Consumer theory practice problems

Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in www.ecopoint.in Preferences (1)[Preferences] Consider bundle X , consisting of 6 cans of beans and 4 boxes of cereal. Use just the assumption of monotonicity ( ―more ( ―more is better‖) to determine whether each of the following bundles are more preferred to bundle X, or less preferred to bundle X, or preference cannot be determined without more information. Briefly justify your answers. a. Bundle A , consisting of 6 cans of beans beans and 3 boxes boxes of cereal. •

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b. Bundle B , consisting of 7 cans of beans and 4 boxes of cereal. c. Bundle C , consisting of 10 cans of beans and 2 boxes of cereal. d. Bundle D , consisting of 4 cans of beans and 6 boxes of cereal. e. Bundle E , consisting of 8 cans of beans and 5 boxes of cereal. (2) [Preferences] Suppose the bundles X and Y are equally preferred, where bundle X consists of 5 units of energy and 8 units of food, while bundle Y consists of 11 units of energy and 6 units of food. Use the assumption of diminishing marginal rate of substitution in consumption (MRSC) to determine whether the following bundles are more preferred to either bundle X or bundle Y, or. less preferred to either bundle X or bundle Y. Briefly explain your answers. [Hint: You may find it useful to plot bundles X, Y, A, and B carefully on a graph.] •

•

a.

Bundle A , consisting of 8 units of energy and 7 units of food.

b.

Bundle B , consisting of 17 units of energy and 4 units of food.

(3) [Utility functions] Suppose a person person has the utility function U(q1, q2) = q1q21/2 , where q1 denotes the quantity of food the person enjoys and q 2 denotes the quantity of clothing. Rank the following bundles from most preferred to least preferred. a.

Bundle A , consisting of 10 units of food and 16 units of clothing.

b.

Bundle B , consisting of 7 units of food and 25 units of clothing.

c.

Bundle C , consisting of 13 units of food and 9 units of clothing.

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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in www.ecopoint.in

(4) [Utility functions] functions] Suppose a person has the the utility function U(q1, q2) = 5q1 + 3q2 . Assume q1 and q2 are positive quantities. a. Find formulas formulas for the marginal marginal utilities MU 1 and MU 2 . b. Determine whether this utility function satisfies the assumption of monotonicity ( ―more ―more is better‖). Explain your reasoning. [Hint: Determine whether the marginal utilities are positive.] c. Find a formula for the marginal marginal rate of substitution substitution in consumption consumption (MRSC) of good 2 for good 1. [Hint: This is the absolute value of the slope of the indifference curve, when good 1 is on the vertical axis and good 2 is on the horizontal axis.] d. Determine whether this utility function satisfies the assumption of diminishing MRSC. Explain your reasoning. [Hint: According to the formula for the MRSC, does it diminish as q1 decreases and q2 increases?] (5) [Utility functions] Suppose a person has the utility function U(q 1, q2) = (q1-5) q22 . Assume q1 > 5 and q2 > 0 . a. Find formulas formulas for the marginal marginal utilities MU 1 and MU 2 . b. Determine whether this utility function satisfies the assumption of monotonicity ( ―more ―more is better‖). Explain your reasoning. [Hint: Determine whether the marginal utilities are positive.] c. Find a formula for the marginal marginal rate of substitution substitution in consumption consumption (MRSC) of good 2 for good 1. [Hint: This is the absolute value of the slope of the indifference curve, when good 1 is on the vertical axis and good 2 is on the horizontal axis.] d. Determine whether this utility function satisfies the assumption of diminishing MRSC. Explain your reasoning. [Hint: According to the formula for the MRSC, does it diminish as q1 decreases and q2 increases?] (6) [Utility functions] Suppose a person has the utility function U(q1, q2) = - (3/q1) – (5/q2). Assume q1 and q2 are positive quantities. a. Find formulas formulas for the marginal marginal utilities MU 1 and MU 2 .


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in b. Determine whether this utility function satisfies the assumption of monotonicity ( ―more is better‖). Explain your reasoning. [Hint: Determine whether the marginal utilities are positive.] c. Find a formula for the marginal rate of substitution in consumption (MRSC) of good 2 for good 1. [Hint: This is the absolute value of the slope of the indifference curve, when good 1 is on the vertical axis and good 2 is on the horizontal axis.] d. Determine whether this utility function satisfies the assumption of diminishing MRSC. Explain your reasoning. [Hint: According to the formula for the MRSC, does it diminish as q1 decreases and q2 increases?]

(7) [Utility functions] Suppose a person has the utility function U(q 1, q2) = 3 q11/2 + 2 q21/2 . Assume q1 and q2 are positive quantities. a. Find formulas for the marginal utilities MU 1 and MU 2 . b. Determine whether this utility function satisfies the assumption of monotonicity ( ―more is better‖). Explain your reasoning. [Hint: Determine whether the marginal utilities are positive.] c. Find a formula for the marginal rate of substitution in consumption (MRSC) of good 2 for good 1. [Hint: This is the absolute value of the slope of the indifference curve, when good 1 is on the vertical axis and good 2 is on the horizontal axis.] d. Determine whether this utility function satisfies the assumption of diminishing MRSC. Explain your reasoning. [Hint: According to the formula for the MRSC, does it diminish as q1 decreases and q2 increases?] (8)

[Utility functions] Consider the utility function U(q1, q2) = q13/4 q21/4 . a. Find a formula for the marginal rate of substitution in consumption (MRSC) of good 2 for good 1. b. Find three different utility functions that yield exactly the same MRSC formula as your answer to part (a). Check your answers by finding the MRSC formulas in each case.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in (9) [Utility functions, finance] In portfolio theory, the utility of investors is often modeled as a function of the expected rate of return (R) of their investment portfolio and the risk associated with that portfolio. Risk is measured as standard deviation ( σ). A typical utility function might be U(R,σ) = R – 0.03 σ2 . a. Find formulas for the marginal utilities MUR and MU σ . b. We usually assume that "more is better" for consumer, and therefore that the marginal utilities should be positive. Explain why it makes sense for MU R to be positive and for MUσ to be negative in this situation. c. Find a formula for the marginal rate of substitution in consumption (MRSC) of σ for R. [Hint: This is the absolute value of the slope of the indifference curve, when R is on the vertical axis and σ is on the horizontal axis.]

Budget And Choice (1) [Budget line] Suppose a consumer has $50 to spend on hamburgers and minipizzas this month. Hamburgers cost $3 and minipizzas cost $4. Consider this consumer ’s budget line. a. Let q1 denote the number of hamburgers and q 2 denote the number of minipizzas. Give an equation for the consumer ’s budget line. b. Which of the following bundles are just affordable? Which are affordable with money left over? Which are not affordable? Bundle (i), consisting of 5 hamburgers and 5 minipizzas. Bundle (ii) consisting of 10 hamburgers and 5 minipizzas. Bundle (iii) consisting of 6 hamburgers and 8 minipizzas. Bundle (iv) consisting of 8 hamburgers and 8 minipizzas. (2) [Budget line] Suppose a consumer has $80 to spend on movie tickets and video rentals this month. Movie tickets cost $5 and video rentals cost $4. Consider this consumer’s budget line. a. Let q1 denote the number of movie tickets and q 2 denote the number of video rentals. Give an equation for t he consumer’s budget line. b. Compute the intercept of the budget line on the ―movie ticket‖ axis.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in c. Compute the intercept of the budget line on the ―video rental‖ axis. d. Find the slope of the budget line when movie tickets are on the vertical axis and video rentals are on the horizontal axis. (3) [Budget line] Consider the impact on a consumer’s budget constraint of each scenario below. Indicate whether the impact is a parallel shift in the budget line, a rotation of the budget line, or no change in the budget line. Also indicate whether the budget line moves closer to the origin or farther away from the origin. a. Income increases by 20%. b. The price of one good increases by 20%. c. The price of both goods increase by 20%. d. The price of one good increases by 20% and income simultaneously increases by 20%. e. The prices of both goods increase by 20% and income simultaneously increases by 20%. (4) [Kinked budget line] Suppose oranges cost $4 per pound for the first 5 pounds, but, due to a special discount program, additional oranges cost only $1 per pound. Assume that apples always cost $2 per pound and that the consumer has $30 income. Plot the consumer’s budget constraint. [Hint: This budget constraint has a kink where the quantity of oranges equals 5 pounds.] (5) [Kinked budget line] Suppose a consumer can enjoy a reduced price on food by paying an up-front annual membership fee at a discount food store. The consumer has a total income of $1000. Without a discount, the usual price of food is $2 per unit. If the consumer pays a fee of $200, then the price of food is only $1 per unit. The price of other goods is always $1 per unit. a. Give an equation for the consumer’s budget line without the discount. Sketch the budget line or describe it in words. Compute the intercepts. Compute the slope when food is on the horizontal axis. b. Give an equation for the consumer’s budget line with the discount. [Hint: Treat the membership fee as a loss of income.] Sketch the budget line or describe it in words. Compute the intercepts. Compute the slope when food is on the horizontal axis.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in (6) [Kinked budget line] Suppose a consumer enjoy a ―frequent-customer‖ discount on DVD rentals. The first five rentals in a month cost $5 each. Additional rentals cost only $3 each. The consumer has an entertainment budget of $100 per month and other entertainment goods cost $1 each. Note that this change in price after the fifth rental causes a kink in the budget line. Consider the graph of this budget line with DVD rentals on the horizontal axis. a. What is the maximum number of other goods this consumer could afford, if they never rented DVDs? b. What is the maximum number of video rentals this consumer could afford, if they spent their entire budget on DVD rentals? c. Compute the coordinates of the kink point. [Hint: How many other goods could the consumer afford if they purchased five DVD rentals?] d. Compute the slope of the budget line, with DVD rentals on the horizontal axis, when the consumer rents fewer than five DVDs . e. Compute the slope of the budget line, with DVD rentals on the horizontal axis, when the consumer rents more than five DVDs. f.

Sketch the budget line or describe it in words.

g. From your sketch of the budget line, do you think anyone would ever choose to rent exactly five DVDs per month? Why or why not? (7) [Choice] Suppose the price of beans is p1=$5 and the price of potatoes is p 2=$2. Find equations for the tangency conditions for each of the following consumers. Which consumers will make the same choices if they have the same income? Explain your reasoning. [Hint: Compare the tangency conditions for these consumers.] a. Anne, whose utility function is U(q1,q2) = q12q2 . b. Bill, whose utility function is U(q1,q2) = q12/3q21/3 . c.

Carol, whose utility function is U(q1,q2) = (q1-2)(q2-1)

(8) [Choice] Suppose a consumer has $150 to spend on food and clothing. Food costs $4 per unit and clothing costs $5 per unit. The consumer ’s utility function is U(q 1,q2) = q12q2 , where q1 denotes the quantity of food and q 2 denotes the quantity of clothing. a. Give an equation for the consumer’s budget line.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of clothing for food. [Hint: This is the slope of the consumer’s indifference curve when food is on the vertical axis and clothing is on the horizontal axis.] c. Compute the quantities of food and clothing that this consumer will choose. (9) [Choice] Suppose a consumer has $310 to spend on energy and other goods. Energy costs $3 per unit and other goods cost $2 per unit. The consumer ’s utility function is U(q1,q2) = q1(q2-5), where q1 denotes the quantity of energy and q 2 denotes the quantity of other goods. a. Give an equation for the consumer’s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of other goods for energy. [Hint: This is the slope of the consumer’s indifference curve when energy is on the vertical axis and other goods is on the horizontal axis.] c. Compute the quantities of energy and other goods that this consumer will choose. (10)[Choice] Suppose the consumer in the previous problem enjoys an increase in income to $370. There is no change in prices. Compute the quantities of energy and other goods that this consumer will now choose. (11)[Choice] Suppose a consumer has $210 to spend on health care and other goods. Health care costs $9 per unit and other goods cost $8 per unit. The consumer ’s utility function is U(q1,q2) = –(5/q1) –(10/q2), where q1 denotes the quantity of health care and q 2 denotes the quantity of other goods. a. Give an equation for the consumer’s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of other goods for health care. [Hint: This is the slope of the consumer’s indifference curve when health care is on the vertical axis and other goods is on the horizontal axis.] c. Compute the quantities of health care and other goods that this consumer will choose

(12) [Choice] Suppose a consumer has $60 to spend on food and other goods. Food costs $2 per unit and other goods cost $4 per unit. The consumer ’s utility function is


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in U(q1,q2) = q11/2 + q21/2, where q1 denotes the quantity of food and q 2 denotes the quantity of other goods. a. Give an equation for the consumer’s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of other goods for food. [Hint: This is the slope of the consumer’s indifference curve when food is on the vertical axis and other goods is on the horizontal axis.] c. Compute the quantities of food and other goods that this consumer will choose. (13) [Choice with corner solutions] Suppose a consumer has income of $60 to spend on sodapop. This consumer has utility function U(q 1,q2) = q1 + 2q2 , where q1 denotes the number of small bottles of sodapop and q 2 denotes the number of large bottles of sodapop consumed. a. Draw the consumer's indifference curves when U = 10, 20, or 30. b. On the same graph, but in a different color, draw the consumer's budget line when p1 = $2 and p2 = $3. How many large bottles and how many small bottles will this consumer choose? [Hint: Calculus is useless for this problem because the solution is not a tangency. Instead, just study your graph.] c. On the same graph, but in a different color, draw the consumer's budget line when p1 = $2 and p2 = $6. How many large bottles and how many small bottles will this consumer choose? (14) [Choice, finance] Suppose an investor has the U(R,σ) = R–0.03σ2.

utility function

a. Find a formula for the marginal rate of substitution in consumption (MRSC) of σ for R. [Hint: This is the absolute value of the slope of the indifference curve, when R is on the vertical axis and σ is on the horizontal axis.] According to the Capital Asset Pricing Model, if there is a risk-free asset with a return of 4 percent, and if the market return is 10 percent with a standard deviation of 20 percent, then the investor faces a constraint of R = 4 + (10-4)/20 σ or R = 4 + 0.3 σ . b. Compute the slope of the constraint, dR/dσ .


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in c. Compute the values of R and σ that this investor will choose. [Hint: Set your answer to (a) equal to your answer to (b) and solve for σ. Then insert this into the constraint to find R.] In reality, the investor cannot purchase R and σ directly. Instead, the investor purchases the risk-free asset and the market portfolio. The investor's R and σ are then weighted averages of the values for the two assets. In particular, let w equal the fraction of the investor's wealth invested in the risk-free asset, and thus (1-w) equal the fraction of the investor's wealth invested in the market portfolio. Then R = 4 w + 10 (1-w)

and

σ = 0 w + 20 (1-w).

d. Compute w, the fraction of wealth that the investor will devote to the risk-free asset, and (1-w), the fraction that the investor will devote to the market portfolio (15) [Choice, finance] Suppose an investor has the utility function U(R, σ) = R–0.01σ2. a. Find a formula for the marginal rate of substitution in consumption (MRSC) of σ for R. [Hint: This is the absolute value of the slope of the indifference curve, when R is on the vertical axis and σ is on the horizontal axis.] According to the Capital Asset Pricing Model, if there is a risk-free asset with a return of 4 percent, and if the market return is 10 percent with a standard deviation of 20 percent, then the investor faces a constraint of R = 4 + (10-4)/20 σ or R = 4 + 0.3 σ . b. Compute the slope of the constraint, dR/dσ . c. Compute the values of R and σ that this investor will choose. [Hint: Set your answer to (a) equal to your answer to (b) and solve for σ. Then insert this into the constraint to find R.] In reality, the investor cannot purchase R and σ directly. Instead, the investor purchases the risk-free asset and the market portfolio. The investor's R and σ are then weighted averages of the values for the two assets. In particular, let w equal the fraction of the investor's wealth invested in the risk-free asset, and thus (1-w) equal the fraction of the investor's wealth invested in the market portfolio. Then R = 4 w + 10 (1-w)

and

σ = 0 w + 20 (1-w).

d. Compute w, the fraction of wealth that the investor will devote to the risk-free asset, and (1-w), the fraction that the investor will devote to the market portfolio.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in Budget Constraint, Homogeneity and Demand Functions [Budget constraint] Assume there are only two goods and that the demand for good #1 is given by . Substitute this into the equation for the budget line to find the formula for the demand for good #2. 1.

[Budget constraint] If a consumer always spends one-fourth of her or his income on housing—regardless of income, the price of housing, or the price of other goods —then what must be the demand function for housing? 2.

[Budget constraint and homogeneity] Consider whether the following functions might be legitimate demand functions for an individual consumer. 3.

a. Is the budget constraint satisfied by this demand system? (Assume there are only two goods.) Show your work, step by step. b. Are these functions homogeneous of degree zero in income and prices? Show your work, step by step. 4. [Budget constraint and homogeneity] Consider whether the following functions might be legitimate demand functions for an individual consumer.

a. Is the budget constraint satisfied by this demand system? (Assume there are only two goods.) Show your work, step by step. b. Are these functions homogeneous of degree zero in income and prices? Show your work, step by step. 5. [Budget constraint and homogeneity] Consider whether the following functions might be legitimate demand functions for an individual consumer. a. Is the budget constraint satisfied by this demand system? (Assume there are only two goods.) Show your work, step by step. b. Are these functions homogeneous of degree zero in income and prices? Show your work, step by step.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in 6. [Homogeneity] Which the following functions are homogeneous of degree zero? Show your work, step by step.

7. [Homogeneity] Are the following functions homogeneous of degree zero in income and prices? Show your work, step by step.

8. [Finding demand functions] Suppose a consumer has utility function U(q 1,q2) = 1/3 2/3 q1 q2 and has income I. The price of good #1 is p 1 and the price of good #2 is p 2. a. Give the equation for the consumer’s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumer’s indifference curve when good #1 is on the vertical axis and good #2 is on the horizontal axis.] c. Find an expression for the consumer’s demand for good #1 ( q 1* ) as a function of p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p 2 /p1. Solve this equation for q2. Substitute the resulting expression in the budget line and solve for q1.] d. Find an expression for the consumer’s demand for good #2 ( q 2* ) as a function of p1 , p2, and I . 9. [Finding demand functions] Suppose a consumer has utility function U(q 1,q2) = 1/4 q1 q23/4 and has income I. The price of good #1 is p 1 and the price of good #2 is p2. a. Give the equation for the consumer’s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumer’s indifference curve when good #1 is on the vertical axis and good #2 is on the horizontal axis.]


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in c. Find an expression for the consumer’s demand for good #1 ( q 1* ) as a function of p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p 2 /p1. Solve this equation for q2. Substitute the resulting expression in the budget line and solve for q1.] d. Find an expression for the consumer’s demand for good #2 ( q 2* ) as a function of p1 , p2, and I . 10. [Finding demand functions] Suppose a consumer has utility function U(q 1,q2) = (q1-15) q22 and has income I. The price of good #1 is p 1 and the price of good #2 is p 2. a. Give the equation for the consumer’s budget line. b. Give a formula for the consumer’ s marginal rate of substitution in consumption (MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumer’s indifference curve when good #1 is on the vertical axis and good #2 is on the horizontal axis.] c. Find an expression for the consumer’s demand for good #1 ( q 1* ) as a function of p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p 2 /p1. Solve this equation for q2. Substitute the resulting expression in the budget line and solve for q1.] d. Find an expression for the consumer’s demand for good #2 ( q 2* ) as a function of p1 , p2, and I . 11. [Finding demand functions] Suppose a consumer has utility function U(q 1,q2) = (q1-5)(q2-4) and has income I. The price of good #1 is p 1 and the price of good #2 is p2. a. Give the equation for the consumer’ s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumer’s indifference curve when good #1 is on the vertical axis and good #2 is on the horizontal axis.] c. Find an expression for the consumer’s demand for good #1 ( q 1* ) as a function of p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p 2 /p1. Solve this equation for q2. Substitute the resulting expression in the budget line and solve for q1.]


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in d. Find an expression for the consumer’s demand for good #2 ( q 2* ) as a function of p1 , p2, and I . 12. [Finding demand functions] Suppose a consumer has utility function U(q 1,q2) = 1/2 q1 + q21/2 and has income I. The price of good #1 is p 1 and the price of good #2 is p2. a. Give the equation for the consumer’s budget line. b. Give a formula for the consumer’s marginal rate of substitution in consumption (MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumer’s indifference curve when good #1 is on the vertical axis and good #2 is on the horizontal axis.] c. Find an expression for the consumer’s demand for good #1 ( q 1* ) as a function of p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p 2 /p1. Solve this equation for q2. Substitute the resulting expression in the budget line and solve for q1.] d. Find an expression for the consumer’s demand for good #2 ( q 2* ) as a function of p1 , p2, and I . 13. [Finding demand functions] The following three utility functions must yield exactly the same demand functions: Explain why, without solving explicitly for the demand functions.

14. [Expansion path] Suppose a consumer has the utility function U(q 1,q2) = -(1/q1) – (1/q2) and faces prices p1=$2 and p2 = $3. What is the equation for the consumer’s income-expansion path? 15. [Properties of demand functions] Suppose the purported demand function for good #1 is supposed to be given by q 1* = (1/2) I p 1-2/3 p2-1/3. a. Is this function homogeneous of degree zero in income and prices? Why or why not? b. Find an expression for ∂q1*/∂I. Is good #1 a normal good or an inferior good? Why?


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in c. Find an expression for ∂q1*/∂p1. Is good #1 an ordinary good or a Giffen good? Why? d. Find an expression for ∂q1*/∂p2. Are goods #1 and #2 complements or substitutes? Why? 16. [Properties of demand functions] Suppose the purported demand function for good #1 is supposed to be given by q 1* = 5 I p 1-4/5 p21/5. a. Is this function homogeneous of degree zero in income and prices? Why or why not? b. Find an expression for ∂q1*/∂I. Is good #1 a normal good or an inferior good? Why? c. Find an expression for ∂q1*/∂p1. Is good #1 an ordinary good or a Giffen good? Why? d. Find an expression for ∂q1*/∂p2. Are goods #1 and #2 complements or substitutes? Why? 17. [Properties of demand functions] Suppose the purported demand function for good #1 is supposed to be given by q 1* = 3 (p1*)-1/2 (I*) , where p 1* = (p1 /CPI), I* = (I/CPI), and CPI = an index of consumer prices. Note that p 2 does not appear in this function, except through the CPI. a. Is this function homogeneous of degree zero in income and prices? Why or why not? b. Find an expression for ∂q1*/∂I. [Hint: Use the chain rule.] Is good #1 a normal good or an inferior good? Why? c. Find an expression for ∂q1*/∂p1. [Hint: Use the chain rule.] Is good #1 an ordinary good or a Giffen good? Why?


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in Slutsky and Hicksian Substitution Effects (1) [Total effect of price change] Suppose a consumer has the demand function q 1* = I/p1 – (p2 /p1) . Suppose initially that income is $1000, the price of good #1 is $20, and the price of good #2 is $5. a. Calculate exactly the change in quantity demanded as the price of good #1 rises from $20 to $21. b. Find a formula for the partial derivative of q 1* with respect to p 1. c. Compute the value of the partial derivative of q 1* with respect to p 1 when income is $1000, the price of good #1 is $20, and the price of good #2 is $5. d. Use the approximation formula (6.1) to calculate the change in quantity demanded as the price of good #1 rises from $20 to $21. (2) [Slutsky substitution effect] The graph below shows a consumer's response to a rise in the price of energy. The consumer's income remains constant at $30.

a. What was the old price of energy, according to the old budget line? b. How much energy was purchased with the old budget line? c. What is the new price of energy, according to the new budget line?


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in d. Did the Slutsky substitution effect of the energy price change cause the consumer to buy more or less energy? How much more or less? e. Did the income effect of the energy price change cause the consumer to buy more or less energy? How much more or less? f.

Did the total effect of the energy price change cause the consumer to buy more or less energy? How much more or less?

(3) [Slutsky substitution effect] The graph below shows a consumer's response to a fall in the price of clothing. The consumer's income remains constant at $20.

a. What was the old price of clothing, according to the old budget line? b. How much clothing was purchase with the old budget line? c. What is the new price of clothing, according to the new budget line? d. Did the Slutsky substitution effect of the clothing price change cause the consumer to buy more or less clothing? How much more or less? e. Did the income effect of the clothing price change cause the consumer to buy more or less clothing? How much more or less? f.

Did the total effect of the clothing price change cause the consumer to buy more or less clothing? How much more or less?


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in (4) [Slutsky equation] Suppose a consumer's monthly demand for telephone calls (in minutes) is given (approximately) by q 1 = 50 – 1000 p1 + 0.5 I. Monthly income is originally I=$1,000 and the price of telephone minutes is p 1=$0.05. Then the price of telephone minutes rises to p 1=$0.10. Compute the following: a. the original amount purchased (q1). b. the total effect (Δq1) of the price change. c. the income adjustment required to keep the old bundle affordable. d. the income effect of the price change. e. the substitution effect of the price change. (5) [Slutsky equation] Suppose at a particular point, the partial derivatives of a consumer’s weekly demand for gasoline take the f ollowing values. The partial derivative with respect to the price of gasoline is ∂q*/∂p = -4. The partial derivative with respect to the consumer’s (weekly) income is ∂q*/∂I = 0.05. The consumer currently buys 20 gallons of gasoline per week. a. Is gasoline a normal good or an inferior good for this consumer? Why? b. Is gasoline an ordinary good or a Giffen good for this consumer? Why? c. Compute the approximate total change in the amount of gasoline purchased if the price rose by $0.50 (fifty cents). d. Compute the income-effect component of this change. e. Compute the substitution-effect component of this change. (6)[Slutsky equation] Suppose the price of gasoline rose by fifty cents per gall on, but the government awarded tax credits to compensate for the increase. In particular, if the average consumer previously bought 500 gallons per year, then each consumer would be given a tax credit equal to $250. Would consumption of gasoline by the average consumer increase, decrease, or stay the same? Explain your answer using an indifference curve diagram, if possible. (7)[Slutsky equation] Suppose the government is concerned that poor people are having trouble paying their electric power bills. Assume the price of electricity is currently $0.10 per kilowatt hour and the typical poor family uses about 2000 kilowatt -hours per month. Thus, the typical poor family pays about $200 per month for electricity. The government i s


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in considering two alternative programs to help poor people, programs which to noneconomists might seem equivalent.

Lump-sum payment: Poor families would continue to pay the rate of $0.10 per kilowatt hour, but the government would mail a check for $100 per month to all poor famili es that could be applied toward their utility bills. Rate subsidy: Poor families would enjoy a reduced el ectricity rate of $0.05 per kilowatt hour and the government would pay electricity companies the difference. Now consider two alternative formulas for the change in electricity use by poor families. •

•

a. Is formula (i) positive or negative? Why? b. Is formula (ii) positive or negative? Why? c. Which program would cause the change in electricity consumption given by formula (i)? Why? [Hint: See equation 6.4.] d. Which program would cause the change in electricity consumption given by formula (ii)? Why? e. Which program would cause the larger increase in electricity usage by poor families—the lump-sum payment or the rate subsidy? Why? f

Which program would be more costly for the government? Why?

Elasticity (1) [Price elasticity] Suppose a consumer always spends a total of $100 on CDs every year, no matter what the price and no matter what her income. a. Find a formula for this person's demand function q = f(p). b. Compute this person's price elasticity of demand.


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in (2) [Price and income elasticities] Suppose a consumer always spends 25% of his income on housing, no matter what the price of housing and no matter what his income. a. Find a formula for this person's demand function q = f(p,I). b. Compute this person's price elasticity of demand. c. Compute this person's income elasticity of demand.

(3) [Price and income elasticities] Suppose the consumer has the particular utility function U = q1 q22 and faces budget constraint I = p 1q1 + p2q2 . a. Find the consumer’s demand function for good 1. b. Is the consumer’s own-price elasticity of demand for good #1 constant? If so, what is its value? c. Is the consumer’s cross-price elasticity of demand for good #1 with respect to the price of good #2 constant? If so, what is its value. d. Is the consumer’s income elasticity of demand for good #1 constant ? If so, what is its value?

(4) [Price elasticity of demand] Indicate for each of the following demand functions whether the function has a constant price elasticity of demand. If the price elasticity is constant, give its value.

(5) [Price elasticity and revenue] Determine whether the following statement is true, false, or uncertain, and justify your answer using the concept of elasticity. ―If College X raises tuition, it will get more tuition revenue.‖


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in To answer the next two problems, use the following definition: Price elasticity of demand = (% change in quantity) ÷ (% change in price), and the following formula, derived from the approximation formula for products: % change in revenue ≈ (% change in price) + (% change in quantity)

(6) [Price elasticity and revenue] Refer to the information in the previous box. Suppose the price of gasoline rises by 10% and the elasticity of demand for gasoline is known to be -0.4 . Assume income and other prices do not change. a. Will the quantity of gasoline demanded increase or decrease? By how much? b. Will the total the total amount of money spent by consumers on gasoline increase or decrease? By approximately how much?

(7) [Price elasticity and revenue] Refer to the information in the previous box. Suppose a company believes that the elasti city of demand for its product is –1.5, and consider what would happen if it decreased its price by 2%. Assume income and other prices do not change. a. Will the quantity sold increase or decrease? By how much? b. Will the total the total amount of revenue generated by the product increase or decrease? By approximately how much?

(8) [Income elasticity of demand] Determine whether the following statement is true, false, or uncertain, and justify your answer. ―If a person’s income elasticity of demand for clothing is one, then as the person’s income rises, the fraction of income spent on clothing remains constant.‖ (9) [Computing income elasticity of demand] The U.S. government's Consumer Expenditure Survey reports the following figures. *

Total annual expenditures Expenditure on eggs Expenditure on car rentals

Low-income consumers $30 thousand $38 $200


High-income consumers $50 thousand $42 $440

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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in

Assume that total annual expenditures equal c onsumer income. Assume that all consumers face the same prices; this implies that the percent change in the quantity of a good is equal to the percent change in the expenditure on that good. a. Compute the income elasticity of demand for eggs using the arc-elasticity formula. b. Compute the income elasticity of demand for car rentals using the arc-elasticity formula. c. Compute the income elasticity of demand for eggs using the difference-inlogarithms formula d. Compute the income elasticity of demand for car rentals using the difference-inlogarithms formula.

(10) [Income elasticity of demand] Indicate for each of the following demand functions whether the function has a constant income ela sticity of demand. If the income elasticity is constant, give its value. To answer the next two

use the following definition:

Income elasticity of demand = (% change in quantity) ÷ (% change in income). and the following formula, derived from the formula for ratios: % change in budget share ≈ (% change in quantity) - (% change in income)

(11) [Income elasticity and budget share] Refer to the information in the previous box. Suppose the income elasticity of demand for travel is 2.5. Now suppose income rises by 2%. a. Will the amount of travel demanded increase or decrease? By how much? b. Will spending on travel, as a fraction of a consumer's total budget, increase or decrease? By approximately how much?


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Consumer Theory Very Basic Question Set Series 001 Microeconomics www.ecopoint.in (12) [Income elasticity and budget share] Refer to the information in the previous box. Suppose the income elasticity of demand for t oothpaste is 0.4. Now suppose income rises by 5%. a. Will the amount of toothpaste demanded increase or decrease? By how much? b. Will spending on toothepaste, as a fraction of a consumer's total budget, increase or decrease? By approximately how much?

(13) [Demand elasticities] Suppose a typical consumer is believed to have the following demand function for electricity: q 1* = 50 p1-0.6 p20.1 I0.5 . Here, p1 denotes the price of electricity, p 2 denotes the price of natural gas, and I denotes the consumer ’s income. a. Is this function homogeneous of degree zero in income and prices? Why or why not? b. Find the price elasticity of demand for electricity. Is electricity an ordinary good or a Giffen good? Why? c. Find the income elasticity of demand for electricity. Is electricity a normal good or an inferior good? Why? d. Find the cross price elasticity of demand for electricity with respect to the price of natural gas. Are electricity and natural gas substitutes, complements, or unrelated goods why?


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Consumer theory practice problems

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