Answer:
The total effect is 35 out of which income effect is 15 and substitution effect is 20.
Explanation:
Ross has an income of $1440.
The price of chocolates (Px) is $10 and donuts (Py) is $9.
The utility function is given as
U = 0.5xy
Before price rise, Budget line:
1440 = 10x + 9y,
Consumption is optimal when
![\frac{MUx }{ MUy} = \frac{Px}{Py} = \frac{10}{9} = 1.11](https://tex.z-dn.net/?f=%5Cfrac%7BMUx%20%7D%7B%20MUy%7D%20%3D%20%5Cfrac%7BPx%7D%7BPy%7D%20%3D%20%5Cfrac%7B10%7D%7B9%7D%20%3D%201.11)
0.5y / 0.5x= 1.11
y = 1.11x
Substituting in budget line,
1440 = 10x + 9y = 10x + 9(1.11x)
1440 = 10x + 9.99x
19.99x = 1440
x = 72
y = 1.11x = 79.92 = 80
After price rise,
Py = 16.
New budget line:
1440 = 10x + 16y,
Price ratio
![\frac{Px}{Py } = /](https://tex.z-dn.net/?f=%5Cfrac%7BPx%7D%7BPy%20%7D%20%3D%20%20%2F)
=![\frac{10}{16}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B16%7D)
= 0.625
And,
![\frac{MUx}{Muy} = \frac{0.5y}{0.5x} = 0.625](https://tex.z-dn.net/?f=%5Cfrac%7BMUx%7D%7BMuy%7D%20%3D%20%5Cfrac%7B0.5y%7D%7B0.5x%7D%20%3D%200.625)
![\frac{y}{x} = 0.625](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%20%20%3D%200.625)
y = 0.625x
Substituting in new budget line: 1440 = 10x + 16y
1440 = 10x + 16(0.625)x
1440 = 20x
X = 72
Y = 0.625x = 45
So, total effect (TE)
= Decrease in consumption of y
= 80 - 45
= 35
With previous (x, y) bundle,
U = 0.5xy
U = 0.5 x 72 x 80
U = 2880
Keeping utility level the same & substituting,
y = 0.625x in utility function:
28800 = 0.5xy
![2880 = 0.5\ \times\ 0.625x](https://tex.z-dn.net/?f=2880%20%3D%200.5%5C%20%5Ctimes%5C%200.625x)
![2880 = 0.3125x^{2}](https://tex.z-dn.net/?f=2880%20%3D%200.3125x%5E%7B2%7D)
![x^{2} = \frac{2880}{0.3125}](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%20%3D%20%5Cfrac%7B2880%7D%7B0.3125%7D%20)
![x^{2} = 9216](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%3D%209216)
![x = \sqrt{9216}](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%7B9216%7D)
x = 96
Now, putting the value of x,
y = ![0.625\ \times\ x](https://tex.z-dn.net/?f=0.625%5C%20%5Ctimes%5C%20x)
y = ![0.625\ \times\ 96](https://tex.z-dn.net/?f=0.625%5C%20%5Ctimes%5C%2096)
y = 60
Substitution effect (SE)
= 80 - 60
= 20
Income effect
= TE - SE
= 35 - 20
= 15