Micro Problem Set 2

4a) His income is €300, so his budget constraint is the line between y=30 and x=75. The optimal choice (utility-maximizing) is shown by the point of tangency with his utility function. This is point E (and x =30).
b) The budget constraint shifts to x = 120. The new optimal choice is point C (and x = 35).
c) ΔY∙py = (30-22,5)∙10 = €75
d) The total effect: Point E C
e) Income effect: F C Substitution effect: E F
f) Income effect < 0, so X is an inferior good.

5a) U(x,y) = x1x2
Normalize the parameters so that α + β = 1
U(x,y) = xɗy1-ɗ, where ɗ = α/(α+β) and 1-ɗ = β/(α+β). The new utility function is U(x,y) = x11/2x21/2
L (x1,x2,λ) = x11/2x21/2 + λ (I – px1x1 – px2x2) ɗL/ɗx1 = (1/2)x1-1/2 x21/2 – λpx1 = 0 ɗL/ɗx2 = (1/2)x11/2 x2-1/2 –λpx2 = 0 ɗL/ɗλ = I – px1x1 – px2x2= 0
=> px1/px2 = x2/x1 => x1 = (px2x2) / px1. Put this in the budget constraint:
I = px1(px2x2) / px1 + px2x2 => x2* = I / 2px2
Using the same way to find the demand function of good 1, we obtain: x1* = I / 2px1
b) x1* = 24 / (2∙1) = 12 x2* = 24 / (2∙2) = 6
c) x1* = 24 / (2∙1) = 12 x2* = 24 / (2∙3) = 4
d) Substitution effect = ɗx/ɗpx = – I/2px2
Income effect = – xɗx/ɗI = – x/2px
The price of good 1 did not change, so there is no effect.
The price of good 2 rose to €3.
Substitution effect = – 24 / (2∙32) = – 1,33
Income effect = – 4 / (2∙3) = –