EC 501/Zenginobuz
Fall 2021
Problem Set #7
Due 23:59, Dec 27, 2021
1) A small economy has only two consumers, Cahit and Derin. Cahit’s utility function is
U(x, y) = x + 154y1/2. Derin’s utility function is U(x, y) = x + 7y. At a Pareto optimal
allocation in which both individuals consume some of each good, how much y does Cahit
consume?
2) Mr. N’s utility function is UN(BN, PN) = BN + 4P1/2N and Mr. I’s utility function is
UI(BI, PI) = BI + 2P1/2I, where BN and BI are the number of good B and PN and PI are the
number of good P consumed by Mr. N and Mr. I respectively. If Mr. N’s initial endowment
is 4 units of B and 25 units of P and if Mr. I’s initial endowment is 2 units of B and 20 units
of P,
a) find the set of Pareto efficient allocations;
b) Find an allocation that will be in the core.
3) There are two goods in the world, milk (x) and honey (y), and two consumers, Milne and
&
Shepard. Milne’s utility function is 𝑢" = 𝑥" 𝑦"
and his endowment is ωm = (4, 4).
Shepard’s utility function is 𝑢' = 𝑥' 𝑦' and his endowment is ωs = (0, 0). Let pX = 1, and let
pY vary.
a) Is the original endowment Pareto optimal? Explain. Suppose we employ a lump-sum
tax on at Tm = − 4, and Shepard’s lump-sum transfer at Ts = 4.
b) Write down Milne’s new budget constraint. Solve for Milne’s optimal consumption
∗
∗
∗
∗
bundle, (𝑥"
, 𝑦"
), with 𝑥"
and 𝑦"
figured as functions of pY.
c) Write down Shepard’s new budget constraint. Solve for Shepard’s optimal consumption
bundle, (𝑥'∗ , 𝑦'∗ ), with 𝑥'∗ and 𝑦'∗ figured as functions of pY.
∗
∗
d) Solve for the competitive equilibrium; that is, the market-clearing (𝑥"
, 𝑦"
) and (𝑥'∗ , 𝑦'∗),
and the equilibrium pY.
e) Prove that the new equilibrium allocation is Pareto optimal.
4) Consider a two-consumer, two-good exchange economy. Utility functions and endowments
are
𝑢* (𝑥* , 𝑥, ) = (𝑥* 𝑥, ),
and 𝑒 * = (18,4),
𝑢, (𝑥* , 𝑥, ) = 𝑙𝑛𝑥* + 2𝑙𝑛𝑥,
and 𝑒 , = (3,6𝑝)
a) Characterise the set of Pareto-efficient allocations as completely as possible.
b) Characterise the core of this economy.
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5) There are two goods and two consumers. Preferences and endowments are described by
𝑢* (𝑥* , 𝑥, ) = 𝑀𝑖𝑛(𝑥* , 𝑥, )
and 𝑒 * = (30,0),
𝑣 , (𝑝, 𝑤) = 𝑤⁄2>𝑝* 𝑝,
and 𝑒 , = (0,20)
a) Find a Walrasian equilibrium for this economy
b) Do the same when 1’s endowment is e1 = (5, 0) and 2’s remains e2 = (0, 20).
6) An exchange economy has two consumers with expenditure functions:
𝑒 * (𝑝, 𝑢) = (3(1.5), 𝑝*, 𝑝, 𝑒𝑥𝑝B )*⁄&
𝑒 , (𝑝, 𝑢) = (3(1.5), 𝑝,, 𝑝* 𝑒𝑥𝑝B )*⁄&
If initial endowments are e1 = (10, 0) and e2 = (0, 10), find the Walrasian equilibrium.
7) There are 100 units of x1 and 100 units of x2. Consumers 1 and 2 are each endowed with 50
units of each good. Consumer 1 says, ‘I love x1 , but I can take or leave x2’. Consumer 2
says, ‘I love x2, but I can take or leave x1’.
a) Draw an Edgeworth box for these traders and sketch their preferences.
b) Identify the core of this economy.
c) Find all Walrasian equilibria for this economy.
8) Consider a simple exchange economy in which consumer 1 has expenditure function
1
(𝑝* + 𝑝, )𝑢
𝑒 * (𝑝, 𝑢) = C3
𝑢𝑝,
𝑢𝑝*
for 𝑝, ⁄2 < 𝑝* < 2𝑝,
for 𝑝* ≥ 2𝑝,
𝑝* ≤ 𝑝, ⁄2
and consumer 2 has expenditure function 𝑒 , (𝑝, 𝑢) = (𝑝* + 𝑝, )𝑢 for all (𝑝* + 𝑝, ).
a) Sketch the Edgeworth box for this economy when aggregate endowments are (1, 1).
Identify the set of Pareto-efficient allocations.
b) Sketch the Edgeworth box for this economy when aggregate endowments are (2, 1).
Identify the set of Pareto-efficient allocations.
9) In a two-good, two-consumer economy, utility functions are
𝑢* (𝑥* , 𝑥, ) =
𝑢, (𝑥* , 𝑥, ) =
Total endowments are (10, 20).
2
𝑥* (𝑥, ),
(𝑥* ), 𝑥,
a) A social planner wants to allocate goods to maximise consumer 1’s utility while holding
consumer 2’s utility at u2 = 8000/27. Find the assignment of goods to consumers that
solves the planner’s problem and show that the solution is Pareto efficient.
b) Suppose, instead, that the planner just divides the endowments so that e1 = (10, 0) and
e2 = (0, 20) and then lets the consumers transact through perfectly competitive markets.
Find the Walrasian equilibrium and show that it the same as the solution in part (a).
10) Suppose that in a single-consumer economy, the consumer is endowed with none of the
consumption good, y, and 24 hours of time, h, so that e = (24, 0). Suppose as well that
preferences are defined over ℝ,L and represented by u(h, y) = hy , and production
possibilities are Y ={ ( − h , y ) | 0 ≤ h ≤ b and 0 ≤ y ≤ √ h }, where b is some large positive
number. Let py and ph be prices of the consumption good and leisure, respectively.
a) Find relative prices py/ph that clear the consumption and leisure markets simultaneously.
b) Calculate the equilibrium consumption and production plans and sketch your results in
ℝ,L .
c) How many hours a day does the consumer work?
11) Consider a simple Robinson Cruose economy. There is an initial endowment of one day of
time, T, per day of calendar time. There is no leisure. Time can be used to produce a good
X or a good Y. The production function for X is given by 𝑥 = >𝑇N , and the production
function for Y is given by 𝑦 = >𝑇O , where TX and TY are time devoted to production of
good X and Y, respectively. Let 𝑇N + 𝑇O = 1. The preferences are represented by the utility
function 𝑈 (𝑥, 𝑦) = 𝑥𝑦
a) Find the Pareto efficient allocation for this economy.
b) What are the equilibrium prices that will support the efficient alloction as a competitive
equilibrium?
12) Let the demand and supply relations involving two markets be as follows:
𝐷* (𝑝* , 𝑝, ) = 8 − 4𝑝, − 𝑝* ;
𝑝, ≤ 2
𝑆* (𝑝* , 𝑝, ) = 3𝑝*
𝐷, (𝑝* , 𝑝, ) = 24 − 6𝑝* − 𝑝, ;
𝑝* ≤ 4
𝑆, (𝑝* , 𝑝, ) = 5𝑝,
Demonstrate that each market has an equilibrium when the other's price is fixed. Show,
however, that no pair of prices exists for the two markets at which they are in equilibrium.
Does this supply-demand system provide a counter example for the existence of general
equilibirum prices?
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