ECON/AMS 11A SPRING 2013 Final Exam Instructions • There are 6 questions worth a total of 54 points. 100%=50 points. • No notes or books. • You may use a simple scientific calculator. No graphing or programmable calculators. • Take your time. Answer each question completely. Check your answers. • For full credit, show all your work and reasoning. Good Luck!!! NAME: Problem 1 2 3 4 5 6 Total Score /8 /10 /10 /8 /8 /10 /50 1 ECON/AMS 11A FINAL EXAM SPRING 2013 1. (8 pts) A firm’s production function is given by Q = 20K 3/5 L2/5 , where Q is the firm’s monthly output, K is the firm’s monthly capital input and L is the firm’s monthly labor input. Each unit of capital input costs the firm $1000 and each unit of labor input costs the firm $600. Find the levels of capital and labor input that the firm should use to minimize the cost of producing Q0 = 2000 units of output. What is the firm’s minimum cost? Be sure to explain how you know that your answer yields the absolute minimum cost of producing 2000 units. 2 ECON/AMS 11A FINAL EXAM 3 SPRING 2013 ECON/AMS 11A FINAL EXAM SPRING 2013 2. The demand equation for a monopolistic firm’s product is given by p = 54 − 0.4q, where p is the price per unit that the firm charges (in dollars) and q is the daily demand for the firm’s product. The firm’s daily cost function is c = 10q + 80, where c is also measured in dollars. (a) (8 pts) Find the price that the firm should set to maximize its daily profit, the profit-maximizing output and the maximum daily profit. Be sure to explain how you know that the profit you found is the firm’s absolute maximum. 4 ECON/AMS 11A FINAL EXAM SPRING 2013 (b) (2 pts) The local government imposes a $1.60 per unit production tax on the firm. How much of this tax should the firm pass on to consumers (in the form of a price increase) to maximize their profit under the new tax? Show your work. 5 ECON/AMS 11A 3. Consider the function f (x) = FINAL EXAM SPRING 2013 4x + 3 . x2 + 1 (a) (8 pts) Find the critical points of f (x) and classify the critical values as relative minima, relative maxima or neither. You may use either the first derivative test or the second derivative test— your choice. (b) (2 pts) Does f (x) attain an absolute maximum value on the interval (0, ∞)? Justify your answer. 6 ECON/AMS 11A FINAL EXAM 7 SPRING 2013 ECON/AMS 11A FINAL EXAM SPRING 2013 4. The demand equation for the monopolistic firm ACME Widgets is given by q = 1.2(600 − 4p)3/2 , where p is the price of a widget and q is weekly demand for widgets. (a) (4 pts) Find the demand, q, and price-elasticity of demand for widgets, ηq/p when p = 125. (b) (2 pts) Use your answer to part a. to find the approximate percentage change in demand if ACME lowers the price of a widget from p = 125 to p = 122.5. 8 ECON/AMS 11A FINAL EXAM SPRING 2013 (c) (2 pts) What is ACME’s marginal revenue when p = 125? Justify your answer briefly. 9 ECON/AMS 11A FINAL EXAM WINTER 2013 5. ACME Widgets’ short-term production function is given by q = 75(5l − 6)2/3 , where q is ACME’s weekly output (which is equal to the weekly demand for their product) and l is ACME’s labor input, measured in 40-hour work weeks. E.g., if l = 10, then ACME’s workers are working for a total of 10 × 40 = 400 hours a week. The demand equation for ACME’s product is given in the previous problem. (a) (4 pts) What is ACME’s output and marginal product, dq/dl, when l = 14? (b) (2 pts) Use your answers to part (a), above, and problem 4(c) to find ACME’s marginal revenue product dr/dl when l = 14. (You may assume that the demand for the firm’s product is equal to their output.) 10 ECON/AMS 11A FINAL EXAM WINTER 2013 (c) (2 pts) If ACME hires a new employee to work 20 hours a week, what is the approximate effect on the firm’s revenue? 11 ECON/AMS 11A FINAL EXAM WINTER 2013 6. The consumption function for a small nation is given by C= 8Y 2 + 25 , 9Y + 5 where Y is the nation’s annual income and C is the nation’s annual consumption, both measured in billions of dollars. (a) (6 pts) What are the nation’s marginal propensities to consume and to save when its annual income is $5 billion? 12 ECON/AMS 11A FINAL EXAM WINTER 2013 (b) (2 pts) Use your answer to part a. to estimate the changes in savings and consumption when the nation’s income increases from $5 billion to $5.2 billion. dC and interpret your answer in economic terms. Y →∞ dY (c) (2 pts) Compute the limit lim 13 ECON/AMS 11A FINAL EXAM 14 WINTER 2013 ECON/AMS 11A Winter, 2015 Final Exam March 17th , 2015 Instructions • There are 6 questions worth a total of 54 points. 100%=50 points. • No notes or books. • You may use a simple scientific calculator. No graphing or programmable calculators. • Take your time. Answer each question completely. Check your answers. • For full credit, show the steps in your work, and explain what you are doing. Good Luck!!! NAME: Problem 1 2 3 4 5 6 Total Score /10 /8 /10 /8 /10 /8 /50 1 ECON/AMS 11A FINAL EXAM Winter, 2015 1. The demand equation for a monopolistic firm’s product is given by p = 150 − 0.75q and the firm’s cost function is c = 0.05q 2 + 14q + 800. (a) (6 pts) Find the price that the firm should set to maximize its profit and find the maximum profit. Explain how you know that the price you found yields the absolute maximum profit. 2 ECON/AMS 11A FINAL EXAM Winter, 2015 (b) (4 pts) The state imposes a production tax on the firm of $4.80 per unit. By how much should the firm increase the price of its product to maximize its profit under this new condition? 3 ECON/AMS 11A FINAL EXAM Winter, 2015 2. (8pts) The Jones Family’s utility function is given by U = 8 ln x + 17 ln y, where U is their monthly utility, x is the quantity of essential goods that they consume per month and y is the quantity of luxury goods that they consume per month. The average price per unit of essential goods is px = $10 and the average cost per unit of luxury goods is py = $40. Find the quantities of essential and luxury goods that the Jones family should consume per month to maximize their utility, given that their monthly budget for these goods is B = $5000. What is their maximum utility? Be sure to justify your claim that the utility you find is the absolute maximum. 4 ECON/AMS 11A FINAL EXAM 5 Winter, 2015 ECON/AMS 11A 3. Consider the function f (x) = FINAL EXAM Winter, 2015 4x − 3 . x2 + 1 a. (6 pts) Find the critical points of f (x) and classify the critical values as relative minima, relative maxima or neither. b. (2 pt) Does f (x) have an absolute maximum on the interval (0, ∞)? Justify your answer. c. (2 pts) Does f (x) have an absolute maximum on the interval (−∞, ∞)? Justify your answer. 6 ECON/AMS 11A FINAL EXAM 7 Winter, 2015 ECON/AMS 11A FINAL EXAM Winter, 2015 4. The short-run production function of ACME Widgets Inc. is given by q = 50(5` − 8)2/3 , where q is the number of Widgets ACME produces in a month, and ` is ACME’s monthly labor input, measured in $1000’s. a. (4 pts) Find ACME’s labor-elasticity of output as a function of `. b. (2 pts) What is the labor-elasticity of output when monthly labor input is ` = 7? c. (2 pts) Use your answer to b. to estimate the percentage change in output if ACME increases its labor input from $7,000.00 to $7,400.00. 8 ECON/AMS 11A FINAL EXAM 9 Winter, 2015 ECON/AMS 11A FINAL EXAM Winter, 2015 5. ACME Widgets’ marginal revenue function is given by dr 30q + 47 = , dq 0.1q 2 + 10 where r is the monthly revenue, measured in $1000’s. ACME’s production function is given in the previous problem. a. (5 pts) What is ACME’s marginal-revenue product when labor input is ` = 7? b. (3 pts) What is the approximate change to the firm’s revenue if ACME increases its labor input to $7,400.00? c. (2 pts) Assuming that labor input is the only variable that is changing, what is the approximate effect on ACME’s profit? Explain. 10 ECON/AMS 11A FINAL EXAM 11 Winter, 2015 ECON/AMS 11A FINAL EXAM Winter, 2015 6. The savings function for a small nation is given by S= 2Y 2 + 3 , 11Y + 20 where Y is the nation’s annual income and S is the nation’s annual savings, both measured in billions of dollars. a. (4 pts) What are the nation’s marginal propensities to consume and to save when its annual income is $7 billion. b. (2 pts) Use your answer to part a. to estimate the changes in consumption and savings when the nation’s income increases from $7 billion to $7.3 billion. dS c. (2 pts) Compute lim and interpret the result in economic terms. Y →∞ dY 12 ECON/AMS 11A FINAL EXAM 13 Winter, 2015 ECON/AMS 11A FINAL EXAM 14 Winter, 2015 ECON/AMS 11A Winter, 2016 Final Exam March 15th , 2016 Instructions • There are 6 questions worth a total of 48 points. 100%=45 points. • No notes or books. • You may use a simple scientific calculator. No graphing or programmable calculators. • Take your time. Answer each question completely. Check your answers. • For full credit, show the steps in your work, and explain what you are doing. Good Luck!!! NAME: Problem 1 2 3 4 5 6 Total Score /8 /8 /8 /8 /8 /8 /45 1 ECON/AMS 11A FINAL EXAM Winter, 2016 1. The demand equation for a monopolistic firm’s product is given by p = 120 − 0.65q and the firm’s cost function is c = 0.05q 2 + 15q + 1000. (a) (6 pts) Find the price that the firm should set to maximize its profit and find the maximum profit. Explain how you know that the price you found yields the absolute maximum profit. (b) (2 pts) The state imposes a production tax on the firm of $4.20 per unit. By how much should the firm increase the price of its product to maximize its profit under this new condition? 2 ECON/AMS 11A FINAL EXAM 3 Winter, 2016 ECON/AMS 11A 2. Consider the function f (x) = FINAL EXAM Winter, 2016 2x − 3 . x2 + 4 a. (6 pts) Find the critical points of f (x) and classify the critical values as relative minima, relative maxima or neither. b. (1 pt) Does f (x) have an absolute maximum on the interval (0, ∞)? Justify your answer. c. (1 pts) Does f (x) have an absolute maximum on the interval (−∞, ∞)? Justify your answer. 4 ECON/AMS 11A FINAL EXAM 5 Winter, 2016 ECON/AMS 11A FINAL EXAM Winter, 2016 3. (8 pts) The production function for a certain firm is given by Q = 20K 0.7 L0.3 where Q is the firm’s annual output, K is the firm’s capital input and L is the firm’s labor input. The price per unit of capital is pk = 10000 and the price per unit of labor is pl = 5000. Find the levels of capital and labor input that maximize the firm’s output given an annual production budget of B = 1, 000, 000, and find the maximal output. Justify your claim that the output you found is the firm’s absolute maximum. 6 ECON/AMS 11A FINAL EXAM 7 Winter, 2016 ECON/AMS 11A FINAL EXAM Winter, 2016 4. The short-run production function of ACME Widgets Inc. is given by q = 40(2` − 8)3/4 , where q is the number of Widgets ACME produces in a month, and ` is ACME’s monthly labor input, measured in $1000’s. a. (4 pts) Find ACME’s labor-elasticity of output as a function of `. b. (2 pts) What is the labor-elasticity of output when monthly labor input is ` = 12? c. (2 pts) Use your answer to b. to estimate the percentage change in output if ACME increases its labor input from $12,000.00 to $12,600.00. 8 ECON/AMS 11A FINAL EXAM 9 Winter, 2016 ECON/AMS 11A FINAL EXAM Winter, 2016 5. The consumption function for a small nation is given by C= 9Y 2 + 25 , 10Y + 3 where Y is the nation’s annual income and C is the nation’s annual consumption, both measured in billions of dollars. a. (4 pts) What are the nation’s marginal propensities to consume and to save when its annual income is $5 billion. b. (2 pts) Use your answer to part a. to estimate the changes in consumption and savings when the nation’s income increases from $5 billion to $5.2 billion. dC c. (2 pts) Compute lim and interpret the result in economic terms. Y →∞ dY 10 ECON/AMS 11A FINAL EXAM 11 Winter, 2016 ECON/AMS 11A FINAL EXAM Winter, 2016 6. (8 pts) The Smith Family’s utility function is given by U = 7 ln x + 13 ln y, where U is their monthly utility, x is the quantity of essential goods that they consume per month and y is the quantity of luxury goods that they consume per month. The average price per unit of essential goods is px = $10 and the average cost per unit of luxury goods is py = $30. Find the quantities of essential and luxury goods that the Smith family should consume per month to maximize their utility, given that their monthly budget for these goods is B = $3600. What is their maximum utility? Be sure to justify your claim that the utility you find is the absolute maximum. 12 ECON/AMS 11A FINAL EXAM 13 Winter, 2016 ECON/AMS 11A FINAL EXAM 14 Winter, 2016 
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