Question

Refer to the information in Homework 2 Question 2: Ross derives utility from only two goods, chocolates (x) and donuts (y). His utility function is as follows: U(x,y) = 0.5xy. His marginal utility from chocolates (x) and donuts (y) are given as follows: MUx = 0.5y and MUy = 0.5x. Ross has an income of $1440 and the price of chocolates (Px) is $10 and donuts (Py) is $9. b. Suppose price of donuts (Py) increase to $16, price of chocolates and income remain unchanged. How much is the total effect of this price change on Ross's consumption of donuts? Show your work. How much of this total effect is due to income effect and how much is due to substitution effect (round up your answer up to two decimal places)? Show your work. (2+6+4+4 = 16 points)

Answer 1

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

$\frac{MUx }{ MUy} = \frac{Px}{Py} = \frac{10}{9} = 1.11$

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

$\frac{Px}{Py } = /$

=$\frac{10}{16}$

= 0.625

And,

$\frac{MUx}{Muy} = \frac{0.5y}{0.5x} = 0.625$

$\frac{y}{x} = 0.625$

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

$2880 = 0.5\ \times\ 0.625x$

$2880 = 0.3125x^{2}$

$x^{2} = \frac{2880}{0.3125}$

$x^{2} = 9216$

$x = \sqrt{9216}$

x = 96

Now, putting the value of x,

y = $0.625\ \times\ x$

y = $0.625\ \times\ 96$

y = 60

Substitution effect (SE)

= 80 - 60

= 20

Income effect

= TE - SE

= 35 - 20

= 15