Q: A person has utility u(x,y)=x0.8y0.2 and income of I=100. The market prices are Px=10 and Py=20. Find the MRS and the optimal bundle for this person to purchase.
A: The MRS = MUx/MUy=0.8 x -0.2y0.2 /0.2x0.8y-0.8
To find the optimal bundle for a person with Cobb-Douglas preference we know that
1) MRS=Px/Py which in this problem is 4y/x=10/20 or 8y=x.
2) I=x*Px+y*Py which in this problem is 100= 10x+20y.
Using these two equations we can solve for x and y. Since 8y=x we have 100=10(8y)+20y or 100=100y or y=1. Since y=1 and 8y=x, we have that x=8. Therefore the optimal bundle is (8,1).