Answer: Jed's utility is approximately 6.93 at his utility-maximizing point.
Explanation: Let x be the number of pizzas and y be the number of burgers that Jed consumes per week. Then we have the following constraints: The total amount spent on food is less than or equal to $24: 6x + 3y ≤ 24
Jed's utility function is U(x, y) = 2u(x) + 4u(y) = 8, where u(x) is the marginal utility of pizza and u(y) is the marginal utility of burgers. Since Jed gets equal marginal utility per dollar spent at the utility level of 8, we have: MUx/Px = MUy/Py, where MUx is the marginal utility of pizza, Px is the price of pizza, MUy is the marginal utility of burgers, and Py is the price of burgers.
Substituting the utility function into this equation, we get:
2u'(x)/6 = 4u'(y)/3
u'(x)/u'(y) = 2
This equation tells us that the marginal utility per dollar spent on pizza is twice that of burgers at the utility-maximizing point. To find Jed's utility-maximizing point, we can use the Lagrangian method. The Lagrangian function is: L(x, y, λ) = 2ln(x) + 4ln(y) + λ(24 - 6x - 3y)
Taking partial derivatives with respect to x, y, and λ, and setting them equal to zero, we get:
2/x - 2λ = 0
4/y - 3λ = 0
6x + 3y = 24
Solving these equations, we get:
x = 2
y = 4
λ = 1/6
Therefore, Jed's utility-maximizing point is to consume 2 pizzas and 4 burgers per week, which costs him $18 (2 pizzas at $6 each and 4 burgers at $3 each). His utility at this point is: U(2, 4) = 2u(2) + 4u(4) = 2ln(2) + 4ln(4) = 2ln(2) + 8ln(2) = 10ln(2) ≈ 6.93. Therefore, Jed's utility is approximately 6.93 at his utility-maximizing point.