1.  Consider the following numerical version of the manufacturing/fishing externality problem that was discussed in class. Let:

❼ gm , Pm  denote the quantity produced and price of manufactures ❼ gf , Pf  denote the quantity caught and price of fish

❼ z denote the amount of pollution created by the manufacturer.           Let the technologies for the two firms be represented by the cost functions:

❼ Manufacturing: cm (gm | z) = 3gm(2) + (z _ 2)2

❼ Fishing: cf (gf | z) = 3gf(2) + 3z

That is, cp (gp | z) is the total cost of producing gm  manufactured goods when the level of pollution is z.  Similarly, cf (gf | z) is the cost of catching gf  fish when the level of pollution is z. Profit (which firms maximize) is defined as the difference between total revenue and total cost.

Let Pm  = 13!5, Pf  = 12, and assume that pollution is generated through manufacturing production according to the function : z = !5gm .

a. How much pollution, z* , will be produced if the firms are independent enterprises?

b. How much pollution will be produced if the firms merge? If your answer is different from that derived in part a., what accounts for the difference?

c. Find the Pigouvian tax, π , that induces the independent manufacturer to produce the same amount of pollution as it would if it were merged with the fishing firm.

2.  Consider an economy with one private good (z), one public good (y), and two households with the following utility functions:

vh (zh | y) = ←zh + (1 _ ←) ln y              于or h = 1| 2!

Each household is endowed with Th units of the private good, and there is a technology for producing the public using the private good:

y = 礻zy(γ)                礻 2 0   | α 卡 1

where zy  is the amount of the private good used to produce the public good, rather than consumed.

Derive the “Samuelson” condition for this economy (i.e. the condition satisfied by any Pareto efficient allocation).

3. There are two drivers on a public highway. Driver h chooses speed sh , with his/her utility increasing in how fast he/she drives.  The higher either driver’s speed, however, the more likely that they are involved in a mutual accident.

Let δ(s1 | s2 ) denote the probability of an accident if the drivers choose speeds s1  and s2 , where

aδ

In the event of an accident, each driver incurs monetary cost, dh . Drivers have quasi- linear preferences over speed and money, and maximize expected utility:

E ┌vh (sh | dh )┐ = u (s )hh _ δ(s | s )d12h |

h = 1| 2!