University Microeconomics Assignment: ECON3000 Solution Analysis

This document provides a comprehensive solution to a microeconomics assignment, addressing key concepts such as price elasticity of demand, Lagrangian functions, and various market structures. The assignment covers topics including Bertrand price competition, monopoly, Cournot, and the Stackelberg model. Solutions are presented for multiple-choice questions and problem-solving exercises, with detailed calculations and explanations. The document analyzes profit maximization conditions, reaction functions, and the effects of different market structures on output, prices, and profits. Furthermore, it includes calculations for deadweight loss, consumer surplus, and producer surplus under different scenarios. The assignment also addresses the application of microeconomic principles to real-world scenarios, such as revenue maximization for a university parking system. Overall, the document serves as a valuable resource for students studying microeconomic theory, offering insights into problem-solving techniques and the application of economic concepts.

Running head: MICROECONOMICS
Microeconomics
Name of the Student
Name of the University
Course ID

1MICROECONOMICS
Table of Contents
Question 1........................................................................................................................................2
Question 7........................................................................................................................................3
Question 2........................................................................................................................................4
Question 10......................................................................................................................................4
Question 11......................................................................................................................................5
Question 12......................................................................................................................................7
Question 15......................................................................................................................................7
Question 16......................................................................................................................................7
Question 17......................................................................................................................................7
Question 18......................................................................................................................................8
Question 19......................................................................................................................................8
Question 20....................................................................................................................................10
Question 21....................................................................................................................................11
Question 22....................................................................................................................................14
Question 23....................................................................................................................................14

2MICROECONOMICS
Question 1
Demand curve
x=400−0.4 p
¿ , p= 400−x
0.4
When, x = 100,
p= 400−100
0.4
¿ 750
When, x = 120,
p= 400−120
0.4
¿ 700
Price elasticity of demand = dx
dp × p0
x0
¿−0.4 × 750
100
¿−0.4 ×7.5
¿−3
b) -3

3MICROECONOMICS
Question 7
X =2000−2 p
¿ , p= 2000− X
2
¿ , p=1000− X
2
Total Revenue ( TR )=X × p
¿ X × (1000− X
2 )
¿ 1000 X− X2
2
First order condition for maximizing revenue
d ( TR )
dX =0
¿ , 1000− X=0
¿ , X =1000
p=1000− X
2
¿ 1000− 1000
2
¿ 1000−500
¿ 500

4MICROECONOMICS
d. p = 500$/year
Question 2
b. The slope of the line that is tangent to y=f(x) at the point (2,f(2)) in the (x,y) Cartesian space.
Question 10
MinC=40000 x1 +10000 x2
Subject ¿ y =f ( x1 , x2 ) =48 x1 x2=3072
Lagrangian function
L=40000 x1 +10000 x2+ λ (48 x1 x2−3072)
First order conditions
∂ L
∂ x1
=40000+ λ 48 x2=0∨, λ=−40000
48 x2
… … … … … … … (i)
∂ L
∂ x1
=10000+λ 48 x1=0∨, λ=−10000
48 x1
… … … … … …(ii)
∂ L
∂ x1
=48 x1 x2−3072=0 … … … … … … … …(iii )
From (i) and (ii)
−40000
48 x2
=−10000
48 x1
¿ , 40000 x1=10000 x2

5MICROECONOMICS
¿ , x2=4 x1
Putting in (iii)
( 48 x1 ×4 x1 ) −3072=0
¿ , 192 x1
2=3072
¿ , x1
2=16
¿ , x1=4
x2=4 x1
¿ ( 4 ×4 )
¿ 16
Therefore,
C¿=40000 x1 +10000 x2
¿ ( 40000 × 4 ) + ( 10000× 16 )
¿ 160000+160000
¿ 320000
The minimum cost at which the output level is achieved is 320000.00
Question 11
Max π = py−w1 x1−w2 x2
¿ 34 y −4 x1−4

6MICROECONOMICS
Subject ¿ y =f ( x1 , x2 )=2 x1
2
3 x2
1
3
¿ , y=2 x1
2
3
Lagrangian function,
L=34 y−4 x1−4+ λ ( y−2 x1
2
3 )
First order conditions for maximization,
∂ L
∂ y =34+ λ=0∨, λ=−34 … … … … … … … . ( i )
∂ L
∂ x1
=−4−λ 2× 2
3 x
−1
3 =0
¿ ,−4− 4
3 λ x
−1
3 =0
¿−4− 4
3 ×−34 × x
−1
3 =0
¿ , 136
3 x
1
3
=4
¿ , 3 x
1
3 =136
4
¿ , 3 x
1
3 =34
¿ , x
1
3 =11.33
¿ , x=1454.42

7MICROECONOMICS
∂ L
∂ λ = y−2 x1
2
3 =0
¿ , y−2 × ( 1454.42 )
2
3 =0
¿ , y−2 × ( 11.333 )
2
3 =0
¿ , y−2 ×128.37=0
¿ , y=256.74
Maximum Profit =py −w1 x1−w2 x2
¿ ( 34 × 256.74 )− ( 4 ×1454.42 ) − ( 4 × 1 )
¿ 8729.16−5817.68−4
¿ 2907.48
Question 12
c. Is usually solved using the quantity supplied as a control variable, and using the price as
control variable would produce different numerical results that are appropriate for describing
markets in which the monopolist fixes the price rather than the quantity supplied.
Question 15
c. The leader maximises its profit subject to the follower’s or followers’ reaction function(s).
Question 16
b. There is intense price competition, in the sense that consumers can switch from one supplier to
another at no, or a very low, switching cost.

8MICROECONOMICS
Question 17
c. Each firm in the market makes an output decision that is a best-response to what the other
firm(s) in the market do
Question 18
Bertrand price competition
Marginal cost for each firm is 50.
Under Bertrand competition price equals marginal cost.
Therefore,
p=50
The missing value in cell (i.d) is 50.
Question 19
Monopoly
Demand : p=550−50 y1
Total Revenue ( TR ) = ( 550−50 y1 ) × y1
¿ 550 y1−50 y1
2
Marginal Revenue ( MR ) = d ( TR )
d y1
¿ d ( 550 y1 −50 y1
2 )
d y1

9MICROECONOMICS
¿ 550−100 y1
Cost function : C1=50 y1
Marginal cost ( MC ) = d C1
d y1
¿ 50
Profit maximization condition
MR=MC
¿ , 550−100 y1=50
¿ , 100 y1=500
¿ , y1=5.00
p=550−50 y1
¿ 550− (50 × 5.00 )
¿ 550−250.00
¿ 300.00
TR=5.00× 300
¿ 1500.00
TC=50.00 ×5
¿ 250.00
π1=TR−TC

10MICROECONOMICS
¿ 1500.00−250.00
¿ 1250.00
The missing value in (ii.e) of Table 2 is 1250.00
Question 20
Cartel
p=550−50 yT
TR= ( 550−50 yT ) × yT
¿ 550 yT−50 yT
2
TC=C1 +C2
¿ 50 y1+ 50 y2
¿ 50 ( y1 + y2 )
¿ 50 yT
πT =TR−TC
¿ 550 yT−50 yT
2 −50 yT
Profit maximization condition,
d ( πT )
d yt
=0
¿ , 550−100 yT −50=0

11MICROECONOMICS
¿ , 100 yT=500
¿ , yT =5.00
p=550−50 yT
¿ 550− (50 × 5.00 )
¿ 550−250.00
¿ 300.00
CS=1
2 × ( 550−300.00 ) ×5
¿ 1
2 ×250.00 ×5
¿ 625.00
The missing value in cell (iii_j) is 625.00
Question 21
Stackelberg model
p=550−50 y1−50 y2
Profit function of firm 2
π2=TR2−TC2
¿ p y2−50 y2
¿ ( 550−50 y1 −50 y2 ) × y2−50 y2