Question

[SESSION 4] (Sept 2017] Charlie has preferences u = xy and income m = 24. In the base year, he paid px = 1 and pr = 1. In the current year, he pays px = 2 and Py = 1. Compute the LQ index. Compute the PQ index. Compute the LP index. Compute the PP index.

Answer 1

Solution:

Utility function: u = x*y; income, m = 24

Base year prices: px0 = 1, py0 = 1

Current year prices: px1 = 2, py1 = 1

Given the utility function of Cobb-Douglas preferences, demand function for a good can be defined as:

X = (a/(a+b))*(m/Px), where a and b are the preference weights on two goods. From the utility function, we can see that a = b =1

So, x* = (1/(1+1))*24/px = 12/px

and y* = (1/(1+1))*24/py = 12/py

Now, at base year, the consumption levels were: x0 = 12/1 = 12 units

and y0 = 12/1 = 12 units

While at current year prices, x1 = 12/2 = 6 units and y1 = 12/1 = 12 units

Calculating the required indices:

Laspeyres quantity index (LQ) (fixed price at base period, comparing quantities) = (x1*px0 + y1*py0)/(x0*px0 + y0*py0)

LQ index = (6*1 + 12*1)/(12*1 + 12*1) = 0.75

Paasche quantity index (PQ) (fixed price at current period, comparing quantities) = (x1*px1 + y1*py1)/(x0*px1 + y0*py1)

PQ index = (6*2 + 12*1)/(12*2 + 12*1) = 0.667

Laspeyres price index (LP) (fixed quantity at base period, comparing prices) = (x0*px1 + y0*py1)/(x0*px0 + y0*py0)

LP index = (12*2 + 12*1)/(12*1 + 12*1) = 1.5

Paasche price index (PP) (fixed quantity at current period, comparing prices) = (x1*px1 + y1*py1)/(x1*px0 + y1*py0)

PP index = (6*2 + 12*1)/(6*1 + 12*1) = 1.33