Problem 07-04 (algo)

Columns 1 through 4 in the following table show the marginal utility, measured in utils, that Ricardo would get by purchasing various amounts of products A, B, C, and D. Column 5 shows the marginal utility Ricardo gets from saving. Assume that the prices of A, B, C, and D are, respectively, $18, $6, $4, and $12 and that Ricardo has an income of $142.

Instructions: Enter your answers as a whole number.

What quantities of A, B, C, and D will Ricardo purchase in maximizing his utility?

units of A

units of B

units of C

units of D

How many dollars will Ricardo choose to save?

$

Check your answers by substituting them into the algebraic statement of the utility-maximizing rule.

($ × + $ × + $ × + $ × + $ ) = $

BlackTom题目解析

Question

Problem 07-04 (algo)

Columns 1 through 4 in the following table show the marginal utility, measured in utils, that Ricardo would get by purchasing various amounts of products A, B, C, and D. Column 5 shows the marginal utility Ricardo gets from saving. Assume that the prices of A, B, C, and D are, respectively, $18, $6, $4, and $12 and that Ricardo has an income of $142.

Instructions: Enter your answers as a whole number.

What quantities of A, B, C, and D will Ricardo purchase in maximizing his utility?

units of A

units of B

units of C

units of D

How many dollars will Ricardo choose to save?

$

Check your answers by substituting them into the algebraic statement of the utility-maximizing rule.

($  ×  + $  ×  + $  ×  + $  ×  + $  ) = $

BlackTom题目解析

BlackTom AI · Accepted Answer

Here is how to approach this problem step by step, given that the data in the table are incomplete.

- Restate the situation and what is missing:
  The problem provides prices for four goods A, B, C, and D (18, 6, 4, and 12 dollars respectively) and an income of 142 dollars, plus a column for marginal utility (MU) for different quantities of each good and for saving. However, the actual marginal utility values for each unit of A, B, C, D, and saving are not given in the prompt you provided. Without those MU values, you cannot determine which purchases maximize Ricardo’s total utility for the given budget.

- What you would need to compute the optimal quantities:
  1) The marginal utility per dollar spent for each good, which is MU of an additional unit of that good divided by its price: MU_A / 18, MU_B / 6, MU_C / 4, MU_D / 12. For saving, you would also need the marginal utility per dollar of saving, MU_S / 1 (if saving is measured in dollars and MU_S is the MU from saving one additional dollar; the exact interpretation can vary by context).
  2) A rule for the optimization: the standard utility-maximizing condition (under a budget constraint) in such discrete MU tables is to keep purchasing units for which MU per dollar is at least as high as MU per dollar for any alternative option, and to equalize MU per dollar across all chosen options as you exhaust the budget. In many textbook problems like this, you would allocate the budget so that MU_A/18 = MU_B/6 = MU_C/4 = MU_D/12 = MU_S/1 for the marginal (last) unit of each option that is consumed, until you run out of income.
  3) The budget constraint: 18*A + 6*B + 4*C + 12*D + Savings = 142, where Savings represents how much Ricardo chooses to save. The sum of expenditures on A, B, C, D plus the amount saved must equal 142.

- How to proceed once you have the MU values:
  a) Compute MU per dollar for each option at the relevant quantity levels (A, B, C, D, and saving). 
  b) Identify the option with the highest MU per dollar and purchase one additional unit of that option, updating the total expenditure and the remaining income. 
  c) Recompute MU per dollar after each purchase, since MU typically changes with quantity in these marginal utility tables. 
  d) Continue this process until you can no longer afford any additional unit of any option without exceeding income, or until MU per dollar for all affordable options is equalized at the margin.

- About the specific prompt line “Cannot be determined without the marginal utility values from the table.”
  This is a correct assessment given the data provided here: without the actual MU numbers for each quantity level, you cannot compute MU per dollar, compare options, or determine the exact quantities of A, B, C, D, and the amount to save that maximize utility. Any attempt to fill in quantities or savings would be speculative.

- What to do if you had the table (quick checklist):
  1) List MU_A, MU_B, MU_C, MU_D, and MU_S for successive units as shown. 2) For each unit, compute MU/p for A, B, C, D, and saving. 3) Starting from zero quantities, repeatedly add the unit with the highest MU/p that does not violate the budget, updating remaining income each time. 4) When no affordable unit has MU/p greater than any other option’s MU/p, stop. 5) Record the final quantities of A, B, C, D, and the dollar amount saved, and verify the algebraic condition by plugging into the equation given in the prompt.

- Final note on the given student answer: 
  The provided answer fragment states that the quantities cannot be determined without the table’s marginal utility values. That assessment is correct, and the next step for a student is to use the actual MU values from the table to perform the allocation process described above.

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