ECON 290 - CANADIAN MICROECONOMIC POLICY
IRYNA DUDNYK
Tutorial 7. Free-Riding, Congestible Public Goods, Pricing of Public
Services.
Solutions.
Problem 1. The purpose of this problem is to help you understand the free-rider problem.
Recall Problem 2 from Tutorial 6 about small town and trees:
5 people with M BH = 100 − .5Q - consumers with high benefits
10 people with M BL = 50 − .25Q, consumers with low benefits from trees
M SB = 1, 000 − 5Q,
M C = 25, efficient output of the public good QE = 195.
Suppose that the people in the small town want to implement Lindahl scheme for the provision of the efficient number of trees as an equilibrium situation1 . Check whether the efficient
Q = 195 is achievable in Lindahl equilibrium.
Equilibrium quantity of the public good under Lindahl scheme must satisfy the following
conditions
1. Each person’s contribution should be such that each person demands the
same amount of the public good. Since we only want to check whether Q = 195
can be achieved in equilibrium, simply find the per unit contributions (individual prices)
at which all people demand 195 trees. Substitute Q = 195 into the demands to find the
prices:
PH = 100 − .5 · 195 = 2.5
PL = 50 − .25 · 195 = 1.25
If people with high benefits have to pay 2.50 per tree and people with low value have to
pay 1.25 per tree, all people will demand the same quantity of 195 trees.
2. Sum of the contributions per unit is equal to the MC. Of course if prices are
sufficiently low, people will demand any quantity, but that is of little help with the finance: what people are willing to pay together for each unit of the public good must be
sufficient to cover the actual cost. Check whether sum of the contributions per tree covers
the marginal cost of a tree.
5 · 2.5 + 10 · 1.25 = 25 - these contributions will be sufficient to cover the marginal cost
of the trees. Notice the following, if there was a fixed cost, for example in order to plant
any trees at all the city would have to purchase a tractor, the contributions would not
suffice to cover the entire cost.
3. Participation in the cost sharing agreement is voluntary - if somebody does
not want to pay his share, he cannot be forced to participate. Which, in other words
also means that is somebody prefers to play a free-rider strategy, he will not participate
in sharing the cost. So let’s check whether anybody has an incentive to free-ride in this
1
Keep in mind the intuition behind the concept of equilibrium: a situation in which nobody wants to
change their behaviour. In simple words, what we are going to verify is whether it is possible to share the
cost and maintain efficient number of trees as a situation in which no one wants to change behavior - that is
free-ride.
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ECON 290 - CANADIAN MICROECONOMIC POLICY
IRYNA DUDNYK
case. Let’s examine whether a person with M BL has an incentive to free-ride. To check
whether he will choose to free ride or share the cost we have to evaluate his net benefits
in each case:
Share cost if the person shares the cost, 195 trees will be provided. In this case the net benefits
of sharing the cost are represented simply by the consumer surplus at Lindahl price
and Q = 195, which is equal to CS = .5(50 − 1.25)195 = 4, 753.125.
Free Ride if the person chooses to free ride while the remaining 14 people will provide trees
according to their true marginal benefits, 194 trees will be planted. The demand by
the remaining 14 people is D = 5 · M BH + 9 · M BL = 950 − 4.75Q, which equals to
MC=25 at Q=194.7, for simplicity suppose that since the tree is indivisible, they
have not enough money to make it to 195 trees, so they will purchase 194. The
net benefits under free riding are equal to the total value of 194 trees: the person
enjoys the benefits of 194 trees, and does not pay a penny of the cost. Marginal
benefit of the 194th tree is M BL = 50 − .25 · 194 = 1.5 and the Total value of 194
· 194 = 4, 995.5. When you look at a diagram with M BL you
trees is T V = 50+1.5
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will notice that these numbers make sense. If the person shares the cost the total
contribution to the public good= 1.25 · 195, which is over 200 dollars. In case the
person does not share the cost, he saves more than $200! What does he lose? he
loses the marginal value of the 195th tree, which is only 1.25. In this case it is clear
that the benefits of free riding outweigh the loss from lower number of trees in town.
Based on our calculations we can conclude that it makes sense to free ride for at least
one person with low marginal benefits of the trees. This means that 195 trees is not an
equilibrium quantity even when costs are shared and Lindahl scheme will be prone to
free-rider problem.
Problem 2. This problem deals with a special case of goods. Let’s consider such things as:
public transit system, bridges, ferries, highways.
(a) Can we characterize goods above as public goods? Explain your answer.
These goods can be characterized as congestible public goods.
(b) City council asks you to give a recommendation on the optimal capacity for a new highway; council also wants to know whether there should be a toll and if yes, what should
be the price. Use a demand-supply diagram to give a recommendation. The city council
also points out to you that demand for the highway is cyclical during the day: it is high
during rush hours and low during off-peak hours. City council wants to use public funds
efficiently: highway should not be too large to be idle at any time, but of course should
not be unreasonably narrow. City council also wants to provide efficient service: the
highway should not be congested at any time. To simplify the analysis assume that MC
of expanding highway to accommodate additional cars is zero.
Demands draw two demand curves, one for rush hours and one for off-peak.
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ECON 290 - CANADIAN MICROECONOMIC POLICY
IRYNA DUDNYK
Supply the supply of the ‘highway’ can be thought of as its capacity. After the highway is
installed the capacity is fixed, also in contrast to public transit, the physical MC
of another person taking the highway is zero. The question is where would you
recommend to place this fixed supply. To simplify the analysis assume that MC
of accommodating an extra car on the highway is zero. In this case The efficient
capacity is at the horizontal intercept of the off-peak demand: MV of an extra car
on the highway is zero and the highway is wide enough to accommodate all people
during off-peak hours. Making highway any larger would be wasteful as there would
be some idle capacity during off-peak time (which is clearly a problem when MC is
not zero, which is more realistic).
Pricing If we install the capacity at the quantity demanded off-peak at zero price, there is
no point to charge the toll during off-peak hours - there is no congestion. If there is
no toll during the rush hour - there will be a congestion, which is clearly inefficient.
During rush hour the price should be such that demand for using the highway is
equal to the capacity and price should be zero during off-peak, this mechanism is
called peak-load pricing. This price scheme is better than charging a flat fee
throughout the day: there is no need to discourage use of highway during off-peak
hours. A question might arise: is such user fee fair? Toll fee during the rush hour
is a good mechanism to allocate the capacity to drivers who value it the most: when
toll is charged only people who’s MB is greater than the price will choose to use
highway; people who’s MBs are lower than the price will take alternative route or
will stay at home, which is efficient because they will not impose the negative externality on users with high MBs. This pricing scheme will also encourage people who
have flexible schedule to reallocate use of highway from the rush hour to off-peak.
In conclusion out of three alternative ways to price the highway: zero price, flat
price and peak-load pricing, the latter improves on efficiency compared to the first
two.
Discussion Question. Consider government provision of the the following services: transit
and utilities - these are price excludable public goods that are financed through sales revenues.
Things such as water, electricity and some means of transportation can be thought of as life
necessities. Do you think government should allow low-income individuals use these services
for free. Explain why yes or why not. Use a diagram to support your answer.
Draw a demand (downward sloping) and a supply (upward sloping) curves for water. Show
efficient usage of water. Show how much water will be used if it is free (P=0), show the DWL
in this case. If water is not priced, people fail to take into account the marginal costs and
use too much (to the point where MSC of providing the good is higher than the MSB created).
Recall the externalities: when people do not bear the full cost of their decision they do not take
it into account which leads to inefficiencies.
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