Final exam – Winter 2019 – Econ 10A – Answers March 16, 2019 Test Form A Name (Please Print) : ______________________________________ Seat:_____________ Has the person sitting to YOUR RIGHT given you a valid photo ID that matches the one written on the top of their cover sheet If the seat to your right is empty, check the ID of the next person to the right. ____ Yes ____ No ____ There is no person to my right Has the person sitting to YOUR LEFT given you a valid photo ID that matches the one written the top of their cover sheet If the seat to your left is empty, check the ID of the next person to the left. ____ Yes ____ No ____ There is no person to my left Name (Please Print):____________________________________________ Signature_____________________________________________________ Final exam – Winter 2019 – Econ 10A Test Form A March 16, 2019 Name (Please Print): ______________________________________ UCSB Perm Number:______________________________________ Seat:_____________ Academic Integrity: On midterm exams and the final exam, the standard rules apply as outlined in “Academic Integrity at UCSB: A Student’s Guide.” On those exams, you are cheating if you copy from another student, consult notes or a textbook during the exam, have another student take an exam for you, and communicate with another student about answers to the exam. Electronic devices of any kind (calculators, cell phones, computers, etc.) are not permitted. Any student who cheats during an exam will receive an F for the course, and the incident will be reported to the Office of Judicial Affairs for further sanctions. Possible sanctions include suspension and dismissal. The Economics Department has established a hotline at which you can anonymously report cheating incidents. The URL is https://econ.ucsb.edu/cheating/ This exam is 150 minutes in duration. You may turn your test in early, except during the last 10 minutes. If you need to use a restroom, you will need permission from one of the proctors. Each multiple choice question is worth 2 points, there is no penalty for wrong answers. In the long answer section possible points for each question (or sub-question) are listed at the beginning of the question. In the long answer section you must completely prove all answers – unless otherwise stated, undefended answers receive zero points. Answers that are not completely proven and defended will not receive full credit. You are required to turn in the whole test, including the multiple choice questions. (1) Wilhelmina has fully insured so that she will consume $10,000 in both the good and bad state of the world. Without insurance, she would have received $15,000 in the good state and $0 in the bad state. How much does each dollar of insurance cost, rounded to the nearest cent (Assume that the per-dollar price of insurance is constant.) (a) $0.67 (b) $0.50 (c) $0.33 (d) $0.25 (e) None of the above Consumption in both the good and bad state of the world under full insurance is c = M γL. From the consumption in good and bad state under no insurance, we can tell that income is 15,000 (cg = = 15000) and the loss is 15,000 (cb = M L = 0). Therefore, 10000 = 15000 15000γ → 15000γ = 5000 → γ = 1 3 ≈ 0.33 (2) In the long run, a first order condition for unconditional factor demand equates the value of the marginal product of labor equal to _____. Assume diminishing marginal product and that the number of hours of labor used is positive. (a) The wage rate (b) The hourly rental cost of capital (c) The cost per machine of capital (d) The wage rate divided by the hourly cost of capital (e) None of the above For unconditional factor demand, the first order condition sets MPL equal to w and MPK equal to r. (3) Josephina’s utility is given by () = √. She receives $3,600 today but there is a 50% chance that she will bump her head and have to pay a $2,000 medical bill. If Josephina is an expected utility maximizer, what is the largest amount she is willing to pay to fully insure for the cost of the medical bill (Pick the closest answer.) (a) $800 (b) $900 (c) $1,000 (d) $1,100 (e) $1,200 To calculate the maximum willingness to pay to fully insure, we need set the expected utility under no insurance equal to utility under full insurance. The amount paid to fully insure that makes the two utilities equal is the maximum willingness to pay. U(no insurance) = πb√ + √ = 1 2 √3600 2000 + 1 2 √3600 = 1 2 √1600 + 1 2 √3600 = 1 2 (40) + 1 2 (60) = 50 Set the U(full insurance) equal to 50 and solve for γL: U(full insurance) = πb√ + √ = √ = 50 → 3600 = 2500 → = (4) In the market for calcium-infused water, the demand is given by D(p) = 100 – p and the supply is given by S(p) = p. Quantities are expressed in gallons, and prices are in dollars. Calcium-infused water can be supplied in any non-negative amount, including fractional amounts. If the government imposes a price ceiling of $10 per gallon on calcium-infused water, how much will the producer surplus be in this market (a) $50 (b) $100 (c) $1,250 (d) $2,500 (e) None of the above Price ceiling limits how high price can rise. First, we need to check whether price ceiling is binding (price ceiling is binding if market equilibrium price is higher than the price ceiling of $10). D(p) = S(p) → 100 p = p → 100 = 2p → p = 50 Therefore, the price ceiling is binding. This means that the price of calcium-infused water will be $10 per gallon. We need to calculate the size of the area between p = 10 and S(p) = p. When p = 10, S(p = 10) = 10. So the amount of supply will be 10 under the price ceiling. This is the base of the triangle. To get the height of the triangle, we need to measure supply at p = 0. As S(p = 0) = 0, the height of the triangle is 10 0 = 10. Producer surplus is 10 × 10 × 1 2 = 50. (5) Which of the following best describes an indifference curve (a) All points of a curve with equal cost (b) All points of a curve with equal utility (c) The demand of each commodity (d) The marginal utility of one of the commodities (e) All points in the weakly preferred set An indifference curve is collection of all bundles with equal utility. (6) Kamala will receive $120 today and $120 one year from today. If the annual interest rate is 20%, how much is the total present value of the two payments (a) $200 (b) $220 (c) $240 (d) $244 (e) None of the above The present value of payment in each time period is given by PV = x (1+r)t In this case, PV = 120 + 120 1.2 = 120 + 100 = 220. (7) Which of the following best describes marginal product of input 2 Assume only two inputs. (a) Holding constant the amount of input 2 used, how much extra output is produced when one more unit of input 1 is used (b) Holding constant the amount of output produced, how many additional units of good 2 is needed for production if the number of units of good 1 is reduced by one (c) Holding constant the amount of input 1 used, how much extra output is produced when one more unit of input 2 is used (d) Holding constant the amount of input 2 used, how much extra revenue is generated when one more unit of input 1 is used (e) Holding constant the amount of input 1 used, how much extra revenue is generated when one more unit of input 2 is used Marginal product is the amount of extra output produced when an additional unit of input is used holding the amount of other inputs constant. (c) is consistent with the definition. (8) If economic rent is positive and economic profit is zero, which of the following is true at the profit- maximizing quantity (AVC = average variable cost) (a) The AVC for all factors except land is less than price (b) The AVC for all factors except land is more than price (c) The AVC for all factors except land is equal to price (d) The positive economic rent drives economic profits to always be negative (e) None of the above If economic profit is zero, then the average cost is equal to price. Under this condition, if economic rent is positive, the cost of valuable input (which is land in this case) must be positive. AC = Economic Rent Q + Therefore, AC > AVC when economic rent is positive is positive. (9) A company’s short-run total cost function is () = 42 + 5000. If the price of output sells for $120 each, how many units will this company sell to maximize profit (a) 24 (b) 12 (c) 6 (d) 0 (e) None of the above To calculate the output that maximizes profit, first construct a profit function: π = py C(y) = 120y 4y2 5000 Take the first order condition and set it equal to zero in order to find the critical value of y. π y = 120 8 = 0 → 8 = 120 → = (10) Paris has the following utility function for two goods: (1, 2) = 1 22 3. If she has $100 to spend and the cost of each good is $1 each, how many units of good 2 will she purchase to maximize utility (a) 30 (b) 40 (c) 50 (d) 60 (e) None of the above The utility function is representative of Cobb-Douglas preferences. Set MRS equal to price ratio to find the optimality condition: MRS = 2×12 3 31 22 2 = 22 31 = 1 1 = 1 2 2 3 x2 = 1 Plug into the budget constraint: 100 = x1 + 2 → 100 = 2 3 2 + 2 → 100 = 5 3 2 → = Alternatively, according to the utility function the income will be split in a 2 to 3 ratio if Paris is consuming optimally. This means that Paris will spend $40 on good 1 and $60 on good 2. As the price of each good is $1, she will buy 40 units of good 1 and 60 units of good 2. (11) If a utility function for Steve is (1, 2) = 1 + 2 5, which of the following utility functions is also consistent with Steve’s preferences (a) (1, 2) = (1 + 2 5) 2 (b) (1, 2) = 1 + 22 (c) (, ) = + + (d) (1, 2) = 1 2 + 2 2 (e) None of the above We can rule out (b) because only good 2 is multiplied by 2. (d) can be eliminated as well, because 1 2 + 2 2 can’t be constructed easily from 1 + 2 5. With (a), for values of x1 and x2 such that x1 + x2 < 5, the utility is negative in the original function. Squaring the negative value reverses the preferences ordering. Therefore, (a) is incorrect. Finally, (c) can be constructed by first adding 5 to the original utility function and squaring it. Once 5 is added to the function ((1, 2) = 1 + 2), utility won’t be negative for any value of good 1 and good 2. Therefore, squaring it does not reverses the ordering. (12) Sammie has an intertemporal utility function of (1, 2) = 12, with c1 the amount of consumption this year and c2 the amount of consumption next year. She earns $100,000 this year, and will earn $77,000 next year. She can save and borrow as much as she wants at a 10% annual rate, as long as she pays back any borrowed money by next year. What is the largest amount of consumption possible for next year, given this budget constraint (a) $85,000 (b) Between $86,000 and $150,000 (c) $187,000 (d) $197,000 (e) None of the above The largest amount of consumption possible is the combination of the incomes this year and next year in the future value. FV = (1 + )m1 + 2 = 100000(1.1) + 77000 = 110000 + 77000 = 187000 Solve each problem in the space provided on that page. Solutions written elsewhere will not be graded. Your solutions must be completely and clearly justified for full credit. (13) (6 points) One of the common assumptions about consumption in any time period is that it is a normal good. For the utility function (1,2) = 5000 ln(1) + 2, determine whether or not consumption in time period 1 and time period 2 are normal for this particular utility function. Assume = 100%, 1 > 20,000, and 2 = 0. To check whether consumptions are normal, we first need to calculate the consumptions in both periods in terms of income. Income in this case is m1 + 2 1+ = 1 because m2 = 0. First, set MRS equal to price ratio: MRS = 5000 c1 = 2 = 1 + c1 = 5000 2 = 2500 At c1 = 2500, the consumption in period 1 does not depend on income. Also, for any income over 20,000, consuming 2,500 in the first period is always possible. Therefore, is not normal. Now solve for c2 using the budget constraint: m1 = 1 + 2 1 + → 1 = 2500 + 2 2 → 2 = 21 5000 As m1 increases, c2 increases as well ( c2 m1 = 2 > 0). Therefore, is normal. (14) (3 points) Elizabeth has non-labor income each week of $1,000. She can work up to 100 hours per week at an hourly wage of $30 per hour. Her utility function for recreation (R) and consumption (C) is given by the equation (, ) = . What is Elizabeth’s reservation wage Reservation wage is the wage at which any increase in wage would make Elizabeth work. At reservation wage, L = 0. Therefore, R = 100 and C = 1000. MRS(R = 100, C = 1000) = → = → = (15) (4 points) A company’s production function is (, ) = 4 + 5, with K denoting the number of hours of capital used, and L the number of labor hours used. If the hourly cost of capital is $2 and the hourly wage rate is $3 per hour, what is the minimum amount that would need to be spent to produce 60 units of output The production function implies that labor and capital are perfect substitutes. We use whichever input factor that are more productive per each dollar spent on it. MPK r = 4 2 = 2 . = 5 3 Therefore, each dollar spent on capital produces more outputs than each dollar spent on labor. Only use capital (L = 0): 60 = 4K + 5(0) → K = 15 For 15 units of capital, the company must spend = = () = . (16) (3 points) Fifty firms each have a total long-run cost function of () = 1002. If the price of the output is $1,000 per unit, how many units are supplied in total by the 50 firms in the long run If price of the output is $1,000, each firm has a profit function: π = 1000y 100y2 Profit maximizing output for each firm is: π y = 1000 200 → = 5 With fifty firm each producing 5 units of output, the total supply is 5 × 50 = (17) (4 points) If (, ) = 10 + 200√, how much labor will be demanded to maximize profits if the cost of capital is $40, the wage rate is $10, and each unit of output sells for $4 To find the profit maximizing amount of labor, take the first order conditions of the profit function (π = 4(10K + 200√L) 40 10): π K = 40 40 = 0 π L = 400 1 2 10 = 0 The first order conditions imply that the firm is indifferent about using capital because using each unit of capital results in no change in profit. Assume K=0, and maximize the profit with respect to labor. 400L 1 2 10 = 0 → 400 1 2 = 10 → 1 2 = 40 → = (18) (6 points) Rick receives 50 rocks (commodity 1) and 80 clocks (commodity 2) each week. The price to buy and sell each rock and each clock is currently $1 each, and Rick’s utility function for these two commodities is consistent with (1, 2) = min (1, 2). If the price of clocks increases to $2 each, what is the change in rocks demanded due to the substitution effect (SE) and income effect (IE) You need to include a graph to justify your answer. To find substitution and income effect, we must find points A, B, and C. First, calculate point A and C. Point A (initial consumption point): Set x1 and x2 equal since rocks and clocks are perfect complements with utility function U = min(1, 2). Plug into the budget constraint. p11 + 22 = 11 + 22 → 50 + 80 = 1 + 2 130 = 2×1 → 1 = 65, 2 = 65 Point C (final consumption point): Again set x1 and x2 equal and plug into the budget constraint. p11 + 22 = 11 + 22 → 50 + 2(80) = 1 + 22 210 = 3×1 → 1 = 70, x2 = 70 Now, graph points A and C and find where point B lies. For perfect complements the indifference curves are L- shaped, and the optimal consumption points are found at the kink. Therefore, each level of utility only has one optimal consumption points for prices greater than 0. Therefore, points A and B are the same and substitution effect is zero for both goods. Income effect is the entire change in consumption from point A to point B. Therefore, income effect for good 1 is = and the income effect for good 2 is 5 as well.
