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