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Unformatted text preview: University of California, Davis Date: June 23, 2008
Department of Economics Time: 5 hours
Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE mANSVVRKEYS
Question 1 1(3). Equations deﬁning (fcl(pl,p2, w,E‘,.),)Ac2 (p1.,p2, w,E_i),E(pl,p2,w,E_i))
Problem PG. max”l "695) u(x1 ,x2,E) subject to plx] +p2x2 5 w and E  E4  gxz 5 0. (Plus
any nonnegativity constraints, ignored in what follows). Lagrangian. u(x, , x2, E) — k1(p1x] + pzx2 —— w) — k2(E — E_,. — gxz) KT Conditions
aux,x,E aux,x,E
( 18x2 )_)\'1pl=09 _—( 5x2 )'—)\.1p2+)\.2g=0,
l 2
au(xl,x2,E) 8E —?t2 =0, 7t1(p1x1+p2x2 —w)=0, X2(E—E_, ~gx2)=0. Substituting the values for the multipliers found in the ﬁrst and third condition into the second
condition, and appealing to the assumption that constraints (1) and 2 are satisﬁed with equality, we obtain the following system of three equations in the three unknowns (x1, x2, E)
8u(xl,x2,E) _ 811(xl,x2,E)p2 + au(xl,x2,E) :: 0,
8x2 8x1 pl 8E g
plxl +p2x2 ‘WZO, (3)
E—E_, —gx2 =0. Because PG is a concave program (concave objective function and linear constraints), the KT
conditions are sufﬁcient. Hence, a solution to the above system of equations solves Problem PG. 1(b). For the rest of this part we consider the utility function 7 x7 1 X
:ERXSR‘m , 7,E =x+ ,,a “ —*x,EB 2 ,
Ll + (xl x ) 1 [0 E][E: 2: 2 ] [E] b b . . .
where (a2, aG) >> 0 and B 2[ 22 25 IS a symmetric, pos1tive deﬁnite matrix.
El EE l(b)(i). Specialize to this utility function the system of equations obtained in 1(a). it!
b b
We compute Qﬂ =1 and 8x2 = a2 ] — 22 25 [x2] . Therefore the system of
8x‘ 3—11 aE bE2 bEE E
8E equations (3) specializes to (612 _b22x2 _b2£E)_£2_+(a5 _b52x2 _b51:E)g = 0,
pl plxl +p2x2 ——w=0,
E—Egi—gx2 =0. A 8x2
8E demand for good 2 increases as the amount of the extemality created by everybody else increases.
Because the contribution of our consumer to the total amount of the public good is gig , the rate at which her contribution to the public good responds to an increase in the amount contributed by 8x2 . If %— > 0, then our consumer increases her demand
8E . BE for good 2, together with her contribution to the public good, as the amount contributed by
everybody else increases. . It is the rate at which our consumer’s 1(b)(ii). Verbal interpretation of everybody else has the same sign as A . . 8E . . .
Verbal interpretation of a? It IS the rate at which our consumer’s des1red amount of the —i public good increases as the amount of the extemality created by everybody else increases. 8_E: + 3x2 ,i.e., ﬁrZ—>0:> a—E>0.Butwhen 3x2
8E_,. 8E_,. all, 8E_,. 8E_,.
31;— can in principle be positive or negative. If, say, 8x2 > 0 anda—E < 0, then our consumer
8E . 8E . 8E decreases her demand for good 2 but ends up increasing enjoying a higher amount of the public
good as the amount contributed by everybody else increases. Computation of 8x2 and 35
8E». 8E . I —l Because E = E_,. +ng2 , < 0, then The partial derivatives can found either by computing the explicit solutions or by implicit
differentiation. This Answer Key follows the ﬁrst approach. The first and third equations of 1(b) (i) do not involve x; (this is a consequence of quasilinearity).
Hence, we can solve these two equations for x2 and E, i. e., (‘72 “bzzxz ‘szE) _%+(05 ‘bszz “bEEE)g = 0’
1 E—EAI. —gx2 =0,
or in matrix notation
’bzz —bE2g _b25 —bEEg x2 = _a2 +(P2 /p1) —aEg i e
—g 1 E E ’ i I, —i —i szz'l'bgzg bZE+bEEg x2 = az—(pZ/p1)+a£g
L g —1 E —E ’ which can be solved as x2 = b22+b52g bZE+bEEg '1 LIZ—(P2 /P1)+a5g
E +g —1 —E . “I
9 z; ‘1 4725‘ EEg aZ—(pz/p1)+a5g
‘A ‘g b22+b52g _E 9 Where _A : —(b22 + bEzg) _ (bu + bEEg)g = _b22 _ bEZg _' bug — bEEg2 ’ Ai yi elding 3x2 BE .1725 _bEEg
:’ = A . (4)
8E bzz + bazg
3E4 A 1(b)(iii). The positive deﬁniteness of the matrix B, implies b22 > 0, 555 > 0, (5)
as well as
2 b22 bZE 1
A=b22+b52g+b2Eg+bEEg =[l,g] >0. (6)
bEZ bEE g
Let by; Z 0. Using (5) and (6), the signs of (4) can be ascertained as:
8552 _bZE — bEEg :M
3E4 _ A . ,{+} _ *
, — of Sign pattern — ,
8E b22 + bEzg i+} + {+ or 0} +
8E4 A {+)
. an, . aé . . . .
1. e., 8E < 0 and 1. e., f > 0. In words, an increase of the contributions to the public good by everybody else induces our consumer to decreases her contribution (and her consumption of good
2), at a rate that still allows her to enjoy a higher level of the public good. 1(b)(iv). Let b2E < 0. Now the two terms in the numerators of the expressions 8x2 _bzg _ bEEg
aE—i A . . . . . .
A = are of oppos1te Sign, and hence the partial derivatives cannot be Slgned
8E bzz + bEZg
aE_, A
without additional assumptions on the entries of the matrix B and g. as . . . a)?
8E2. IS then negative when ~ by; — bggg < 0, while 3E2. _, z 1922 + bEzg > 0. In this case, sz is negative but relative small in absolute value, and the signs are
those of 1(b)(iii), where by; _>_ 0. If ~ [)2]; — bggg < 0 but 1);; + bEzg < 0 (i. e.,  bEEg < bEz <  b22/g < 0, meaning that bgz takes an
intermediate value) then both derivatives are negative. (9)22 . . .
> 0, which 1m lies that
313 . p 319 _, ~i Maintain 1725 < 0. is positive if 8x2 . If A bzg ~ bEEg > 0, then IS also positive, because E = E_,. + gﬁz . 1(c) (i) We ﬁrst deﬁne the Walrasian demand functions
(we, , Z53, p5, w),)EZ (pl , p2,pE, w),E(pl,p2,1—95, M) by the maximization ofu subject to the budget constraint falxl + ﬁzxz + [75E S W. Compute (22 (51,52,55, W),E('ﬁl diva” W» .
It is the usual quasilinear Walrasian demand, with Lagrangian x 1 x _ _ _ _
x1 +[a2,a,5][Ez]—E[x2,E]B[g]—p(plx1 +p2x2 +pEEw) and ﬁrstorder conditions l—pﬁl :0, a x 7 0 _ _ _ _
‘: 2]—B: 2]—P[fu]=[ :a p(p1x1+pzx2+pEE_W)=O
ab. E 195 0 From the ﬁrst equation, we obtain p 21/ 131. Hence, (22,1?) can be found by solving [a2] [x2]_[1.72/l—71] . [x2] [—02 +(l—72 /El): [x2]_[a2 —(P2 /I—91):
—B —_ _ ,1.e.,—B = _ _ orB — _ _ .
a5 E pg /p1 E _aE+pE/p1 E aE_pE /pl Multiplying both sides by the inverse B'1 we obtain [x2] = _1_[ bEE _b25][az _(I_72 /I—91)]
E A “b5: b2: a5 ‘05:; /51)
Hence, neither the Walrasian for good 2 nor that for E depends on wealth, and therefore [52: 52.5] 852 3135 1 [’4ng /pl) @215 /51) :__ 1 [‘bEE bZE ]
(b52/1—71) ‘(bzz /Ei) bgz "[722 , 1313 35 8E _ Z
3.53 317i—
where K E bZZbEE — szbE2 > 0 by the positivedeﬁniteness of B. 1(c) (iii). It follows that goods 2 and E are net (and gross) complements if szE< 0, i.e., b2); <
0, and net (and gross) substitutes ifsz5> 0, i. e., bzg > 0. 1(c) (iv). Use 1(c) (iii) to verbally discuss the conditions analyzed in 1(b) (iii)(v). 3x2 <: 0 and i. e., _8§_ > o. In view of l(c)(iii), this is
8E . BE . “I —I As seen in l(b)(iii), if sz 3 0, then the case where goods 2 and E are net substitutes (or borderline substitutecomplements). An
increase in the amount E_,~ of the public good exogenously made available to the consumer leads
her to decrease her demand for the substitute good 2. If by; < 0, then goods 2 and E are net complements, and the signs of the partial derivatives
88:2 and 887 are ambiguous, depending on the magnitudes of the various Slutsky terms. If the l complementarity is strong (1)25 < 0 and large in absolute values), then it may be that the consumer
ends up demanding a higher amount of good 2 when E_,~ increases.
The table summarizes the comparative statics results. Goods 2 and E Goods 2 and E complements
substitutes (S25 < 0, 1925 < 0)
(S25 2 0, sz 2 0)  1’55. 1725 <  bEEg < 0
b22<b2£g<0 b25<b22g<0
822
BE», _ —
8E Answer key June, Questions 2,3 2(a) ll ultr‘) > all, b7’(.1:'i. 11") is the maximum number of units of the bundle g which can be
subtracted from 1:" still keeping agent i at the utility level if. If ui(;ri) < 01', and bi(x'i,vi) is
negative and 16%;“. v1) is the minimum units of bundle: g which must be added to the consumption
.13" in order that agent 2' reaches the utility level vi. Formally, if ui(:ci) 2 vi and bl(:ci,vi) < O, by
strict monotonicity ’u,i(:ci — bi(xi., Ui)g > ui(:ri) 2 vi and by continuity lit(1‘1.  (bi(zi, '01) + 6)g > vi
for e sufficiently small, which contradicts the maximum property of him", 12‘). Thus bi(:ci, 1)") Z 0 The same reasoning shows that when Mimi) > vi then bi(a:l, 11‘) > 0. 80°“ <‘wdAIMCL CumL V“ k" (b) Let 3’ : b1(i",vi). By definition,
we — 319) 2; W)
011 the other hand the allocation (1? — 3i “(1)11:1 does not use all the available resources since
Zea — 31' gr) s 22:3; 3 2w;
1' 1 i where the second inequality comes from the fact that i is feasible, and where the ﬁrst inequality
is strict if 915 > 0. Let E be such that Ic g 2131. Since the preference are strongly monotone, the allocation (if —« Big + cgﬂgl is feasible and strictly preferred by each agent to xi, so that xi is not Pareto optimal l“) \ ((1) Since Mac”, 1)“) z 0, and 23* satisﬁes the constraint of the program (1), the maximum in (1)
is larger or equal to 0. If the maximum is strictly positive, then there exist an allocation 5: which
satisﬁes the constraint7 i.e. which is feasible, and which is such that :1 bi(:Ei, 11“) > 0. By (b) this contradicts that 1'" is Pareto optimal. (0) Suppose x* is solution of (1) (with a maximum value of zero for the objective) and that 15* is not
Pareto optimal. There exists a feasible allocation 5% such that iﬂi’) Z ui(a:*i) with at least a strict
inequality. By (a)7 bi(fi. v” 2 0 with at least one strict inequality. The allocation :2“ satisﬁes the constraints of (1), since it is feasible, and gives a positive value to the objective, which contradicts that .7,” is a solution of (1). (f) Let mi and ii be two consumption bundles, vi a utility level, ,3" = (14,111.), 51 : bi(ii,vi) and
t E [0, 1]. By quasi—concavity of U1. ui’(t$i’ + (1 — t);i‘Z — (15,6i + (1 — t)ﬁi)g) 2 Inin{ui(xi  ﬂig), 111(53‘ — 319)}2 vi Thus bi(t.17i + (1 A tﬂi, U") 2 tﬁi + (1 — 25m”, so that U is concave in mi. 3(a)(i). If t, > 0 and (E, > m, then the charity could increase m by decreasing E by e and distributing
e to the agents who have the minimum consumption. So only agents at the minimum consumption level receive transfers from the charity.
(a)(ii) There are two cases to consider: (a) If Z 3 mp [(wR — 2R) — (cup ~ 2P)], the charity does not have enough resources (or at the plimit
has just enough resources) to equalize the consumption of all agents. By (i) only the poor receive a transfer, thus Z Z tR(Z) : 01 7511(2) 2 —a m<Z> : WP _ 2P + — TIP
(/3) If Z > np[(wR * zR) ~ (1.0,, — zp)] and the charity give transfers only to the poor agents, they
will have more consumption than the rich, which contradicts (i). Thus both types of agents
must receive transfers and by (i) they must all have the minimum (same) consumption. Thus the transfers )5 must satisfy
tuﬂ~zR+tR=uJP~zP+tP7 nRtR+nPtP=Z Solving the two linear equations give tR(Z) : kw 23(2) 2 We);
m<Z> = MUM nR+nP The functions are linear in Z but there is a change of slope when Z = nPKwR — ZR) — (cup — zP )]. (b)(i) In an equilibrium with voluntary contribution, Z, Z 0 must maximize ln(wi + f, — z, +
yln(7n(Z‘l + 20), with obvious notation. The marginal cost of increasing zi is MC: % and the
I A _
:52. Since "y < l and 771’ (Z) S 1 (the minimum consumption cannot increase by more than the additional contribution), if 27 = m(Z) the marginal cost exceeds the marginal marginal beneﬁt is ”y beneﬁt. If 2, was positive agent 2' could increase his utility by decreasing his contribution. Thus 2,
must be equal to 0. (b)(ii) Consider an equilibrium with voluntary contributions (33R, ER, tR), (sip, 5P, tP) in which agents
of the same type are treated equally. The minimum consumption is either that of the poor, or
that of the rich, or that of both types of agents who then have identical consumption. Let us
show that the last two cases are not possible. Suppose all agents are at minimum consumption. Then by (i) nobody contributes, and it is thus impossible to equalize the consumption of the rich and the poor. Thus this is not possible in equilibrium. Suppose that the rich have the mini
mum consumption. Then by (i) they do not contribute and only the poor contribute. But then
:1": R Z wR > (up w 2P 2 EP, which contradicts the assumption that the rich have the lower con—
sumption. Thus it must be that the poor have the minimum consumption and we are in the case of question (a)(ii)((v,) with Z : nRER > 0.
The FOCs for the maximum problem of a rich agent maxln(wR — z) + 'yln(m(Z“i + 2)), z 2 0 1 IZ—i
_ +7_~m( ,+Z>§0, :ifz>0
wR 7,2 m(Z’1+z)
By <a)(ii)
. Z‘i . 1
m(Z"+z)=wP+——+Z, ml(ZTZ+Z)=*
nP np Thus if z > 0 and Z‘i + Z 2 715,2 (equal contributions) 4:) (71R + 'y)z : wa — nPwP Thus if ’wa e nPwP > 0, the rich contribute, and if wa ~ npwp S 0, the equilibrium involves
no redistribution. Note that the case where there are positive contributions to the charity is very
restrictive: it must be that the total income of the poor is less than the income of one rich agent. (b)(iii) Suppose that the equilibrium has no redistribution. Suppose that a planner takes 6
from each rich agent and distributes nRe/nP to each poor agent. The utility of the poor agents
increase since both their own consumption and m increase. If e is small the change is utility of a
rich agent is approximately il/wRe + (l/cup)(nRe/nP)7 which is positive if “/anR > nPwP. Thus it is possible to increase the utility of all agents by redistributing from the rich to the poor. (0) A representative rich agent will choose the tax rate If 2 0 which maximizes ln(wR(l A t) +
e," 111(0111, + "3th ). Assuming t > 0. writing the FCC, and solving for t leads to HP A ’j/TIRLUR — rupwp (1 + 7)nrtwR which is positive if 77LRwR — nPwP > 0. Thus this system where the rich tax themselves involves
more redistribution than the voluntary contribution equilibrium. One can show that actually the
equilibrium with tax is Pareto optimal. The reason why it works better is that when computing
the marginal beneﬁt of increasing the tax to choose the optimal tax rate7 a representative of the
rich takes into account that each rich person will contribute twR, which in good case like this one amounts to summing the marginal beneﬁts for all rich agents of one agent’s contribution. This is different from a voluntary contribution equilibrium where each rich agent only considers his
own marginal beneﬁt, forgetting the marginal beneﬁt of his contribution for all other agents. The
marginal beneﬁt considered in choosing the tax being closer to the true social marginal beneﬁt, the level of the contributions is closer to being optimal. Microeconomics Prelim June 2008
Answer Keys for Questions 4 and 5 4. (a.1) We need to be able to ﬁnd a p such that p 2 SA and p S 2 pib, . Thus a necessary and
ieQ sufﬁcient condition is s,4 S 2 pib, .
iEQ
45A (a.2) a2————.
SA+SB+SC+SD (b.1) We need to be able to ﬁnd a p such that SE S p < s, (this is always possible since, by hypothesis, SB < SA) and p S 2 [—p’Ww—jb, . Thus a necessary and sufﬁcient ie{B.C,D} p3 +pc +170
condition is SE S [—J 2 p,b,. (11 this condition 18 satisfied an appropriate
p3 + pc + PD iE{B,C,D}
price exists).
3s
(b.2) a 2 —3.
SB + SC + SD (c.1) We need SD S p < SC and p S bD. Thus a necessary and sufﬁcient condition is SD S D ,
which is one of our hypotheses. (c.2) 0721. ((1) Any situation where there are unsold cars is Pareto inefﬁcient, since buyers value cars more
than the sellers. Thus, while the equilibria of part (a) are Pareto efﬁcient, those of part (b),
and (c) are Pareto inefﬁcient. (e) If 1),, 2 s, + q AR then the owner of an A car is willing to sell with warranty at price pW; if,
furthermore, pW < s, + qiR for all i e {8, C, D} then the owners of cars of qualities B, C and
D are not willing to sell with warranty at price pW. Only cars of quality D are offered for
sale without warranty at price pN if sD S pN and. pN < s). for all i E {A,B, C} and such a
price exists since, by hypothesis, SD < Si for all its {A,B, C} . On the buyer side we need
pW S V for the buyer to be willing to buy with warranty and pN S bD for the buyer to be
willing to buy a D car without warranty; the latter condition is always possible since, by hypothesis, SD < 1),), Thus a sufﬁcient condition is SA + q AR < min {V, min {5, +q,R}} . iE{B,C,D}
(f) In this case, SA +qAR = 16.5, SB + qBR = 17.5, SC +ch = 20, SD +qDR = 21 so that the
sufﬁcient condition of part (f) is satisﬁed. Any pair (pW,pN) with 16.5 SpW < 17.5 and
6 S pN S 8 = min{bC = V — qCR = 9, SC = 8} will yield the desired equilibrium.
(g) First we compute b, = V—qiR: bl, = 22.5, bl, = 16.5, bC =12, bD = 9. SincepW= 17 > s, + qAR = 16.5 , while pN = 11 < SA = 15, cars ofquality A will be sold with warranty.
Since PW = 17 < s]. + qiR for all ie {B,C,D} (respectively, 17.5, 20 and 21: see part (0), no other qualities will be sold with warranty. Since pN = 11 > SB = 10, all other qualities p3 +PC +190 ie{B,C.D}
are willing to buy a car without warranty at price pN = 11, realizing that it will be of either . l 1
will be offered for sale. Since [_—] Z pibi = §(16.5 +12 +9) = 12.5 buyers quality B or C or D. 5. (a) The extensive form is as follows. L H
.4" "2 A.
L H L H
’1 $1 ’1 £1
p p p p p p p p
.5 2 t .5 2 ﬂ 0* 2 ﬂ .5 2 {.
P D P P p P p p p p p p p ’p’. p H p i‘i‘i‘l‘i‘i‘i‘ii (b) (L, p, p ’, p ’, p) (going from left to right). Firm 1 has 25 = 32 strategies. 80‘8p1 if p1<p2
(C) D1(P1’p2)= %(80_8P1) ifpizpz
0 If A >192 (d) In the subgame where they both choose H as well as in the subgame where they both choose
L, by Bertrand’s theorem the unique Nash equilibrium is p1 = p2 = O with corresponding proﬁts of zero for both ﬁrm. Now consider a subgame where one ﬁrm chooses H and the
other chooses L. The proﬁt functions are ﬂ'H = pH (80—40pH +40pL) and 7:14 = pL (4OpH —50pL ). To ﬁnd the Nash equilibrium solve 8”” = 0 and GEL = 0. The
am, 31% . . 5 1 . . 125
solutlon IS pH = Z = l.25,pL = —2— = 0.5 w1th corresponding proﬁts IZ'H = 7 = 62.5 and 25 .
IZ'L = 7 = 12.5 . Thus the game can be reduced to the followmg onestage game: Firm 2
H L 12.5 , 62.5 62.5 , 12.5 Thus there are two subgameperfect equilibria: (H.ifHle =0, ifHLpl =1.25, ifLle =05, ifLLpl=O),(L,ifI1Hp2 =0, ifHLp, =05, ifLsz =1.25, ifLLpz =0) ﬁmi l's strategy ﬁrm 2's strategy where ﬁrm 1 chooses H and sets a price of 1.25 and ﬁrm 2 chooses L and sets a price of 0.5,
and (L,ifHH p1 = 0. ifHLpl =1.25, ifLHp‘ = 0.5, ifLLpl = 0),(H,if Hsz = 0, ifHLp, = 0.5, ifLHp2 =1.25, ifLLpz = 0)
gm—ﬂﬂﬂ—ﬁ _—__.__—_,_—__.—.——._—_1
ﬁrm 1's strategy ﬁmt 2's strategy where ﬁrm 1 chooses L and sets a price of 0.5 and ﬁrm 2 chooses H and sets a price of 1.25. (e) There are many. For example, pick one of the above equilibria and change both prices in the
unreached secondstage games to zero. (f) In the subgame where they both choose H, inverse demand is P =10 ~% so that the proﬁt
functions are 7:1 = q1 [10 — q] :q2 j and 72‘2 = £1410 — q] ‘18"612 ) . To ﬁnd the Nash equilibrium
9 a . . 80 . . solve 5i : 0 and 8% : 0. The solut10n1s q1 : q2 : ? = 26.67 w1th corresponding proﬁts q1 q2
7r, =7:2 =§2g=88.89. 9 In the subgame where they both choose L, inverse demand is P = 8 —1% so that the proﬁt
functions are 7:] = q] [8 — (1112612 J and 7:2 = (12(8 — q, 12612 ). To ﬁnd the Nash equilibrium
solve gti = 0 and 3% = 0 . The solution is (1] = q2 = 83—0 = 26.67 with corresponding proﬁts q1 q2 640 72'} =ﬂ'2 :T=71.11
Now consider a subgame where one ﬁrm chooses H and the other chooses L. The proﬁt functions are 7:” = qH (I —1”~—q—L) and IQ = qL( —g’i—q—L). To ﬁnd the Nash 8 10 10 10
. . . 875, {ML . . .
equ1libr1um solve = 0 and = 0. The solution IS (qH = 30, qL = 25) w1th
aql—l aqL
. 225 125
corresponding proﬁts 72'” = T = 112.5 and ﬂ'L == 7 = 62.5. Thus the game can be reduced to the following onestage game:
Firm 2 Firm 112.5 , 62.5 71.11 ,71.11 Thus there is a unique subgame—perfect equilibrium where they both choose H and produce
26.67 units each: strategy of ﬁrm 1: (H,if HH q] = 26.67, ifHL q1 r: 30, if LH q1 = 25, if LL q1 = 26.67)
strategy of ﬁrm 2: (H,if HH q2 = 26.67, if HL q2 = 25, ifLH q2 = 30, ifLL q2 = 26.67) (g) The main difference is that in scenario 1 the ﬁrms choose to differentiate their products,
while in scenario 2 they choose to produce a homogeneous product. ...
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