Microeconomics, monopoly, mathematical final exam multi-question practice problem sets
(The attached PDF file has better formatting.)
** Exercise 11.1: Monopoly Pricing (algebra)
This exercise shows the algebra for linear dem and curves and m arginal cost curves. The next exercise is a
num erical exam ple. The final exam uses num erical exam ples and m ultiple choice questions.
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An industry has one firm which sets prices to m axim ize profits.
The m arket dem and curve is P = " – $ × Q.
The firm ’s m arginal cost curve is MC = ( + * × Q.
[Note: Final exam problem s m ay use either form of the dem and curve.]
A.
B.
C.
D.
E.
F.
G.
H.
W hat is the firm ’s total revenue curve?
W hat is the firm ’s m arginal revenue curve?
W hat is the m onopoly quantity?
W hat is the m onopoly price?
W he price elasticity of dem and at the m onopoly price and quantity?
W hat is consum ers’ surplus?
W hat is producers’ surplus?
W hat is the dead weight loss?
Part A: Total revenue is price tim es quantity = P × Q = Q × (" – $ × Q) = " Q – $ × Q 2.
Jacob: This exercise gives the dem and curve as P = " – $ × Q. Som e exam problem s gives the dem and
curve as Q = " – $ × P. W hich form should we use?
Rachel: If an exam problem asks to derive the total population dem and curve from the dem and curves of two
sub-populations, Q = " – $ × P. The two dem and curves from the sub-populations are Q 1 = " – $ × P and Q 2
= " – $ × P. The total population dem and curve is Q 1 + Q 2 = " – $ × P.
For com petitive pricing, we use the relation that price = m arginal cost. Marginal cost is expressed as a function
of quantity, so we express price as a function quantity. The m arginal cost depends on the production function
– how production varies with the num ber of item s produced (quantity). It does not depend on the price.
For m onopoly pricing, we use the relation that m arginal revenue = m arginal cost. Marginal cost is expressed
as a function of quantity, so we express price as a function quantity. Marginal revenue is price × quantity, so
we express m arginal revenue as f(Q) × Q, not as P × f(P).
Jacob: W hat if the exam problem gives the reverse form of the dem and curve? For exam ple, what if the exam
problem gives a dem and curve Q = " – $ × P and asks to derive the m onopoly price?
Rachel: A m arket dem and curve P = " – $ × Q is the sam e as Q = " / $ – 1/$ × P. Sim ilarly, A m arket dem and
curve Q = " – $ × P is the sam e as P = " / $ – 1/$ × Q. Many exam problem s give the reverse form and want
you to convert one form to the other.
Part B: Marginal revenue is the partial derivative of total revenue with respect to quantity =
M(" Q – $ × Q 2) / MQ = " – 2$ Q.
Jacob: W hy not express m arginal revenue as the partial derivative of total revenue with respect to quantity:
M(" P – $ × P 2) / MP = " – 2$ P.
Rachel: To solve for the m onopoly price and quantity, we set m arginal revenue equal to m arginal cost. Since
m arginal cost is a function of quantity, not of price, we express m arginal revenue as a function quantity, not
of price.
Exception: If m arginal cost is constant, we can express m arginal revenue as a function of either price or
quantity. Both m ethods give the sam e solution.
Part C: Let Q* = equilibrium quantity and equate m arginal cost and m arginal revenue:
" – 2$ Q* = ( + * × Q* A Q* = (" – () / (* + 2 $).
The next exercise shows a num erical exam ple that solves for the equilibrium quantity and price.
Part D: Derive the equilibrium price from the dem and curve, not the m arginal cost curve. Let P* = equilibrium
price, so
P* = " – $ × Q* = " – $ × (" – () / (* + 2 $)
Part E: The price elasticity of dem and at the m onopoly price is (MQ/Q) / (MP/P) = MQ/MP × P/Q.
At the m onopoly price and quantity, this is P*/Q* × –1/$.
The elasticity is negative: as P increases, Q decreases.
Part F: Consum ers’ surplus is a right triangle with base Q* and height = the price at Q = zero m inus the
equilibrium price: " – P*.
Use the m onopoly price and quantity, the dem and curve, and the values for P* and Q* derived above.
Note: If the m arket dem and curve is perfectly elastic, $ is zero and consum ers’ surplus is zero.
Part G: Producers’ surplus is the sum of
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a right triangle with base Q* and height = (the m arginal cost at Q*) – the m arginal cost at Q = 0 (()
a rectangle with base Q* and height = (P* – the m arginal cost at Q*)
Part H: The dead weight loss can be derived two ways.
Method #1: The dead weight loss = the decrease in net social gain from m onopoly = the social gain from
com petition m inus the social gain in m onoply = (consum ers’ surplus + producers’ surplus) in com petition –
(consum ers’ surplus + producers’ surplus) in m onopoly.
Method #2 (for linear dem and curve and m arginal cost curve): The dead weight loss = ½ × (com petitive
quantity – m onopoly quantity) × (m onopoly price – m arginal cost at the m onopoly price)
Method #2 follows from the geom etry of consum ers’ surplus and producers’ surplus with linear dem and and
m arginal cost curves.
** Exercise 11.2: Monopoly pricing (num erical exam ple)
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An industry has one firm which sets prices to m axim ize profits.
The m arket dem and curve is P = 120 – ½ Q (equivalent to Q = 240 – 2 P)
The firm ’s m arginal cost curve is MC = 10 + Q.
A.
B.
C.
D.
E.
F.
G.
H.
W hat is the firm ’s total revenue curve?
W hat is the firm ’s m arginal revenue curve?
W hat is the m onopoly quantity?
W hat is the m onopoly price?
W he price elasticity of dem and at the m onopoly price and quantity?
W hat is consum ers’ surplus?
W hat is producers’ surplus?
W hat is the dead weight loss?
Part A: Total revenue is P × Q = (120 – ½ Q) × Q = 120Q – ½ Q 2
Part B: Marginal revenue MR = MTR/MQ = 120 – Q
Part C: The m onopoly quantity is at the intersection of the m arginal revenue and m arginal cost curves.
MR = MC A 120 – Q = 10 + Q A 2Q = 110 A Q = 55
Part D: The m onopoly price is derived from the m onopoly quantity and the dem and curve:
120 – ½ Q = 120 – ½ × 55 = $92.50
Part E: The price elasticity of dem and at the m onopoly price is (MQ/Q) / (MP/P) = MQ/MP × P/Q.
At the m onopoly price and quantity, this is 1/$ × P*/Q* = –2 × $92.50 / 55 = -3.364
Alternative m ethod: m arginal revenue = price × (1 – 1/|ç|)
At Q = 55, P = 92.5 and m arginal revenue = 120 – 55 = 65.
Marginal revenue / price = 0.70270 = 1 – 1 / | 0 | A | 0 | = 1 / (1 – 0.7027) A 0 = –3.364
Jacob: A 1% reduction in price causes a 3.364% increase in quantity sold. Shouldn’t the m onopolist reduce
the price to gain the higher quantity?
Rachel: The m onopolist loses not just the 1% on the last item sold but 1% on every item sold. The revenue
lost m inus the profits from the 3.364% increase in quantity offsets the m arginal cost of the last item sold.
Part F: Consum ers’ surplus is the area under the dem and curve, down to the equilibrium price, and out to the
equilibrium quantity. Evaluate the area of this right triangle.
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P* = 92.5; Q* = 55.
From the dem and curve, if Q = 0, P = 120.
The vertices of the right triangle are (0, 120), (0, 92.5), (55, 92.5).
The area of the right triangle = ½ × 55 × (120 – 92.5) = $756.25.
Part G: Producers’ surplus is the area under the equilibrium price, down to the m arginal cost curve, and out
to the equilibrium quantity. This is a rectangle on top of a right triangle. The base of both the rectangle and
the right triangle is Q* = 55. The m arginal cost at a quantity of 55 is $55 + $10 = $65.
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The height of the right triangle is the m arginal cost at Q* – the m arginal cost at 0: $10 + $55 – $10 = $55.
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The height of the rectangle is P* m inus the m arginal cost at Q*: $92.5 – $65 = $27.50.
Producers’ surplus is (92.5 – 65) × 55 + ½ × (65 – 10) × 55 = $3,025.00.
Alternative: Producers’ surplus is a taller rectangle m inus the right triangle.
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The taller rectangle has a height of $92.5 – $10 = $82.5 and a base of 55 A area = $4,537.50
The right triangle has a base of 55 and a height of $65 – $10 = $55 A area = ½ × 55 × 55 = $1,512.50
Producers’ surplus is $4,537.50 – $1,512.50 = $3,025.00
Part H: If the m arket were com petitive, the equilibrium price and quantity are at the intersection of the supply
and dem and curves:
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P* = 120 – ½ Q = 10 + Q A Q* = 110 × b = 73.333
P* = 10 + 73.333 = 83.333 or P* = 120 – ½ Q* = 83.333
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Consum ers’ surplus is ½ × 73.333 × (120 – 83.333) = $1,344.44.
Producers’ surplus is ½ × 73.333 × (83.333 – 10) = $2,688.89.
Total social gain from com petition is $1,344.44 + $2,688.89 = $4,033.33.
Dead weight loss = $4,033.33 – $3,025.00 – $756.25 = $252.08.
" Alternative derivation: Dead weight loss = ½ × (73.333 – 55) × ($92.50 – $65) = $252.08.
Take heed: Som e exam problem s give the m arket dem and curve as quantity in term s of price. Do not use
this form to derive total revenue in term s of price.
Jacob: How do we solve problem s if we are given the dem and curve as Q = f(P)?
Rachel: Convert the dem and curve to the form P = f–1 (Q). The exercise gives both form s of the dem and curve.
Final exam problem s give only one form .
Jacob: The m arginal revenue curve here has twice the slope of the dem and curve and the sam e intercept (on
the vertical axis); is this always true?
Rachel: It is true for linear dem and curves. The Y-intercept is always the sam e, but the relative slopes depend
on the form of the dem and curve.
Jacob: The m arginal revenue curve lies below the dem and curve here; is that always true?
Rachel: It is true as long as the m arginal revenue curve is upward sloping.
Jacob: How do fixed costs affect the equilibrium price?
Rachel: In the short run, fixed costs do not affect the equilibrium price. In the long-run, if fixed costs exceed
producers’ surplus, the net profit is negative, and the m onopolist exits the industry.
Jacob: Is consum ers’ surplus lower with m onopoly than in a com petitive m arket?
Rachel: In general yes, but with caveats:
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If the industry is a natural m onopoly, there is no com petitive equilibrium .
A m onopoly m ay produce econom ies of scale; if they are large enough, consum ers’ surplus m ay increase.
In som e industries, m arket power is negligible even for the largest firm s, since barriers to entry are sm all,
but econom ies of scale are large. In these industries, m ergers are socially beneficial, since they do not
create m onopoly pricing but raise consum ers’ surplus. Exam ples are superm arkets, departm ent stores
(and m ost retail stores), groceries, ratio stations, and insurance.
Jacob: Is producers’ surplus greater for a com petitive m arket or a m onopolistic m arket?
Rachel: Producers’ surplus is greater for a m onopoly. If producers’ surplus were greater at the com petitive
price, the m onopolist would charge the com petitive price, not the m onopoly price.
Jacob: Is producers’ surplus + producers’ surplus greater for a com petitive m arket or a m onopolistic m arket?
Rachel: The sum of both surpluses is sm aller for a m onopoly. The decrease is the dead weight loss.
** Exercise 11.3: Dem and curve, m arginal cost curve, consum ers’ surplus, and producers’ surplus
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ABC is the only supplier in its m arket. Barriers to entry are not m aterial, so the m arket is contestable and
ABC acts as a com petitive frm . The com petitive price is $95 and the com petitive quantity is 500.
The governm ent grants ABC the right to supply the whole m arket, so ABC becom es a m onopoly. The
m onopoly price is $100, and the m onopoly quantity is 400.
The m arket dem and curve is linear: P = " – $ × Q. The m arginal cost curve is linear: MC = ( + * × Q. The
m arginal cost curve is the sam e for the m onopoly as for a com petitive firm .
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C.
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E.
F.
G.
H.
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the dem and curve (what are " and $)?
the m arginal cost curve (what are ( and *)?
consum ers’ surplus in the com petitive m arket?
producers’ surplus in the com petitive m arket?
net social gain in the com petitive m arket?
consum ers’ surplus in the m onopolized m arket?
producers’ surplus in the m onopolized m arket?
net social gain in the m onopolized m arket?
Part A: The dem and curve is P = " – $ × Q. The two m arkets show that
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Com petitive m arket: $95 = " – $ × 500
Monopoly m arket: $100 = " – $ × 400
This pair of linear equations gives
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$100 – $95 = $5 = –$ × (400 – 500) A $5 = $100 × $ A $ = 0.05
$100 = " – 0.05 × 400 A " = $100 + $20 = $120
The dem and curve is P = $120 – 0.05 × Q.
Part B: The m arginal cost curve is MC = ( + * × Q.
In the com petitive m arket, price = m arginal cost, so $95 = ( + * × 500.
For a m onopoly m arket, we set m arginal cost = m arginal revenue. For a linear dem and curve P = " – $ × Q,
m arginal revenue = " – 2$ × Q A m arginal revenue = $120 – 2 × 0.05 × 400 = $80, so $80 = ( + ä × 400.
This pair of linear equations gives
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$95 – $80 = $15 = * × (500 – 400) A $15 = $100 × * A * = 0.15
$80 = ( + 0.15 × 400 A ( = $80 – $60 = $20
The m arginal cost curve is MC = $20 + 0.15 × Q.
Part C: Consum ers’ surplus is a right triangle whose
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base is the equilibrium quantity and
height is the price at Q = 0 m inus the price for the equilibrium quantity.
For the com petitive m arket, consum ers’ surplus = ½ × 500 × ($120 – $95) = $6,250.
Consum ers’ surplus considers only the dem and curve, not the m arginal cost curve (supply curve). It is the
area above the equilibrium price and below the dem and curve.
Part D: Producers’ surplus is the area above the m arginal cost curve and below the equilibrium price. For a
com petitive m arket with a linear m arginal cost curve, it is a right triangle whose
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base is the com petitive quantity, and
height is the m arginal cost at the com petitive quantity m inus the m arginal cost at Q = 0.
Producers’ surplus = ½ × 500 × ($95 – $20) = $18,750.
Jacob: Is producers’ surplus always less than consum ers’ surplus in com petitive m arkets?
Rachel: The size of consum ers’ surplus and producers’ surplus depend on the elasticities of the dem and curve
and the m arginal cost curve. In this exercise, both curves are linear and the slope of the dem and curve is 0.05
/ 0.15 = one third the slope of the m arginal cost curve, so consum ers’ surplus is one third of producers’
surplus. The m ore elastic the curve, the less surplus is created. A horizontal dem and curve (perfectly elastic)
gives no consum ers’ surplus; a horizontal m arginal cost curve gives no producers’ surplus.
Part E: Social gain from trade = consum ers’ surplus + producers’ surplus = $6,250 + $18,750 = $25,000. W e
can also com pute the social gain from trade directly, as the area between the dem and curve and the m arginal
cost curve. W ith linear curves, the area is a triangle whose
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base is the equilibrium quantity, and
height is the difference between price and m arginal cost at a quantity of zero:
½ × 500 × ($120 – $20) = $25,000.
Part F: Consum ers’ surplus is com puted the sam e way in m onopolies as in com petitive m arkets: the area of
a right triangle whose
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base is the equilibrium quantity and
height is the price at Q = 0 m inus the price for the equilibrium quantity.
For the m onopoly, consum ers’ surplus = ½ × 400 × ($120 – $100) = $4,000.
Part G: Producers’ surplus for a m onopoly with a linear m arginal cost curve is a right triangle plus a rectangle.
For the right triangle:
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The base is the equilibrium quantity in the m onopolized m arket.
The height is the m arginal cost at the equilibrium quantity m inus the m arginal cost at Q = 0.
The area of this triangle = ½ × 400 × ($80 – $20) = $12,000.
For the rectangle:
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The base is the equilibrium quantity in the m onopolized m arket.
The height is the m onopoly price m inus the m arginal cost at the equilibrium quantity.
The area of this triangle = 400 × ($100 – $80) = $8,000.
Producers’ surplus = $12,000 + $8,000 = $20,000.
Part H: Social gain from trade = consum ers’ surplus + producers’ surplus = $4,000 + $20,000 = $24,000.
W e verify by com puting the dead weight loss, which is a triangle with
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height equal to the difference between the m onopoly price and the m arginal cost at the m onopoly quantity
($100 – $80)
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and width equal to the difference between the com petitive quantity and the m onopoly quantity (500 – 400).
The area of this triangle is ½ × ($100 – $80) × 100 = $1,000. The dead weight loss is the difference between
the social gains in the com petitive m arket and the m onopoly m arket: $1,000 = $25,000 – $24,000.
Graphic: consumers’ surplus and producers’ surplus for competitive and monopolistic markets.
The graphic shows consum ers’ surplus and producers’ surplus for com petitive and m onopolistic m arkets.
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A + B + C + D + E: consum ers’ surplus in com petitive m arket
F + G + H + J: producers’ surplus in com petitive m arket
A + B: consum ers’ surplus in m onopolized m arket
C + D+ F + G + J: producers’ surplus in m onopolized m arket
E + H: dead weight loss from m onopoly
** Exercise 11.4: Contestable m arket
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B.
C.
D.
E.
F.
G.
H.
Firm ABC is the only supplier in its m arket. Barriers to entry are insignificant, so the m arket is contestable
and ABC acts as a com petitive frm . The com petitive price is $100 and the com petitive quantity is 1,000.
The governm ent grants ABC the right to supply the whole m arket, so ABC becom es a m onopoly. The
m onopoly price is $120, and the m onopoly quantity is 800.
If the dem and curve is P = " – $ × Q, what are " and $?
If the m arginal cost curve is MC = 0 + ( × Q, what is (?
In the contestable m arket, what is consum ers’ surplus?
In the contestable m arket, what is producers’ surplus?
In the contestable m arket, what is the social gain from trade?
In the m onopolistic m arket, what is consum ers’ surplus?
In the m onopolistic m arket, what is producers’ surplus?
In the m onopolistic m arket, what is the social gain from trade?
Part A: The dem and curve is P = " – $ × Q. The two m arkets show that
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$100 = " – $ × 1,000
$120 = " – $ × 800
This gives
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$20 = $ × 200 A $ = 0.100
$100 = " – 0.1 × 1,000 A " = $200
The dem and curve is P = $200 – 0.100 × Q.
Part B: The m arginal cost curve is MC = 0 + ( × Q.
In the com petitive m arket, price = m arginal cost, so $100 = ( × 1,000 A ( = 0.100.
Take heed: Som e problem s say that m arginal costs are constant. If that were the case here, the m arginal cost
curve would be $100 + 0 × Q. This exercise assum es the m arginal cost curve passes through the origin.
Jacob: W hat is the constant $100?
Rachel: In a com petitive m arket, price equals m arginal cost.
Part C: Consum ers’ surplus is a right triangle:
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The base is the com petitive quantity.
The height is the price at Q = 0 m inus the com petitive price in the m arket.
Consum ers’ surplus = ½ × 1,000 × ($200 – $100) = $50,000.
Part D: Producers’ surplus is a right triangle:
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The base is the com petitive quantity.
The height is the m arginal cost at the com petitive quantity m inus the m arginal cost at Q = 0.
Producers’ surplus = ½ × 1,000 × ($100 – $0) = $50,000.
Jacob: Producers’ surplus = consum ers’ surplus in this exercise; is that usual?
Rachel: This occurs because the dem and curve and the m arginal cost curve are both linear and the m arginal
cost is zero at Q = 0. In practice, m arginal cost rises slowly with Q (the m arginal cost curve is relatively flat)
and the dem and curve is relatively steep. Consum ers’ surplus is generally larger than producers’ surplus in
a com petitive m arket. If m arginal cost is constant (does not vary with Q), producers’ surplus is zero.
Part E: Social gain from trade = consum ers’ surplus + producers’ surplus = $50,000 + $50,000 = $100,000.
Part F: Consum ers’ surplus is a right triangle:
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The base is the equilibrium quantity in the m onopolized m arket.
The height is the price at Q = 0 m inus the equilibrium price in the m onopolized m arket.
Consum ers’ surplus = ½ × 800 × ($200 – $120) = $32,000.
Part G: Producers’ surplus is a right triangle plus a rectangle. For the right triangle:
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The base is the equilibrium quantity in the m onopolized m arket.
The height is the m arginal cost in the m onopolized m arket m inus the m arginal cost at Q = 0.
The area of this triangle = ½ × 800 × ($80 – $0) = $32,000.
For the rectangle:
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The base is the equilibrium quantity in the m onopolized m arket.
The height is the m onopoly price m inus the m arginal cost in the m onopolized m arket.
The area of this triangle = 800 × ($120 – $80) = $32,000.
Producers’ surplus = $32,000 + $32,000 = $64,000.
Part H: Social gain from trade = consum ers’ surplus + producers’ surplus = $32,000 + $64,000 = $96,000.
W e verify by com puting the dead weight loss: a triangle with height ($120 – $80) and width ($1,000 – 800):
½ × $40 × 200 = $4,000 = $100,000 – $96,000.