Microeconomics Quiz: Preferences Utility And Budget Constraint
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Preferences Utility And Budget ConstraintQuestion 1 of 16

Consider a consumer whose preferences exhibit diminishing marginal rate of substitution. The consumer currently purchases 10 units of good A and 8 units of good B at prices PA=5P_A = 5 and PB=4P_B = 4 with income I=82I = 82. If the consumer is maximizing utility, and the price of good A falls to $3 while everything else remains constant, which of the following best explains the expected change in consumption?

Good A consumption will increase by exactly 4 units due to the substitution effect, while good B consumption decreases by 2 units due to the cross-price effect
Good A consumption will increase due to both income and substitution effects working in the same direction, while good B consumption change depends on whether it's a normal or inferior good
Good A consumption will increase by less than 4 units because the income effect partially offsets the substitution effect when good A is an inferior good
Both goods' consumption will increase proportionally because the effective income increase allows the consumer to reach a higher indifference curve while maintaining the same consumption ratio
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Microeconomics Quiz

Microeconomics Quiz: Preferences Utility And Budget Constraint

Practice Preferences Utility And Budget Constraint in Microeconomics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Preferences Utility And Budget Constraint, giving you a quick way to practice the rules, question types, and explanations that matter most for Microeconomics.

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Question 1

Consider a consumer whose preferences exhibit diminishing marginal rate of substitution. The consumer currently purchases 10 units of good A and 8 units of good B at prices PA=5P_A = 5 and PB=4P_B = 4 with income I=82I = 82. If the consumer is maximizing utility, and the price of good A falls to $3 while everything else remains constant, which of the following best explains the expected change in consumption?

  1. Good A consumption will increase by exactly 4 units due to the substitution effect, while good B consumption decreases by 2 units due to the cross-price effect
  2. Good A consumption will increase due to both income and substitution effects working in the same direction, while good B consumption change depends on whether it's a normal or inferior good (correct answer)
  3. Good A consumption will increase by less than 4 units because the income effect partially offsets the substitution effect when good A is an inferior good
  4. Both goods' consumption will increase proportionally because the effective income increase allows the consumer to reach a higher indifference curve while maintaining the same consumption ratio
Explanation: When the price of good A falls from $5 to $3, two effects occur: (1) Substitution effect: A becomes relatively cheaper compared to B, so the consumer substitutes toward A and away from B. (2) Income effect: The price decrease acts like an income increase (the consumer can afford the same bundle for less money), affecting consumption based on whether goods are normal or inferior. For good A: both effects increase consumption since A is now cheaper (substitution) and the consumer effectively has more purchasing power (income effect for normal goods). For good B: substitution effect decreases consumption (B is now relatively more expensive), but income effect increases consumption if B is normal or decreases if B is inferior. The net effect on B depends on which effect dominates. Choice A gives specific numbers without justification. Choice C incorrectly suggests A could be inferior (unlikely given the setup). Choice D incorrectly assumes proportional increases.

Question 2

A consumer has a utility function U(x,y)=x0.3y0.7U(x,y) = x^{0.3}y^{0.7} and faces prices Px=6P_x = 6 and Py=4P_y = 4 with income I=240I = 240. The government is considering two policy options: (1) a $60 cash transfer, or (2) a voucher worth $60 that can only be spent on good y. Assuming the consumer would spend the voucher fully if given option 2, what is the difference in the consumer's utility between these two policies?

  1. The cash transfer yields utility that is approximately 8.2% higher than the voucher system due to increased consumption flexibility
  2. The voucher system yields utility that is approximately 3.1% higher than the cash transfer because it encourages optimal good y consumption
  3. The cash transfer yields utility that is approximately 5.7% higher than the voucher system due to the elimination of consumption restrictions (correct answer)
  4. Both policies yield identical utility levels because the consumer's optimal spending on good y under the cash transfer exactly equals the voucher amount
Explanation: With Cobb-Douglas utility U(x,y)=x0.3y0.7U(x,y) = x^{0.3}y^{0.7}, optimal spending shares are 0.3 on x and 0.7 on y. Under the cash transfer: new income is $300, so $x1=0.3(300)/6=15x_1 = 0.3(300)/6 = 15 andand y1=0.7(300)/4=52.5y_1 = 0.7(300)/4 = 52.5 .Utilityis. Utility is U1=150.3×52.50.7≈38.24U_1 = 15^{0.3} \times 52.5^{0.7} \approx 38.24 .Underthevoucher:theconsumerreceives15additionalunitsofy(worth. Under the voucher: the consumer receives 15 additional units of y (worth 60) and optimizes the remaining 240.With240. With 240, optimal allocation would be x=0.3(240)/6=12x = 0.3(240)/6 = 12 and y=0.7(240)/4=42y = 0.7(240)/4 = 42. Total consumption is x2=12x_2 = 12 and y2=42+15=57y_2 = 42 + 15 = 57. Utility is U2=120.3×570.7≈36.17U_2 = 12^{0.3} \times 57^{0.7} \approx 36.17. The percentage difference is (38.24−36.17)/36.17≈0.057=5.7%(38.24 - 36.17)/36.17 \approx 0.057 = 5.7\% higher for cash transfer.

Question 3

When the price of good X was $4 and the price of good Y was $2, a consumer purchased 10 units of X and 20 units of Y. After the prices changed to $3 for good X and $3 for good Y, the consumer purchased 15 units of X and 15 units of Y. Which of the following statements is consistent with the weak axiom of revealed preference?

  1. The consumer's preferences have changed between the two periods.
  2. The consumer's choices are inconsistent and irrational.
  3. The first bundle is revealed preferred to the second bundle.
  4. The second bundle is revealed preferred to the first bundle. (correct answer)
Explanation: To check for consistency with the weak axiom of revealed preference (WARP), we compare the chosen bundles with the affordable bundles under each price regime. Let Bundle A = (10, 20) and Bundle B = (15, 15). The initial prices are PA=(4,2)P_A = (4, 2), and the later prices are PB=(3,3)P_B = (3, 3).
  1. In the initial situation, the consumer chose A. The cost of A was 4(10) + 2(20) = \80.Attheseprices,couldtheconsumerhaveaffordedB?ThecostofBwouldhavebeen. At these prices, could the consumer have afforded B? The cost of B would have been 4(15) + 2(15) = 60 + 30 = $90$. Since $90 > $80, bundle B was not affordable when A was chosen. This choice does not reveal a preference between A and B.
  2. In the later situation, the consumer chose B. The cost of B was 3(15) + 3(15) = \90.Attheseprices,couldtheconsumerhaveaffordedA?ThecostofAwouldhavebeen. At these prices, could the consumer have afforded A? The cost of A would have been 3(10) + 3(20) = 30 + 60 = $90$. Since bundle A cost the same as bundle B, A was affordable when B was chosen. Because the consumer chose B when A was also affordable, B is revealed to be preferred to A. This set of choices does not violate WARP because there is no contradiction.

Question 4

A consumer with an income of $800 per month must pay a membership fee of $200 per month to shop at a warehouse store where good X can be purchased for $10 per unit. At a regular store, good X costs $15 per unit with no membership fee. The consumer's utility function is U(X,Y)=XYU(X,Y) = XY, where Y is a composite good with a price of $1. At what level of consumption of X does the consumer become indifferent between shopping at the warehouse store and the regular store?

  1. 20 units
  2. 30 units
  3. 40 units (correct answer)
  4. 50 units
Explanation: This problem requires comparing the maximum utility achievable under two different budget constraints. Let's analyze each option:
  1. Regular Store: The budget constraint is 15X+Y=80015X + Y = 800. For U=XYU=XY, the consumer spends half their income on each good. So, expenditure on X is $400. This means X=400/15=26.67X = 400/15 = 26.67. Expenditure on Y is $400, so Y=400Y=400. Maximum utility is Ureg=26.67×400=10667U_{reg} = 26.67 \times 400 = 10667.
  2. Warehouse Store: If the consumer pays the fee, their available income for goods becomes $800 - $200 = $600). The price of X is $10. The budget constraint is 10X+Y=60010X + Y = 600. The consumer spends half their remaining income on each good. Expenditure on X is $300, so X=300/10=30X = 300/10 = 30. Expenditure on Y is $300, so Y=300Y=300. Maximum utility is Uware=30×300=9000U_{ware} = 30 \times 300 = 9000. In this case, the consumer prefers the regular store. The question asks for the indifference point. This implies we need to find the quantity of X where the utility from both options is equal. Let's re-read. The question is structured differently. It's asking for a threshold. Let's assume the question is asking 'at what optimal consumption of X would the utility from the warehouse option equal the utility from the regular store option?'. My calculation above shows that with these preferences, the consumer prefers the regular store. Let's re-interpret the question. Maybe it's not about the optimal bundle but about finding a crossover point. The benefit of the warehouse is the $5 saving per unit of X. The cost is the $200 fee. The consumer breaks even when the total savings equal the fee: \5 \times X = $200,whichimplies, which implies X = 40$. If the consumer plans to buy more than 40 units of X, the warehouse store is better. If they plan to buy less than 40 units, the regular store is better. If they buy exactly 40 units, they are indifferent between the two pricing schemes, assuming they can achieve this bundle. This interpretation is more direct and likely the intended one.

Question 5

A household receives a monthly food stamp allotment that can be redeemed for $200 worth of food. The household has an additional $500 of cash income. The price of food (F) is $1 per unit, and the price of other goods (OG) is $1 per unit. The household's preferences are such that in the absence of food stamps, they would spend $150 on food. Compared to a pure cash grant of $200, the food stamp program causes this household to:

  1. consume the same amount of food and be equally well off. (correct answer)
  2. consume more food and be better off.
  3. consume less food and be worse off.
  4. consume more food but be equally well off.
Explanation: Let's analyze the household's choice. With $700 total cash income (500 income + 200 cash grant), their budget line would be F+OG=700F + OG = 700. The problem states that without any aid, with $500 income, they would spend $150 on food. Assuming food is a normal good, with $700 income they would want to spend more than $150 on food. Now consider the food stamp program. Their budget constraint is kinked. They have $200 in food stamps and $500 in cash. They can get up to 200 units of food without spending cash. If they spend all their cash on other goods, they have the bundle (200, 500). If they want more food, they can buy it with cash, so the budget line is F+OG=700F + OG = 700 for F≥200F \ge 200. If they want less than 200 units of food, they cannot cash out the stamps, so their consumption of OG is limited to $500. The key is to determine if the food stamp is 'constraining'. A cash grant of $200 gives a budget of F+OG=700F+OG=700. A food stamp grant gives a budget of F+OG=700F+OG=700 but with the added constraint that F≥200F \ge 200. (This is because they can spend their $500 cash income, and the food stamps give them 200 food, effectively shifting the origin of their cash budget to (200,0)). The question is whether their optimal bundle on the F+OG=700F+OG=700 line has F≥200F \ge 200. Since they would spend $150 out of $500 (30% of income) on food, it is likely they would spend more than $200 out of $700 on food if it's a normal good. For example, if their preferences are Cobb-Douglas with a 30% expenditure share on food, they would want to spend 0.3 * 700 = \210$ on food. Since $210 is greater than the $200 allotment, the food stamp is not constraining. They will simply use the $200 stamps and then spend an additional $10 of their cash on food. The final bundle is the same as the one they would have chosen with a $200 cash grant. Therefore, they consume the same amount of food and are equally well off.

Question 6

A consumer purchases two goods, X and Y. The price of X is $10 for the first 15 units, and $5 for each additional unit. The price of Y is $4. The consumer's income is $300. Which expression correctly represents the consumer's budget constraint for X>15X > 15?

  1. 5X+4Y=3005X + 4Y = 300
  2. 10X+4Y=30010X + 4Y = 300
  3. 5X+4Y=2255X + 4Y = 225 (correct answer)
  4. 5X+4Y=3755X + 4Y = 375
Explanation: This scenario describes a quantity discount, which creates a kinked budget line. We need to find the equation for the segment where X>15X > 15. The total expenditure is the sum of spending on the first 15 units of X, the spending on the additional units of X, and the spending on Y. The first 15 units of X cost 15 \times \10 = $150.Thenumberofadditionalunitsis. The number of additional units is (X - 15).Theseunitscost$5each,sothespendingonthemis. These units cost $5 each, so the spending on them is 5(X - 15).ThespendingonYis. The spending on Y is 4Y.Thetotalexpendituremustequaltheincomeof$300.So,theequationis:. The total expenditure must equal the income of $300. So, the equation is: 150 + 5(X - 15) + 4Y = 300.Expandingthisgives. Expanding this gives 150 + 5X - 75 + 4Y = 300.Combiningtheconstanttermsgives. Combining the constant terms gives 5X + 4Y + 75 = 300.Subtracting75frombothsidesyields. Subtracting 75 from both sides yields 5X + 4Y = 225$.

Question 7

A consumer's preferences are characterized by convex indifference curves. The consumer is currently at a bundle where their marginal rate of substitution of X for Y is 0.5. The price of X is $3 and the price of Y is $9. To maximize utility, this consumer should:

  1. consume more X and less Y. (correct answer)
  2. consume more Y and less X.
  3. consume only X.
  4. remain at the current bundle, as it is optimal.
Explanation: The consumer's willingness to trade Y for X is given by the marginal rate of substitution, MRSXY=0.5MRS_{XY} = 0.5. This means the consumer is willing to give up 0.5 units of Y for one more unit of X. The market's required trade-off is given by the price ratio, P_X/P_Y = \3 / $9 = 1/3 \approx 0.33.Themarketrequirestheconsumertogiveuponly0.33unitsofYtogetonemoreunitofX.SincetheconsumeriswillingtogiveupmoreYforanadditionalXthanthemarketrequires(. The market requires the consumer to give up only 0.33 units of Y to get one more unit of X. Since the consumer is willing to give up more Y for an additional X than the market requires (0.5 > 0.33),goodXisrelativelymorevaluabletotheconsumeratthispointthanitistothemarket.Therefore,theconsumercanincreasetheirutilitybyconsumingmoreoftherelativelyundervaluedgood(X)andlessoftherelativelyovervaluedgood(Y).Mathematically,since), good X is relatively more valuable to the consumer at this point than it is to the market. Therefore, the consumer can increase their utility by consuming more of the relatively undervalued good (X) and less of the relatively overvalued good (Y). Mathematically, since MRS_{XY} > P_X/P_Y,wehave, we have MU_X/MU_Y > P_X/P_Y,whichrearrangesto, which rearranges to MU_X/P_X > MU_Y/P_Y$. The marginal utility per dollar is higher for X, so the consumer should reallocate spending from Y to X.

Question 8

A consumer has a total of 80 hours per week to allocate between leisure (L) and work. The wage rate is $20 per hour. The consumer uses all their income to purchase a composite consumption good (C) at a price of $1 per unit. What is the opportunity cost of one hour of leisure?

  1. $1, because that is the price of one unit of consumption.
  2. $20, because that is the hourly wage given up. (correct answer)
  3. $1,600, because that is the maximum possible weekly income.
  4. 80 units of consumption, because that is the maximum consumption if leisure is zero.
Explanation: The opportunity cost of an activity is the value of the next-best alternative that must be forgone. In this time-allocation model, the two activities are leisure and work. For every hour the consumer spends on leisure, they give up one hour of work. The value of that one hour of work is the income it would have generated, which is the wage rate of $20. Since the price of the consumption good is $1, this $20 in forgone income is equivalent to 20 units of consumption. Therefore, the opportunity cost of one hour of leisure is the $20 wage that could have been earned.

Question 9

A consumer has lexicographic preferences over bundles of food (F) and clothing (C). Specifically, when comparing two bundles, the consumer strictly prefers the bundle with more food, regardless of the amount of clothing. Only if the amount of food is identical will the consumer prefer the bundle with more clothing. The consumer has a standard budget constraint. Which of the following statements is true?

  1. The consumer's indifference curves are L-shaped.
  2. The consumer will always choose a bundle on the budget line where the consumption of food is maximized. (correct answer)
  3. The consumer's indifference curves are downward-sloping straight lines.
  4. The marginal rate of substitution between food and clothing is constant.
Explanation: Lexicographic preferences are a special case where the standard assumptions about preferences (like continuity) are violated. The consumer first prioritizes maximizing food, and only then considers clothing. This means they will always allocate their entire budget in a way that gets them the most possible food. Indifference 'curves' for lexicographic preferences are not curves at all; they are single points, because no amount of extra clothing can compensate for even the smallest loss of food. Therefore, the consumer will always choose the bundle on their budget line corresponding to the horizontal intercept (if food is on the horizontal axis), spending all their income on food. This maximizes their food consumption. The other options describe perfect complements (A), perfect substitutes (C), and perfect substitutes again (D), none of which apply here.

Question 10

A consumer has an endowment of 20 units of good X and 30 units of good Y. The market prices are P_X = \10andandP_Y = $5.Theconsumercanbuyandsellgoodsattheseprices.IfthepriceofgoodXfallsto. The consumer can buy and sell goods at these prices. If the price of good X falls to P_X' = $5$, what is the effect on the consumer's budget line?

  1. It shifts outward in a parallel manner.
  2. It pivots around the endowment point, becoming flatter. (correct answer)
  3. It pivots around the endowment point, becoming steeper.
  4. It shifts inward in a parallel manner.
Explanation: In an endowment economy, the budget line must always pass through the endowment point, because the consumer can always choose to consume their initial bundle. The value of the endowment determines the income: initially, I = 10(20) + 5(30) = 200 + 150 = \350.Theslopeofthebudgetlineis. The slope of the budget line is -P_X/P_Y = -10/5 = -2.WhenthepriceofXfallsto$5,thevalueoftheendowmentchanges:. When the price of X falls to $5, the value of the endowment changes: I' = 5(20) + 5(30) = 100 + 150 = $250.Thenewslopeis. The new slope is -P_X'/P_Y = -5/5 = -1$. Since the magnitude of the slope has decreased from 2 to 1, the budget line has become flatter. Because the line must still pass through the endowment point (20, 30), the change is a pivot around this point. The consumer is a net seller of Y and a net buyer of X if they consume more than 20 X. With the lower price of X, their ability to purchase X increases, while the purchasing power of their Y endowment decreases.

Question 11

A consumer with income II faces prices PXP_X and PYP_Y for goods X and Y. The government decides to impose a quantity constraint, such that the consumer cannot purchase more than XmaxX_{max} units of good X. If the consumer's original optimal bundle (without the constraint) contained more X than XmaxX_{max}, what will be true of the new optimal bundle?

  1. The consumer will be on a lower indifference curve, and the MRS will equal the price ratio.
  2. The consumer will be on a lower indifference curve, and the MRS will be greater than the price ratio. (correct answer)
  3. The consumer will be on the same indifference curve, but at a different bundle.
  4. The consumer will be on a lower indifference curve, and the MRS will be less than the price ratio.
Explanation: The quantity constraint cuts off a portion of the original budget set. Since the original, unconstrained optimal bundle is now unaffordable, the consumer must choose a new bundle from the smaller feasible set. This will necessarily place them on a lower indifference curve. The new optimal bundle will be at the 'corner' of the new budget set, where X=XmaxX = X_{max}. At this point, the indifference curve will not be tangent to the original budget line. Since the consumer was originally choosing more X, it means they want to substitute towards X. The constraint prevents this. At X=XmaxX = X_{max}, the indifference curve will be steeper than the budget line, meaning the consumer values an additional unit of X more than the market does. A steeper indifference curve means a higher slope, so MRSXY>PX/PYMRS_{XY} > P_X/P_Y. The consumer would prefer to buy more X, but is constrained from doing so.

Question 12

A consumer has preferences for goods X and Y with non-convex indifference curves (bowed out from the origin). If this consumer is faced with a standard linear budget constraint, where will the utility-maximizing bundle be located?

  1. At a point of tangency between an indifference curve and the budget line.
  2. At a point where the consumption of both goods is equal.
  3. At a corner solution, where only one of the two goods is consumed. (correct answer)
  4. At a point inside the budget constraint, not on the line itself.
Explanation: Non-convex indifference curves (bowed out from the origin) represent preferences where the consumer prefers extremes to averages. The diminishing MRS assumption is violated; the MRS is increasing. While a point of tangency between an indifference curve and the budget line may exist, this point represents a local utility minimum along the budget line. The consumer can increase utility by moving away from the tangency point in either direction along the budget line. Utility will be maximized by moving to one of the two endpoints of the budget line, consuming only good X or only good Y. This is known as a corner solution.

Question 13

A consumer's preferences for goods X and Y can be represented by the utility function U(X,Y)=X0.2Y0.8U(X,Y) = X^{0.2}Y^{0.8}. The consumer has an income of $400. Currently, the price of X is $2 and the price of Y is $8. If the price of X increases to $4, what will be the consumer's total expenditure on good Y?

  1. $80
  2. $160
  3. $320 (correct answer)
  4. $400
Explanation: This is a Cobb-Douglas utility function of the form U(X,Y)=XaYbU(X,Y) = X^a Y^b. A key property of Cobb-Douglas preferences is that the consumer spends a constant fraction of their income on each good. The fraction of income spent on good Y is given by the exponent of Y divided by the sum of the exponents: b/(a+b)b / (a+b). In this case, the share of income spent on Y is 0.8/(0.2+0.8)=0.8/1=0.80.8 / (0.2 + 0.8) = 0.8 / 1 = 0.8. This expenditure share does not depend on the prices of the goods or the consumer's income. Therefore, the consumer will always spend 80% of their income on good Y. Total expenditure on Y will be 0.8 \times \400 = $320$. The change in the price of X is irrelevant to the total amount spent on Y.

Question 14

Consider a consumer choosing between a specific good, X, and a composite good, Y, representing all other goods (with P_Y = \1).Ifthegovernmentintroducesaper−unitsubsidyof). If the government introduces a per-unit subsidy of songoodX,howdoesthisaffecttheconsumer′sbudgetconstraint,whichinitiallywason good X, how does this affect the consumer's budget constraint, which initially wasP_X X + Y = I$?

  1. The budget line shifts outward in a parallel manner.
  2. The budget line's Y-intercept increases, and the slope remains the same.
  3. The budget line's X-intercept increases, and the line becomes flatter. (correct answer)
  4. The budget line's X-intercept increases, and the line becomes steeper.
Explanation: A per-unit subsidy on good X reduces the effective price that the consumer pays. The new price for the consumer is PX′=PX−sP_X' = P_X - s. The initial budget constraint is PXX+Y=IP_X X + Y = I, with a slope of −PX/PY=−PX-P_X/P_Y = -P_X (since PY=1P_Y=1) and an X-intercept of I/PXI/P_X. The new budget constraint is (PX−s)X+Y=I(P_X - s)X + Y = I. The Y-intercept (where X=0X=0) is still Y=IY=I, so it does not change. The new X-intercept (where Y=0Y=0) is I/(PX−s)I / (P_X - s), which is larger than the original X-intercept. The new slope is −(PX−s)/PY=−(PX−s)-(P_X - s)/P_Y = -(P_X - s). Since PX−s<PXP_X - s < P_X, the magnitude of the new slope is smaller, meaning the budget line has become flatter. Thus, the budget line pivots outward around the Y-intercept, becoming flatter and increasing the X-intercept.

Question 15

A consumer's income is $120. The price of good X is $8 and the price of good Y is $5. If the government imposes a quantity tax of $2 per unit on good X and provides a lump-sum income subsidy of $40, what is the equation of the new budget line?

  1. 10X+5Y=16010X + 5Y = 160 (correct answer)
  2. 10X+5Y=12010X + 5Y = 120
  3. 6X+5Y=1606X + 5Y = 160
  4. 8X+5Y=1608X + 5Y = 160
Explanation: The initial income is I = \120.Theinitialpricesare. The initial prices are P_X = $8andandP_Y = $5.Aquantitytaxof$2ongoodXincreasesthepricetheconsumerpaysforX.ThenewpriceofXis. A quantity tax of $2 on good X increases the price the consumer pays for X. The new price of X is P_X' = $8 + $2 = $10.Alump−sumincomesubsidyof$40increasestheconsumer′stotalincome.Thenewincomeis. A lump-sum income subsidy of $40 increases the consumer's total income. The new income is I' = $120 + $40 = $160.ThepriceofYremainsunchanged.Thenewbudgetlineisgivenby. The price of Y remains unchanged. The new budget line is given by P_X' X + P_Y Y = I',whichis, which is 10X + 5Y = 160$.

Question 16

The utility function U(X,Y)U(X,Y) and the function V(X,Y)=[U(X,Y)]2V(X,Y) = [U(X,Y)]^2 both represent the same consumer's preferences. The marginal rate of substitution derived from V (MRSVMRS_V) will be:

  1. twice the marginal rate of substitution derived from U (MRSUMRS_U).
  2. half the marginal rate of substitution derived from U (MRSUMRS_U).
  3. the square of the marginal rate of substitution derived from U (MRSUMRS_U).
  4. equal to the marginal rate of substitution derived from U (MRSUMRS_U). (correct answer)
Explanation: The function V is a positive monotonic transformation of the function U. Specifically, V is the square of U, and since utility is typically assumed to be positive, squaring it preserves the order of preferences. A key property of monotonic transformations is that they do not change the marginal rate of substitution (MRS). The MRS is the ratio of marginal utilities. Let's show this mathematically. MRSU=MUX,U/MUY,UMRS_U = MU_{X,U} / MU_{Y,U}. For V, the marginal utilities are found using the chain rule: MUX,V=dV/dX=2⋅U(X,Y)⋅(dU/dX)=2⋅U⋅MUX,UMU_{X,V} = dV/dX = 2 \cdot U(X,Y) \cdot (dU/dX) = 2 \cdot U \cdot MU_{X,U}. Similarly, MUY,V=dV/dY=2⋅U(X,Y)⋅(dU/dY)=2⋅U⋅MUY,UMU_{Y,V} = dV/dY = 2 \cdot U(X,Y) \cdot (dU/dY) = 2 \cdot U \cdot MU_{Y,U}. The new MRS is MRSV=MUX,V/MUY,V=(2⋅U⋅MUX,U)/(2⋅U⋅MUY,U)MRS_V = MU_{X,V} / MU_{Y,V} = (2 \cdot U \cdot MU_{X,U}) / (2 \cdot U \cdot MU_{Y,U}). The 2U2U terms cancel out, leaving MRSV=MUX,U/MUY,U=MRSUMRS_V = MU_{X,U} / MU_{Y,U} = MRS_U. Therefore, the MRS is unchanged by this transformation.