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Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 William M. Boal Signature: Printed name: MIDTERM EXAMINATION #1 VERSION B “Mathematical Tools” September 9, 2009 INSTRUCTIONS: This exam is closed-book, closed-notes, and calculators are NOT permitted. Point values for each question are noted in brackets. As usual in this course, “exp(x)” denotes the exponential function (also written ex ) while “ln(x)” denotes the natural logarithm function (logarithm to base e). I. MULTIPLE CHOICE: Circle the one best answer to each question. Use margins for scratch work. [3 pts each—39 pts total] (1) Suppose y = 5x2 + 3x + 2. Then the derivative of y with respect to x is a. dy/dx = 3 . b. dy/dx = 5 . c. dy/dx = 10x + 3 . d. dy/dx = 5x + 3 . e. dy/dx = 3x + 2 . f. dy/dx = 5x2 + 3x + 2 . (4) If x increases by 4 percent, then ln(x) increases by about a. 4 units. b. 0.04 units. c. ln(4), or about 1.39 units. d. 0.04 percent. e. 4 percent. (2) Suppose y = 4 (3x+7) 2. Then the derivative of y with respect to x is a. dy/dx = 3 . b. dy/dx = 4 . c. dy/dx = 8 (3x+7) . d. dy/dx = 24 (3x+7) . e. dy/dx = 4 (3x+7) . f. dy/dx = 24 (2x+3) 2 . (5) Suppose we wish to maximize the function y = f(x), which is continuously differentiable. Assuming there are no restrictions on the possible values of x, the maximizing value x* must satisfy a. x* = 0. b. f(x*) = 0. c. dy/dx = 0, if x = x*. d. d2y/dx2 = 0, if x = x*. e. All of the above. (3) Consider the following functional forms. Which form has constant slope (or derivative)? a. y = ln (3x) . b. y = exp (6x) . c. y = 12 + 7 x . d. y = 2 + (5/x) . e. y = 4 x2 . f. y = 3 + 5 x + 9 x2 . Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 The next question refers to the following graph of y = f(x) . Midterm Examination #1 Version B Page 2 of 8 The next two questions refer to the following graph of a level curve, or contour, of the function y = f(x1,x2) . y=f(x) x1 x x2 (6) In this graph, the derivative of y with respect to x (that is, df/dx) equals zero at a. no point on the graph. b. one point on the graph. c. two points on the graph. d. three points on the graph. e. four points on the graph. f. more than four points on the graph. (7) Suppose y and x are strictly positive variables. If the derivative dy/dx is negative, then the elasticity of y with respect to x a. can be positive, negative, or zero. b. can be positive or negative but not zero. c. must be positive. d. must be negative. e. must be zero. (8) Consider the following functional forms. Which form has constant elasticity? a. y = ln (3x) . b. y = exp (6x) . c. y = 12 + 7 x . d. y = 2 + (5/x) . e. y = 4 x2 . f. y = 3 + 5 x + 9 x2 . (9) By definition, all points along the curve in this graph have identical values of a. the marginal rate of substitution. b. y . c. x1 . d. x2 . e. both x1, and x2 . f. all of the above. (10) According to this graph, if x2 increases and y is to be held constant, then x1 must a. equal zero. b. increase. c. decrease. d. remain constant. e. cannot be determined from the information given. (11) Suppose y = (x1–2)3 (x2–5)2 . Then the partial derivative of y with respect to x2 is given by the formula a. y/x2 = 3(x1–2)2. b. y/x2 = 2(x2–5) . c. y/x2 = 3(x1–2)2 2(x2–5). d. y/x2 = 3(x1–2)2 (x2–5)2 . e. y/x2 = (x1–2)3 2(x2–5) . f. y/x2 = 3(x1–2)2 (x2–5)2 + (x1–2)3 2(x2–5) . Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 (12) Consider the following functional forms for y = f(x1,x2). Which form has constant partial derivatives (y/x1 and y/x2)? a. y = 12 + 2 x1–1 + 4 x2–1 . b. y = 16 + 8 x11/2 + 9 x21/2 . c. y = 1 + 7 x1 + 2 x2 . d. y = 5x1 + 3x2 + 1.5 (x1x2)1/2 . e. y = 14 x13 x22 . f. y = 17 (x1–4)3 (x2–2) . Midterm Examination #1 Version B Page 3 of 8 (13) Consider the following functional forms for y = f(x1,x2). Which form has constant partial elasticities (1 and 2) ? a. y = 12 + 2 x1–1 + 4 x2–1 . b. y = 16 + 8 x11/2 + 9 x21/2 . c. y = 1 + 7 x1 + 2 x2 . d. y = 5x1 + 3x2 + 1.5 (x1x2)1/2 . e. y = 14 x13 x22 . f. y = 17 (x1–4)3 (x2–2) . II. SHORT ANSWER: Please write your answers in the boxes on this question sheet. Use margins for scratch work. (1) [4 pts] Suppose the derivative of the function y = f(x) equals 5 at a particular value of x . Moreover, the elasticity of y with respect to x equals 1.8. Further suppose that x increases by 0.3. a. Will y increase or decrease? b. By about how much? units (2) [4 pts] Suppose the derivative of the function y = f(x) equals 2.5 at a particular value of x . Moreover, the elasticity of y with respect to x equals 0.6. Further suppose that x increases by 5 percent. a. Will y increase or decrease? b. By about how much? % (3) [4 pts] Consider the function y = f(x1,x2) . Suppose at a particular point, y/x1 = 4, and y/x2 = 3, and that the partial elasticities are 1 = 1.2 and 2 = 0.8. Further suppose that x1 increases by 0.5 and simultaneously x2 increases by 2. a. Will y increase or decrease? b. By about how much? units Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 Midterm Examination #1 Version B Page 4 of 8 (4) [4 pts] Consider the function y = f(x1,x2) . Suppose at a particular point, y/x1 = 6, and y/x2 = 3, and that the partial elasticities are 1 = 0.5 and 2 = 0.7. Further suppose x1 increases by 2, but suppose we want to keep the value of y constant. a. Must x2 increase or decrease? b. By about how much? units (5) [4 pts] Revenue equals price times quantity sold. Suppose price increases by 4 percent and the quantity sold decreases by 5 percent. a. Will revenue increase or decrease? b. By about how much? % (6) [4 pts] The capital-labor ratio equals the quantity of capital divided by the quantity of labor. Suppose the quantity of capital increases by 5 percent and the quantity of labor increases by 3 percent. a. Will the capital-labor ratio increase or decrease? b. By about how much? % (7) [4 pts] Consider the function y = f(x1,x2) , where y denotes output, x1 denotes capital input, and x2 denotes labor input. Suppose at a particular point, the partial elasticity of output with respect to capital equals 0.3, and the partial elasticity with respect to labor equals 0.6. Further suppose capital increases by 5 percent and simultaneously labor increases by 2 percent. a. Will output increase or decrease? b. By about how much? % (8) [2 pts] Consider the function y = f(x1,x2) . Suppose at a particular point, the y/x1 = 4, and y/x2 = 7. Then the value of the marginal rate of substitution of x2 for x1 (that is, the |slope| of the level curve with x1 on the vertical axis and x2 on the horizontal axis) equals Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 Midterm Examination #1 Version B Page 5 of 8 III. PROBLEMS: Please write your answers in the boxes on this question sheet. Show your work and circle your final answers. (1) [Optimization: 12 pts] Consider the function y = f(x) = 3 x2 24 x + 5. Assume that x cannot be negative. a. Find a formula (in terms of x ) for the derivative of y with respect to x ( dy/dx ). b. Compute the value x* that maximizes this function, subject to the restriction that x cannot be negative. [Hint: sketch the curve.] c. Compute the maximum value y* = f(x*), subject to the restriction that x cannot be negative. Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 Midterm Examination #1 Version B Page 6 of 8 (2) [Marginal rate of substitution: 8 pts] Consider the following three functions. (i) (ii) (iii) y = 5 (x1 x2)1/2 . y = 2 ( x1 + x2 ) . y = 2 (x1 x2) . a. Which two functions have exactly the same formula, in terms of x1 and x2, for the marginal rate of substitution of x2 for x1 ? (Recall that the MRS of x2 for x1 is the |slope| of the level curve with x1 on the vertical axis and x2 on the horizontal axis.) b. What is that formula? Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 Midterm Examination #1 Version B Page 7 of 8 (3) [Partial elasticities: 8 pts] Suppose y = (x1+7)2 x23. a. Find a formula for the partial elasticity of y with respect to x1. Express your answer in terms of x1 and x2 alone, not y. b. Find a formula for the partial elasticity of y with respect to x2. Express your answer in terms of x1 and x2 alone, not y. Intermediate Microeconomic Analysis (Econ 173) Drake University, Fall 2009 Midterm Examination #1 Version B Page 8 of 8 IV. CRITICAL THINKING: Answer just one of the questions below (your choice). [3 pts] (1) Suppose y = f(x1,x2). Further suppose that the sum of the partial elasticity of y with respect to x1 and the partial elasticity of y with respect to x2 equals 1. That is, 1 + 2 = 1. If x1 and x2 both simultaneously increase by 3 percent, does y increase by more than 3 percent, exactly 3 percent, or less than 3 percent? Justify your answer. (2) Suppose y = f(x1,x2). Further suppose y/x1 is positive, but y/x2 is negative. Do the level curves of f(x1,x2) slope up or down? Justify your answer. Circle the question you are answering. Please write your answer below. Full credit requires good grammar, accurate spelling, and correct reasoning. [end of exam]