Microeconomics Assignment: Analyzing Consumer Preferences and Choices

This microeconomics assignment explores consumer behavior through a series of questions analyzing utility functions, budget constraints, and demand analysis. The student addresses topics such as utility maximization, the impact of income on demand (normal and inferior goods), and the relationship between goods (substitutes and complements). The assignment includes calculations and interpretations of consumer choices, and the application of these concepts to scenarios involving price changes. Furthermore, it examines the concepts of compensating and equivalent variation as measures of consumer welfare. The solution uses mathematical models and graphical representations to explain the economic principles involved, providing a comprehensive understanding of consumer behavior in microeconomics.

Running head: MACROECONOMICS
Macroeconomics
Name of the Student
Name of the University
Author note

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Table of Contents
Answer a..........................................................................................................................................2
Answer b..........................................................................................................................................2
Answer c..........................................................................................................................................4
Answer d..........................................................................................................................................5
Answer e..........................................................................................................................................5
Answer f...........................................................................................................................................6
Answer g..........................................................................................................................................7
Answer h..........................................................................................................................................9
Answer i.........................................................................................................................................10
Answer j.........................................................................................................................................10
Answer k........................................................................................................................................11
References......................................................................................................................................15

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Answer a
The utility function is given as U ( X , Y )=5 ln ( X +1 )+Y
X: Roses
Y: Chocolate
The budget constraint is PX X +PY Y =M
M: Income
Consumer maximize utility function subject to the budget constraint
Therefore, Rose optimization problem is given as
Max: U ( X , Y )=5 ln ( X +1 )+Y
Subject to, PX X + PY Y =M
X, Y are endogenous variables.
PX, PY and M are exogenous variables
Answer b
If Rose spends all income on rose then then M/PX units of rose can be brought. If Rose
spends all income on chocolate then M/ PY units can be brought

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Figure 1: budget constraint of Rose
Therefore,
M/PX units of rose = M/PY units of chocolates
1 unit of rose = (M/PY)/( M/PX ) = PX/PY
PX X + PY Y =M
Or, Y = M
PY
− PX
PY
∗X

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dY
dX =−PX
PY
The opportunity cost of rose is same as the absolute value of slope of the budget constraint.
The slope of the budget constraint is the relative price of two goods, which in turn is the
opportunity cost of the goods. Therefore, the two are same.
Answer c
U ( X , Y )=5 ln ( X +1 )+Y
Budget constraint
PX X + PY Y =M
The relevant Lagrange function for solving utility maximization problem is
L=5 ln ( X+1 )+Y + λ( M−PX X−PY Y )
The first order condition for maximizing utility is
1. ∂ L
∂ X = 5
X +1 −λ PX =0
2. ∂ L
∂Y =1−λ PY =0
3. ∂ L
∂ λ =M −PX X−PY Y =0
From 1 and 2,
5
X +1 = PX
PY
¿ , PX ( X +1 ) =5 PY

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¿ , X +1= 5 PY
PX
¿ , X =5 PY
PX
−1
Putting the value of X in (3)
M −PX X−PY Y =0
¿ , M−PX (5 PY
PX
−1)−PY Y =0
¿ , M−5 PY + PX =PY Y
¿ , Y = M
PY
−5+ PX
PY
¿ , Y = M + PX
PY
−5
Rose’s ordinary (Marshallian) demand for rose X M=¿
Answer d
Rose’s ordinary (Marshallian) demand for Chocolate Y M = M + PX
PY
−5
Answer e
A good is said to be normal good if demand for the good increases with increase in
income. For this good income, effect is positive. For inferior goods, people decrease demand for
such goods with rise in income. A negative relation is found between demand of these goods and
income.

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For rose the demand function is given as
X¿= 5 PY
PX
−1
For rose, no relation is found between income and demand. Rose consumes a fixed quantity of
this good.
For Chocolate, the demand function is obtained as
Y ¿= M + PX
PY
−5
The income effect is,
d Y ¿
dM = 1
PY
>0
Therefore, income has a positive impact on chocolate demand. Therefore, Chocolate is a normal
good.
Answer f
For two goods to be substitute, an increase in price of one good leads to increase in demand for
its substitute good. The cross price elasticity measures the effect of price change in one good on
quantity demanded of the related good (1). The cross price elasticity of substitute goods is
positive. For complementary goods, an increase in price of one good leads to a decrease in
demand for the complementary good. The cross price elasticity for complementary goods are
negative.
Cross price elasticity between rose and chocolate

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If rise in price of chocolate positively affect the demand of rose then rose is a substitute of
chocolate. If the price effect is negative then the goods are considered as complementary good.
X M = 5 PY
PX
−1
d X¿
d PY
= 5
PX
Cross price elasticity=
d X¿
d PY
∗PY
X¿
¿ 5
PX
. PY
X¿ >0
The positive cross price elasticity implies rose is a substitute of chocolate.
Chocolate is considered as a substitute or complementary of good of rose if price of rose effect
affect the demand for chocolate positively or negatively.
Y M = M + PX
PY
−5
d Y ¿
d PX
= 1
PY
Cross price elasticity=
d Y ¿
d PX
∗PX
Y ¿
¿
1
PY
∗PX
Y ¿ >0
The computed cross price elasticity is positive. This implies chocolate is a substitute for rose.

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Answer g
Given, income (M) = $100, Price of Chocolate (PY ) = $1, Price of rose (PX ) = $1
At the utility maximizing level optimal choice of rose is
X¿= 5 PY
PX
−1
¿ 5∗1
1 −1
¿ 5−1
¿ 4
The utility maximizing level of chocolate is
Y ¿= M + PX
PY
−5
¿ 100+ 1
1 −5
¿ 101−5
¿ 96

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Answer h
Level of utility at the optimum basket
U ( X , Y )=5 ln ( X +1 )+Y
¿ 5 ln ( 4 +1 ) +96
¿ 5 ln(5)+96
¿ 104.0472 104

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Figure 2: optimal utility of at initial price
Answer i
M = $100, Price of chocolate (PY) = $1, now price of roses increases to become PX = $6
X¿= 5 PY
PX
−1
¿ 5∗1
6 −1
¿−0.17
Which is not possible, therefore if price of rose increases to 6 then Rose will not consume rose at
all and all the income is spent on purchasing chocolate.
Y ¿= M + PX
PY
−5
¿ 100
Answer j
New level of utility
U =5 ln ( X +1 ) +Y
¿ 5 ln ( 0+1 ) +100
¿ 5 ln 1+100

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¿ 100
Figure 3: Optimal utility after price change
Answer k
Equivalent and compensating variation is a measure of consumer welfare. Compensating
variation measures how much compensation should be given to the consumer to attain initial
level of utility when price changes. Equivalent variation on the other hand measures the variation
in the wealth level to achieve new welfare level when price changes (2).
Before calculating compensating and equivalent variation Hicksian demand function needs to be
derived. Hicksian demand function is derived from minimizing the expenditure.

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Min PX X + PY Y
Subject to,
U0 =5 ln ( X +1 ) +Y
The Lagrange multiplier is given as
PX X + PY Y +λ(U −5 ln ( X +1 ) −Y )
1. ∂ L
∂ X =PX −λ 5
x +1 =0
2. ∂ L
∂Y =PY −λ=0
3. ∂ L
∂ λ =U −5 ln ( X +1 ) −Y =0
From (1) and (2)
5
X +1 = PX
PY
¿ , PX ( X +1 ) =5 PY
¿ , X +1= 5 PY
PX
¿ , X H = 5 PY
PX
−1
Putting the value of X in (3)
U −5 ln ( X +1 )−Y =0
Y H =U−5 ln ( X +1 )

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¿ U −5 ln ( 5 PY
PX
−1+1)
¿ U −5 ln ( 5 PY
PX
)
Initial Utility Level (U0) = 104
With increase in price of rose, utility level reduces to 100
PX
0 =1 , PX
1 =6 , M =100
Compensating variation ( CV ) =¿ e ( P0 ,U0 ) −e ( P1 , U0 ) ∨¿
e ( P0 ,U0 ) = ( PX
0 ∗XM + PY∗Y M )
¿(1∗4 +1∗96)
¿ 100
e ( P1 , U0 ) =(PX
1 ∗XH + PY∗Y H )
XH and YH at initial utility level and new price level
X H = 5
6 −1=0
Y H =U−5 ln ( 5
6 )
¿ 104.91 105
However, given the income constraint the consumption of chocolate can never exceed
100.
Therefore, YH =100
e ( P1 , U0 ) =(PX
1 ∗XH + PY∗Y H )

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¿ ( 6∗0+1∗100 )
¿ 100
Therefore, CV =|100−100|=0
Equivalent variation ( EV ) =e ( P0 , U1 ) −e (P1 , U1 )
XH and YH at new utility level and old price level
X H = 5 PY
PX
−1=5−1=4
Y H =U−5 ln ( 5 PY
PX
)
¿ 104−5 ln 5
¿ 95.95 96
e ( P0 ,U1 ) = ( PX
0 ∗X H + PY∗Y H )
¿ ( 1∗4+1∗96 )=100
e ( P1 , U1 )=( PX
1 ∗X M + PY∗Y M )
¿ ( 6∗0+1∗100 )=100
Therefore,
EV =100−100=0
CV =EV =0

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References
1. Fine B. Microeconomics. University of Chicago Press Economics Books. 2016.
2. Baumol WJ, Blinder AS. Microeconomics: Principles and policy. Cengage Learning;
2015 Mar 20.