MICROECONOMICS II - REVIEW QUESTIONS I

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2 7. Isoquants can be convex, linear, or -shaped. What does each of these shapes tell you about the nature of the production function? What does each of these shapes tell you about the MRTS? Convex isoquants imply that within some range, some number of units of one input can be substituted for a unit of the other input, and output can be maintained at the same level. In this case, the MRTS is diminishing as we move down along the isoquant. inear isoquants imply that the slope, or the MRTS, is constant. This means that the same number of units of one input can always be exchanged for a unit of the other input and output can be maintained. The inputs are perfect substitutes. -shaped isoquants imply that the inputs are perfect complements, or that the firm is producing under a fixed proportions type of technology. In this case the firm cannot give up one input in exchange for the other and still maintain the same level of output. For example, the firm may require exactly 4 units of capital for each unit of labor, in which case one input cannot be substituted for the other. 8. Can an isoquant ever slope upwards? Explain. No. This would mean that if you increased both inputs then output would stay the same. As a general rule, if the firm has more of all the inputs they can produce more output. 9. Explain the term marginal rate of technical substitution? What does a MRTS=4 mean? MRTS is the amount by which the quantity of one input can be reduced when the other input is increased by one unit, while maintaining the same level of output. If the MRTS is 4 then the one input can be reduced by 4 units as the other is increased by one unit and output will be the same. 0. Explain why the marginal rate of technical substitution is likely to diminish as more and more labor is substituted for capital. As the quantities of the inputs are changed the marginal product of each input will change. As more and more labor is added, the marginal product of labor is likely to diminish. Because capital has been reduced, each unit of capital remaining is likely to be more productive. Therefore, more units of labor will be required to replace each unit of capital. Alternatively, as we move down and to the right along an isoquant along which the MRTS is diminishing, we have to give up less capital for each unit of labor added to keep output constant.. Is it possible to have diminishing returns to a single factor of production and constant returns to scale at the same time? Discuss. Diminishing returns to a single factor are observable in all production processes at some level of inputs. This fact is so pervasive that economists have named it the law of diminishing marginal productivity. By definition, the marginal product of an input is the additional output generated by employing one more unit of the input, all other inputs held fixed. The extra output, or returns, to the single input diminish because all other inputs are held fixed. For example, when holding the level of capital constant, each additional unit of labor has less capital to work with. Unlike the returns to a single factor, returns to scale are proportional increases in all inputs. While each factor by itself exhibits diminishing returns, output may more than double, less than double, or exactly double when all the inputs are doubled. The distinction again is that with returns to scale, all inputs are increased in the same proportion and no input is held fixed.. Can a firm have a production function that exhibits increasing returns to scale, constant returns to scale, and decreasing returns to scale as output increases? Discuss. Most firms have production functions that exhibit first increasing, then constant, and ultimately decreasing returns to scale. At low levels of output, a proportional increase in all inputs may lead to a larger-than-proportional increase in output, based on an increase in the opportunity for each factor to specialize. For example, if there are now two people and two computers, each person can specialize by completing those tasks that they are best at, which allows output to more than double. As the firm grows, the opportunities for specialization may diminish and a doubling of all inputs will lead to only a doubling of output. When there are constant returns to scale, the firm is replicating what it is already doing. At some level of production, the firm will be so large that when inputs are doubled, output will less than double, a situation that can arise from management diseconomies.

3 3. Suppose that output q is a function of a single input, labor (). Describe the returns to scale associated with each of the following production functions: a. q=/ b. q = c. q = log() 4. Fill in the gaps in the table below. Quantity of Variable Input Total Output Marginal Product of Variable Input Average Product of Variable Input The marginal product of labor in the production of computer chips is 50 chips per hour. The marginal rate of technical substitution of hours of labor for hours of machine-capital is /4. What is the marginal product of capital? The marginal rate of technical substitution is defined at the ratio of the two marginal products. Here, we are given the marginal product of labor and the marginal rate of technical substitution. To determine the marginal product of capital, substitute the given values for the marginal product of labor and the marginal rate of technical substitution into the following formula: MP MP MP = 00 computer chips per hour. 50 = MRTS, or = MP 4, or 6. Do the following functions exhibit increasing, constant, or decreasing returns to scale? What happens to the marginal product of each individual factor as that factor is increased, and the other factor is held constant? a. q= 3+ This function exhibits constant returns to scale. For example, if is and is then q is 0. If is 4 and is 4 then q is 0. When the inputs are doubled, output will double. Each marginal product is constant for this production function. When increases by q will increase by 3. When increases by q will increase by. b. q= (+ ) This function exhibits decreasing returns to scale. For example, if is and is then q is.8. If is 4 and is 4 then q is 4. When the inputs are doubled, output will less than double. 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 q = (+ ) Since is in the denominator, as gets bigger, the marginal product gets smaller. If you do not know calculus, then you can choose several values for, find q (for some fixed value of ), and then find the marginal product. For example, if =4 and =4 then q=4. If =5 and =4 then q=4.4. If =6 and =4 then q= Marginal product of labor falls from 0.4 to 0.3..

4 c. q= 3 This function exhibits increasing returns to scale. For example, if is and is then q is 4. If is 4 and is 4 then q is 9. When the inputs are doubled, output will more than double. Notice also that if we increase each input by the same factor λ then we get the following: q'= 3(λ)(λ) =λ 3 3 =λ 3 q. Since λ is raised to a power greater than, 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, when is increased by unit, q will go up by 3 units, which is a constant number. Using calculus, the marginal product of capital is MP=*3**. As increases, MP will increase. If you do not know calculus then you can fix the value of, choose a starting value for, and find q. Now increase by 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. q= This function exhibits constant returns to scale. For example, if is and is then q is. If is 4 and is 4 then q is 4. When the inputs are doubled, output will exactly double. Notice also that if we increase each input by the same factor λ then we get the following: q'= (λ) (λ) =λ =λq. Since λ is raised to the power, 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 MP= For any given value of, as increases, MP will increase. If you do not know calculus then you can fix the value of, choose a starting value for, and find q. et =4 for example. If is 4 then q is 4, if is 5 then q is 4.47, and if is 6 then q is The marginal product of the 5 th unit of is =0.47, and the marginal product of the 6 th unit of is =0.4. Hence we have diminishing marginal product of capital. You can do the same thing for the marginal product of labor. e. q=4 + 4 This function exhibits decreasing returns to scale. For example, if is and is then q is If is 4 and is 4 then q is 4. When the inputs are doubled, output will less than double. The marginal product of labor is decreasing and the marginal product of capital is constant. For any given value of, when is increased by unit, q will go up by 4 units, which is a constant number. To see that the marginal product of labor is decreasing, fix = and choose values for. If = then q=8, if = then q=9.65, and if =3 then q=0.93. The marginal product of the second unit of labor is =.65 and the marginal product of the third unit of labor is =.8. Marginal product of labor is diminishing..

5 7. Graph the short-run total product curves for each of the following production functions if is fixed at 0=4. a. b. Q Q Q=6 Q= Do the two production functions obey the law of diminishing returns? The production function in (a) has a constant marginal product of labor for all levels of labor input, and the production function in (b) an increasing marginal product of labor for all levels of labor input. So neither production function obeys the law of diminishing returns. 9. True or false: If the marginal product is decreasing, then the average product must also be decreasing. Explain. 0. A factory adds a worker and subsequently discovers that the average product of its workers has risen. True or false: The marginal product of the new worker is less than the average product of the plant s workers before the new employee s arrival.

6. The law of diminishing marginal returns begins to take effect at labor input level: a. 0 b. X c. Y d. Z

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