Instructor' s Manual \ to accompany The Structure of Economics A Mathematical Analysis Third Edition Eugene Silberberg University ofWashington Wing Suen University of Hang 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
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Instructor' s Manual \
to accompany
The Structure of Economics A Mathematical Analysis
Third Edition
Eugene Silberberg University ofWashington
Wing Suen University of Hang 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' J dt < 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(2D2 - ~1r( J?-J') = lcD2(2- ~) :: kD2a, where
a = 2 - ~, and where 0 ::; k :5 1 indicat~s the ability to recycle this material. The
objective function is thus
minimize
subject to
rrhD + 2D2 - kD 2a
rrD2h=l.
4
Using the constraint to eliminate h,
Therefore
minimize ~ + D2(2 ;_ ka) "D
41r - rrD• +2D(2- ka) = 0 or
- rrh + 2D(2- ka) = 0.
h 4-2ka 4-k(4-rr)
v= " = "
When k = 0 (no recycling), -jj = t; when k = 1 (no waste), -jj = 1.
(b) If y2 is held fixed, then this essentially becomes the one-variable monopalist of
Chapter 1; hence, ( 8yif8t)y, < 0.
8. Unless the revenue and cost functions can be measured, these two models are obser
vationally 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
-----~---------------------------------
The first and second-order conditions are
.. , = pft - w, = 0,
", .. -pf" < 0 u- u '
1r2 = pfa- (1 + t)w2 = 0
D = p{!u/22 -1[2) > 0.
(c) Differentiating the first-order identities with respect to w2 and t, ~ = -w17f = 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' j8w2 == -8:t2f8p.
[) • [) ' (~)2 .J!_ := .J!_ - _P_
ap - ilp (~) ·
Since 8:t2jow2 < 0, oy' fop > oy' f8p.
5
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 ;<h first-order relation, but not a constraint, then ßxi jßa; = - 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 = -H11f H > 0. Since ß enters two first-order equations, no refutable propositions are