Question

On the planet Homogenia every consumer who has ever lived consumes only two goods, x and y, and has the utility function U(x, y) = xy. The currency in Homogenia is the fragel. In this country in 1900, the price of good 1 was 1 fragel and the price of good 2 was 2 fragels. Per capita income was 84 fragels. In 2000, the price of good 1 was 3 fragels and the price of good 2 was 4 fragels. The Laspeyres price index for the price level in 2000 relative to the price level in 1900 is A) 2.50 B) 3.50 C) 2.33 D) 4 can you the show work

Answer 1

Option C) 2.33

On the planet Homogenia, the utility function U(x, y) = xy.

To maximize utility, the equilibrium condition is,

MU_x / P_x = MU_y / P_y

MU_x = = y

MU_{y = = x}

_Then, MU_x / P_x = MU_y / P_y  

In 1900, price of good 1 (which we have considered x) = 1 fragel and

The price of good 2 (which we have considered y) =  2 fragels.

_or, y / 1 = x / 2

_or, 2 y = x

We see that the consumer will buy double quantities of the good y than good x.

Per capita income =  84 fragels.

So, 3x = 84 fragels.

or, x = 28 fragels, i.e the consumer will spend 28 fragels in good x.

x = 2 y

So, y = 28 * 2 = 56 fragels, i.e. the consumer will spend 56 fragels in good y.

Since the price of good x is 1 fragel, the consumer will buy 28*1 = 28 units in 1900 and since the price of good y is 2 fragel, the consumer will buy 56/2 = 28 units in 1900.

Now, to calculate the Laspeyres price index, lets see the following table below:

Price1900	Quantity1900	Price2000	Price1900*Quantity1900	Price2000*Quantity1900
1	28	3	28	84
2	28	4	56	112
Total			84	196

Laspeyres price index for the price level in 2000 relative to the price level in 1900

= (Price2000*Quantity1900 / Price₁₉₀₀*Quantity₁₉₀₀ )  * 100

= (196/84) * 100

= 2.33 * 100

= 233.

The Laspeyres price index multiplies 100 so the index is not 2.33 instead it is 100 * 2.33 which is 233.