1) Anni Has the Following Utility Function: U = X1/2Y1/2. Suppose Initially That Px = Py

1) 
Anni has the following utility function: u = x1/2y1/2. Suppose initially that Px = Py = $20 and that I = $4000. Suppose the price of x now increases to Px = $80.

a)  Note that this is a Cobb-Douglas utility function with α = ½, so we know that x*=I/(2*Px) and y*=I/(2*Py). So: x0*=100, y0*=100 and u0*=100 while x1*=25, y1*=100 and u1*=50.

b)  You need to solve two equations simultaneously to get the Hicks bundle. First: (x*y)1/2 = 100 so that you are on the same indifference curve. You also need MRS = the new ERS. The new ERS = 4, and the MRS = (y/x). So, y/x = 4 or y = 4x. By substitution, (4x2)1/2 = 100. Solving you get: xH*=50 and yH*=200 and uH*=100.

c)  See graphs below.

2)  So: XB = K – sPB and XG = F - tPG and we are given that K = F.
This means that ηB = s*PB/[ K – sPB] and ηG = t*PG/[ F - tPG]. Note that if we are at prices so that XG = XB then K – sPB = F - tPG . And since we know that K=F, it must be true that sPB = tPG. If that is the case, then by simple substitutions, ηB = ηG.

3)  Ralph was better off in period 0. In period 0 his income was $2*10 + $1*80 = $100. Note in period 0 he could have afforded the bundle he consumes in period 1, it would cost $2*15 + $1*65 = $95, but he didn’t, so by direct revelation of preference, we know that the bundle (10, 80) is preferred to the bundle (15, 65) so in period 0 he consumed a bundle he likes better than one the he ends up consuming in period 1.

4)  Note: with a wage increase for all hours, the laborer can be at either point A or point B, depending on his indifference curve map. If overtime wages are offered for hours only after the normal working day, the laborer must choose a bundle between points Eo and F. This guarantees less leisure consumed so more labor supplied. It’s only the substitution effect now.