ECONOM 43514351.f2015.pracprob2. Intermediate Microeconomics Economics 4351/7351 Fall 2015 Answers for Practice Problem Set 2, Chapters 6-8

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Chapter 6 8. Do the following functions exhibit increasing, constant, or decreasing returns to scale? What happens to the marginal product of each individual factor as the factor is increased, and t... he other factor is held constant? a. q = 3L + 2K This function exhibits constant returns to scale. When the inputs are doubled, output will double. Show this explicitly by substituting 2L for L and 2K for K. Then demonstrate the analogous result for any multiple  1and positive. Each marginal product is constant for this total product function. When L increases by 1, q will increase by 3. When K increases by 1, q will increase by 2. b. q =(2L + 2K) 2 1 This function exhibits decreasing returns to scale. When the inputs are doubled, output will less than double. Demonstrate this explicitly. Then demonstrate with positive  1 . The marginal product of each input is decreasing. This can be determined using calculus by differentiating the production function with respect to either input, while holding the other input constant. For example, the marginal product of labor is 1 2 2(2 2 ) 2 L K q L     Since L is in the denominator, as L gets bigger, the marginal product gets smaller. If you do not know calculus, then you can choose several values for L, find q (for some fixed value of K), and then find the marginal product. For example, if L=4 and K=4 then q=4. If L=5 and K=4 then q=4.24. If L=6 and K=4 then q=4.47. Marginal product of labor falls from 0.24 to 0.23. c. q = 3LK2 This function exhibits increasing returns to scale. 1 Notice that if we increase each input by the same factor  then we get the following: q3(L)(K)2 33LK 2 3q . Since  is raised to a power greater than 1, we have increasing returns to scale. The marginal product of labor is constant and the marginal product of capital is increasing. For any given value of K , when L is increased by 1 unit, q will go up by 3K 2 units, which, for a given value of K , is a constant number. Using calculus, the marginal product of capital is MPK 23L K 6LK . As K increases, MPK will increase. If you do not know calculus then you can fix the value of L , choose a starting value for K , and find q . Now increase K by 1 unit and find the new q . Do this a few more times and you can calculate marginal product. This was done in part b above, and is done in part d below. d. 2 1 1 2 q L K This function exhibits constant returns to scale. Notice that if we increase each input by the same factor  then we get the following: q L K L K 2 q 1 1 2 1 2 1 2 ( ) ( ) . Since  is raised to the power of 1, we have constant returns to scale. The marginal product of labor is decreasing, and the marginal product of capital is decreasing. Using calculus, the marginal product of capital is 1 2 1 2 2K L MPK  . For any given value of L, as K increases, MPK will increase. If you do not know calculus then you can fix the value of L, choose a starting value for K, and find q. Let L=4 for example. If K is 4 then q is 4, if K is 5 then q is 4.47, and if K is 6 then q is 4.89. The marginal product of the 5th unit of K is 4.47-4=0.47, and the marginal product of the 6th unit of K is 4.89-4.47=0.42. Hence we have diminishing marginal product of capital. You can do the same thing for the marginal product of labor. e. q 4L2 4K 1   This function exhibits decreasing returns to scale. When the inputs are doubled, output will less than double. Demonstrate this explicitly, and then demonstrate the analogous result for all positive  1 . The marginal product of labor is decreasing and the marginal product of capital is constant. For any given value of L, when K is increased by 1 unit, q will go up by 4 units, which is a constant number. To see that the marginal product of labor is decreasing, fix K=1 and choose values for L. If L=1 then q=8, if L=2 then q=9.65, and if L=3 then q=10.93. The marginal product of the second unit of labor is 9.65-8=1.65 and the marginal product of the third unit of labor is 10.93-9.65=1.28. Marginal product of labor is diminishing. 2 11. Suppose life expectancy in years (L) is a function of two inputs, health expenditures (H) and nutrition expenditures (N) in hundreds of dollars per year. The production function is L  cH0.8N0.2. a. Beginning with a health input of $400 per year (H  4) and a nutrition input of $4900 per year (N  49), show that the marginal product of health expenditures and the marginal product of nutrition expenditures are both decreasing. When H  4 and N  49, L  c(40.8)(490.2)  6.602c. Holding N constant at 49, when H  5, L  7.893c, and when H  6, L  9.132c. The marginal product of H drops from 1.291c (7.893c  6.602c) to 1.239c (9.132c  7.893c). Therefore the marginal product of health expenditures is decreasing. Now hold H constant at 4 and increase N to 50. L  c(40.8)(500.2)  6.629c. Increasing N to 51, L  6.655c. The marginal product of N drops from 0.027c (6.629c  6.602c) to 0.026c (6.655c  6.629c), so the marginal product of nutrition expenditures is decreasing. b. Does this production function exhibit increasing, decreasing, or constant returns to scale? There are constant returns to scale. If both inputs are doubled, the new output is L  c(2H)0.8(2N)0.2  cH0.8N0.2(20.8)(20.2)  2cH0.8N0.2  2L. So when both inputs are doubled, life expectancy also doubles. Hence there are constant returns to scale. c. Suppose that in a country suffering from famine, N is fixed at 2 and that c  20. Plot the production function for life expectancy as a function of health expenditures, with L on the vertical axis and H on the horizontal axis. The production function becomes L  20(H0.8)(20.2)  22.974H0.8. To plot this, find the life expectancies for various levels of H, plot those points and draw a smooth curve through them. Here are some points: H  1 and L  22.97; H  2 and L  40.00; H  3 and L  55.38; H  4 and L  69.64; H  5 and L  83.26. This production function is plotted below as a dashed line. d. Now suppose another nation provides food aid to the country suffering from famine so that N increases to 4. Plot the new production function. The production function becomes L  20(H0.8)(40.2)  26.390H0.8. Points to plot are H  1 and L  26.39; H  2 and L  45.95; H  3 and L  63.55; H  4 and L  80.00; H  5 and L  95.63. This production function is plotted below as a solid line. 3 e. Now suppose that N  4 and H  2. You run a charity that can provide either food aid or health aid to this country. Which would provide a greater benefit: increasing H by 1 or N by 1? If N  4 and H  2, life expectancy is 45.95 years as calculated in part d. If N remains at 4 and H increases by 1 (so H  3), life expectancy increases to 63.55 years as shown in part d. On the other hand, if H remains at 2 and N increases by 1 (so N  5), life expectancy rises to 20(20.8)(50.2)  48.05. It is clearly much more beneficial to increase H by 1, because life expectancy increases to 63.55 years. An increase in N by 1 raises life expectancy to only 48.05 years. Chapter 7 3. A firm has a fixed production cost of $5,000 and a constant marginal cost of production of $500 per unit produced. a. | What is the firm’s total cost function? Average cost? The variable cost of producing an additional unit, marginal cost, is constant at $500, so VC=$500q, and   |  | q | . Fixed cost is $5,000 and average fixed cost is | q | . The total $500 $500 q VC AVC | $5,000 qcost function is fixed cost plus variable cost or TC=$5000+$500q. Average total cost is the sum of average variable cost and average fixed cost: q ATC $500  $5000 . b. If the firm wanted to minimize the average total cost, would it choose to be very large or very small? Explain. The firm should choose a very large output because average total cost will continue to decrease as q is increased. As q becomes infinitely large, ATC will equal $500. 4 4. Suppose a firm must pay an annual tax, which is a fixed sum, independent of whether it produces any output. a. | How does this tax affect the firm’s fixed, marginal, and average costs? Total cost, TC, is equal to fixed cost, FC, plus variable cost, VC. Fixed costs do not vary with the quantity of output. Because the franchise fee, FF, is a fixed sum, the firm’s fixed costs increase by this fee. Thus, average costs, equal to | q FC VC | , and average fixed cost, equal to q FC , increase by the averagefranchise fee q FF . Note that the franchise fee does not affect average variable cost. Also, because marginal cost is the change in total cost with the production of an additional unit and because the fee is constant, marginal cost is unchanged. b. Now suppose the firm is charged a tax that is proportional to the number of items it produces. Again, how does this tax affect the firm’s fixed, marginal, and average costs? Let t equal the per unit tax. When a tax is imposed on each unit produced, variable costs increase by tq . Average variable costs increase by t, and because fixed costs are constant, average total costs also increase by t. Further, because total costs increase by t with each additional unit, marginal costs increase by t. Note how different the results are for the “lump sum” tax in part (a) versus the “proportional” tax in part (b). Write out in mathematical form the complete list of cost functions for the two cases to see this very clearly. 8. You manage a plant that mass produces engines by teams of workers using assembly machines. The technology is summarized by the production function. q = 5KL where q is the number of engines per week, K is the number of assembly machines, and L is the number of labor teams. Each assembly machine rents for r=$10,000 per week and each team costs w=$5,000 per week. Engine costs are given by the cost of labor teams and machines, plus $2,000 per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design, i.e. K is fixed at K=5 in the short run. Note: First do this problem, ignoring the raw materials input. Then go over it with the raw materials input. a. What is the cost function for your plant – namely, how much would it cost to produce q engines? What are average and marginal costs for producing q engines? How do average costs vary with output? K is fixed at 5. The short-run production function is therefore q=25L. This implies that for any level of output q, the number of labor teams hired (the “labor requirement function”) will be 25 q | L  | . The totalcost function is thus given by the sum of the costs of capital, labor and raw materials: 5 TC q q q q TC q rK wL q ( ) 50,000 2,200 ) 2,000 25 ( ) 2000 (10,000)(5) (5,000)(         For this linear cost function, the average cost function is given by: . ( ) 50,000 2,200 ( ) q | q and the marginal cost function is given by: ( ) MC q 2200.  TCq TC q AC q      q If you do not know calculus, simply compute TC(q 1)  TC(q) MC(q) . Marginal costs are constant and average costs will decrease as quantity increases (due to the fixed cost of capital). [Show More]

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