Question

06 Question (3 points) e See page 149 Douglas consumes two goods, x and y. His utility function is u(x, y) = (x + y. In the questions below, give your answers to two decimal places. 1st attempt Part 1 (1 point) See Hint Let the price of good x be $2 and the price of good y be $10. Furthermore, assume that Douglas has $360.00 to spend on these two goods. How much of good x does Douglas demand? Part 2 (2 points) See Hint Now suppose that the price of good x decreases to $1.00. What is the income effect of this price change on the demand for x? d for x? What is the substitution effect of this price change?

Answer 1

1. The consumer's problem is:
$\dpi{80} \fn_phv \small max \ \sqrt{x}+y \ s.t. \ 2x+10y=360$
At equilibrium, marginal rate of substitution is equal to the price ratio:
$\dpi{80} \fn_phv \small MRS=\frac{1}{2\sqrt{x}}=\frac{2}{10}\rightarrow x=\frac{25}{4}$
substituting this into the consumer''s budget equation:
$\dpi{80} \fn_phv \small 2*\frac{25}{4}+10y=360\rightarrow y=\frac{695}{20}$

2. After the price increase, at equilibrium:
$\dpi{80} \fn_phv \small MRS=\frac{1}{2\sqrt{x}}=\frac{1}{10}\rightarrow x=25$
Substituting this into the consumer's budget equation:
$\dpi{80} \fn_phv \small 25*1+10y=360\rightarrow y=33.5$

Income required by the consumer to purchase the original bundle at new prices is

$\dpi{80} \fn_phv \small \frac{25}{4}*1+\frac{695}{20}*10=353.75$
For the intermediate bundle, the consumer's problem is:
$\dpi{80} \fn_phv \small max \ \sqrt{x}+y \ s.t. \ x+10y=353.75$
At equilibrium, marginal rate of substitution is equal to price ratio:
$\dpi{80} \fn_phv \small \frac{1}{2\sqrt{x}}=\frac{1}{10}\rightarrow x=25$
Substituting this into the consumer's budget equation:
$\dpi{80} \fn_phv \small 25+10y=353.75\rightarrow y=32.875$

Substitution effect for demand for x is the difference between the demand for x at the intermediate bundle and the demand for x at the original bundle:
$\dpi{80} \fn_phv \small SE=25-\frac{25}{4}=18.75$