Lecture notes
Microeconomics 1 Block 3 Market Failures Lecture Notes 20/21
- Module
- Microeconomics 1
- Institution
- The University Of York (UOY)
Microeconomics 1 Block 3 Market Failures Lecture Notes 20/21
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Micro Block 3: Market Failures3.1 Monopoly
De nition
A market is a monopoly when there is a single rm o ering the good. Thus, the rm is facing the
full market demand, and consequently can easily manipulate the market price by changing its
output level.
Origins of monopolies
Whenever the total cost of producing any amount of output y is minimal when production is done
by one rm instead of several, we say that this rm is a natural monopoly. Natural monopolies
often occur in network industries.
Monopolies may occur due to control of a rare resource, input or patent; in this case, the rm can
be the only one to produce the good. Furthermore, the government can grant a rm monopoly
position on the market; often, post services are an institutional monopoly. Finally, predatory
behaviour can lead to monopolies as a rm can prevent other rms from entering the market by
using aggressive pricing strategies (e.g. selling at a very low price for some time, even if it means
incurring losses). It could also be the case that a few rms decide to collude and e ectively
behave as a monopoly.
Demand and inverse demand
The market (or aggregate) demand D(p) for some good is simply the sum of all individual demands
for that good. The amount y of good that can be sold on the market is therefore y = D(p). Recall
that we can invert the demand function to get the inverse demand function denote p(y). The
inverse demand function de nes the maximum price at which y units of a good can be sold.
Revenue function
The revenue function of the monopoly rm is denote r(y) = p(y)y. It de nes the revenue of the
monopoly for producing y units of a good, which can be sold at price p(y). Note that in perfect
competition revenue would be py, because p is given (the rm would be a price taker); in the
monopoly case, the rm can alter the market price by adjusting output.
1
fi fi fi fi fifi fifi fffi fi fi fi fifi ff fi
, Marginal revenue function
The marginal revenue function, MR(y) = r(y)/ y de nes the additional revenue the rm gets from
selling one additional unit of output. Marginal revenue can also be de ned by MR(y) = p(y) + [ p(y)/
y]y. Whenever the monopoly decides to increase its output by one unit, it has two e ects on the
revenue: selling one more unit of output gets the rm p(y), and also pushes the price down on all
the units it is selling.
Pro t maximisation
The monopoly’s pro t maximisation problem can be written maxy p(y)y - c(y) where c(y) is the total
cost function. The rst order condition for this problem is simply r(y)/ y - c(y)/ y = 0. This implies
that at the optimal choice of output y*, the marginal revenue should be equal to the marginal cost,
that is MR(y*) = MC(y*).
Figure 39: pro t maximisation
Note: In this gure, we show the optimal output choice for a monopoly with the cost curve c(y) = F
+ y + βy2 and facing the inverse demand pD = a - by. We chose F = 1, = 1, β = 0.5, a = 10 and b
= 1.
EXAMPLE The optimality condition MR = MC makes sense. Suppose the monopoly is choosing
to produce y units of output. Suppose that for this level of output, MR = 5 and MC = 4, so MR >
MC. Can y be pro t maximising? No. We see that if the monopoly was selling one more unit of
output, it would get £5 extra of revenue, and would have to pay only £4 to produce this extra unit.
Thus the pro t would increase, and so y cannot possibly pro t maximising. With this example, we
see that at the pro t maximising level of output, it must be that MR = MC. Graphically, we can
represent the optimal output choice of the monopoly as in gure 40.
EXAMPLE Consider the example from gure 40. The cost function is c(y) = 1 + y + 0.5y2, hence
the marginal cost MC(y) = 1 + y. The demand function is D(p) = 10 - p, from which we conclude
that the inverse demand function is p(y) = 10 - y. The revenue is thus r(y) = 10y - y2 and so the
marginal revenue is MR(y) = 10 - 2y. We know that the monopoly is choosing output such that
MR(y) = MC(y), that is 10 - 2y = 1 + y. Thus, the monopoly choose the optimal quantity of output
y* = 3, which he can sell for p* = 7.
2
𝝏 𝛼 fi fi fi fifififi fi𝝏 𝝏 fi fi fi 𝝏 fi𝝏 𝛼 𝝏 𝝏 fi ff 𝝏
, Figure 40: the monopolist behaviour
Note: In this gure, we show the optimal output choice for a monopoly with the cost curve c(y) = F
+ y + βy2 and facing the inverse demand pD = a - by. We chose F = 1, = 1, β = 0.5, a = 10 and b
= 1.
Marginal revenue and elasticity
The marginal revenue is MR(y) = p(y) + [ p(y)/ y]y = p(y)(1 + [y/p(y)][ p(y)/ y]). After some algebra,
we get that MR(y) = p(y)[1 + 1/ (y)] where (y) is the own-price elasticity of demand (y) = [ D(p)/ p]
[p/D(p)]. The own-price elasticity of demand measures the percentage change in demand after a
percentage change in the price of the good.
Usually, the own-price elasticity of demand is negative (since when the price increases, usually
demand decreases). Thus, we have (y) > 0. In addition, the monopoly won’t operate at output
levels where demand is too inelastic, i.e. when 0 > (y) > -1. The reason is that in this case, for
some level of output, the marginal revenue is negative, meaning that the rm can increase its
revenue by decreasing output. Since reducing output also decreases the cost, the rm will
increase its pro t. Thus, that level of output could not be optimal.
The optimal choice of output y* satis es p(y*)[1 + 1/ (y*)] = MC(y*). This is another way of writing
the MC = MR optimality condition. Rearranging the terms, we get p(y*) = [MC(y*)] / [1 + 1/ (y*)].
The optimal pricing strategy of the monopoly is to charge a markup over the marginal cost
(remember: (y) < -1).
Consider the example from gure 40. We saw that the monopoly will choose y* = 3 and sell at a
price p* = 7. Lets just check that the price it chose is consistent with the formulae above. The
elasticity of demand at y* is = -7/3. The marginal cost at y* is MC(y*) = 1 + 3 = 4. According to
3
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𝜖fi fi fi𝜖 𝜖 𝜖fi 𝝏 𝜖 𝝏 𝜖𝜖 𝝏 𝛼𝝏 fi fi𝜖 𝝏𝜖 𝝏
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