Columbia Business School
Problem Set 7: Solution
Part 1:
Part a:
The definition of elasticity is:
∆Q
εd =
Q
∆P
P
=
∆Q P
∆P Q
The term ∆Q/∆P is simply the slope of the demand function Q(p). (Note that this is the
reciprocal of the slope of the inverse demand, p(Q), that we usually show in the plane
with quantity on the x-axis and dollars on the y-axis.)
Since Qd = 15,000 – 10p, the term ∆Q/∆P is –10 and the elasticity is:
εd =
∆Q P
1200
= − 10 *
=−4
∆P Q
3000
The marginal revenue at the price of 1200 and quantity of 3000 is most easily found
starting from the general form of the revenue function:
R(Q) = p(Q) Q
The marginal revenue function is obtained using the product rule for taking derivatives:
MR(Q ) =
dp (Q )
Q + p (Q )
dQ
Now, we pull a p(Q) from each term:
 1 dp (Q )
MR(Q ) = p (Q )
Q+
 p (Q ) dQ


dp(Q) Q
+ 1
1 = p(Q) 

 dQ p (Q ) 
The first term in the brackets is just the reciprocal of the elasticity of demand:

1
MR(Q ) = p (Q ) + 1
ε d

2
Using this formula, MR(3000) = 1200(-1/4 + 1) = 1200(3/4) = 900.
Marginal revenue is less than price the monopolist perceives a downward sloping
demand curve. While selling an extra unit of output brings new revenue, the sale of this
unit is only possible when the monopolist reduces the price not only of the marginal unit,
but of all units, of output. So the revenue gain on the last unit is accompanied by a
revenue loss on the other units, and because of that loss marginal revenue is less than
the price of the last unit.
Part b:
Repeating the above calculations for the point where quantity is 10,000 and price is 500,
ηd = -10 * 500/10,000 = -10* 0.05 = -0.5. Marginal revenue at this point is MR(10,000) =
500(1/-0.5 + 1) = 500(-2 + 1) = -500. At this point, the revenue gain from the sale of the
last unit is less than the revenue loss on all the other units. Clearly the monopolist would
be better off producing in the elastic region of the demand curve, as in part (a), and not
in the inelastic region of the demand curve.
Part c:
Repeating the calculation again, ηd = -20 * 1200/3000 = -20* 0.4 = -8. At this single
point, the “flatter”demand curve here in part (c) is more elastic than the “steeper”
demand curve in part (a).
Part d:
The demand curve D2 is more elastic than the demand curve D1 at a price of $1200 and
a quantity of 3,000.
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Same Point on Two Demand Curves
P
1,500
1,350
D1
1,200
D2
3,000
15,000
27,000
Q
Part 2:
Part a:
We will want to find the quantity at which marginal revenue is equal to marginal cost.
Beginning on the cost side, we find a cost function C(Q) = 500 + 1.5Q. Thus marginal
cost is constant at 1.5.
The revenue function will be of the general form R(Q) = p(Q) Q. Thus we need a rule
which gives price as a function of the quantity produced by the monopolist. We have the
demand function, a rule that gives quantity as a function of price. So we will transform
the demand function into the inverse demand function:
Q = 55,000 – 10,000P è 10,000P = 55,000 – Q è P = 5.5 – 0.0001Q
The revenue function becomes R(Q) = (5.5 – 0.0001Q)Q = 5.5Q – 0.0001Q2. And
marginal revenue is MR(Q) = 5.5 – 0.0002Q. Finding the Q* which sets marginal
revenue equal to marginal cost:
MR(Q) = MC(Q)
5.5 – 0.0002Q = 1.5
Q* = 20,000
To find the price at which the monopolist can sell 20,000 units, we substitute Q = 20,000
into the inverse demand schedule:
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P = 5.5 – 0.0001Q = 5.5 – 0.0001(20,000) = 3.5
To find profits, we subtract total cost from total revenue:
π(Q) = R(Q) – C(Q) = 3.5(20,000) – [500 + 1.5(20,000)] = 39,500
Part b:
The maximum franchise fee the management can charge is thus 39,500 per day. For
the year, the maximum fee is $3,199,500 (= 81*$39,500). Management can charge a
fixed fee that extracts all of the profit from the potential vendor. The underlying
assumption is that the vendor competes against many other potential vendors with the
same cost structure. This fixed fee changes the total cost of the vendor but does not
change the vendor’s marginal cost.
Part c:
Since this capacity constraint of 10,000 is less than the monopolist’s desired output of
20,000, one would expect that the monopolist would sell the entire capacity. However,
with this diminished capacity, the monopolist will charge a higher price.
One way to approach this problem is to look at marginal revenue at a quantity of 10,000:
MR(10,000) = 5.5 – 0.0002(10,000) = 3
Since MR(10,000) = 3 and MC(10,000) = 1.5, the monopolist should produce all 10,000
units. The 10,000th unit costs only 1.5 to produce, but can be sold generating marginal
revenue (gain minus loss from lowering price on the other units) of 3. The price at which
10,000 units can be sold is again obtained from the inverse demand function:
P = 5.5 – 0.0001Q = 5.5 – 0.0001(10,000) = 4.5
After the 10,000th unit, the marginal cost “jumps.” With a literal interpretation of the
capacity constraint, the cost of producing another unit is infinite.
Part d:
The figure below shows the marginal revenue function in the presence of the price cap.
Because the price can never rise about 3, there is no revenue loss associated with
increasing output from 0 to 25,000. There is only a revenue gain of 3 per unit. In other
words, since the price of the 24,999th unit cannot be above 3, selling the 25,000th unit
does not require lowering the price on the 24,999th or any other units. But with
production of the 25,001th unit, the price not just of that unit, but of all the previous units
falls below 3. Hence the big drop in marginal revenue at that point.
After the quantity of 25,000, the marginal revenue is the same as it was in part (a).
Therefore, the marginal revenue curve is discontinuous. For quantities less than 25,000,
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marginal revenue is 3.00; for quantities above 25,000, marginal revenue is 5.50 –
0.0002q.
Marginal Revenue with Price Ceiling
$/unit
MR
3.00
MR
0.50
25,000
Q
Part e:
Since MC is constant, the markets can be treated separately. If the marginal cost were
increasing in quantity, then we would have to worry about increasing sales in one market
increasing the marginal cost for producing for the other market. Setting marginal cost
equal to marginal revenue in each market gives the monopolist’s desired quantity in
each market. For the bleachers, marginal revenue is 3.50 – QB/3000. Thus, the quantity
in the bleachers is 6,000 and price is $2.50. Notice that this price is lower than the price
charged in part (a). For the grandstands, marginal revenue is 8.50 – QG/2000. Setting
marginal revenue equal to marginal cost leads to a quantity of 14,000 and a price of
$5.00.
Producing the total quantity of 20,000 has a cost of $30,500 (same as above). The
revenue is $15,000 in the bleachers and $70,000 in the grandstand. Profit is $54,500.
The ability to price discriminate raises the firm’s profits.
If marginal cost had depended on the quantity produced, then the solution method would
be more complicated. This more complicated method leads to the same answer in this
example.
The first step in the process is to express quantity in each market as a function of
marginal revenue. In the bleachers, QB = 10,500 - 3000MR. The bleacher quantity is
positive when the marginal revenue is less than $3.50. In the grandstands, QG = 17,000
- 2000MR, where this quantity is positive when the marginal revenue is less than $8.50.
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The second step is to “add”these marginal revenue curves (segments, to be technical).
Maximum marginal revenue is $8.50 so the aggregate marginal revenue curve starts at
$8.50. For marginal revenue between $3.50 and $8.50, the firm can only sell in the
grandstands so the relationship between quantity and marginal revenue is Q = 17,000 2000MR. For marginal revenue below $3.50, the vendor can sell in both markets so the
relationship between marginal revenue and quantity is the sum of the two segments or Q
= QB + QG = 10,500 - 3000MR + 17,000 - 2000MR = 27,500 – 5000MR. It is important to
note that the vendor chooses quantities such that the marginal revenue is the same in
the two markets.
The third step is to equate marginal revenue and marginal cost. In order to solve for
quantity, marginal revenue must be expressed as a function of quantity. Therefore, the
marginal revenue expression must be inverted. The expression above can be inverted
to find:

 8.50 − Q / 2000 if Q ≤10,000
MR(Q ) = 

 5.50 − Q / 5000 if Q > 10,000
Setting the first of these segments to be equal to marginal cost leads to a quantity of
14,000. However, this violates the inequality associated with the first segment. Setting
the second of these segments to be equal to marginal cost leads to a quantity of 20,000
which satisfies the inequality. Thus, the total quantity is 20,000 and the marginal
revenue is equal to $1.50.
To find the allocation of the 20,000 across the two markets, find the quantity in each
market that has marginal revenue of $1.50. The allocation described above satisfies this
condition – 6,000 in the bleachers and 14,000 in the grandstand.