A Consumer Has the Following Utility Function for Goods X

question 77

Essay

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.

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Definitions:

Sloped Surface

An area that inclines at an angle relative to a horizontal plane, significant in design for drainage, aesthetics, and functional purposes.

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A surface that forms a right angle (90 degrees) with another surface or the base line of reference.

Rolled Edge

A finishing feature where the edge of a metal sheet, part, or component is curved or folded over to eliminate sharp edges or increase strength.