EconS_305__Homework_3.pdf
EconS 305: Intermediate Microeconomics w/o Calculus
Homework 3:
Market Analysis, Monopoly and Perfect Competition
Due: Friday, June 5th, 2020 at 5:00pm via Blackboard
- Please submit all homework solutions in the order the questions are presented and as one
.PDF.
- Please show all calculations as these exercises are meant to refine your quantitative tool
set. If I can not follow your calculations or it seems as you just “copy and pasted” answers
from the internet, I will be deducting half the points from that solution.
1. A Market Welfare Analysis of a Tax using Linear Demand and Supply Curves
Consider a demand curve for rice QD = 22 � 2P and a supply curve for rice QS = 3P � 23,
where both quantities are measured in pounds.
(a) Find the before tax market equili
ium (i.e. Q⇤ and P ⇤).
(b) Find the pre-tax consumer surplus.
(c) Find the pre-tax producer surplus.
(d) What is the total social welfare in the market of rice?
(e) Now, assume that we are going to apply a $.50 tax to each pound of rice sold, which means that
each rice producer will have pay $.50 to the government for every pound sold. This means that
the price the buyer will pay is Pb = Ps + $.50. What is the equili
ium price received by the
uyer (Pb) and producer (Ps), respectively? (Hint: the new demand curve is QD = 22� 2(Pb)
and the new supply curve for rice QS = 3Ps � 23. Solve for Ps first, then plug in for Pb.)
(f) What is the new quantity demanded and supplied in the new equili
ium with the tax? Show
that they are equivalent using the prices you just found.
(g) What is the Consumer Surplus after the tax?
(h) What is the Producer Surplus after the tax?
(i) What is the Total Social Welfare after the tax? What is the loss in Social Welfare because of
the tax?
(j) What is the Government Revenues?
(k) Calculate the Dead Weight Loss (DWL) from the tax.
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2. The Basic Case of a Monopoly with Fixed Costs
Consider a monopolist facing a linear inverse demand function of p(Q) = a�b(Q), where a > c,
and a total cost function of TC(Q) = cQ + F . We interpret a as the intercept, or the choke
price consumers are willing to pay for Q = 0, of the inverse demand curve, and b as the slope
of the inverse demand curve. Graphically, it can be represented as
Figure 1: The Linear Demand Curve
We can interpret the fixed cost (F ) as perhaps some “entry” fee, and we interpret c as the
marginal cost the firm has to pay according to how much output they produce. We can
epresent the Profit Maximization Problem (PMP ) for firm as:
max
Q�0
⇡ = p(Q)Q� (cQ+ F )
=) max
Q�0
⇡ = [a� bQ]Q� (cQ+ F )
CALCULUS PART:
From here, we can take our derivatives and set them equal to zero
@⇡(Q)
@Q
= a� 2bQ� c = 0 (1)
where we now have one equation ((1)), and one choice variable (Q) to solve for.
CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE.
(a) Find the firm’s optimal allocation of production (Q) to maximize its profit in equili
ium (i.e.
find Q⇤).
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(b) What is the equili
ium price the firm will receive (i.e. find p(Q⇤)?
(c) What is the optimal profit function of the firm (i.e. ⇡⇤(Q⇤))?
(d) What is the level of fixed cost in which the firm will choose to continue to operate?
(e) Will the firm produce if a < c? Careful when answering this question.
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3. A Cournot Game of Competing in Quantities w/ Fixed Costs
Consider two firms competing a la Cournot in a market with an inverse demand function
of p(Q) = a� b(Q) where Q = qi + qj and a > c, and total cost function of TCi(qi) = F + ciqi.
Notice that each firm has the same fixed cost (F ) but their marginal costs (ci) are not equal
to each other (i.e. ci 6= cj). This means these homogeneous product producing firms have
asymmetric costs, and we can represent the Profit Maximization Problem (PMPi) for firm i
as:
CALCULUS PART:
max
qi�0
⇡i = [a� b(qi + qj)] qi � (F + ciqi)
@⇡i(qi, qj)
@qi
= a� 2bqi � bqj � ci = 0 (2)
And through symmetry we know that firm j’s PMP is
max
qj�0
⇡j = [a� b(qi + qj)] qj � (F + cjqj)
@⇡j(qi, qj)
@qj
= a� 2bqj � bqi � cj = 0 (3)
where we now have two equations ((2) and (3)), and two choice variables (qi and qj) to solve for.
CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE.
(a) Before you solve for the optimal equili
ium allocations, find the Best Response Functions
(BRFs) for each firm (i.e. find qi(qj) and qj(qi)). How does the firm respond in their own
quantities with respect to an increase in a, b, ci, and qj?
(b) Find the optimal equili
ium allocation for each firm when they are competing a la Cournot.
That is, find q⇤i and q
⇤
j . How does firm i’s equili
ium allocation change with respect to an in-
crease in their own marginal costs (ci) and their opponents marginal cost (cj)? Which increase
is larger in absolute magnitude?
(c) Now, consider that the firm’s have symmetric costs (i.e. ci = cj = c) in the competitive equi-
li
ium and for all analyses from here on out. Find the competitive equili
ium quantities (i.e.
find q⇤i and q
⇤
j ).
(d) Find the equili
ium price (i.e. p(Q⇤) = a� b(Q⇤)).
(e) Find the equili
ium profits (i.e. ⇡⇤).
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4. A Cournot Game with N Firms Competing in Quantities w/ Fixed Costs
Consider N firms competing a la Cournot in a market with an inverse demand function of
p(Q) = a � b(Q), where Q =
PN
i=1 qi and a > c, and total cost function of TCi(qi) = F + cqi.
Notice that each firm has the same fixed cost (F ) and, for simplicity, their marginal costs (c)
are equal to each other (i.e. ci = cj = · · · = cN = c). This means these homogeneous product
producing firms have symmetric costs, and we can represent the Profit Maximization Problem
(PMPi) for firm i as:
CALCULUS PART:
max
qi�0
⇡i =
"
a�
NX
i=1
qi
!#
qi � (F + cqi)
=) max
qi�0
⇡i =
"
a�
qi +
NX
i 6=j
qj
!#
qi � (F + cqi)
@⇡i(qi, qj)
@qi
= a� 2bqi �
NX
i 6=j
qj � c = 0 (4)
where we now have a symmetric equation ((4)) and one choice variable for each firm i (qi) to
solve for.
CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE.
(a) Before you solve for the optimal equili
ium allocation for firm i, find the Best Response Func-
tion (BRFs) for firm i (i.e. find qi
⇣PN
i 6=j qj
⌘
). How does the firm respond in their own quantity
with respect to an increase in a, b, c and all other quantities (i.e. qj)?
(b) Find the optimal equili
ium allocation for each firm i when they are competing a la Cournot.
That is, find q⇤i for all i 2 {1, 2, . . . , N}. To do this, please invoke the assumption that the
firms are symmetric in output (i.e. qi = qj), and that the sum of a constant is equal to the
multiplying by the number of constants in the sum (i.e.
PN
i=1 qi = Nqi when qi = qj).
(c) Find the Aggregate Quantity Demanded (i.e. Q⇤ =
PN
i=1 q
⇤
i )
(d) Find the equili
ium price (i.e. P (Q⇤))
(e) Find the equili
ium profits for each firm (i.e. ⇡⇤).
(f) Assuming that we are operating in a perfectly competitive equili
ium (i.e. set ⇡⇤ = 0, find
the optimal number of firms in the industry (i.e. solve for N⇤). Does the equili
ium numbe
of firms increase or decrease as the demand curve becomes more inealastic?
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5. Comparing Outputs and Profits across Market Structures
- Please assume a > c throughout the analysis.
(a) Take each optimal quantity produced from Questions 2-4, and compare them mathematically
(i.e. qMonopolyi (< or >) q
Duopoly
i (< or >) q
Perfect Competition
i ). Please rank them in terms of high-
est quantities to lowest, assuming that N � 3. What happens as the number of firms increases?
(b) Take each optimal price you found from Questions 2-4, and compare them mathematically
(i.e. p(Q⇤)Monopoly (< or >) p(Q⇤)Duopoly (< or >) p(Q⇤)Perfect Competition). Please rank them in
terms of highest prices to lowest, assuming that N � 3. Which price is the greatest and which
is the least? Is this di↵erent than the quantities ranking? If so, why is this?
(c) Take each profit you found from Questions 2-4, and compare them mathematically
(i.e. ⇡Monopoly (< or >) ⇡Duopoly (< or >) ⇡Perfect Competition). Please rank them in terms of
highest profits to lowest, assuming that N � 3.
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