Utility Maximization & Demand Derivation - Microeconomics

A worker has preferences over consumption (C) and leisure (L). The worker earns a wage w for each hour worked (H), receives non-labor income N, and has 24 hours available per day. The price of consumption is 1. If the government imposes a proportional tax at rate t on labor earnings only, what is the effect on the worker's budget constraint in the (L, C) space?
Answer: The budget constraint pivots, becoming flatter. The original budget constraint is $C = wH + N$ . Since $H = 24 - L$ , this is $C = w(24 - L) + N$ , which can be written as $C + wL = 24w + N$ . The slope is $-w$ , which is the opportunity cost of leisure. With the tax, the after-tax wage is $w(1-t)$ . The new budget constraint is $C = w(1-t)H + N$ , or $C = w(1-t)(24 - L) + N$ . This can be written $C + w(1-t)L = 24w(1-t) + N$ . The new slope is $-w(1-t)$ . Since $t>0$ , $w(1-t) < w$ , meaning the slope has a smaller absolute value. The budget constraint becomes flatter, pivoting around the point where L=24 (no labor income), so the C-intercept from non-labor income is unaffected if it's drawn that way.
A consumer makes choices over consumption in period 1 ( $C_1$ ) and period 2 ( $C_2$ ). They can save or borrow at a single interest rate, $r$ . If $C_1$ is plotted on the horizontal axis and $C_2$ on the vertical axis, what is the slope of the consumer's lifetime budget constraint?
Answer: $-(1+r)$ The lifetime budget constraint equates the present value of consumption to the present value of income: $C_1 + \frac{C_2}{1+r} = M_1 + \frac{M_2}{1+r}$ . Let W be the present value of wealth on the right side. To find the slope of the budget line on a graph with $C_2$ on the vertical axis, we solve for $C_2$ : $\frac{C_2}{1+r} = W - C_1 \implies C_2 = W(1+r) - C_1(1+r)$ . This equation is in the form $y = b + mx$ , where $y=C_2$ , $x=C_1$ , and the slope $m$ is the coefficient on $C_1$ . The slope is $-(1+r)$ . This represents the opportunity cost of period 1 consumption: for every unit of $C_1$ consumed, the consumer gives up $1+r$ units of $C_2$ .
Consider a consumer with utility function $U(x,y) = \min(2x, 3y)$ facing prices $P_x = 6$ and $P_y = 8$ , with income $I = 240$ . If the price of $x$ falls to $P_x = 4$ , what portion of the total change in demand for $x$ is due to the substitution effect?
Answer: The substitution effect accounts for 0% of the total demand change With perfect complements $U = \min(2x, 3y)$ , optimal consumption requires $2x = 3y$ regardless of prices (as long as both goods are consumed). The demand functions depend only on income, not relative prices. Initially: $x_1 = \frac{3I}{2P_x + 3P_y} = \frac{3(240)}{2(6) + 3(8)} = 20$ . After price change: $x_2 = \frac{3(240)}{2(4) + 3(8)} = 22.5$ . Since the optimal ratio is fixed, all demand change is due to income effect. Choices A, B, C incorrectly assume substitutability exists.
A consumer's utility function is given by $U(X,Y) = XY + X$ . From the consumer's utility maximization problem, the resulting Marshallian demand function for good X is $X^* = \frac{M+P_Y}{2P_X}$ . Based on this demand function, what is the relationship between good X and good Y?
Answer: They are substitutes. To determine the relationship between two goods, we examine the sign of the cross-price elasticity, or simply the sign of the partial derivative of the demand for one good with respect to the price of the other. We take the partial derivative of the demand for X with respect to $P_Y$ : $\frac{\partial X^*}{\partial P_Y} = \frac{1}{2P_X}$ . Since $P_X$ must be positive, this derivative is positive. This means that as the price of good Y increases, the quantity demanded of good X increases. This defines the goods as substitutes.
A consumer's preferences are described by the utility function $U(X,Y) = \min(2X, Y)$ . If income is $120, the price of X is $5, and the price of Y is $10, what is the optimal quantity of good X consumed?
Answer: 4.8 This utility function represents perfect complements. The consumer will always choose a bundle where the arguments of the min function are equal, so $2X = Y$ . This is the 'kink' in the L-shaped indifference curves. To find the optimal bundle, substitute this condition into the budget constraint: $P_X X + P_Y Y = M$ . This gives $5X + 10(2X) = 120$ . Simplifying, $5X + 20X = 120$ , which means $25X = 120$ . Solving for X gives $X = 120/25 = 4.8$ . The optimal quantity of Y would be $Y = 2(4.8) = 9.6$ .
A consumer maximizes the utility function $U(X,Y) = \ln(X) + Y$ subject to a budget constraint. Given prices $P_X = \$2$ , $P_Y = \$4$ , and income $M = \$20$ , what is the marginal utility of income at the optimal consumption bundle?
Answer: 0.25 First, find the optimal bundle. The marginal utilities are $MU_X = 1/X$ and $MU_Y = 1$ . The utility-maximizing condition is $MRS = MU_X/MU_Y = P_X/P_Y$ . So, $1/X = 2/4 = 0.5$ , which gives $X = 2$ . The expenditure on X is $P_X \cdot X = 2 \cdot 2 = \$4$ . The remaining income to be spent on Y is $M - P_X X = 20 - 4 = \$16$ . The quantity of Y is $Y = 16 / P_Y = 16/4 = 4$ . The optimal bundle is (2, 4). The marginal utility of income (the Lagrangian multiplier, $\lambda$ ) at the optimum is equal to $MU_X/P_X$ and $MU_Y/P_Y$ . Using good Y: $\lambda = MU_Y/P_Y = 1/4 = 0.25$ .
The demand function for a good X is observed to be $X = \frac{M}{P_X + 2P_Y}$ , where M is income, $P_X$ is the price of good X, and $P_Y$ is the price of good Y. What can be inferred about good X and its relationship with good Y?
Answer: X is a normal good and a complement to Y. We can analyze the properties of the good by taking partial derivatives of the demand function. First, with respect to income M: $\frac{\partial X}{\partial M} = \frac{1}{P_X + 2P_Y} > 0$ , so X is a normal good. Next, with respect to the price of good Y, $P_Y$ : $\frac{\partial X}{\partial P_Y} = -\frac{M}{(P_X + 2P_Y)^2} \cdot 2 < 0$ . Since the quantity demanded of X decreases when the price of Y increases, X and Y are complements.
A consumer's preferences for goods X and Y are represented by the utility function $U(X,Y) = 3X + 2Y$ . The consumer has an income of $60, and the prices are $P_X = \$6$ and $P_Y = \$5$ . Which of the following bundles will the consumer choose to maximize utility?
Answer: (X=10, Y=0) The utility function represents perfect substitutes. The marginal rate of substitution is $MRS = MU_X / MU_Y = 3/2 = 1.5$ . The price ratio is $P_X / P_Y = 6/5 = 1.2$ . Since $MRS > P_X / P_Y$ , the consumer values good X more at the margin than the market does. To check this, compare the marginal utility per dollar: $MU_X/P_X = 3/6 = 0.5$ and $MU_Y/P_Y = 2/5 = 0.4$ . Since $MU_X/P_X > MU_Y/P_Y$ , the consumer gets more utility per dollar from X and will spend their entire income on good X. This is a corner solution. The quantity of X purchased is $M/P_X = 60/6 = 10$ . The optimal bundle is (10, 0).
A consumer with utility function $U(X,Y) = X^{0.4}Y^{0.6}$ has an income of $200. Initially, prices are $P_X = \$2$ and $P_Y = \$3$ . The government imposes a $1 per-unit tax on good X. By how much does the consumer's optimal consumption of good Y change?
Answer: It remains unchanged. For a Cobb-Douglas utility function $U(X,Y) = X^a Y^b$ , the demand for good Y depends only on income M, its own price $P_Y$ , and the exponents, according to the formula $Y^* = (\frac{b}{a+b})\frac{M}{P_Y}$ . Here, $a=0.4$ and $b=0.6$ , so $Y^* = (\frac{0.6}{1})\frac{M}{P_Y} = \frac{0.6M}{P_Y}$ . Since the demand for Y does not depend on $P_X$ , a tax on good X (which increases $P_X$ to $3) will not change the quantity of Y consumed. The initial and final quantities of Y are both $Y = (0.6 \cdot 200) / 3 = 120 / 3 = 40$ . The change is 0.
A consumer with an income of $40 faces prices $P_X=2, P_Y=2$ and chooses bundle A = (10, 10). When prices change to $P_X=1, P_Y=2$ , the consumer chooses bundle B = (20, 10). Based on the Weak Axiom of Revealed Preference (WARP), what can be concluded?
Answer: Bundle B is revealed preferred to bundle A. When you encounter revealed preference questions, you're analyzing what a consumer's actual choices tell us about their preferences, regardless of what they might say they prefer. The Weak Axiom of Revealed Preference (WARP) works like this: if a consumer chooses bundle X when bundle Y was affordable, then X is "revealed preferred" to Y. Let's check what was affordable in each situation. In situation 1: Income = $40, prices (2,2), consumer chose A = (10,10) - Cost of bundle A: $2(10) + $2(10) = $40 - Cost of bundle B: $2(20) + $2(10) = $60 Since bundle B costs $60 but the consumer only has $40, bundle B wasn't affordable when A was chosen. In situation 2: Income = $40, prices (1,2), consumer chose B = (20,10) - Cost of bundle B: $1(20) + $2(10) = $40 - Cost of bundle A: $1(10) + $2(10) = $30 Here's the key: when the consumer chose bundle B, bundle A was affordable (costing only $30) but was rejected in favor of B. This reveals that B is preferred to A. Choice A is wrong because A was not chosen when both were available. Choice B is incorrect because we can draw a clear conclusion from situation 2. Choice C is wrong because the choices are perfectly consistent with rational preferences—there's no violation. Therefore, D is correct: Bundle B is revealed preferred to bundle A. Study tip: In revealed preference problems, always check both directions—what was affordable but not chosen in each scenario? The choice made when both bundles were affordable reveals the true preference.
For a good to be classified as a Giffen good, which of the following conditions is necessary?
Answer: The good must be inferior, and the income effect must outweigh the substitution effect. A Giffen good is a good for which quantity demanded increases as its price increases, violating the law of demand. According to the Slutsky equation, the total effect of a price change is the sum of the substitution and income effects. The substitution effect is always negative for a price increase (quantity falls). For the total effect to be positive (quantity rises), the income effect must be positive and larger in magnitude than the substitution effect. A positive income effect for a price increase (real income falls, quantity demanded rises) means the good must be inferior. Therefore, a Giffen good must be an inferior good for which the income effect is strong enough to dominate the substitution effect.
A consumer always buys coffee (C) and sugar (S) in a fixed ratio of one cup of coffee to two spoons of sugar. The price of coffee is $P_C$ and the price of sugar is $P_S$ . How can this consumer's utility maximization problem be simplified for analysis?
Answer: By treating coffee and sugar as a composite good with a price of $P_C + 2P_S$ . When you encounter goods consumed in fixed proportions, you're dealing with perfect complements - a special case where utility maximization can be dramatically simplified through the composite good approach. Since this consumer always buys coffee and sugar in a 1:2 ratio (one cup coffee to two spoons sugar), they're essentially purchasing bundles rather than individual goods. Each "consumption unit" contains exactly 1C + 2S. To find the price of this composite good, you add up the cost of all components in the fixed bundle: one unit of coffee ( $P_C$ ) plus two units of sugar ( $2P_S$ ), giving you $P_C + 2P_S$ per bundle. This transforms a two-good optimization problem into a simple one-good problem where the consumer chooses how many bundles to buy based on their budget and the bundle price. Option A incorrectly adds $P_C + P_S$ , which would be correct if the ratio were 1:1, but ignores that two spoons of sugar are needed per cup. Option B is wrong because perfect complements are actually the easiest case to simplify - the fixed ratio eliminates the substitution complexity that makes other problems difficult. Option C suggests treating the goods as independent, which contradicts the premise that they're consumed in fixed proportions. Study tip: Whenever you see "fixed ratio" or "always consumed together," immediately think composite good. The bundle price always equals the sum of (quantity of each good in the bundle × its price). This approach works for any perfect complements, from left shoes and right shoes to coffee and cream.
A consumer has the utility function $U(X,Y) = X^{0.5}Y^{0.5}$ . The consumer's income is $100, the price of good X is $2, and the price of good Y is $4. If the price of good X decreases to $1, what is the resulting change in the quantity of good Y consumed?
Answer: The quantity of Y does not change. For a Cobb-Douglas utility function of the form $U(X,Y) = X^a Y^b$ , the Marshallian demand for good Y is given by $Y^* = \frac{b}{a+b} \frac{M}{P_Y}$ . In this case, $a=0.5$ and $b=0.5$ , so the demand for Y is $Y^* = \frac{0.5}{1} \frac{M}{P_Y} = \frac{M}{2P_Y}$ . Notice that the demand for Y does not depend on the price of X ( $P_X$ ). Therefore, a change in $P_X$ will not affect the quantity of Y demanded. Initially, $Y = 100 / (2*4) = 12.5$ . After the price change, $Y = 100 / (2*4) = 12.5$ . The change is zero.
For a consumer with standard convex preferences, the price-consumption path (PCC) for good X is observed to be upward-sloping as the price of X varies. What does this imply about the price elasticity of demand for good X?
Answer: Demand is price inelastic. When analyzing consumer behavior, the price-consumption curve (PCC) traces out how a consumer's optimal consumption bundle changes as the price of one good varies while holding income and other prices constant. The slope of this curve reveals crucial information about demand elasticity. An upward-sloping PCC for good X means that as the price of X increases, the consumer purchases more of the other good (typically plotted on the vertical axis). This pattern indicates that when X becomes more expensive, the consumer doesn't reduce their consumption of X very much – they mainly substitute by buying more of the other good rather than dramatically cutting back on X. This behavior signals that demand for good X is price inelastic (D). When demand is inelastic, consumers are relatively unresponsive to price changes, so quantity demanded falls less than proportionally to price increases. Option A is incorrect because elastic demand would produce a downward-sloping PCC – consumers would sharply reduce X consumption when its price rises. Option B represents perfectly inelastic demand, which would create a vertical PCC where X consumption never changes regardless of price – but we observe the consumer does adjust their behavior somewhat. Option C describes unit elastic demand, which would typically produce a PCC with a specific slope relationship that doesn't match the general upward-sloping pattern described. Remember this connection: upward-sloping PCC signals inelastic demand. The consumer "absorbs" the price increase mainly by adjusting other goods rather than dramatically cutting the expensive good, revealing that good's necessity or lack of close substitutes.
When the price of an inferior good falls, what are the respective directions of the substitution and income effects on the quantity demanded?
Answer: The substitution effect is positive, and the income effect is negative. Let's analyze the effects of a price fall for good X. The substitution effect describes the change in consumption due to the change in relative prices, holding real income constant. A fall in $P_X$ makes X relatively cheaper, so the substitution effect always leads to an increase in quantity demanded (a positive effect). The income effect describes the change in consumption due to the change in purchasing power. A fall in $P_X$ increases the consumer's real income. Since the good is specified as inferior, an increase in real income leads to a decrease in the quantity demanded (a negative effect). Thus, the two effects work in opposite directions.
A consumer is choosing between pizza (P) and soda (S). Her marginal rate of substitution of pizza for soda ( $MRS_{PS}$ ) is 3. The price of a slice of pizza is $4 and the price of a soda is $1. To move towards her utility-maximizing consumption bundle, what action should she take?
Answer: Consume less pizza and more soda. The consumer's marginal rate of substitution, $MRS_{PS} = MU_P/MU_S = 3$ , represents her subjective willingness to trade: she is willing to give up 3 sodas for 1 more slice of pizza. The market price ratio, $P_P/P_S = 4/1 = 4$ , represents the market's trade-off: she must give up 4 sodas for 1 more slice of pizza. Since her personal valuation of pizza (3 sodas) is less than its market cost (4 sodas), i.e., $MRS < P_P/P_S$ , she should consume less pizza. Equivalently, the marginal utility per dollar for pizza ( $MU_P/\$4$ ) is less than that for soda ( $MU_S/\$1$ ), since $MU_P/MU_S = 3$ implies $MU_P/4 < MU_S/1$ . She should reallocate spending from the lower marginal utility per dollar good (pizza) to the higher one (soda).
A consumer has a quasi-linear utility function $U(X,Y) = 4\sqrt{X} + Y$ . Her income is $30, the price of X is $1, and the price of Y is $5. What is her optimal consumption of good Y?
Answer: 0 First, find the interior solution condition by setting $MRS = P_X/P_Y$ . Here, $MU_X = 2/\sqrt{X}$ and $MU_Y = 1$ , so $MRS = 2/\sqrt{X}$ . The price ratio is $1/5$ . Setting $2/\sqrt{X} = 1/5$ implies $\sqrt{X} = 10$ , so $X=100$ . The cost of this bundle would be $P_X X = 1 \cdot 100 = \$100$ . However, the consumer's income is only $30. This means the interior solution is not affordable, and the consumer must choose a corner solution. We must compare the utility from spending all income on X versus all on Y. If all is spent on X, $X=30, Y=0$ , and $U = 4\sqrt{30} \approx 21.9$ . If all is spent on Y, $X=0, Y=6$ , and $U = 0+6 = 6$ . The consumer achieves higher utility by consuming only good X. Therefore, the optimal consumption of Y is 0.
A consumer has the utility function $U(x,y) = x^{0.4}y^{0.6}$ and income $I = 120$ . If the price of good $x$ increases from $P_x = 2$ to $P_x = 3$ while $P_y = 4$ remains constant, what is the change in the consumer's demand for good $x$ ?
Answer: Demand decreases by 8 units For Cobb-Douglas utility $U = x^{0.4}y^{0.6}$ , optimal consumption is $x = \frac{0.4I}{P_x}$ and $y = \frac{0.6I}{P_y}$ . Initially: $x_1 = \frac{0.4(120)}{2} = 24$ . After price increase: $x_2 = \frac{0.4(120)}{3} = 16$ . The change is $16 - 24 = -8$ units. Choice A uses incorrect expenditure shares. Choice C miscalculates the new quantity. Choice D assumes linear demand.
A low-income consumer with standard convex preferences receives a government subsidy. Compared to receiving $100 in food stamps (which can only be spent on food), the consumer's utility from receiving a $100 cash grant will be:
Answer: at least as high, and possibly higher. A cash grant gives the consumer the most flexibility. The choice set available with a $100 cash grant contains all the consumption bundles available with $100 in food stamps, plus additional bundles where the consumer spends less than $100 of the subsidy on food. If the consumer, given $100 in cash, would have chosen to spend $100 or more on food anyway, then the food stamps are not a binding constraint, and their utility is the same as with cash. If they would have spent less than $100 on food, the food stamps force them to a suboptimal consumption bundle, making their utility lower than it would be with cash. Therefore, the consumer's utility is at least as high with cash, and strictly higher if the food stamp constraint is binding.
A consumer's preferences are represented by the utility function $U(X,Y) = \min(X + 2Y, 2X + Y)$ . The consumer will choose to consume only good Y (i.e., the demand for X is zero) if and only if the price ratio $P_X/P_Y$ satisfies which condition?
Answer: $P_X/P_Y > 2$ The indifference curves for this utility function have a kink where $X + 2Y = 2X + Y$ , which simplifies to $Y = X$ . To the left of this line (where $Y > X$ ), $U = 2X + Y$ , and the MRS is $MU_X/MU_Y = 2/1 = 2$ . To the right of this line (where $Y < X$ ), $U = X + 2Y$ , and the MRS is $MU_X/MU_Y = 1/2$ . A corner solution where the consumer buys only Y (X=0) occurs when the budget line is steeper than any portion of the indifference curve. The steepest portion has a slope (MRS) of 2. Therefore, if the price ratio $P_X/P_Y$ is greater than 2, the consumer will maximize utility by consuming only good Y.

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