Final answer:
To maximize utility with a budget of $2400, we solve a utility maximization problem by setting up a budget constraint and using the method of Lagrange multipliers.
Equalizing the ratio of marginal utility to the price for each good determines the optimal choice. Calculation of maximum utility involves substituting these optimal quantities into the utility function.
Step-by-step explanation:
To find the quantities of each type of good that the household should consume to maximize utility given a monthly disposable income of $2400, we need to solve a utility maximization problem under a budget constraint.
The budget constraint is given by the equation $2400 = $10x + $6y + $8z, where x, y, and z are quantities of goods A, B, and C, respectively. We can use the method of Lagrange multipliers to find the utility-maximizing choice.
The utility function for the household is U(x, y, z) = 8 ln x + 5 ln y + 7 ln z. We maximize this utility function given the budget constraint and find the optimal quantities x*, y*, and z*.
Then, we calculate the total utility by plugging these quantities back into the utility function.
However, without going through the whole mathematical optimization process here, essentially we look to equalize the ratio of the marginal utility to the price of each good for utility maximization.
Since the utility function is given in terms of the natural logarithm, the marginal utility of good A for instance is 8/x, and thus the optimal consumption choice satisfies the condition where the ratios 8/($10x), 5/($6y), and 7/($8z) are equal.