Suppose a consumer has M = $200 to spend on two goods, X and Y.If the per-unit prices of X and Y are respectively given by P_X = $2 and P_Y = $4, then to maximize utility subject to a budget constraint can be solved by form which of the following Lagrangian? A)? = U(X,Y) + $\lambda$(200 - 2X - 4Y). B)? = U(X,Y) + $\lambda$(200 - 2P_X - 4P_Y). C)? = (200 - 2X - 4Y) + $\lambda$U(X,Y). D)? = U(X,Y) + $\lambda$(2X - 4Y + 200).

Question

Quizplus · Accepted Answer

The Lagrangian for this optimization problem should have the form: 
ℒ = U(X,Y) + λ(200 - P1X - P2Y)
where λ is the Lagrange multiplier, and P1 and P2 are the prices of goods X and Y, respectively. 
Substituting the given prices, we get:
ℒ = U(X,Y) + λ(200 - 2X - 4Y)
Option A is the only one that matches this form. The correct Lagrange multiplier to use with this Lagrangian is λ/2, but this is not specified in any of the answer choices.

Suppose a Consumer Has M = $200 to Spend on Two