Prof.
Bryan Caplan
bcaplan@gmu.edu
http://www.bcaplan.com
Econ
849
HW
#2 (please
type; answer any TWO essay questions and ALL of the other questions)
4. Using the two-equation S&D system from
the notes, prove/show that Q falls by [bd/b+d]t whether buyers or sellers
legally pay the tax.
With
zero tax, we have:
S:
Q=a+bP
D:
Q=c-dP
P=[-a+c]/[b+d]
Q=[ad+bc]/[b+d]
Case
1: "Sellers Pay"
S:
Q=a+b(P-t)
D:
Q=c-dP
So: P=[-a+c+bt]/[b+d]
Substituting
in: Q=
Simplifying: Q=
Compared
to no taxes, therefore, Q falls by bd/[b+d]t.
Case
2: "Buyers Pay"
S:
Q=a+bP
D:
Q=c-d(P+t)
So: P=[-a+c-dt]/[b+d]
Substituting
in: Q=
Simplifying: Q=
So
again, compared to no taxes, Q falls by bd/[b+d]t.
5. Using the two-equation S&D system from
the notes, prove/show that a profit-maximizing monopolist in a market without
taxes produces a larger quantity than a competitive market under a
tax-revenue-maximizing government. To
simplify the problem, assume that a=0.
Hint: The monopoly maximizes PQ(P)-TC, with TC=Q2/2b. The government just maximizes tQ(t).
The
Monopolist
The
demand curve is given by: Q=c-dP, so the monopolist maximizes:
P(c-dP)-(c-dP)2/2b
cP-dP2-(d2P2-2cdP+c2)/2b
-(d+d2/2b)P2+(c+cd/b)P-c2/2b
Differentiating
wrt P:
-2(d+d2/2b)P+(c+cd/b)=0
So
P=
So
Q=c-d
Simplifying:
Q=bc/[2b+d]
The
Tax-Maximizing Government
From
the previous problem, we know that Q=. Setting a=0:
tQ(t)=
Simplifying:
Differentiating
wrt t:
t=
, so Q=
=bc/2[b+d]<bc/[2b+d].
QED.
6. Give examples of:
a. Single-peaked preferences
Preferences
over drug-legalization. People who want
to ban, say, alcohol, rarely want to legalize anything. Most people prefer the status quo. Some people want to legalize marijuana, and
these people rarely want to ban anything currently legal. Others want to legalize everything currently
illegal, and they are even less likely to want to ban anything currently legal.
b. Non-single-peaked transitive preferences
Preferences
over national defense. A person might
ideally want to have a small military and an isolationist party, but
conditional on having interventionist policies, prefers a strong military. Their last choice would be an
intermediate-sized military that enters foreign adventures it is unable to
win. There is nothing intransitive about
these preferences - there are clear 1st, 2nd, and 3rd
choices.
c. Intransitive preferences
Preferences
over careers. A person working as a
professor might decide it would be better to work in industry; working in
industry, might decide it would be better to work in government; working in
government, decide it would be better to work as a professor.
7. Imagine a simple median voter situation with
the following twist: party A is more popular than party B, such that every voter
is indifferent between their shared ideal point p* provided by party B and a
deviation of e from their ideal point provided by party
A. Specifically:
U(A)=U(B)
iff 
Assume
further that if voters are indifferent, they vote for party A.
Then
intuitively answer the following questions: (no math needed)
a. If both parties maximize their votes, what
platforms do they offer and who wins?
, and party A definitely wins, getting 100% of the vote. When both parties offer the same ideal
platform, but one is more popular in everyone's eyes, the more popular party
wins all the votes.
b. If party A has policy preferences (it wants
to make p as large as possible conditional on winning), what platforms are
offered and who wins?
Party
A would set pA>p*. It
would stop once it drove its share of the votes down to 50%+1. Party B's vote-maximizing (but still losing)
strategy would be to offer p*. The
larger the value of e, the more party A could
safely deviate.
c. How would uncertainty about the position of
the median voter affect the results?
Party
A would have to deviate from p* by less because driving its vote share down to
50%+1 leaves no margin for error. So it
would have to target a greater vote share, and even then, party B would
occasionally get lucky and win.