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Tutorial Assignment 4: Eco 3020F_Advanced Microeconomics
The submission deadline is Monday 17 March at 12:00.
1. Determine whether externality is present and why under the following situations;
a. Upon visiting a doctor, a TB patient happened to walk through dense crowd of
people in hospital wards in Sera-Leone
In his health production decision (seeing a doctor), the patient transmit TB to others
in medical ward, consequently the latter suffer from sickness, incur medical care
cost, loss of production and leisure time, all of which are negative externality.
b. In driving you from home to UCT, your parent(dad/mom) takes the highway (main
road).
By taking highway, your dad increase congestion for other drivers, which represents
negative externality
c. Your parents had your primary school siblings received flu shot during last winter
By vaccinating your siblings against flu, your parents are not only protecting them,
but also protecting their classmates/schoolmates from the flu, which is positive
externality
d. A doctor and a baker share an office building. The baker’s machine is louder when it
runs.
The baker’s loud machine disturbs the doctor’s medical practice in which case the
doctor cannot treat patients when the baker’s machine is running. Consequently, the
doctor’s medical services production decreases, leading to a fall in his earnings. This
is a negative production externality.
e. A BMW automotive dealer trained its logistic department staff on new skill of
advanced logistic management shortly before its decision to close its branch in South
Africa. Soon after its close down, the entire logistic department staff joined Toyota
dealers in South Africa.
The new skill gained via training by BMW, meant increased productivity of the staff, who latter
joined Toyota dealer. Thus, Toyota dealer enjoys such increased productivity or reduced
production cost without incurring any cost of capacity building (training) on these staff. This is
also positive production externality.
2. Suppose that in one of suburbs of Durban, South Africa, a coal producing factory is
located upstream from fishery. The factory produces coal using labour with the
following production function;
𝑦 = 𝑓 𝑙! = 𝑙!
Where 𝑦 the quantity of coal is produced and 𝑙! is the quantity of labour employed to produce
the coal.
In producing each unit of coal, the factory, unavoidably, generates waste, 𝑤 which it dumps into
the river downstream. The waste generation function of the factory is given as;
𝑒 = 𝑔 𝑦 = 3𝑦!
On other hand, the fishery produces fish according to the following production function
𝑓 = ℎ 𝑙! , 𝑒 = 4 𝑙! − 𝑒
Where 𝑙! is the quantity of labour used to produce the fish. Suppose again that both coal
factory and fishery face completive respective product markets and labour market such that coal
factory charges a unit price 𝑝! = 4, fishery charges a unit price 𝑝! = 4 and both of them pay
wage rate of 𝑤 = 2
Based on this set of information workout the following problems;
a. If the firms are to operate individually, what are the quantity of coal and waste that
maximize the profit of coal company? Are they socially optimal quantities? Why or why
not?
Solution(i) 𝑀𝑎𝑥!! 𝜋! = 𝑝!𝑦 𝑙! − 𝑤𝑙! = 𝑝! 𝑙! − 𝑤𝑙! = 4 𝑙! − 2𝑙!
F.O.C → !!!!!!
= 2𝑙!!!! = 2→ 𝑙!
∗ = 1
→ 𝑦∗ = 1=1 → 𝑒∗ = 3𝑦∗! = 3
Profit: 𝜋!∗ == 4 ∗ 1− 2 ∗ 1 = 𝟐
b. In the same token workout the optimal quantity of fish that maximise profit of fishery.
Solution (ii). 𝑀𝑎𝑥!! 𝜋! = 𝑝!𝑓 − 𝑤𝑙! = 𝑝! 4 𝑙! − 𝑒 − 𝑤𝑙! = 4 ∗ 4 𝑙! − 𝑒 − 2𝑙!
F.O.C → !!!!!!
= 8𝑙!!!! = 2→ 𝑙!
∗ = 16
→ 𝑓∗ = 4 16− 𝑒,𝑤ℎ𝑒𝑟𝑒 → 𝑒∗ = 3 → 𝑓∗ = 4 ∗ 16− 3 = 13
Profit: 𝜋!∗ = 4 ∗ 4− 12− 2 ∗ 16 = 𝟐𝟎
c. Suppose now that the fishery buys out coal company. In that case show(prove) that the
new set of optimal quantities for coal, waste and fish that maximize the joint profit is
social optimal outcome. Is the joint profit obtained here greater than the sum of profits in
(a) and (b)?
Solution (iii)
𝑚𝑎𝑥!! !!,𝜋 = 𝜋! + 𝜋! = 𝑝! 𝑙! − 𝑤𝑙! = 4 𝑙! − 2𝑙! + 4 ∗ 4 𝑙! − 𝑒 − 2𝑙!
Note here that 𝑒 can be given as
𝑒 = 𝑔 𝑦 = 3𝑦! = 3𝑙!!∗!! = 3𝑙!
This implies that
𝑚𝑎𝑥!! !!,𝜋 = 𝜋! + 𝜋! = 𝑝! 𝑙! − 𝑤𝑙! = 4 𝑙! − 2𝑙! + 4 ∗ 4 𝑙! − 3𝑙! − 2𝑙!
F.O.C → !"!!!
= 2𝑙!!!! − 14 = 0 → 𝑙!
∗ = 1/49
𝑦∗ = 1/49 = 1/7 → 𝑒∗ = 3𝑦∗! = 3/49
F.O.C → !"!!!
= 8𝑙!!!! = 2→ 𝑙!
∗ = 16
→ 𝑓∗ = 4 16− 𝑒,𝑤ℎ𝑒𝑟𝑒 → 𝑒∗ = !!"→ 𝑓∗ = 4 ∗ 4− !
!"= 15 46/49
Profit: 𝜋∗ = 4 ∗ !!− 2 ∗ !
!+ 16 ∗ 4− 12 ∗ !
!"− 2 ∗ 16 = 𝟑𝟐.𝟐𝟗
This profit level is greater than the sum of profits from coal production and fishery when
the two firms operate individually i.e. 𝟑𝟐.𝟐𝟗 > 𝟐𝟐
d. Suppose now that the fishery has the right to clean river water and willing to let coal
company dump the waste into the river for price 𝑞 per unit of waste. What is the
equilibrium price of waste and how much compensation would this scheme earn the
fishery?
e. Instead suppose that coal company has the right to pollute the river and is willing to cut
waste discharge (to abate waste) at price 𝑞 per unit of the waste. Workout the
equilibrium price that coal company charges fishery to cut the waste
f. Suppose now that the government intends to levy tax on coal production thereby protect
fishery. What would be tax per unit of waste to cut coal production to its socially optimal
level? Does this change if the government doesn’t observe the waste quantity directly, but
opted to levy tax per unit of coal? Also work out tax revenue that accrues to the
government under both scenarios.
Solution (iv) 𝑀𝑎𝑥 𝜋! = 𝑝!𝑦 𝑙! − 𝑤𝑙! − 3𝜏𝑙! = 4 𝑙! − 2𝑙! − 3𝜏𝑙!
where 𝜏 is the tax rate on externality;
𝐹.𝑂.𝐶 → !"!!!
= 2𝑙!!!! − 2− 3𝜏 = 0. From (c), we know that optimal level of labor in
coal production being 𝑙!∗ = 1/49, thus, we can solve for 𝜏 as follows
2 ∗149− 2− 3𝜏 = 0 → 𝜏∗ = 4
Tax revenue under this scenario, will be 𝑒∗ ∗ 𝜏∗ = 3/49 ∗ 4 = 0.48, where 𝑒∗ is obtained from
(c)
g. In light of Coase theorem, will there be any differences between solutions in (d), (e) and
(f) in terms of the quantity of coal produced?
3. Now suppose that the productions costs in outputs for these firms are given as follow;
𝑐 𝑦 = 𝑦! coal producing firm and 𝑐 𝑓 = 𝑓! + 4𝑦. The pollution function is given as
𝑒 = 𝑔 𝑦 = 𝑦. Suppose again that each of them faces a prevailing market price 𝑝! = 40
and 𝑝! = 20 respectively. Using this information set;
a. Determine the level of coal production and fishery production that maximize the joint
profit if the two firms operate under merger. Also compute the joint profit.
Solution (v) 𝑀𝑎𝑥 𝜋 = 40𝑦 − 𝑦! + 20𝑓 − 𝑓! − 4𝑦
F.O.C → !"!"= 40− 2𝑦 − 4 = 0→ 𝑦∗ = 18
F.O.C → !"!"= 20− 2𝑓 = 0→ 𝑓∗ = 10
Profit : 𝜋 = 40 ∗ 18− 18! + 20 ∗ 10− 10! − 4 ∗ 18 = 𝟒𝟐𝟒
b. Determine per unit Pigovian tax to obtain optimal outputs in (a)
Solution(vi)
𝑀𝑎𝑥 𝜋! = 40𝑦 − 𝑦! − 𝜏𝑒, 𝑠. 𝑡.𝑦 = 𝑒
F.O.C; !!!!"
= 40− 2𝑦 − 𝜏 = 0 → 40 = 2𝑦 + 𝜏
F.O.C, uses the fact that 𝑒 = 𝑔 𝑦 = 𝑦 . Note also from (a) we know that 𝑦∗ = 18
which implies 𝜏 = 4 and this tax equals marginal external cost of coal productions.
c. Suppose now that the fishery has the right to clean river water and willing to let coal
company dump the waste into the river for price q per unit of waste. What is the
equilibrium price of waste and the corresponding waste pollution level?
Solution(vii)
Coal firm: 𝑀𝑎𝑥 𝜋! = 40𝑦 − 𝑦! − 𝑞𝑒, 𝑠. 𝑡:𝑦 = 𝑒
!!!!"
= 40− 2𝑒 − 𝑞 = 0→ 𝑞 = 40− 2𝑒 (𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑜𝑟 𝑝𝑜𝑙𝑙𝑢𝑡𝑖𝑜𝑛) (*)
Fishery: 𝑀𝑎𝑥 𝜋! = 20𝑓 − 𝑓! − 4𝑒 + 𝑞𝑒
!!!!"
= −4+ 𝑞 = 0 → 𝑞∗ = 4(supply of pollution right)
The solution implies that supply of pollution right=demand for pollution right at the optimal
outcome i.e. 4 = 40− 2𝑒 → 𝑒∗ = 18