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14.9 Since consumers only value
,XQ
firms can be treated as selling that commodity
(i.e., batteries of a specific useful life). Firms seek to minimize the cost of
producing
XQ
for any level of that output. Setting up the Lagrangian,
( ) ( )C X Q K XQ
L
Analytical Problems
14.10 Taxation of a monopoly good
The inverse elasticity rule is
.
11
MC
Pe
When the monopoly is subject to an ad valorem tax,
,t
this becomes
1.
1 1 1
MC
Pte
b. With constant elasticity demand, the inequality in part (a) becomes an equality so
pre tax
after tax .
1
P
Pt
c. If the monopoly operates on a negatively sloped portion of its marginal cost curve
we have (in the constant elasticity case)
after tax
after tax
pre tax
pre tax
1
1 1 1
1
1 1 1
.
1
MC
Pte
MC
te
P
t
d. The key part of this question is the requirement of equal tax revenues. That is
,
a a s
tP Q Q
where the subscripts refer to the monopoly’s choices under the two
tax regimes. Suppose that the tax rates were chosen so as to raise the same
revenue for a given output level, say Q. Then
, hence
aa
tP tMR
. But in
general under an ad valorem tax
(1 ) ,
a
MR t MR MR tMR
whereas under a
specific tax,
s
MR MR
. Hence, for a given
,Q
the specific tax that raises the
same revenue reduces
MR
by more than does the ad valorem tax. With an
upward sloping
,MC
less would be produced under the specific tax, thereby
dictating an even higher tax rate. In all, a lower output would be produced, at a
higher price than under the ad valorem tax. Under perfect competition, the two
equal-revenue taxes would have equivalent effects.
14.11 Flexible functional forms
a. Writing the monopoly profit function as
( ) [ ( ) ( )] ,Q P Q AC Q Q
substituting
the given functional forms yields, after rearranging,
0 0 1 1
( ) ( ) ( ) .
s
Q a c a c Q Q
Looking ahead to part (c), where it will be important to simplify, we could have
alternatively made the substitution
s
xQ
in the first-order condition and solved
for
.x
Let’s try that, as well as substituting
i i i
d a c
to further simplify. The
first-order condition becomes
01
1
(1 ) 0,d s d x
b. Constant average and marginal cost corresponds to
10.c
Substituting into the
solution from part (a) gives
1/
1
00
( 1) .
s
msa
Qac
c. Monopoly profit with this yet more flexible specification is
0 0 1 1 2 2
0 1 2
( ) ( ) ( ) ( )
.
ss
ss
Q a c a c Q a c Q Q
d d Q d Q Q
d. Here is a graph showing possible shapes for the average cost curve.
14.12 Welfare possibilities with different market segmentations
a. The perfectly competitive outcome with marginal-cost pricing (
0
c
p
) yields the
b. With just two consumer types, the monopolist can achieve perfect price
v
c. The single-price monopolist can choose from one of two pricing strategies, either
selling at the high types’ willingness-to-pay and just serving them, earning profit
,qv
or selling at the low types’ willingness-to-pay and serving all consumers,
earning profit
( ) .q q v
The assumed inequality means that the high-price
strategy is more profitable.
i. The monopoly price is
,v
quantity is
,q
and profit is
.qv
There is no
consumer surplus because the whole valuation is extracted from the high-
value consumers who end up buying. Welfare equals the profit,
.qv
ii. The profit from serving just high-value consumers in segment
B
is
,bqv
and from serving all consumers in that segment is
( ) .bq q v
At
*,b
these
profits are equal:
**
( ) ,b qv b q q v
or solving,
*.
()
qv
bq v v
iii.
d. The inequality assumed in this part means that the profit-maximizing single price
for the monopolist now equals
.v
The graph is identical to that in part (c) except that the labels on the
corners have been swapped because price discrimination across this
segmentation destroys rather than creates consumer surplus.
14.13 Shrouded prices
a. Monopoly profit is
( ) (10 )( 6).Q P AC P P
Solving the first-order
16 2 0P
8.
m
P
2,
m
Q
4,
m
b. Monopoly profit is
( ) (10 )( 6).Q P s AC P P s
Solving the first-order
16 2 0Ps
8 / 2.
m
Ps
2 / 2
m
Qs
c. Gross consumer surplus can be computed as the area of the trapezoid under the
demand curve up to the quantity sold:
11
(10 ) 18 2 .
2 2 2 2
m m m ss
GCS P Q
11
18 2 8 2 (4 3 )(4 ).
2 2 2 2 2 8
ms s s s
CS s s
d. Welfare is
211
2 4 3 4 (12 )(4 ),
2 8 8
m m m s
W CS s s s s
e. The solution for the monopoly price is exactly as in part (b). The difference here
m
sQ
