Answer to Question #273381 in Microeconomics for Ryan

Question

Do each of a-d, both geometrically (you need not be precise) and using calculus. There are only two goods; x is the quantity of one good and y of the other. Your income is I and u(x,y) = xy + x + y.

(a) P_x = $2; P_y = $1; I = $15. Suppose P_y rises to $2. By how much must I increase in order that you be as well off as before?

(b) In the case described in part (a), assuming that I does not change, what quantities of each good are consumed before and after the price change? How much of each change is a substitution effect? How much is an income effect?

(c) P_x = $2; I =$15. Graph the amount of Y you consume as a function of P_y , for values of P_y ranging from $0 to $10 (your ordinary demand curve for Y).

(d) With both prices equal to $1, show how consumption of each good varies as I changes from $0 to $100.

Answer 1

Solution:

a.). Budget constraint: I = PxX + pyY

15 = 2X + Y

U (x,y) = xy + x + y

MUx = y + y

MUy = x + x

$\frac{MUx}{MUy} = \frac{Px}{Py}$

$\frac{y + y}{x + x} = \frac{2}{1}$

$\frac{2y}{2x} = \frac{2}{1}$

Y = 2x

15 = 2X + 2X

15 = 4X

X = 3.75

Y = 2(3.75) = 7.5

Py rises to 2:

Income must increase by 7.5 to 22.5 from 15.

 

b.). $\frac{MUx}{MUy} = \frac{Px}{Py}$

$\frac{2y}{2x} = \frac{2}{2}$

Y = x

15 = 2X + X

15 = 3X

X = 5

Y = 2(5) = 10

Quantities produced before the price change = U(x,y) = 3.75, 7.5

Quantities produced after the price change = U(x,y) = 5, 10