ECON 203Principles of MicroeconomicsProblem Set 3Answer KeyThis answer key also provides sketches of solutions and you should use these as a guide tofully develop your answers. Ask your TA if you need help. Problem-solving, as opposed to“solution-reading,” is an essential skill that you’re going to build in this course.Page 1 of 11Question 1Suppose that a consumer’s utility function is given by the function u(x, y) = 2x+3y, definedfor all bundles such that x ≥ 0 and y ≥ 0. Then, the consumer is indifferent between thebundle (5, 5) and the bundle(A) (9, 2)(B) (10, 2)(C) (11, 1)(D) (12, 1)Answer:CSolution sketch: The utility provided by the bundle (5, 5) equals u(5, 5), which is((2 × 5) + (3 × 5)) = 25. Verify that, according to this utility function, the bundle(11, 1) also provides a utility of 25.Page 2 of 11Question 2Suppose that a consumer’s utility function is u(x, y) = 6x + 6y, defined for all bundles suchthat x ≥ 0 and y ≥ 0.(Good X is associated with the horizontal axis and Good Y is associated with the verticalaxis.)The equation of the indifference curve corresponding to a utility of 6 is(A) y = −x + 1(B) y = −x + 6(C) y = −6x + 1(D) y = −6x + 6Answer:ASolution sketch: The equation is derived by setting 6x + 6y = 6, i.e., y = −x + 1.Page 3 of 11Question 3Suppose that a consumer’s utility function isu(x, y) = 10 +xy,defined for all bundles such that x > 0 and y > 0.(Good X is associated with the horizontal axis and Good Y is associated with the verticalaxis.)The indifference curve corresponding to a utility of 11 is(A) a vertical line.(B) a horizontal line.(C) an upward-sloping line.(D) a downward-sloping line.Answer:CSolution sketch: We set11 = 10 +xy,orxy= 1,which yields the equation y = x, which is an upward-sloping line.Page 4 of 11Question 4Suppose that a consumer’s utility function isu(x, y) = 1 + 6y,defined for all bundles such that x ≥ 0 and y ≥ 0.(Good X is associated with the horizontal axis and Good Y is associated with the verticalaxis.)The indifference curve corresponding to a utility of 3 is(A) a vertical line.(B) a horizontal line.(C) an upward-sloping line.(D) a downward-sloping line.Answer:BSolution sketch: We set 1 + 6y = 3, i.e., y = (1/3), which is a horizontal line.Page 5 of 11Question 5Consider a consumer with income equal to $20. Furthermore, pX= $3 and pY= $4.(Good X is associated with the horizontal axis and Good Y is associated with the verticalaxis.)The equation of the consumer’s budget line is(A) y = −(3/4)x + 5(B) y = −(3/4)x + 20(C) y = −(4/3)x + (20/3)(D) y = −(4/3)x + 20Answer:ASolution sketch: To get the equation of the budget line, set pXx + pYy = I, i.e.,3x + 4y = 20. Thus, the equation is y = −(3/4)x + 5.Page 6 of 11Question 6Consider a consumer with income I and with preferences over two goods X and Y . Supposethat the following two facts are known: (i) pX= pYand (ii) the bundle (5, 0) exhausts theconsumer’s income. (Note that prices and income are always positive real numbers.)(Good X is associated with the horizontal axis and Good Y is associated with the verticalaxis.)(You are not provided with any further information.)The equation of the consumer’s budget line is(A) y = −x + 5(B) y = −x + 10(C) y = −5x + 5(D) y = −5x + 10Answer:ASolution sketch: Since pX= pY, the absolute value of the slope of the budgetline equals 1. Moreover, the vertical-axis intercept equals 5. Thus, the equation isy = −x + 5.Page 7 of 11Question 7Consider a consumer with income I and with preferences over two goods X and Y . SupposethatpX= pY= I = 2.(Good X is associated with the horizontal axis and Good Y is associated with the verticalaxis.)(You are not provided with any further information.)The equation of the consumer’s budget line is(A) y = −x + 1(B) y = −x + 2(C) y = −2x + 1(D) y = −2x + 2Answer:ASolution sketch: To get the equation of the budget line, set pXx + pYy = I, i.e.,2x + 2y = 2. Thus, the equation is y = −x + 1.Page 8 of 11Question 8Consider a consumer with preferences over two goods X (represented on the horizontal axis)and Y (represented on the vertical axis). Assume that the consumer’s indifference curvesare strictly convex to the origin. The consumer’s income equals $20, pX= $2, and pY= $4.At the bundle (x = 4, y = 3), the absolute value of the consumer’s MRS (the marginal rateof substitution of good Y for good X) equals 1.Select the correct answer from the options below.(A) To maximize utility the consumer should buy less of good X and more of good Y .(B) To maximize utility the consumer should buy more of good X and less of good Y .(C) The bundle x = 4 and y = 3 maximizes the consumer’s utility.Answer:BSolution sketch: First verify that the bundle (x = 4, y = 3) lies on the budgetline. Next, observe that the MRS (= 1) is greater than the absolute value of the slopeof the budget line (= 0.5). Thus, it follows from the geometry of utility maximizationthat the consumer should buy more of good X and less of good Y (verify this foryourself). Intuitively, the market offers the consumer a better deal than what theywere willing to accept to stay at the same utility level.Page 9 of 11Question 9Consider a consumer with preferences over two goods X (represented on the horizontal axis)and Y (represented on the vertical axis). Assume that the consumer’s indifference curvesare strictly convex to the origin.The consumer’s income equals $10, pX= $1 , and pY= $1 . At the bundle (x = 5, y = 5),the absolute value of the consumer’s M RS (the marginal rate of substitution of good Y forgood X) equals 1.Select the correct answer from the options below.(A) To maximize utility the consumer should buy less of good X and more of good Y .(B) To maximize utility the consumer should buy more of good X and less of good Y .(C) The bundle x = 5 and y = 5 maximizes the consumer’s utility.Answer:CSolution sketch: First verify that the bundle (x = 5, y = 5) lies on the budgetline. Next, observe that the MRS (= 1) equals the absolute value of the slope of thebudget line (= 1). Thus, it follows from the geometry of utility maximization thatthe bundle (x = 5, y = 5) maximizes the consumer’s utility (subject, of course, tothe budget constraint).Page 10 of 11Question 10Consider a consumer with preferences over two goods X (represented on the horizontal axis)and Y (represented on the vertical axis). Assume that the consumer’s indifference curvesare strictly convex to the origin.The