Ch3_Q&A

1 Chapter 3: Consumer Theory -- Suggested Questions Question 1: Jane’s preferences for Good X and Good Y are represented in the indifference curve below. Determine Jane’s marginal rate of substitution of Good Y for Good X and her utility level in the table below. Combination (Bundle) Y X MRS YX  MRS YX  Utility level (Given U = XY) A 2 10 B 4 5 C 5 4 D 10 2 Quantity of Good X (units/period) Quantity of Good Y (units/period) Jane’s Indifference curve (IC) A (Y = 2, X = 10) B (Y = 4, X = 5) C (Y = 5, X = 4) D (Y = 10, X = 2) • • • •

2 Solution: Combination (Bundle) Y X MRS YX  MRS YX  Utility level (Given U = XY) A 2 10 - - 20 B 4 5 -2.5 2.5 20 C 5 4 -1 1 20 D 10 2 -0.4 0.4 20 • The indifference curve of Question 1 is convex towards the origin. This shape of the indifference curve reflects the law of diminishing marginal rate of substitution. • The law of diminishing marginal rate of substitution states that  Y   MRS YX in the absolute value (  MRS YX  ). • Jane is willing to give up a smaller quantity of Good X for one more unit of Good Y as we move down and to the right along the indifference curve (A  B  C  D). • In general, consumers ’ preferences follow this type of indifference curve, representing imperfect substitutes between Good Y and Good X.

4 Solution: Combination (Bundle) Y X MRS YX  MRS YX  Utility level (U) (Given U = Y + 2X) A 2 9 - - 20 B 4 8 -0.5 0.5 20 C 5 7.5 -0.5 0.5 20 D 10 5 -0.5 0.5 20 • Jane is willing to substitute 0.5 units of Good X for one additional unit of Good Y as we move down and to the right along the indifference curve (A  B  C  D). • This indicates that Jane’s marginal rate of substitution of Good Y for Good X remains constant along her indifference curve. • This indifference curve does not follow the law of diminishing marginal rate of substitution. • The constant marginal rate of substitution of Good Y for Good X is a characteristic of perfect substitutes between Good Y and Good X.

5 Question 3: Jane views two units of Good Y as perfect substitute for one unit of Good X. (a) Draw Jane’s indifference curve s. (b) Determine Jane’s marginal rate of substitution of Good Y for Good X ( MRS YX ). (c) Derive Jane’s utility function s and indifference curve functions of each level of utility. Solution: (a) (b) MRS YX = Slope of an indifference curve = -0.5. Quantity of Good X (units/period) Quantity of Good Y (units/period) Jane’s Indifference curve s (IC) U = Utility level IC 1 < IC 2 < IC 3 IC 1 (U = 2) IC 2 (U = 4) IC 3 (U = 6)

7 Question 4: Determine the marginal rate of substitution of Good X for Good Y (MRS XY ) for the given utility functions. a) U = 2X 0.5 Y. b) U = 0.5X 2 Y 4 . c) U = 2X + Y. d) U = X + 4Y. Solution: a) MRS XY = − 0.5Y X . b) MRS XY = − Y 2X . c) MRS XY = -2. d) MRS XY = − 1 4 . Question 5: Determine the marginal rate of substitution of Good Y for Good X (MRS YX ) for the given utility functions. a) U = 2X 0.5 Y. b) U = 0.5X 2 Y 4 . c) U = 2X + Y. d) U = X + 4Y. Solution: a) MRS XY = − 2X Y . b) MRS XY = − 2X Y . c) MRS XY = − 1 2 . d) MRS XY = -4.

8 Question 6: Based on the given consumer’s preferences below, derive the utility function. a) Jack views one unit of Good A as perfect substitute for one unit of Good B. b) Olivia always consumes one unit of Good A with one unit of Good B. c) Taylor views ten unit of Good A as perfect substitute for one unit of Good B. d) Billie always consumes four units of Good A with one unit of Good B. Solution: a) U = A + B , with A is on the horizontal axis and B on the vertical axis for the indifference curve, or U = B + A with B is on the horizontal axis and A on the vertical axis for the indifference curve. b) U = min {A, B} , with A is on the horizontal axis and B on the vertical axis for the indifference curve, or U = min {B, A} with B is on the horizontal axis and A is on the vertical axis for the indifference curve. c) U = A + 10B , with A is on the horizontal axis and B on the vertical axis for the indifference curve, or U = 10B + A with B is on the horizontal axis and A on the vertical axis for the indifference curve. d) U = min {A, 4B} , with A is on the horizontal axis and B on the vertical axis for the indifference curve, or U = min {4B, A} with B is on the horizontal axis and A on the vertical axis for the indifference curve.

10 Solution: a) b) T = 10 – 2C. c) The slope of the budget line = -2. d) P C = $2 per cup. P T = $1 per cup. e) P C C + P T T = M  2C + T = 10. T, Tea (cups) C, Coffee (cups) Budget line, T = 10 – 2C.

11 Question 8: The diagram below represents one of Olivia’s indifference curves and her budget line. a) If the price of a Timbit (P T ) is $1, how much is Jane ’ s budget (M)? b) What is the price of a Donut? c) Derive Olivia’s budget line function. d) Derive Olivia’s budget constraint function. e) What is the slope of Olivia’s budget line? f) What is Olivia’s marginal rate of substitution of Donuts for Timbits at her optimal bundle? T, Timbit (units) D, Donut (units) Olivia’s Budget line Olivia’s indifference curve

13 Question 10: Ning views two units of Potato as a perfect substitute for one unit of Meat . a) What is her marginal rate of substitution of Potato for Meat ( MRS PM )? b) Plot Ning’s indifference curve s. c) What is her utility function? Meat costs $4 per unit and Potato costs $2 per unit. Ning has a monthly income of $16 to spend on Meat and Potato. d) What is her budget constraint? e) What is her budget line function? Plot the budget line. f) What is the slope of the budget line? g) What combination of Meat (M*) and Potato (P*) should Ning buy to maximize her utility under her budget? Solution: a) MRS PM = -0.5. b) c) U = P + 2M. M, Meat (units) P, Potato (units) IC = Indifference curve IC 1 < IC 2 < IC 3 < IC 4 < IC 5 IC 1 IC 2 IC 3 IC 4 IC 5

14 d) M = P P P + P M M.  16 = 2P + 4M. e) M = 4 – 0.5P. f) Since the slope of the indifference curve (MRS PM ) equals the slope of the budget line, any combination of Potato and Meat along this line is her optimal bundle. This represents an interior solution for perfect substitutes. M, Meat (units) P, Potato (units) Ning’s budget line M = 4 – 0.5P. f) Slope of the budget line = -0.5. IC 1 IC 2 IC 3 IC 4 IC 5 P, Potato (units) M, Meat (units) IC 4 is the same as the budget line . MRS PM = the slope of the budget line = -0.5 Ning’s budget line , M = 4 – 0.5P.

16 Question 14: Olivia ’s utility function representing her preferences is: U = 5X 0.4 Y 2 . The price per unit of Good X is $20, and the price per unit of Good Y is $10. If her budget to spend on Good X and Good Y is $600, what is her optimal bundle (X*, Y*)? Solution: X* = 5 units. Y* = 50 units. Question 15: Jane ’ s demand function for Good Y is Q = 25 – 0.5P. If the market price of Good Y is $10, what is her consumer surplus (CS)? Solution: CS = $400.