Department of Economics
Trent University
ECON4000H – Advanced Microeconomics
Winter Semester, 2021
Pete
orough Campus
Assignment #2
Due Fe
uary 28, 2021 at 11:59pm EST
Instructions:
This assignment has 4 questions for a total of 50 marks, and is worth 25% of the course grade.
Marks for each question are as indicated, and are evenly divided among the components of the
question, except if indicated otherwise. Answers to the questions must be word-processed or hand-
written and scanned, and be legible and concise; written explanations, where requested, should
have no more than a
ief paragraph of content. While students may work together on the exam,
they must develop, write up and submit answers independently. Answers must be submitted as a
PDF file via the assignments page of the course website.
Questions:
1. (12 marks) Consider the production function ?(?, ?) = ????? where ? > 0, ? > 0, ? > 0
and ? + ? < 1 and where ? and ? denote labour and capital, respectively. Let ?, ? and ?
denote the prices of labour, capital and output, respectively, and let ? denote output.
a. Solve the profit maximization problem to show that the factor demand and supply
functions are:
?(?, ?, ?) = [
???1−???
?1−???
]
1
1−?−?
?(?, ?, ?) = [
?????1−?
???1−?
]
1
1−?−?
?(?, ?, ?) = [
???+?????
????
]
1
1−?−?
. Solve the cost minimization problem to show that the conditional factor demand
functions are:
?(?, ?, ?) = (
??
??
)
?
?+?
(
?
?
)
1
?+?
?(?, ?, ?) = (
??
??
)
?
?+?
(
?
?
)
1
?+?
Show that all cost-minimizing production plans line on a common ray from the
origin in ?-? space.
c. Show that the cost function is:
?(?, ?, ?) = (? + ?) [
?????
?????
]
1
?+?
Derive the average cost function ??(?, ?, ?) and marginal cost function ??(?, ?, ?).
d. Use the results of Part (c) to derive the supply function ?(?, ?, ?), and then use this
along with the results of Part (b) to derive the factor demand functions ?(?, ?, ?) and
?(?, ?, ?). {Hint: To verify co
ectness, check that the functions derived in this part
are the same as those derived in Part (a)}.
2. (16 marks) Consider a 2-agent, 2-good pure exchange economy, where agents are denoted
y superscripts and goods by subscripts. The endowments for the economy are ?1 = ?1
1 +
?1
2 and ?2 = ?2
1 + ?2
2 for goods 1 and 2, respectively, where ??
? is the endowment of good
? ∈ {1,2} for agent ? ∈ {1,2}. Consumption of goods 1 and 2 are denoted by ?1 and ?2,
espectively, where the Edgeworth box for the economy is drawn with ?2 on the vertical
axis. Let �̅�1 and �̅�2 denote the allocations of goods 1 and 2, respectively, for agent 1,
implying the co
esponding allocations for agent 2 are ?1 − �̅�1 and ?2 − �̅�2. The utility
functions are ?1 = ?1
??2
1−? for agent 1 and ?2 = ?1
?
?2
1−?
for agent 2, where ?, ? ∈ (0,1).
a. Show that the contract curve is given by the function:
�̅�2(�̅�1) =
(1 − ?)??2�̅�1
?(1 − ?)?1 + (? − ?)�̅�1
Verify that the contract curve connects the lower left and upper right corners of the
Edgeworth box.
. Let ??
? (?) denote the demand function of agent ? for good ?, where ? is the price of
good 2 and the price of good 1 is set equal to 1 without loss of generality. Derive the
four demand functions.
c. Derive the competitive equili
ium allocation ?1
1, ?2
1, ?1
2 and ?2
2.
d. Show that the First Welfare Theorem holds for the economy.
3. (10 marks) Consider a risk-averse agent with utility ?(?) where ? is annual income. The
agent faces a probability ? of an income loss of ? in any given year. Insurance against the
loss is obtainable at a premium of ? = ?? per year where ? is the benefit paid out in the
event of a loss and ? ∈ (0,1) is a constant. Show that the agent fully insures against the loss
if the insurance premium for the benefit ? is actuarially fair.
4. (12 marks) A risk-averse agent with utility ?(?) over lifetime income ? considers whether
to live honestly or live a life of crime. If he lives honestly, the agent earns a lifetime income
of ?ℎ with certainty. As a criminal, the agent earns a lifetime income of ?(?) if he is caught
and prosecuted by the justice system, which depends negatively on ?, the associated prison
sentence he serves in years; otherwise, as a criminal he earns a lifetime income of ?(0)
?(?) for any ? > 0. The probability of a criminal being caught and prosecuted is ?(?), which
depends negatively on ? because the justice system operates under a fixed budget (i.e. more
esources dedicated to imprisoning criminals leaves fewer resources available for catching
and prosecuting criminals). The gamble presented by criminal life is associated with an
expected lifetime income of ?(?), a certainty equivalent of �̅�(?) and an expected utility
function of ?[?(?)]. The government seeks to determine whether harsher sentences (i.e. an
increase in ?) can be expected to reduce crime, given that would necessarily result in a
lower chance of criminals being caught and prosecuted (i.e. a decrease in ?) due to the fixed
udget for the justice system. The answer will suggest whether the government should
direct more resources to prisons or to police stations and courthouses.
a. State the expressions for ?(?) and ?[?(?)], and state the equation that defines �̅�(?).
Very
iefly explain how the agent’s decision to adopt a life of crime depends on
how �̅�(?) compares to ?ℎ.
. Suppose ? increases from ?0 to ?1 such that ?(?) remains unchanged. Illustrate the
impact of this change on the agent in an appropriate diagram with utility on vertical
axis and lifetime income on the horizonal axis. On the propensity of the agent to
adopt a life of crime, what is the net effect of the increased sentence and the
associated decreased probability of being caught and prosecuted? {Hint: Consider
the impact of the increase in ? on �̅�(?)}
c. Repeat Part (b) but under the assumption that the agent is risk-loving.