Charlie has a utility function U(x, y)= (x + 3)(y + 4).The price of x is $1 and the price of y is $1.When he maximizes his utility subject to his budget constraint, he consumes positive amounts of both goods.Charlie consumes
A)2 more units of x than he consumes of y.
B)1 more unit of x than he consumes of y.
C)exactly as many units of x as of y.
D)1 more unit of y than he consumes of x.
E)None of the above.

Question

Quizplus · Accepted Answer

First, we take the partial derivatives of the utility function with respect to x and y:

Ux = 10(x + 3) and Uy = 10(y + 4)

Then, we can set up the following optimization problem, where M is Charlie's income:

Maximize U(x,y) = 10(x+3)(y+4)

Subject to:

M = Px*x + Py*y, where Px = $1 and Py = $1

We can rewrite the budget constraint as:

M = x + y

We can now use the Lagrangian method to solve for the optimal values of x and y:

L(x,y,λ) = 10(x+3)(y+4) - λ(x+y-M)

Taking the partial derivatives of L with respect to x, y, and λ and setting them equal to 0, we get:

10(y+4) - λ = 0

10(x+3) - λ = 0

x + y = M

Solving for λ in the first two equations and setting them equal to each other, we get:

10(y+4) - λ = 10(x+3) - λ

y + 4 = x + 3

y = x - 1

Substituting this into the budget constraint, we get:

2x - 1 = M

Therefore, Charlie consumes 1 more unit of x than he consumes of y, as in choice B.

Charlie Has a Utility Function U(x, Y)= (X + 3)(y