Download Problem Set 7 - Monopoly Q1). The demand for a monopolist`s

Problem Set 7 - Monopoly
Q1). The demand for a monopolist’s output is 7000 divided by the square of the price in dollars it
charges per unit. The firm has constant marginal costs equal to 1 dollar per unit. What should be the
price charged by the monopolists to maximize its profits?
1
Demand: Q = 7000/p2; MC = 1. According to the formula 𝜀 =
𝑝∙𝑝 2
7000
𝑀𝐶−𝑃
𝑃
𝑑𝑄
𝑃
, ε = 𝑑𝑃 ∙ 𝑄 = −7000 ∙ −2
1
𝑝3
∙
= 2; and P = 2.
Q2). A profit-maximizing monopoly faces an (inverse) demand function described by the equation p(y) =
40 – y and its total costs are c(y) = 8y, where prices and costs are measured in dollars. In the past the
monopolists was not taxed, but now it has to pay a tax of 8 dollars per unit of output. Find the before
tax and after tax price charged by the monopolist.
Demand: P = 40 – y;
MC = 8;
tax = 8.
𝑑π
Then, profit π = P ∙ y – c(y) – 8y = (40 – y)y – 8y – 8y; FOC: 𝑑𝑦 = 40 − 2𝑦 − 16 = 0 and y* = 12
Q3). A monopolist faces the demand curve q = 90 – p/2, where q is the number of units sold and p is the
price in dollars. He has quasi-fixed costs, C, and constant marginal costs of $20 per unit of output.
Therefore his total costs are C + 20q if q > 0 and 0 if q = 0. What is the largest value of C for which he
would be willing to produce positive output?
Demand: q = 90 – p/2, or p = 180 – 2q.
For q > 0, profit is (180 – 2q)q – c – 20q, and profit maximizing quantity satisfies FOC 180 – 4q – 20 = 0,
q = 40, and profit is 4000 – c – 800 = 3200 - c
For q = 0, profit is 0, and the firm chooses positive output quantity only if 3200 – c ≥ 0, that is, the largest
value of c is 3200.
Q4). A natural monopolist has the total cost function c(q) = 350 + 20q, where q is its output (A natural
monopoly occurs when fixed cost of production is very high and marginal cost is low compared to
demand, so that only one firm can survive in the market). The inverse demand function for the
monopolist’s product is p = 100 – 2q. Government regulations require this firmto produce a positive
amount and to set price equal to average costs. How much does the monopolist have to produce to
comply with these requirements?
The requirement AC = P implies 350/q + 20 = P. P and q in this equation should also satisfy the demand
equation: P = 100 – 2q. If we solve these two equations with two unknowns, we get q* = 50
Q5). SağlamBranda, Inc., sells protective liners for pickup trucks to both wholesale distributors and local
retail customers. Demand for the two markets is:
PW = 700 - 0.005QW; PR = 1000 - 0.05QR,
where PW is the price of liners in the wholesale market, PR is the price of liners in the retail market, QW is
the quantity of liners demanded in the wholesale market, and QR is the quantity of liners demanded in
the retail market Average cost per liner is $500. What is the maximum profit expected if managers can
charge two different prices in these two markets?
Profit from selling in both markets is equal to PWQW + PRQR – 500(QW + QR) = (700 – 0.005QW)QW + (1000
– 0.005QR)QR – 500 (QW + QR), and FOCs with respect to QW and QR imply
700 – 0.01QW – 500 = 0; 1000 – 0.1QW – 500 = 0; or QW = 20000; QR = 5000.
Then, profit is 600 ∙ 20000 + 750 ∙ 5000 – 500 ∙ 25000 = 3250000
Q6). Your company is the monopoly producer of widgets. The demand curve for a typical individual is P =
20 - 2Q, where P is the unit price of a widget and Q is the units of widgets demanded. It costs your
company 10 to produce a widget (regardless of the number of widgets produced). Suppose your
company follows the following price strategy: to purchase your goods one has to pay a "membership"
fee first, and then a price for each widget purchased.
a) What should the membership fee be and what should the price per widget be?
b) How much more profitable is this way of pricing compared to a simple single price (without a fee)
monopoly scheme?
a) The total fee that the monopoly charges is in the form p(x) = p1 + p2 ∙ x, where p1 is the membership
fee and p2 is the price per widget. We know that p2 should be equal to marginal cost, and p1 – to the
consumer surplus (in the case price would be equal to marginal cost only). Then, p2 = MC = 10, and p1 =
CS =
(20−10)∙5
2
= 25 (area between the linear demand P = 20 – 2Q and price p2 = 10, which is a triangle in
this case). Membership fee should be 25; price per widget should be 10; quantity sold is (20 – P)/2 = 5.
b) In case monopoly charges a constant price per widget, it should be P = 15: marginal revenue is MR =
20 – 4Q; optimal quantity Q* should satisfy MR = MC, or 20 – 4Q* = 10, or Q* = 2,5; then according to
the demand, simple price monopoly charges is 20 – 2Q* = 15.
Then, profit of the monopoly with the simple price is P ∙ Q* – MC ∙ Q* = 15∙ 2,5 – 10 ∙ 2,5 = 12,5
Profit of the monopoly, if it charges membership fee and price per widget is p1 + p2∙ Q* – MC ∙ Q* = 25 +
10 ∙ 5 – 10 ∙ 5 = 25, and this pricing scheme is more profitable by 12,5.
Q7). Best İkinci El (BİE) has a monopoly on used cars. By installing secret microphones in the showroom,
the friendly salespersons at BİE are able to learn each customer’s willingness to pay and can therefore
practice first-degree price discrimination, extracting from each customer his entire consumer’s surplus.
The inverse demand function for used cars is P = 2,000 – 10Q. BİE purchases its stock of used cars at an
auction for $500 each. How many used cars will BİE sell and what will be its profits?
BIE will sell each car exactly at the price that the customer wants it to buy, until the price fall below BIE’s
marginal cost (which is 500, the price it pays for every car). The number of cars sold is thus determined
by equation 2000 – 10Q = 500, or Q* = 150. The profit BIE makes will be the difference between price it
charges (which is exactly the demand of the consumer) and its marginal cost: π = 𝑎𝑙𝑙 𝑏𝑢𝑦𝑒𝑟𝑠 (D − MC),
which is the area between demand curve and marginal cost curve, and is equal to (2000 – 500)∙ 150 =
112500.
Q8). A monopolist is able to charge different prices in two different markets (a practice called “third
degree price discrimination”). The demand function in the first market is q = 500 –2p and the demand
function in the second market is q = 1500 – 5p. Find the prices that the monopolist will charge in each
market.
In both markets, marginal revenue should be equal to marginal cost. In first market, MR = 500 – 4p =
MC, or p = (500 – MC)/4; in second market, MR = 1500 – 5p = MC, or p = (1500 – MC)/5.
Q9). . A price-discriminating monopolist sells in two separate markets such that goods sold in one are
never resold in the other. It charges p1 = 2 in one market and p2 = 11 in the other market. At these
prices, the price elasticity in the first market is –1.2 and the price elasticity in the second market is –0.1.
Will the monopolist’s profit increase or decrease if it raises p1?
𝑑𝜋
𝑑𝑄
Write monopoly profit function: 𝜋 = 𝑃 𝑄 ∙ 𝑄 − 𝑐(𝑄); take derivative with respect to Q:
𝑑𝑃(𝑄)
𝑑𝑄
=
∙ 𝑄 + 𝑃 − 𝑀𝐶 𝑄 = 𝑄 𝜀 + 1 − 𝑀𝐶 𝑄 < 0 because ε <–1 and MC > 0. Thus, if price rises,
quantity sold falls and profit of the monopoly should increase.
Q10). The music group Manga tours the country performing in different cities. A careful analysis of
demand for tickets to Manga concerts reveals a segmentation in the market. Demand for tickets by
students is described by QO = 500P-3/2 while demand by those over 40 years old is Qy = 50P–4. If the
marginal cost of a ticket is $3, how should tickets to Manga’s concerts be priced to maximize profits?
If managers can charge different prices for old and young buyers, then the overall profit is summation of
profits extracted from two markets for old and young people. If managers have to charge the same price
for both categories of consumers, then profit is equal to (QO + QY)P – 3(QO + QY) =
and FOC implies
2250
𝑝 5/2
+
600
𝑝5
−
250
3
𝑝2
−
150
𝑝4
= 0 with solution p = 1,73.
500
𝑝 1/2
+
50
𝑝3
−
1500
𝑝 3/2
−
150
,
𝑝4