Suppose that Eugene cares only about action figures and ice cream. His utility function is U = A^0.33C^0.67, where A is the number of action figures he owns and C is the number of scoops of ice cream he eats. The price of action figures is $10, and the price of ice cream is $4. Eugene has a budget of $100. How can Eugene's utility maximization problem be expressed as a Lagrangian equation?

Question

Quizplus · Accepted Answer

To write the Lagrangian equation, we can use the following formula: L = U - λ(PaA + PcC - I) where U = A^0.33C^0.67 Pa = $10 (price of action figures) Pc = $4 (price of ice cream) I = $100 (income) After substituting the values, we get: L = A*S1*U1*P1^0.33*S1*P0^C*S1*U1*P1^0.67*S1*P0 - λ(10A + 4C - 100) To find the optimal values of A and C, we need to take partial derivatives of L with respect to A, C, and λ and set them equal to zero. ∂L/∂A = S1*U1*P1^0.33*S1*P0^C - 10λ = 0 ∂L/∂C = S1*U1*P1^0.33*S1*P0^C - 4λ = 0 ∂L/∂λ = 10A + 4C - 100 = 0 Solving these three equations, we get: A = 2S1*P0^(-1)*P1^(0.33)λ^-1 C = 8S1*P0^(-1)*P1^(0.33)λ^-1 Substituting the values of A and C in the budget constraint, we get: 10A + 4C = 100 => 20S1*P0^(-1)*P1^(0.33)λ^-1 + 32S1*P0^(-1)*P1^(0.33)λ^-1 = 100 => 52S1*P0^(-1)*P1^(0.33)λ^-1 = 100 => λ = 0.26S1*P0*P1^(-0.33) Substituting the value of λ in the expressions for A and C, we get: A = 5.2S1*P1^1.33*P0^(-1) C = 20.8S1*P1^0.67*P0^(-1) Therefore, the optimal values of A and C are A = 5.2 and C = 20.8. The Lagrangian equation is: L = A*S1*U1*P1^0.33*S1*P0^C*S1*U1*P1^0.67*S1*P0 - λ(10A + 4C - 100) L = 112.15 - λ(52) Substituting the value of λ, we get: L = 112.15 - 13.48S1*P0*P1^(-0.33) The correct choice is B, which matches this result.

Suppose That Eugene Cares Only About Action Figures and Ice