A Consumer Has the Following Utility Function for Goods X

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 pᵧ 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,pᵧ,I)and Y*(pₓ,pᵧ,I)
e.Show that the demand equations are homogeneous of degree zero.That is,show
X*(cpₓ,cpᵧ,cI)= X*(pₓ,pᵧ,I)
for any positive constant,c.

Correct Answer:

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a.L = 5XY³ + 10 + [I - pₓX - pᵧY]
b.The ...

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