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

Armando Parra · Accepted Answer

Final Answer : B, Explanation :  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 = $1We can rewrite the budget constraint as:M = x + yWe 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 = 010(x+3) - λ = 0x + y = MSolving for λ in the first two equations and setting them equal to each other, we get:10(y+4) - λ = 10(x+3) - λy + 4 = x + 3y = x - 1Substituting this into the budget constraint, we get:2x - 1 = MTherefore, Charlie consumes 1 more unit of x than he consumes of y, as in choice B.

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