Q: A person has utility given by u(x,y)= x2y and
I=300. Initially the market prices are Px=2 and Py=4.
1) Py increases to Py=10. Are x and y compliments, substitutes or neither?
2) Find demand for good x.
1) For this type of utility function we know at the optimal bundle MRS=price ratio. Initially we have 2y/x= 2/4 or 4y=x. We also know the BC is 300=2x+4y. Substituting in for y we have300=2x+x or x=100. When the price changes we have 2y/x=2/10 and 300=2x+10y. Substituting for y we have 300=2x+x or x=100. Since the quantity demanded of x does not change when Py changes the goods are neither compliments nor substitutes.
2) To find the demand we solve the optimization problem for a general Px. Hence we have, 2y/x=Px/Py and 300=xPx+yPy. 2y/x=Px/Py is the same as 2yPy=xPx so we can rewrite the BC as 300=xPx+(xPx)/2 or 300=xPx (1+1/2) or x=200/Px.