Question

Consider John who is considering purchasing red socks (R) and green socks (G). John, who has red-green colorblindness, considers them entirely interchangeable. That is, for John the socks are perfect substitutes.

(Note: Odd number of socks is fine as the laundry machine always eats some anyways. Fractional number of socks is also okay, they can be used to knit holes in other socks perhaps.)

1. (a) Initially the price of red socks is $2 per sock, and the price of green socks is $3 per sock. If John has $60 to buy socks, how many red socks does he purchase? How many green socks does he purchase?

2. (b) During Lunar New Year, the price of red socks increases to $5 per sock. Green socks remain $3 per sock and his sock budget is still $60. How many red socks does he purchase? How many green socks does he purchase?

3. (c) John decides he wants to take his purchasing behavior to the company and discuss how much the price changes affected him. If John calculates the Laspeyres index, how much would it indicate prices have risen by? If John calculates the Paasche index, how much would it indicate prices have risen by?

4. (d) Which of the two indices above is larger (representing a bigger price increase)? Intuitively, why does that make sense?

5. (e) Jane is also a big fan of colored socks. However, unlike John, she believes every green sock should be worn along side a red sock, in the spirit of the holidays. That is, they are perfect complements for Jane. With the initial prices of $2 per red sock and $3 per green sock, and a budget of $100 for socks, how many red socks does Jane buy? How many green socks does she buy?

6. (f) When the price increases for red socks to $5, what happens to Jane’s consumption – How many red socks does Jane buy? How many green socks does she buy?

7. (g) Jane is intrigued by John’s decision to talk to the company about how much the price changes affected him. If Jane calculates the Laspeyres index for her consumption, how much would it indicate prices have risen by? If she calculates the Paasche index, how much would it indicate prices have risen by?

8. (h) How do Jane’s price indices compare to John’s? What’s the intuition for this comparison?

Answer 1

In Case of perfect substitute:

• If P1>P2, then consumer will buy 0 units of good 1 and Whole income will be spend on good 2.
• If P1<P2, then consumer will buy 0 units of good 2 and Whole income will be spend on good 1.
• If P1=P2, then consumer can buy any bundle on its budget line.

a)

The price of red socks= Pr=  $2 per sock

The price of green socks=Pg= $3 per sock

Income= 60

Here Pr<Pg, so John will buy only red socks.

Units of green socks= 0

Units of red socks= Income/Pr= 60/2= 30

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b)

The price of red socks increases to $5 per sock:

Pr'= 5

Pg= 3

Here , Pr>Pg, so John will buy only green socks.

Units of green socks= 60/3= 20

Units of red socks= Income/Pr= 0

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c)

Laspeyres index=

Where Pn is the new price and P0 initial price and Q0 is the initial quantity

Laspeyres index= [Pr' (30) + Pg(0)] / [ Pr (30)+Pg(0)] x 100= [ 150+0]/[60+0] x 100= 250%

Price have risen by 250%

Paasche index=

Where Pn is the new price and P0 initial price and Qn is the new quantity

Paasche index= [Pr' (0) + Pg(20)] / [ Pr (0)+Pg(20)] x 100= [ 60]/[60] x 100= 100%

Price have risen by 100%

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d)

The laspeyres index shows the greater change because the initial quantity included in this index and the initial quantity for red socks is positive whose price have increased while in paasche new quantity is used where units of red socks is zero while units of green socks is positive but its price remain unchanged.