A consumer has the following utility function for goods X and Y: U(X,Y) = 5XY3 + 10

The consumer faces prices of goods X and Y given by px and py and has an income given by I.
a. Write out the Lagrangian expression for the consumer's utility maximization problem.
b. Write out the first order conditions necessary for maximizing utility subject to the budget constraint.
c. Show that the first order conditions imply the budget constraint and MRS condition. Provide the economic (i.e. non-mathematical) interpretation of these conditions – specifically, why are they necessary for the consumer to be at the optimal bundle?
d. Solve for the Demand Equations, X*(px,py,I) and Y*(px,py,I)
e. Show that the demand equations are homogeneous of degree zero. That is, show
X*(cpx,cpy,cI) = X*(px,py,I)
for any positive constant, c.

a. L = 5XY3 + 10 + [I – pxX – pyY]
b. The necessary conditions are
∂L/∂X = 5Y3 – px = 0
∂L/∂Y = 15XY2 – py = 0
∂L/∂ = I - pxX - pyY = 0
c. The first two conditions together become:
Y/3X = px / py
The left-hand side of this is the MRS, or slope of the indifference curve. The right-hand side is the slope of the budget constraint. These must be equal or the consumer will benefit by giving up more of one of the goods for less of the other. The third condition says that the optimal bundle must lie on the BC, which must occur because more is better.
d. Solve to get:
X*(px,py,I) = I/4px
Y*(px,py,I) = 3I/4py
e. Simply plug into demand equations:
X*(cpx,cpy,cI) = cI/(3cpx) = I/3px = X*(px,py,I)