University Economics: Price Theory Assignment Solution and Analysis

This document presents a detailed solution to a price theory assignment. The solution addresses key concepts such as optimal baskets, the marginal rate of substitution, and budget constraints. It analyzes whether a given basket satisfies the conditions for optimality, exploring the relationship between the marginal rate of substitution and price ratios. The assignment further delves into calculating optimal baskets using budget constraints and utility maximization. It examines the characteristics of Giffen goods, and the impact of price changes on demand, including the substitution and income effects. The analysis includes detailed calculations and explanations, providing a comprehensive understanding of consumer behavior and price theory principles. The solution also investigates the decomposition of price changes into substitution and income effects, offering a complete analysis of how changes in price affect consumer choices.

Running head: PRICE THEORY 1
Price Theory
Professor’s Name:
Name:
Date:

PRICE THEORY 2
Question 1
An optimum basket must lie on the indifference curve that just touches, but does not cross the
budget line. This means that is at the point at which the budget line is a tangent to the
indifference curve. Two conditions must be satisfied by the optimal basket:
1. The marginal rate of substitution must be equal to the ratio of the prices i.e.
MRSx , y = Px
Py
implying that MU x
MU y
= Px
P y
2. The basket must satisfy the budget line with an equality i.e. Px x+ P y y =I
In this case, we test if the given basket meets the conditions for optimal basket.
M U x= 20
√ x M U y=1 I =$ 200
Px=$ 4 Py=$ 1
x=25 y=50
MU x
MU y
= Px
P y
20
√ 25
1 = $ 4
$ 1
20
5 =4
4=4
The marginal rate of substitution is equal to the price ratios hence basket satisfies the first
condition. Next we test for the second condition.
Px x+ P y y =I
$ 4 (25 )+$ 1 ( 50 )=$ 200
$ 100+$ 50=$ 200
$ 150=$ 200; Which is not true hence this basket does
not satisfy the budget constrain with equality.
The basket is not an optimal basket because it doesn’t satisfy the budget constraint with
equality despite meeting the condition of marginal rate of substitution being equal to the price
ratio of goods.
Question 2
MRSx , y = MU x
MU y

PRICE THEORY 3
MU x
MU y
=
( 20
√ x )
1
¿ 20
√ x
Question 3
For an optimal basket,
MRSx , y = Px
Py
20
√ x = $ 4
$ 1
We solve for x to get the number of coffee cups in the optimal basket.
$ 4 √x=$ 20
√ x=5
x=25
Next, we substitute this value of x into the budget constraint equation for the optimal
basket and solve for y to find the liters of milk in the optimal basket.
Px x+ P y y =I
$ 4 (25 )+ $ 1 ( y )=$ 200
100+ y =200
y=200−100
y=1 00
Thus, the optimal basket contains 25 cups of coffee (x) and 100 liters of milk (y).
Question 4
For an optimal basket,
MRSx , y = Px
Py
20
√x = $ 5
$ 1
We solve for x to get the number of coffee cups in the new optimal basket.

PRICE THEORY 4
$ 5 √ x=$ 20
√ x=4
x=16
Next, we substitute this value of x into the budget constraint equation for the optimal
basket and solve for y to find the liters of milk in the new optimal basket.
Px x+ P y y =I
$ 5 ( 16 ) +$ 1 ( y ) =$ 200
80+ y =200
y=200−80
y=120
Thus, the new optimal basket contains 16 cups of coffee (x) and 120 liters of milk (y).
Question5
Coffee is not a Giffen good. At the price of $4, 25 cups of coffee are demanded and at $5, 16
cups of coffee are demanded. The quantity demanded falls from 25 to 16 cups when the price
increases from $4 to $5 and thus coffee is a normal good. For a Giffen good, the quantity
demanded should increase when the price increases. From the two baskets, it was found that
when the price of coffee increases, the quantity of coffee contained in the optimum basket
reduces and the quantity of milk consumed increases. Coffee is easily substituted with Milk and
thus cannot be a Giffen good since Giffen goods are not easily substituted.
Question 6
The information given is:
U x , y=40 √ x + y
M U x= 20
√ x
M U y=1
I =$ 200
Py=$1
PxChanges from $4 to $5
Milk (y) is a composite good for sally because its price is Py=$ 1. In this case, the marginal rate
of substitution should be equal to the price of coffee (x) for the optimal basket.
MRSx , y =Px

PRICE THEORY 5
20
√x =Px
Solving the equation for x gives the demand function for coffee.
Px √ x=20
√ x= 20
Px
x= 400
( Px )2
First, we find the initial consumption Basket (Basket A)
The demand for coffee (XA) at the initial price ($4) is given by:
x A= 400
42 =25
The demand for milk (ya) is obtained by substituting the quantity of coffee on the
budget line since the optimum basket lies on the budget line.
Px x+ y A=I
$ 4 ( 25 ) + y A=$ 200
100+ y A =200
y A =200−100
y A =1 00
Next we find the final consumption basket (basket C)
The demand for coffee (XC) when the price was $5:
x A= 400
52 =16
The demand for milk (yc) is obtained by substituting the quantity of coffee on the
budget line since the optimum basket lies on the budget line.
Px x+ yC=I
$ 5 ( 16 ) + yC=$ 200
80+ yC =200
yC=200−80
yC=12 0
Now we find the decomposition consumption basket (basket B).
The utility of basket B is the same as the initial utility:

PRICE THEORY 6
U xA yA
=40 √ xA + y A
¿ 40 √ 25+100
¿ 300
But we know that for the basket,
MU x
MU y
=Px
Now substitute the final price ($5) and the marginal utilities into the expression
( 20
√ x )
1 =5
5 √ x=20
√ x=4
Substituting for √ x in the utility expression to find the quantity of y that gives the same utility
(300)
U xB yB
=40 √ xB + yB =300
40 ( 4 ) + y B=300
yB =140
Now we get the quantity of coffee in the basket by substituting the value of yBinto the budget
line with the new price ($5)
Px xB + y B=I
5 ( xB ) +140=200
5 xB=60
xB =12
The substitution effect is the difference between the quantity demanded in the decomposition
basket ( xB ¿ and the demand at the initial price (x A ¿, i.e. xB −xA while the income effect is the
difference between the quantity demanded at the final price ( xC ¿ and the demand in the
decomposition consumption basket ( xB ¿ , i. e . xC−xB
The substitution effect = xB −xA =12−25=−13
The income effect = xC−xB =16−12=4
Question 7

PRICE THEORY 7
From question 3 and question 4, it was found that the demand for coffee was 25 cups when the
price was $4 and the demand fell to 16 cups when the price rose to $5
It was found that the substitution effect is negative and this implies that the demand for coffee
falls when the price increases and increases when the price falls. The Income effect was positive
implying that the price of coffee falls, the purchasing power increases and more coffee is
demanded; and when the price increases, the purchasing power decreases and less is
demanded. But purchasing power rises with a rise in income. Thus, we conclude that the
demand for coffee increases with a rise in the income and falls with a decrease of income.