Question 1. (10 marks) Consider a pure exchange economy in which households have strictly
convex preferences represented by
?
?(?1
?
, ?2
?) = (?1
? − 10)
1/2
(?2
? − 10)
1/2
and
?
?(?1
?
, ?2
?) = (?1
?)
1
2(?2
?)
1
2
Consider each of the following initial endowments ((?1
?
,?2
?), (?1
?
,?2
?)) of the two goods,
a) ((30,60), (70,40)) and b) ((10,10), (90,90))
i) (6 marks) For each case, show whether the initial allocation is efficient. Do this both
algebraically, and by sketching the initial allocation in an Edgeworth box including
each households’ indifference curves through the point.
ii) (4 marks) For each case, find the competitive market equilibrium if one is feasible.
What would the new allocation be? Draw a new Edgeworth box with both the initial
allocation and the market equilibrium allocation you found (include the budget line,
and the indifference curves through the allocations).
Question 2. (20 marks) Consider a model of a negative consumption externality. Households
have strictly convex preferences over composite (?
?
, ?
?
respectively) and a level of activity
chosen by household A measured by 0 ≤ ? ≤ 100. The households’ utility functions are
?
?(?
?
,?) = (? − 10)
1/2
(? − 10)
1/2
and
?
?(?
?
,?) = ?
1/2
(100 − ?)
1/2
Suppose the initial endowment of composite is ?
? = 30 and ?
? = 70. Laws currently restrict
Household A’s activity, to the legal maximum is 0 < ?̅< 100.
i) (6 marks) Construct a single Edgeworth box measuring composite horizontally and
household A’s activity level vertically, and represent the initial allocation along with
the mutually beneficial set when the legal limit is
a) ?̅= 20 b) ?̅= 60.
ii) (2 marks) Suppose a competitive market is introduced for the level of household A’s
activity to the extent that it deviates from ?̅. The price p per unit of activity is taken
as given by each household. What would each households’ budget constraint be
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when they are endowed with composite (?
? = 30 and ?
? = 70) and then purchase
or sell units of activity?
iii) (4 marks) Find each households’ optimal choice functions for activity level, ?
?(?)
and ?
?(?).
iv) (4 marks) Find the competitive market price p* and the equilibrium level of
household’s activity A in terms of ?̅.
v) (4 marks) Explain using graphical illustration in an Edgeworth box and calculations
how the assignment of legal rights (?̅= 20 vs. ?̅= 60) impacts
a) The efficiency of the resulting allocation
b) Distribution of composite (or wealth) between the households.
Sample Solution