Question

Income is $3,000, the price of x is $10, and the price of y is $30

1. Using the LaGrangean method, find the utility-maximizing quantities of x and y. (Note:  on this part of the problem only, you MUST write out the Lagrange equation, show all derivatives, and show all other work to receive full credit.)  (20)
2. Suppose that the price of x rises to $20.  Find the substitution and income effects of the change in the price of x.  .

• Be sure to end your answer with something like, “the substitution effect causes you to buy 10 more x and 10 less y, the income effect causes you…. etc.)

3. Carefully graph the results of this problem, being sure to label everything and to show the income and substitution effects on the graph.  (35)
4. Using the income, prices, and the utility that you found in part a. of this problem, graph the ordinary and compensated demand curves for x.  (You found these formulas in the first problem.)

• Be sure to label each axis and all appropriate points on both curves (that is, you should have a point labeled on each curve when the price of x is $10 and when it is $20).  (20)

5. For each of the following, choose “income only”, “substitution only”, or “income and substitution effects.”  (5 each)

1. The ORDINARY demand curves show which effect(s) of a change in price on purchases of X?

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2. The COMPENSATED demand curve shows effect(s) of a change in price on purchases of X?

___________________________________

6. For each of the following, choose “ordinary only”, “compensated only”, or “both ordinary and compensated.”  (5 each)

1. A change in utility will shift which demand curve(s)?

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2. A change in income will shift which demand curve(s)?

___________________________________

Answer 1

a)

Assume U(x,y)=xy

Given,

Demand for x is denoted by x

Demand for y is denoted by y

Income, m= $3000

Price of x, p₁=$10

Price of y₁,p₂=$30

Therefore the budget constraint is as follows:

xp₁+yp₂=m

→ 10x+30y= 3000

Setting up the Lagrangian equation

→ L= U+ T(Budget Constraint)

→ L=xy+ T(10x+30y-3000)

Differentiating with respect to x,y and T separately

?L/?x= y+10T=0

→ y= -10T ----->(Eq.1)

?L/?y= x+30T= 0

→ x=-30T -----> (Eq.2)

?L/?T= 10x+30y-3000=0

→ 10x+30y=3000 ------> (Eq 3)

Dividing Equation 1 by Equation 2

→ y/x= 10T/30T

→ y/x=⅓

→ x= 3y (Put in equation 3 and find the value of y)

Eq 3

→ 10x+30y=3000

→ 10(3y)+ 30y= 3000

→ 30y+30y=3000

→ 60y= 3000

→ y= 3000/60

→ y= 50 (Put in x= 3y to find value of x)

x=3y→ x=3(50)--> x=150

Therefore utility maximizing quantities are

(x,y)= (150,50)

From this we can surmise that the utlitlity maximizing demand function is given by the formula

x= m/2p₁ and y= m/2p_2,

b)

Assume U(x,y)=xy

Given,

Demand for x is denoted by x

Demand for y is denoted by y

Income, m= $3000

Initial Price of x, p₁=$10

New Price of x, p₁’=$20

Price of y₁,p₂=$30

To find the Substitution and income effect the following steps must be followed from the Slutsky method:

1. Calculate the original consumption vector {x,y) at original income m=$3,000 and Original price vector ($10,$30)

As we derived in a) part of this problem

x= m/2p₁= 3000/20= 150

y= m/2p_2,= 3000/60= 50

(x,y)= (150,50)

1. Calculate fictitious level of income m’=4500 so that the original bundle is affordable at new price p₁’

Using the budget constraint and original bundle from 1 (x,y)= (150,50)

p₁’x+p₂y=m’

→ 20(150)+30(50)=m’

→ 3000+1500=m’

→ 4500= m’

Thus, to maintain utility from the original bundle, income must rise from 3000 to 4500 to compensate from price rise.

1. Calculate intermediate level of consumption (x₁,y₁) at the fictitious level of income m’ and new price vector (p₁’,p₂)

x₁= m’/2p₁’= 4500/2(20)= 4500/40= 112.5

y₁= m’/2p₂=4500/2(30)= 4500/60= 75

(x₁,y₁)= (112.5,75)

1. Calculate final demand vector (x₂,y₂) at the original income m=300 and new price vector (p₁’,p₂)= (20,30)

x₂= m/2p₁’= 3000/2(20)= 3000/40= 75

y₂= m/2p₂=3000/2(30)= 3000/60= 50

(x₂,y₂)= (75,50)

Total Effect= x₂-x= 75-150= -75

Substitution Effect= x₁-x= 112.5-150= -37.5

Income Effect= Total Effect- Substitution Effect

→ Income Effect= -75-(-37.5) → -75+37.5= -37.5

Substitution Effect= -37.5

Income Effect= -37.5

Total Effect=-75

Substitution effect reduces your consumption of x by 37.5

Income Effect reduces your purchasing power by -37.5 which in turn reduces demand.

Graph: step by step process

Original Budget line at original prices

New Budget line at the increased price

Utiltity maximizing point where IC meets the budget line

Draw a phantom budget line (marked in dotted red), to meet the Indifference Curve (IC1) to determine the proportion of Substitution effect and Income Effect

Optimal Bundles are marked with the coordinates

TE= Total Effect

SE= Substitution Effect

IE= Income Effect