see the derivation of eq. (10-59)),
8xt'f VPI
8xf UPl
M
PI-;;-+ P2-.- =-XI
Since 8xt'f /ßP1 > 0, 8xf /8Pl < 0 is implied.
17.
subject to PlXl + ... +
P•"'• = M.
The first-order conditions include
Therefore, A = xdp2
= x3/P•·
The budget constraint thus becomes
or
< 0.
26
By symmetry,
Adding,
where K
= [(2PIP2- p~) + (2PaP4 -
p~)].
Also,
Note that &:x;f&p;
'f
0,
i
= 1,2; j = 3,4.
Lastly,
and &/" f&p;
'f
0,
j
= 3,4. Thus "two-stage" budgeting is not implied, even by
strongly separable homothetic utility functions.
27
19.
?'I
(a) This postulates that
8
<-g;[u,)
> 0,
8
<-g;!u,) < 0.
Using the quotient rule,
- [U.(8Utf8xt)- Ut(8U./8zt)]/U'f > 0 - [U.(8Utf8x•)- Ut(au.;ax.)]/8Ul < o or
Using the first-order conditions of the utility maximization problern on the secondorder condition:
Uu
D=
Uto
-Pt
u.! u•.•
-p,
-PI
= l/>.
2
-pa
0
=
Ua
ul2
-~
u.!
u ••
-~
-~ -~
[Ut(UaUat- U1Uo2)- Uo(UaUu- UtUl2)]
0
28
which is positive, using the postulate. (b) Assuming U1 U22 - U2U12
> 0, then using the first-order conditions:
then,
Assuming the opposite signs for [8(-U1 /U2/8o:;] implies D < 0, i.e., concave to the OJ:igin indifference curve8. (c) The converse of part (b). ( d) [8( -U!/U2)/8o:2]
>
0 implies [8( -U2/Ul)j8x2)
<
0. Increasing consumption of
>:2.
(o: 1 constant) will increase wealtb, thus the marginal evaluation of o: 2 relative to >:1 will increase wben
0:2
is normal and
>:1
is inferior.
(3) The postulate only asserts that the Ievel curves in two-dimensions are convex to the origin. This may be true for all pa.irs of goods wbile the three-good indifference surfaces are concave to the origin. 20. This problern utilizes the results of problern 19.
CHAPTER 11 1. Note th<>t
8>.M _ Dn+l,n+l _ 8MD -
(Ul'U; · · · u::J D
'
where >.M is the marginal utility of money income and Dis the usual bordered Hessian determinant for the utility maximization model, except that the off-diagonal terms are
29
zero in the first n rows and columns. We have sign D
= (-1 )n by the second-order
conditions. The sign of the numerator term is ( -1 )n if all
(-1)n-l. Thus, if Uf' < 0, a11 i, then one
UJ' > 0).
(J).M j(}M
UJ' > 0;
otherwise it is
> 0; otherwise it is negative
(at most
Now note that
i
= l, ... ,n
(1)
from which the conclusions ab out the income effects follow. Note also
U,!' 8:1J; -- p,· {)(}). 8pJ Pi
• .J.
•
Jr'·
(2)
Now
(3)
using eq. (10-22). Using this and the results for the income effects yields the results for the pure substitution effects. 2.(a) This is a simple consequence of (1) and (2) above. (b) Substitute (1) into (3). 3. If V(r!, ... , rn)
= V1(r1) + ... + Vn(r;,), then clearly u;,Pi = Vr 1r1 = 0 when i :f: j.
From Roy's identity,
Thus for i
:f: j :f: k,
30
(
and the result follows by division. 4. Using this result and (2),
and, (see eqs. (1G-63) and (10-64) this quickly implies unitary income elasticities and homotheticity. 8. This is of course a revealed preference question. Leo is violating the weak axiom.
9. (a) 10 = p 1z 1 ;:: p 1z 2 = 0,
8
14
z 1 rev. pref x2 •
=p2z2 < p2zl = 10
=p z
3 3
< p3 z 2 = 17
The preferences are also transitive so this is consistent with utility maxmization. (b) 21
=p2z 2 ;:: p 2z 3 =18, z 2 rev. pref z 3, but 28 =P"z3 < p3z 2 =27is false. Therefore,
this is inconsistent with the weak axiom and therefore also inconsistent with utility
maximizati9n.
31
(c) '
"
23
=p2 x 2 ;<: p2 x 1 =22, x 2
18
=p 1 x 1 < p 1x 2 = 19
23
= p2 x 2 ;;: p 2x 3 = 20, x 2
17
=p x
3 3
rev. pref x 1
Consistent
ref. pref x3
< p3 x 2 =20
10.
Since this set of purchase bundles is intransitive, and therefore inconsistent with utility maximization (i.e., irrational), the answer is obvious.
32
11. Xt
P2 =-, Pt
PtXt + P2X2
X2
1
= -MP2 -1, P2 < M
Jl
=Pt(:~)+ P2 [ ( : ) -11
=p2+M-p2 =M.
The budget constraint is satisfied, and the demand functions are homogeneaus of degree zero, so s 12 = s21· Thus, a utility function exists. Solve Xt, x2 for Pt = [ z,(!!"+l)] , P2 =
[(z:!t)], and plug into:
=
1
=-Integrate:
J
8x2 "'2
=-
Ja:,t +
f(U)
=-log (xt) + f(U)
or
U = g(x2 +log Xt) 12. Note that
Pt"t
P2M
]
+ P2"' 2 =[ (Pt + P2) +
[
PtM ] (Pt +P2) E M
thus the budget equation is satisfied (as is homogeneity). Noting that
33
j
Pt"t P2 --=P2"2 Pt
1
we have
Integrating,
or
14. Let " be the good, Iet p0 be some price at which the consumer. buys no " at all, and p 1 the market price. We shall suppress the other prices. If the consumer is allowed · to purchase as much as he or she would like at price p 1, utility U1 is achieved. At zero purchase of ", utility U0 is achieved. The definitions of the first three r'neasures of consumer's surplus are then:
1.
p'
- }po { "u• dp p'
- { "u• dp. }p.
34
The integrals are all positive. Clearly, 2.
ut > U0 ' .,f > .,yo and thus 3.
> 1.
always. If the good is normal, then since
> 2.
There are no unique monetary equivalents of the changes in utility in measures 4. and 5. except under special circumstances. If the indifference curves are vertically parallel (see problern 15), then a given change in utility corresponds to a unique amount of numeraire. 15. Inthis case we have n + 1 goods, x 0 ,x 1 , ... ,xn, with prices
1,p~.
... ,Pn· The line
integral yielding the consumer's surpluses is
j
1 dxo
+
t
p;(x)dx;.
i.=l
In order for this line integral to be path independent, ~
= W_<:1 =0, since p = 1. 8
0
The Pi(x)'s represent the marginal value of x 0 in terms of Xj(M RS;J)· ~ = 0 means that the slope of the indifference curves do not change as xo changes. With xo plotted on the vertical axis and any "'i on the horizontal axis, this says the indifference curves are vertically parallel, and hence ~ = 0, j = 1, ... , n.
CHAPTER 12 1.
(a)
(b)
35
In each case, "1+1 is a constant times o:;; if r
> p,
"1+ 1 > o:;, i.e., consumption increases
over time since the premium for earlier availability exceeds the rate of impatience.
13 2. Initially, your consumption possibilities lie along AB. If the interest rate doubles, your frontier pivots to AG; if half the complex burns, the frontier shifts to DG. Unless you have no heirs and wish to consume your entire wealth immediately, you prefer the cause to be a fall in the interest rate. 6. Suppose prices were to start doubling every year. Then depreciation would be seriously understated, artificially increasing reported corporate profits and therefore corporate taxes. The effects of the present rate of inflation are already capitalized into the price of gold and depreciable assets. With greater than anticipated inflation, tbe relative price of gold will increase. The capital gains exclusion for housing ameliorates this effect for housing. 7. The text answer assumes constant nominal mortgage payments; with severe inflation, we would expect to see indexed mortgages, alleviating this problem.
CHAPTER 14 3. The Kuhn-Tocker conditions are:
