Answer to Question #273336 in Microeconomics for cookie

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

(a) $MU_{x} = u'(x) = y + 1,$

$MU_{y} = u'(y) = x + 1.$

In equilibrium $\frac{MU_{x} } {P_{x} } = \frac{MU_{y} } {P_{y}} ,$ so:

(y + 1)/2 = x + 1,

x = 0.5y - 0.5,

2x + y = 15, so:

y - 1 + y = 15,

y = 8 units, x = 0.5×8 - 0.5 = 3.5 units.

If Py rises to $2, then I must increase at least by $8 in order that you be as well off as before.

(b) After the price change the next equilibrium will occur:

$(x + 1)/2 = (y + 1)/2,$

x = y,

2x + 2y = 15,

x = y = 3.75.

So, x increased by 0.25 and y decreased by 4.25.

The substitution effect is 0.25, the income effect is 4.

(c) $Y = \frac{15 - P_{x} \times x} {P_{y} } = 7.5 - 2x/P_{y} .$

(d) With both prices equal to $1, the consumption of each good x = y varies from 0 to 50 units as I changes from $0 to $100.