Problem Set 4: Externalities

Problem Set 4: Externalities Microeconomics 2 Matthew Robson University of York

Question 1 Consider a two-commodity (and ), two-consumer (and) pure-exchange economy. Suppose consumer ’s utility function is and she is endowed with 100 units of and zero units of . Suppose consumer B’s utility function is and she is endowed with zero units of x and 100 units of y. Derive the conditions for Pareto optimality in the above economy(notice that an externality is present).

Question 1 Choose an allocation to maximise: subject to: Feasible Allocations

Question 1 Form the Lagrangian: Where and are Lagrange multipliers. Derive the FOC’s: (1) (2) (3) (4) (5) (6) (7)

Question 1 (1)=(3) (2)=(4)

Question 1 Condition required for Pareto Optimality

Question 1 (Extension) Generalise the Utility Function • Use as a parameter to enable differences in the preference for by . • We can then see no externalities, positive externalities and negative externalities.

Question 1 Form the Lagrangian: Where and are Lagrange multipliers. Derive the FOC’s: (1) (2) (3) (4) (5) (6) (7)

Question 1 (1)=(3) (2)=(4)

Question 1 Condition required for Pareto Optimality (take this further to get )

Question 1 From (6): From (7): Plug into:

Question 2 For the economy in Question 1, derive the conditions that prevail in a general competitive equilibrium, and thus demonstrate that the general competitive equilibrium allocation is not Pareto optimal.

Question 2 General Competitive Equilibrium • Both consumers are maximising utility subject to their budget constraints • ) subject to • ) subject to • Market Clear (Demand = Supply)

Question 2 For Consumer A: FOC: (1) (2) (3) (1)=(2)

Question 2 For Consumer B: FOC: (1) (2) (3) (1)=(2)

Question 2 • Therefore, in General Competitive Equilibrium, we have: • However, for Pareto Optimality our (generalised) condition is: • Then, GE = PO when (i.e. no externalities)

Problem Set 4: Externalities