**Question - **E321: Intermediate Microeconomics – Fall2013
Homework Assignment 5
Due: Thursday, November 21 (in class)
Remark: This a long homework assignment. For this reason you can get extra points. General
Equilibrium and Welfare Economics are two important topics in Economics. The fundamental
theorems of welfare economics are particularly interesting because we can use them to think in
a systematic way about the role of markets and governments.
Problem 1: (3 points) Pure Exchange Economy
Consider an economy with two consumers (� and �) and only two goods (1 and 2). The utility
functions are given by:
���(�1�,�2
...Read More
627408436;)=(�1�)1/2(�2�)1/2
���(�1�,�2�)=(�1�)1/3(�2�)2/3
The endowment of consumer � is (�̅1�,�̅2�)=(10,5), while the endowment of consumer � is
(�̅1�,�̅2�)=(5,10).
a) (0.50 points) Compute all the feasible allocations. Show them in the Edgeworth box.
b) (0.50 points) Find the excess demand functions for each good. Hint: First solve the
consumer’s problem for each consumer. Remember that the two conditions that
characterize the consumer’s decision are ����=��1
��2 and �1�1�+�2�2�=�1�̅1�+�2�̅2� for
�=�,� where �� is the price of good �=1,2. Denote by ���(�1,�2,�1�̅1�+�2�̅2�) the
quantity of good �=1,2 consumed by consumer �=�,� Then, the excess demand
function of good �=1,2 is given by ���=∑[���(�1,�2,�1�̅1�+�2�̅2�)−�̅��]�=�,�.
c) (0.50 points) Verify Walras’ Law. Hint: Recall that Walras’ Law is given by �1��1+
�2��2=0.
d) (0.50 points) Compute the Walrasian Equilibrium. DO NOT FORGET to compute
equilibrium quantities. Hint: Recall that the market clearing condition in market �=1,2
is given by ���=0.
e) (0.50 points) Compute all the Pareto efficient allocations. Hint: The Pareto efficient
allocations are all feasible allocations that satisfy ����=����.
f) (0.50 points) Show that the Walrasian Equilibrium allocation is Pareto efficient.
Problem 2: (3 points) One Consumer and One Producer Economy
Robinson Crusoe is the only survivor in his island. He produces coconuts employing his labor as
input. His utility function is given by:
��(��,�)=(��)1/2(24−�)1/2
where �� is the amount of coconuts he consumes and � is the time he expends working in a day
(there 24 hours available per day). The production function of coconuts is �=�(�)1/2 , where �
is the amount of coconuts produced and � is a measure of productivity.
a) (0.50 points) Compute the all the feasible allocations.
b) (0.50 points) Compute the excess demand functions of coconuts and labor and verify
Walras’s Law. In order to do so, denote by � the price of coconuts and by � the wage
rate.
c) (0.50 points) Compute the Walrasian Equilibrium of this economy. DO NOT FORGET to
compute equilibrium quantities.
d) (0.50 points) Compute the Pareto efficient allocations.
e) (0.50 points) Show that the Walrasian Equilibrium allocation is Pareto efficient.
f) (0.50 points) Comparative Statics: Compute the effect of an increase in � on the
Walrasian Equilibrium of this economy.
Problem 3: (3 points) 2x2x2 Economy
Consider an economy with two citizens (� and �), two goods (� and �) and two factors of
production (� and �). We will use the following notation:
�� is the amount of capital employed in the production of �,
�� is the amount of capital employed in the production of �,
�� is the amount of labor employed in the production of �,
�� is the amount of labor employed in the production of �,
� is the quantity of good � produced,
� is the quantity of good � produced,
�� is the quantity of good � consumed by citizen �,
�� is the quantity of good � consumed by citizen �,
�� is the quantity of good � consumed by citizen �,
�� is the quantity of good � consumed by citizen �.
The total endowment of capital and labor available in the economy is:
(�̅,�̅)
Production Functions are:
�=(��)1/3(��)2/3
�=(��)1/3(��)2/3
Preferences are:
���(��,��)=(��)1/2(�)1/2
���(��,��)=(��)1/2(��)1/2
The social welfare function is:
�(���,���)=(���)1/2(���)1/2
a) (1 point) Compute the socially optimal allocation. Hint: Start setting the welfare
maximization problem. Then, indicate all the conditions that a socially optimal allocation
should satisfy. Finally, go as far as you can solving the equations you obtain.
b) (1 point) First Fundamental Theorem of Welfare Economics. Let � be the wage rate, � the
rental cost of capital, �� is the price of good �, �� is the price of good �, �� the labor
endowment of citizen �, �� the capital endowment of citizen � , �� the labor endowment
of citizen �, and �� the capital endowment of citizen �. Then, the Walrasian Equilibrium
of this economy is given by:
��=1
3�̅ ,
��=2
3�̅
��=2
3�̅
��=1
3�̅
�
��
=1
3(2�̅
�̅)
23
�
��
=2
3(2�̅
�̅)
−1/3
��
��
=�̅
�̅
��=1
3[(1
2
�̅
�̅)
23
��+(1
2
�̅
�̅)
−23
��]
��=1
3[1
2(2�̅
�̅)
23
��+(2�̅
�̅)
−13
��]
��=1
3[(1
2
�̅
�̅)
23
��+(1
2
�̅
�̅)
−23
��]
��=1
3[1
2(2�̅
�̅)
23
��+(2�̅
�̅)
−13
��]
Show that a market economy leads to a Pareto efficient allocation but not necessarily to
the socially optimal allocation.
c) (1 point) Second Fundamental Theorem of Welfare Economics. Suppose that ��=2
3�̅ and
��=1
3�̅. How can we implement the socially optimal allocation using a market system?
In other words, how should we distribute the endowment of capital between � and � in
order to make the Walrasian Equilibrium allocation identical to the socially optimal
allocation?
Problem 4: (1 point)
a) (0.50 points) Define a Pareto efficient allocation.
b) (0.25 points) Briefly explain the first fundamental theorem of welfare economics.
c) (0.25 points) Briefly explain the second fundamental theorem of welfare economics. ...Read Less

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