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- Michigan
- University of Michigan - Ann Arbor
- Economics
- Economics 401
- Lauermann
- PS 8 solution.pdf

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Problem Set 8 - Solutions 6.1 If each extra worker produces an extra unit of output, how do the total product of labor, the average product of labor, and the marginal product of labor vary with labor? If every worker adds exactly one unit of output then L workers will add L units of output. Total product of labor is thus q = L. Average product of labor is qL = 1. Marginal product of labor is 1. 6.4 Why must isoquants be thin? An isoquant is the set of all input combinations (K;L) that will produce a particular quantity q of output. If an isoquant were thick, it would imply that adding a little of both capital and labor does not increase production. We assume that inputs are always productive so isoquants cannot be thick. 6.5 Suppose that a rm has a xed-proportions production function in which one unit of output is produced using one worker and two units of capital. If the rm has an extra worker and no more capital, it still can produce only one unit of output. Similarly, one more unit of capital does the rm no good. (a) Draw the isoquants for this production function. See the gure that follows. The production function is q = minfL; 12Kg. To verify that this is the case, note that when two units of capital are combined with one unit of labor, one unit of the output good is produced. Isoquants are L-shaped with the kink line L = 12K. (b) Draw the total product, average product, and marginal product of labor curves (you will probably want to use two diagrams) for this production funciton. Again, see the gure that follows. Remember that when considering how labor a ects produc- tion, we think of capital as xed at some amount K. First let?s consider the marginal product of labor. Having xed the level of capital at K, for all quantities of labor L < 12 K adding an additional unit of labor increases output by one unit. Once L > 12 K adding an additional unit of labor does not increase output at all. Therefore, MPL = 1 L < 1 2 K0 L 1 2 K 1 Since every unit of labor adds exactly one unit of output until L = 12 K and then nothing after, TPL = L L < 1 2 K1 2 K L 1 2 K Lastly, the average product of labor is equal to TPLL : APL = ( 1 L < 1 2 K1 2 KL L 12 K 6.10 Mark launders his white clothes using the production function q = B + 2G, where B is the number of cups of Clorox bleach and G is the number of cups of generic bleach that is half as potent. Draw an isoquant. What is the marginal product of B? What is the marginal rate of technical substitution at each point on an isoquant? First let?s begin by noting that the book made a mistake. As written it is the Generic that is twice as potent as the Clorox bleach. To x this, let?s pretend the book wrote q = B + 12G. The isoquant above is linear, with a downward slope of 12 with B on the horizontal axis. The marginal product of B is one and of G is one half. The marginal rate of technical substitution of B for G equals one half, that is you would be willing to give up one half cup of Clorox bleach for one cup of Generic bleach because the generic is half as potent. 2 7.1 \There are certain xed costs when you own a plane," [Andre] Agassi explained during a break in the action at the Volvo/San Francisco tennis tournament, \so the more you y it, the more economic sense it makes... The rst ight after I bought it, I took some friends to Palm Springs for lunch." (Oslter, Scott, \Andre Even Flies like a Champ," San Francisco Chronicle, February 8, 1993, C1.) Discuss whether Agassi?s analysis is reasonable. It depends on how you interpret the statement. If Andre?s decision of how many ights to take is made after the purchase of the airplane, it should not matter what he paid for the airplane because that cost was sunk and the decision should merely be based on the marginal costs and bene ts of ights. If, however, Andre was deciding how many ights he would take prior to the airplane purchase, it would make sense for him to include the xed cost of the plane and compute the average cost per ight that would include that one-time expenditure. 7.6 Suppose that your rm?s production function has constant returns to scale. What is the long-run expansion path? Let?s assume the production function is Cobb-Douglas: q = KaL1 a. The solution to the cost minimization problem gives us the condition MRTS = wr 1 a a K L = w r K = a1 awr L So in a graph with L on the horizontal axis and K on the vertical axis, the expansion path is an upward sloping line starting from the origin with slope a1 a wr . 3 Q1 If the production possibilities are given by f(K;L) =pKL, and if w = r = 2, what is the least cost of producing a quantity q = 4? Please, derive the general cost function C(q). Try to use both, the substitution method and the MRTS=Price Ratio. Substitution Method For a given quantity of production q, rms will choose inputs in a cost-minimizing way: min K;L rK + wL s.t q =pKL Using the constraint, we substitute to get L = q2K and rewrite the optimization problem in terms of K only: min K rK + w q 2 K r w q 2 K2 = 0 )K = w r 1 2 q Now plug this back into the constraint to get L = q2K = q2 (wr )12 q = rw 12 q. When r = w = 2 and q = 4, K = L = 4. The cost function is then C( q) = rK + wL = r w r 1 2 q + w r w 1 2 q = (rw)12 q + (rw)12 q = 2(rw)12 q Note that the cost function is linear, implying that Cobb-Douglas production functions result in constant marginal cost technology. 4 MRTS Method MRTS = MPLMP K = 1 2K 1 2L 1 2 1 2K 12L12 = KL = wr )K = wr L Now use the the fact that we must produce quantity q to get q = K12L12 = w r L 1 2 L12 = w r 1 2 L )L = r w 1 2 q Plug this back into the constraint to get K = wr L = wr rw 12 q = wr 12 q. This is the same answer as above. Multiple Choice M1) Economists typically assume that the owners of rms wish to A) produce e ciently. B) maximize sales revenues. C) maximize pro ts. D) All of the above. Answer: C A rm?s optimization problem is usually written as choosing a combination of inputs to maximize pro ts. Answer A is appealing, but not complete because there could be many e cient production combinations that do not maximize pro ts. Maximizing sales revenues is incorrect because it ignores the issue of the cost of production. High revenues combined with high costs may not be very good for the rm. M2) E cient production occurs if a rm A) cannot produce its current level of output with fewer inputs. B) given the quantity of inputs, cannot produce more output. C) maximizes pro t. D) All of the above. Answer: D That A and B are correct comes from the de nition of \e cient production" given in Perlo p. 171. Answer C is correct because any time a rm is maximizing pro ts, it must be producing e ciently. 5 If it were not producing e ciently, it could produce the same amount of output and thus get the same revenue by using fewer inputs and thus lower costs. M3) Which of the following statements best describes a production function? A) the maximum pro t generated from given levels of inputs B) the maximum level of output generated from given levels of inputs C) all levels of output that can be generated from given levels of inputs D) all levels of inputs that could produce a given level of output Answer: B A production function tells us the maximum amount of output obtained from given amounts of input goods. We are not interested in other levels of output given xed input levels because they entail wastefulness in the production process. M4) With respect to production, the short run is best de ned as a time period A) lasting about six months. B) lasting about two years. C) in which all inputs are xed. D) in which at least one input is xed. Answer: D The term \short run" is somewhat subjective, so A and B cannot be true. In the short run, there is at least one input which cannot be adjusted; something, usually capital in our problems, is \ xed". M5) In the long run, all factors of production are A) variable. B) xed. C) materials. D) rented. Answer: A In contrast to the previous question, in the long run all inputs can be adjusted. For instance, there is enough time to even build new factories and buy new machines. M6) The Average Product of Labor is A) the change in total product resulting from an extra unit of labor, holding other factors con- stant. B) the ratio of output to the number of workers used to produce that output. C) the amount of output that can be produced by a given amount of labor. D) equal to the marginal product of labor when the average product is increasing. Answer: B This is de nitional. The average product of labor is the total amount of output produced divided by the amount of labor used to produce that output. It tells us how productive labor is on average. 6 M7) Homer?s Donut Shoppe has the production function q = 10L + 20L2 5L3. The average product of labor is A) AP = 10 + 20L 5L2 B) AP = 10 + 40L 15L2 C) AP=10L D) AP=10 + 20L Answer: A To nd the average product of labor, divide the output by the amount of labor used to produce that output. AP = qL = 10L + 20L 2 5L3 L = 10 + 20L 5L 2 M8) At any given point on the curve, the slope of the total product curve always equals A) the ratio of the marginal product and the average product. B) the change in input divided by the change in output. C) the average product of the input. D) the marginal product of the input. Answer: D With capital xed, the slope of the total product of labor curve gives the marginal product of labor. It tells us how much output we get from the last bit of labor used. M9) Which situation is most likely to exhibit diminishing marginal returns to labor? A) a factory that obtains a new machine for every new worker hired B) a factory that hires more workers and never increases the amount of machinery C) a factory that increases the amount of machinery and holds the number of worker constant D) None of these situations will result in diminishing marginal returns to labor. Answer: B Diminishing marginal returns means that, as we add additional units of labor, production increases by by smaller and smaller amounts. If there is a xed amount of machinery and more and more workers are hired, they have to share the same amount of machinery. It is then likely that these additional workers add less and less to total productivity. At some point it is possible that hiring more workers will lead to lower productivity if the shop oor becomes so crowded that workers are interfering in each others? jobs. M10) Thomas Malthus? prediction of mass starvation resulting from diminishing marginal re- turns has not been ful lled because A) the law of diminishing marginal returns did not hold in this case. B) Malthus ignored other factors like technological change. C) relative to Malthus? day, larger percentage of today?s labor works in the agricultural sector. D) All of the above. 7 Answer: B See the Application box on pp. 178-80 of Perlo . In fact, marginal returns to labor in the agricul- tural sector have increased dramatically since Malthus. M11) At Joey?s Lawncutting Service, a lawn mower cannot cut grass without a laborer. A laborer cannot cut grass without a lawn mower. Which graph in the above gure best represents the isoquants for Joey?s Lawncutting Service when capital per day is on the vertical axis and labor per day is on the horizontal axis? A) Graph A B) Graph B C) Graph C D) Graph D Answer: A An isoquant shows all combinations of capital and labor which yield the same output. Graph A shows a situation where increasing labor while holding capital xed (or vice versa) does not lead to higher output. This is untrue of all the other graphs. M12) An isoquant represents levels of capital and labor that A) have constant marginal productivity. B) yield the same level of output. C) incur the same total cost. D) All of the above. Answer: B This is simply the de nition of isoquant. M13) Isoquants that are downward-sloping straight lines imply that the inputs A) are perfect substitutes. B) are imperfect substitutes. C) cannot be used together. D) must be used together in a certain proportion. Answer: A This is similar to indi erence curves for a consumer with perfect substitute preferences. These isoquants represent a linear production function. The rm can trade capital for labor in a xed ratio at any point and continue to produce the same amount of output. M14) The slope of an isoquant tells us A) how much output increases when both inputs are increased. B) the increase in MPL when capital increases. C) the decrease in capital necessary to keep output constant when labor increases by one unit. 8 D) the decrease in capital necessary to keep MPL constant when labor increases by one unit. Answer: C From Perlo : the slope of an isoquant shows the ability of a rm to replace one input with another while holding output constant. M15) Returns to scale refers to the change in output when A) all inputs increase proportionately. B) labor increases holding all other inputs xed. C) capital equipment is doubled. D) specialization improves. Answer: A This is de nitional. If doubling all inputs leads to more than double the level of output, this might lead us to believe that the industry will be dominated by a few large companies. In contrast, if doubling all inputs leads to a smaller increase in output then we might expect there to be many small rms competing in the industry. M16) Decreasing returns to scale may occur as increasing the amount of inputs used A) increases specialization. B) always increases the amount of output produced. C) may cause coordination di culties. D) increases e ciency. Answer: C The only answer which makes sense is C. If specialization increases, returns would probably be increasing. If e ciency increases, returns would be increasing. lastly, answer B is the de nition of increasing returns to scale. M17) The above gure shows the isoquants for producing steel. Increasing returns to scale are A) present when producing less than 10,000 tons. B) present when producing less than 20,000 tons. C) present when producing less than 30,000 tons. D) never present. Answer: A When capital and labor increase from 1.5 to 2 each (a percentage change of 33%), output increases from 5,000 to 10,000 (a percentage change of 100%). So we have increasing returns to scale at low output levels. M18) A rm?s marginal cost can always be thought of as the change in total cost if A) the rm produces one more unit of output. B) the rm buys one more unit of capital. C) the rm?s average cost increases by $1. 9 D) the rm moves to the next highest isoquant. Answer: A The marginal cost is the amount by which a rm?s cost changes if the rm produces one more unit of output. It is the rate of change of cost with respect to output. M19) Fixed costs are A) a production expense that does not vary with output. B) a production expense that changes with the quantity of output produced. C) equal to total cost divided by the units of output produced. D) the amount by which a rm?s cost changes if the rm produces one more unit of output. Answer: A This is the de nition of xed cost. Whether 1 unit or 1 million are produced, the xed costs remain the same. An example might be rent payments which are made to a landlord whether or not any production takes place that month. Note that this example focuses on the short run because in the long run the rm could decide not to renew it?s lease. M20) Variable costs are A) a production expense that does not vary with output. B) a production expense that changes with the quantity of output produced. C) equal to total cost divided by the units of output produced. D) the amount by which a rm?s cost changes if the rm produces one more unit of output. Answer: B Again, this is de nitional. An example of variable costs are wage payments (assuming workers can be hired and red). M21) If average cost is decreasing A) Marginal cost equals average cost. B) Marginal cost exceeds average cost. C) Marginal cost is less average cost. D) Not enough information Answer: C When marginal cost is lower than average cost, the cost of producing 1 more unit is less than the average cost of producing all the other units. This means producing this last unit would lower average cost, which means the average cost curve is decreasing. M22) If the average cost of producing a good is increasing as a rm produces more of the good, then which of the following must be TRUE? A) AFC is falling. B) AVC is rising. C) MC > AVC. 10 D) All of the above. Answer: D If the average cost curve is increasing, then the cost of producing the last unit must be greater than the average variable cost curve because the MC curve does not include the xed costs. AFC is falling because xed costs are a smaller proportion of output when output grows. Lastly, since AC=AFC + AVC and we know AC is increasing but AFC is falling, AVC must be increasing. M23) Which of the following will cause the marginal cost curve of making cigarettes to shift? A) A $5 million penalty charged to each cigarette maker. B) A $1 per pack tax on cigarettes. C) A $1 million advertising campaign by the American Cancer Society. D) All of the above. Answer: B If the tax is levied on producers, then each time a pack of cigarettes is sold $1 must be given to the government. This is like increasing the cost of production by $1. Answer A describes a xed cost, and answer C describes something that would probably shift demand. M24) Which of the following will cause the average cost curve of making cigarettes to shift? A) A $5 million penalty charged to each cigarette maker. B) A $1 per pack tax on cigarettes. C) A $1 an hour wage increase paid to all cigarette production workers. D) All of the above. Answer: D The large penalty a ects xed costs. As above, the tax a ects marginal costs. The wage increase a ects variable costs. All of these are components of total costs which is the numerator of the average cost curve. M25) In the long run, xed costs are A) sunk. B) avoidable. C) larger than in the short run. D) not included in production decisions. Answer: B The long run is de ned as the length of time in which all factors of production are avoidable. Even factors which entail xed costs can be avoided by switching to other methods of production. M26) When the isocost line is tangent to the isoquant, then A) MRTS = w/r. B) the rm is producing that level of output at minimum cost. C) the last dollar spent on capital yields as much extra output as the last dollar spent on labor. 11 D) All of the above. Answer: D This is true if isoquants are convex. Answer A holds because the slope of the isoquant is the MRTS and the slope of the isocost line is w/r. Answer B can be seen by picking a point on the isoquant which is not tangent to original isocost line, drawing the new isocost line through it, and noting that this line is further from the origin than the original isocost line. Answer C is a re-interpretation of the MRTS = w/r condition. M27) A rm can minimize cost by A) picking the bundle of inputs where the lowest isocost line touches the isoquant. B) picking the bundle of inputs where the isoquant is tangent to the isocost line. C) picking the bundle of inputs where the last dollar spent on one input gives as much extra output as the last dollar spent on any other input. D) All of the above. Answer: D This is true if isoquants are convex. M28) Suppose that each worker must use only one shovel to dig a trench, and shovels are useless by themselves. In the long run, the rm?s cost function is A) TC = (w/r) * q. B) TC = (w + r)/q. C) TC = (w + r). D) TC = (w + r) * q. Answer: D Each worker costs w in wages and each shovel costs r in rent paid to capital. The production technology exhibits perfect compliments, so to produce 1 trench 1 worker and 1 shovel are needed. Answer D shows the cost of producing q trenches. M29) The total cost of producing one unit is $50. The total cost of producing two units is $75. At a production level of two units, the cost function exhibits A) economies of scale. B) rising average costs. C) increasing marginal costs. D) constant returns to scale. Answer: A A cost function exhibits economies of scale if the average cost of production falls as output increases. In this example, the average cost of production falls from $50 to $37.5. M30) Long-run average cost is never greater than short-run average cost because in the long run A) capital costs equal zero. B) the rm can move to the lowest possible isocost curve. 12 C) wages always increase over time. D) wages always decrease over time. Answer: B One way to see this is to notice that any input-output combination which is possible in the short run is also possible in the long run. Therefore, the average cost of producing any output level cannot be higher in the long run. The rm could always choose the same combination of inputs and achieve the same average cost. Sometimes it might be possible in the long run to alter production technology in some way and achieve an even lower average cost. 13

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