Suppose that Arthur cares only about video games and coffee. His utility function is U = VC, where V is the number of video games that he plays and C is the number of cups of coffee that he drinks. The price of video games is $50, and the price of coffee is $3. Arthur has a budget of $800. What is the Lagrangian equation that can be used to solve Arthur's utility maximization problem?

Question

Quizplus · Accepted Answer

Arthur's utility maximization problem can be expressed as:

max U = VC, subject to PvV + PcC = M

where Pv is the price of video games, Pc is the price of coffee, V is the number of video games, C is the number of cups of coffee, and M is Arthur's budget.

Using Lagrangian method, we can specify the Lagrangian function as:

L = VC - λ(PvV + PcC - M)

where λ is the Lagrange multiplier.

Taking the partial derivative of L with respect to V, C, and λ and setting each equal to zero, we get:

∂L/∂V = C - λPv = 0
∂L/∂C = V - λPc = 0
∂L/∂λ = PvV + PcC - M = 0

Solving for V, C, and λ, we get:

V = λPc
C = λPv
PvV + PcC = M

Substituting the first two equations into the third, we get:

λPvPc + λPcPv = M

Simplifying, we get:

2λPvPc = M

Solving for λ, we get:

λ = M/2PvPc

Substituting λ into the first two equations, we get:

V = Pc(M/2PvPc) = M/2Pv
C = Pv(M/2PvPc) = M/6

Substituting the values of Pv, Pc, and M, we get:

V = 800/2(50)(3) = 2.67
C = 800/6 = 133.33

Therefore, Arthur should play 2 video games and drink 133 cups of coffee to maximize his utility with a budget of $800.

Suppose That Arthur Cares Only About Video Games and Coffee