What combination of goods maximizes the consumer's satisfaction?

Question

ScanSolve AI Tutor · Accepted Answer

| Quantity — Total Utility of Brownies — Marginal Utility of Brownies — MU/P for Brownies (P_B=2) — Total Utility of Cookies — Marginal Utility of Cookies — MU/P for Cookies (P_C=4) — Total Utility of Pieces of Pie — Marginal Utility of Pieces of Pie — MU/P for Pieces of Pie (P_P=8)0 — 0 — - — - — 0 — - — - — 0 — - — - 1 — 9 — 9 — 4.5 — 16 — 16 — 4 — 27 — 27 — 3.375 2 — 17 — 8 — 4 — 33 — 17 — 4.25 — 51 — 24 — 3 3 — 24 — 7 — 3.5 — 49 — 16 — 4 — 75 — 24 — 3 4 — 30 — 6 — 3 — 61 — 12 — 3 — 99 — 24 — 3 5 — 34 — 4 — 2 — 69 — 8 — 2 — 109 — 10 — 1.25 6 — 36 — 2 — 1 — 73 — 4 — 1 — 113 — 4 — 0.5 7 — 36 — 0 — 0 — 72 — −1 — −0.25 — 110 — −3 — −0.375 a) What combination of goods maximizes the consumer's satisfaction? Step 1: Calculate the Marginal Utility (MU) for each additional unit of each good. MU = Δ TU / Δ Q. For example, for the 1st brownie, MU_B = 9−0 = 9. For the 2nd brownie, MU_B = 17−9 = 8. Step 2: Calculate the Marginal Utility per Dollar (MU/P) for each unit of each good. MU/P = MU / Price. Given prices are P_B = $2, P_C = $4, P_P = $8. The table above shows these calculations. Step 3: Allocate the budget of $40 by purchasing the good that offers the highest MU/P at each step, until the budget is exhausted. 1. Budget: $40 Highest MU/P is 4.5 (1st Brownie). Buy 1 Brownie. Cost: $2. Remaining Budget: $38. 2. Budget: $38 Highest MU/P is 4.25 (2nd Cookies). Buy 1 Bag of Cookies. Cost: $4. Remaining Budget: $34. 3. Budget: $34 Highest MU/P is 4 (1st Cookies, 2nd Brownie). Let's buy 1st Cookies. Cost: $4. Remaining Budget: $30. (Now 1 Brownie, 2 Cookies) 4. Budget: $30 Highest MU/P is 4 (2nd Brownie, 3rd Cookies). Let's buy 2nd Brownie. Cost: $2. Remaining Budget: $28. (Now 2 Brownies, 2 Cookies) 5. Budget: $28 Highest MU/P is 4 (3rd Cookies). Buy 1 Bag of Cookies. Cost: $4. Remaining Budget: $24. (Now 2 Brownies, 3 Cookies) 6. Budget: $24 Highest MU/P is 3.5 (3rd Brownie). Buy 1 Brownie. Cost: $2. Remaining Budget: $22. (Now 3 Brownies, 3 Cookies) 7. Budget: $22 Highest MU/P is 3.375 (1st Pie). Buy 1 Piece of Pie. Cost: $8. Remaining Budget: $14. (Now 3 Brownies, 3 Cookies, 1 Pie) 8. Budget: $14 Highest MU/P is 3 (4th Brownie, 4th Cookies, 2nd Pie, 3rd Pie, 4th Pie). Let's pick 4th Brownie. Cost: $2. Remaining Budget: $12. (Now 4 Brownies, 3 Cookies, 1 Pie) 9. Budget: $12 Highest MU/P is 3 (4th Cookies, 2nd Pie, 3rd Pie, 4th Pie). Let's pick 4th Cookies. Cost: $4. Remaining Budget: $8. (Now 4 Brownies, 4 Cookies, 1 Pie) 10. Budget: $8 Highest MU/P is 3 (2nd Pie, 3rd Pie, 4th Pie). Let's pick 2nd Pie. Cost: $8. Remaining Budget: 0. (Now 4 Brownies, 4 Cookies, 2 Pies) At this point, the budget is exhausted, and the MU/P for the last unit purchased of each good is 3. For 4th Brownie: MU_B/P_B = 6/2 = 3. For 4th Cookies: MU_C/P_C = 12/4 = 3. For 2nd Pie: MU_P/P_P = 24/8 = 3$. The equimarginal principle is satisfied. The combination of goods that maximizes the consumer's satisfaction is: 4 Brownies, 4 Bags of Cookies, and 2 Pieces of Pie. b) What is the total utility received from this combination? Step 1: Sum the total utility for each good at the quantities determined in part (a). Total Utility from 4 Brownies (TU_B at Q=4) = 30; Total Utility from 4 Bags of Cookies (TU_C at Q=4) = 61; Total Utility from 2 Pieces of Pie (TU_P at Q=2) = 51 Step 2: Calculate the total utility. Total Utility = TU_B(4) + TU_C(4) + TU_P(2) = 30 + 61 + 51 = 142 The total utility received from this combination is 142. c) If the price of pies fell to 6 a piece, what would be the new combination that would maximize the consumer's satisfaction? Step 1: Recalculate MU/P for Pieces of Pie with the new price P_P = $6. The MU for pie remains the same. Quantity — MU_P — New MU_P/P_P (P_P=6$)1 — 27 — 4.5 2 — 24 — 4 3 — 24 — 4 4 — 24 — 4 5 — 10 — 1.67 6 — 4 — 0.67 7 — −3 — −0.5 Step 2: Re-allocate the budget of $40 using the updated MU/P for pie. 1. Budget: $40 Highest MU/P is 4.5 (1st Brownie, 1st Pie). Let's buy 1st Brownie. Cost: $2. Remaining Budget: $38. 2. Budget: $38 Highest MU/P is 4.5 (1st Pie). Buy 1 Piece of Pie. Cost: $6. Remaining Budget: $32. (Now 1 Brownie, 1 Pie) 3. Budget: $32 Highest MU/P is 4.25 (2nd Cookies). Buy 1 Bag of Cookies. Cost: $4. Remaining Budget: $28. (Now 1 Brownie, 1 Cookies, 1 Pie) 4. Budget: $28 Highest MU/P is 4 (1st Cookies, 2nd Brownie, 2nd Pie, 3rd Pie, 4th Pie). Let's buy 1st Cookies. Cost: $4. Remaining Budget: $24. (Now 1 Brownie, 2 Cookies, 1 Pie) 5. Budget: $24 Highest MU/P is 4 (2nd Brownie, 2nd Pie, 3rd Pie, 4th Pie). Let's buy 2nd Brownie. Cost: $2. Remaining Budget: $22. (Now 2 Brownies, 2 Cookies, 1 Pie) 6. Budget: $22 Highest MU/P is 4 (2nd Pie, 3rd Pie, 4th Pie, 3rd Cookies). Let's buy 2nd Pie. Cost: $6. Remaining Budget: $16. (Now 2 Brownies, 2 Cookies, 2 Pies) 7. Budget: $16 Highest MU/P is 4 (3rd Pie, 4th Pie, 3rd Cookies). Let's buy 3rd Pie. Cost: $6. Remaining Budget: $10. (Now 2 Brownies, 2 Cookies, 3 Pies) 8. Budget: $10 Highest MU/P is 4 (4th Pie, 3rd Cookies). Let's buy 4th Pie. Cost: $6. Remaining Budget: $4. (Now 2 Brownies, 2 Cookies, 4 Pies) 9. Budget: $4 Highest MU/P is 4 (3rd Cookies). Buy 1 Bag of Cookies. Cost: $4. Remaining Budget: 0. (Now 2 Brownies, 3 Cookies, 4 Pies) At this point, the budget is exhausted. For 2nd Brownie: MU_B/P_B = 8/2 = 4. For 3rd Cookies: MU_C/P_C = 16/4 = 4. For 4th Pie: MU_P/P_P = 24/6 = 4$. The equimarginal principle is satisfied. The new combination that maximizes the consumer's satisfaction is: 2 Brownies, 3 Bags of Cookies, and 4 Pieces of Pie. That's 2 down. 3 left today — send the next one.Step 1: Calculate Marginal Utility (MU) and Marginal Utility per Dollar (MU/P) for each good. The prices are: Brownie (P_B) = 2, Bag of Cookies (P_C$) = 4, Piece of Pie (P_P$) = $8. The budget is 40. Quantity — TU_B — MU_B — MU_B/P_B — TU_C — MU_C — MU_C/P_C — TU_P — MU_P — MU_P/P_P$0 — 0 — - — - — 0 — - — - — 0 — - — - 1 — 9 — 9 — 4.5 — 16 — 16 — 4 — 27 — 27 — 3.375 2 — 17 — 8 — 4 — 33 — 17 — 4.25 — 51 — 24 — 3 3 — 24 — 7 — 3.5 — 49 — 16 — 4 — 75 — 24 — 3 4 — 30 — 6 — 3 — 61 — 12 — 3 — 99 — 24 — 3 5 — 34 — 4 — 2 — 69 — 8 — 2 — 109 — 10 — 1.25 6 — 36 — 2 — 1 — 73 — 4 — 1 — 113 — 4 — 0.5 7 — 36 — 0 — 0 — 72 — −1 — −0.25 — 110 — −3 — −0.375 a) What combination of goods maximizes the consumer's satisfaction? Step 1: Allocate the 40 budget by choosing the good with the highest MU/P at each step, ensuring the equimarginal principle (MU_B/P_B = MU_C/P_C = MU_P/P_P) is met as closely as possible when the budget is exhausted. 1. Buy 1st Brownie (MU_B/P_B = 4.5$). Cost: $2. Budget: 38. (B=1); 2. Buy 1st Bag of Cookies (MU_C/P_C = 4.25$). Cost: $4. Budget: 34. (B=1, C=1); 3. Buy 1st Piece of Pie (MU_P/P_P = 3.375$). Cost: $8. Budget: 26. (B=1, C=1, P=1); 4. Buy 2nd Brownie (MU_B/P_B = 4$). Cost: $2. Budget: 24. (B=2, C=1, P=1); 5. Buy 2nd Bag of Cookies (MU_C/P_C = 4$). Cost: $4. Budget: 20. (B=2, C=2, P=1); 6. Buy 3rd Bag of Cookies (MU_C/P_C = 4$). Cost: $4. Budget: 16. (B=2, C=3, P=1); 7. Buy 3rd Brownie (MU_B/P_B = 3.5$). Cost: $2. Budget: 14. (B=3, C=3, P=1); 8. Buy 2nd Piece of Pie (MU_P/P_P = 3$). Cost: $8. Budget: 6. (B=3, C=3, P=2); 9. Buy 4th Brownie (MU_B/P_B = 3$). Cost: $2. Budget: 4. (B=4, C=3, P=2); 10. Buy 4th Bag of Cookies (MU_C/P_C = 3$). Cost: $4. Budget: $0. (B=4, C=4, P=2) The budget is exhausted, and the MU/P for the last unit of each good purchased is 3. The combination of goods that maximizes the consumer's satisfaction is 4 Brownies, 4 Bags of Cookies, and 2 Pieces of Pie. b) What is the total utility received from this combination? Step 1: Sum the total utility for the quantities determined in part (a). Total Utility from 4 Brownies (TU_B at Q=4) = 30; Total Utility from 4 Bags of Cookies (TU_C at Q=4) = 61; Total Utility from 2 Pieces of Pie (TU_P at Q=2) = 51 Step 2: Calculate the total utility. Total Utility = 30 + 61 + 51 = 142 The total utility received from this combination is 142. c) If the price of pies fell to 6 a piece, what would be the new combination that would maximize the consumer's satisfaction? Step 1: Recalculate the Marginal Utility per Dollar (MU/P) for Pieces of Pie with the new price P_P = $6. The Marginal Utility (MU) values for pie remain the same as in the table from part (a). Quantity — MU_P — New MU_P/P_P (P_P=6$)1 — 27 — 27/6 = 4.5; 2 — 24 — 24/6 = 4; 3 — 24 — 24/6 = 4; 4 — 24 — 24/6 = 4 5 — 10 — 10/6 ≈ 1.67 6 — 4 — 4/6 ≈ 0.67 7 — −3 — −3/6 = −0.5 Step 2: Re-allocate the budget of 40 by purchasing the good that offers the highest MU/P at each step, until the budget is exhausted, ensuring the equimarginal principle is met. (Refer to the MU/P values for brownies and cookies from the table in part (a), and the new MU/P values for pie above). 1. Buy 1st Brownie (MU_B/P_B = 4.5$). Cost: $2. Remaining Budget: 38. (Current: 1B); 2. Buy 1st Piece of Pie (MU_P/P_P = 4.5$). Cost: $6. Remaining Budget: 32. (Current: 1B, 1P); 3. Buy 2nd Bag of Cookies (MU_C/P_C = 4.25$). Cost: $4. Remaining Budget: 28. (Current: 1B, 1C, 1P); 4. Buy 2nd Brownie (MU_B/P_B = 4$). Cost: $2. Remaining Budget: 26. (Current: 2B, 1C, 1P); 5. Buy 1st Bag of Cookies (MU_C/P_C = 4$). Cost: $4. Remaining Budget: 22. (Current: 2B, 2C, 1P); 6. Buy 2nd Piece of Pie (MU_P/P_P = 4$). Cost: $6. Remaining Budget: 16. (Current: 2B, 2C, 2P); 7. Buy 3rd Piece of Pie (MU_P/P_P = 4$). Cost: $6. Remaining Budget: 10. (Current: 2B, 2C, 3P); 8. Buy 4th Piece of Pie (MU_P/P_P = 4$). Cost: $6. Remaining Budget: 4. (Current: 2B, 2C, 4P); 9. Buy 3rd Bag of Cookies (MU_C/P_C = 4$). Cost: $4. Remaining Budget: 0. (Current: 2B, 3C, 4P); At this point, the budget is exhausted. The MU/P for the last unit of each good purchased is 4 (MU_B/P_B = 8/2 = 4, MU_C/P_C = 16/4 = 4, MU_P/P_P = 24/6 = 4$). The new combination that maximizes the consumer's satisfaction is 2 Brownies, 3 Bags of Cookies, and 4 Pieces of Pie.

Quantity	$MU_P$	New $MU_P/P_P$ ( $P_P=6$ )
1	27	$27/6 = 4.5$
2	24	$24/6 = 4$
3	24	$24/6 = 4$
4	24	$24/6 = 4$
5	10	$10/6 \approx 1.67$
6	4	$4/6 \approx 0.67$
7	-3	$-3/6 = -0.5$

What combination of goods maximizes the consumer's satisfaction?

Step-by-step solution

Calculate the Marginal Utility (MU) for each additional unit of each good.

Calculate the Marginal Utility per Dollar (MU/P) for each unit of each good.

Allocate the budget of $40 by purchasing the good that offers the highest MU/P at each step, until the budget is exhausted.

Sum the total utility for each good at the quantities determined in part (a).

Calculate the total utility.

Recalculate MU/P for Pieces of Pie with the new price `P_P = \`6$.

Re-allocate the budget of $40 using the updated MU/P for pie.

Allocate the $40 budget by choosing the good with the highest MU/P at each step, ensuring the equimarginal principle ($ MU_B/P_B = MU_C/P_C = MU_P/P_P$) is met as closely as possible when the budget is exhausted.

Sum the total utility for the quantities determined in part (a).

Calculate the total utility.

Recalculate the Marginal Utility per Dollar (MU/P) for Pieces of Pie with the new price `P_P = \`6$.

Re-allocate the budget of $40 by purchasing the good that offers the highest MU/P at each step, until the budget is exhausted, ensuring the equimarginal principle is met.

Still stuck on this one?

What combination of goods maximizes the consumer's satisfaction?

Step-by-step solution

Calculate the Marginal Utility (MU) for each additional unit of each good.

Calculate the Marginal Utility per Dollar (MU/P) for each unit of each good.

Allocate the budget of $40 by purchasing the good that offers the highest MU/P at each step, until the budget is exhausted.

Sum the total utility for each good at the quantities determined in part (a).

Calculate the total utility.

Recalculate MU/P for Pieces of Pie with the new price P_P = \6$.

Re-allocate the budget of $40 using the updated MU/P for pie.

Sum the total utility for the quantities determined in part (a).

Calculate the total utility.

Recalculate the Marginal Utility per Dollar (MU/P) for Pieces of Pie with the new price P_P = \6$.

Re-allocate the budget of $40 by purchasing the good that offers the highest MU/P at each step, until the budget is exhausted, ensuring the equimarginal principle is met.

Still stuck on this one?

Recalculate MU/P for Pieces of Pie with the new price `P_P = \`6$.

Recalculate the Marginal Utility per Dollar (MU/P) for Pieces of Pie with the new price `P_P = \`6$.