The utility function u(x1, x2) = x1 x2, represents Cobbie Douglas\' preferences
ID: 1169766 • Letter: T
Question
The utility function u(x1, x2) = x1 x2, represents Cobbie Douglas' preferences for tickets to baseball games (good x1) and to football games (good x2). Suppose that Cobbie's income (m) is $100, and that the ticket prices are respectively p1 = $2 and p2 = $1. Find Cobbie's optimal choice. Suppose that now p1 increases to $4. Find Cobbie's new optimal choice. What is the size of the substitution effect on tickets for baseball games, when pr increases from $2 to $4? What is the size of the income effect on tickets for baseball games? What is the size of the substitution effect on tickets for football games, when p1 increases from $2 to $4? What is the size of the income effect on tickets for football games?Explanation / Answer
u = x1x2
Budget line equation: M = x1 Px1 + x2 Px2
100 = 2x1 + x2
(a)
MUx1 = du / dx1 = x2
MUx2 = du / dx2 = x1
So, MRS = MUx1 / MUx2 = x2 / x1
In optimal condition, MRS = p1 / p2
Or, x2 / x1 = p1 / p2 = 2
x2 = 2x1
substituting in budget line:
100 = 2x1 + x2
100 = x2 + x2 = 2x2
X2 = 100 / 2 = 50
x1 = x2 / 2 = 25
(b) now, p1 = 4
MRS = p1 / p2 = 4
x2 / x1 = 4
x2 = 4x1
substituting in budget line:
100 = 2x1 + x2
100 = 2x1 + 4x1 = 6x1
x1 = 100 / 6 = 16.67
x2 = 4x1 = 66.67
(c)
When p1 increased from 2 to 4, x1 decreased from 25 to 16.67.
When p1 = 2, u = x1x2 = 25 x 50 = 1250
Slope of new budget constraint = -p1 / p2 = - 4
X2 / x1 = 4 or x2 = 4x1
Consumer will buy that combination of x1 & x2 to keep his utility unchanged, therefore
x1x2 = 1250 = x1(4x1) [since x2 = 4x1]
4(x1)2 = 1250
(x1)2 = 312.5
x1 = 17.68
x2 = 4x1 = 70.71
so, we get for good x1 (baseball):
Total effect (x1) = 16.67 - 25 = - 8.33
Substitution effect = 17.68 - 25 = -7.32
Income effect = Total effect – Substitution effect = - 8.33 + 7.32 = -1.01
(d) As seen in previous parts, when p1 increases from 2 to 4, x2 increases from 50 to 66.67.
When utility is kept constant, x2 = 70.71 [Part (c)]
So, for good x2 (football):
Total effect = 66.67 – 50 = 16.67
Substitution effect = 70.71 – 50 = 20.71
Income effect = Total effect – Substitution effect = 16.67 – 20.71 = - 4.04
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