Chapter 6 production questions
5. What is the difference between a production function and an isoquant?
A production function describes the maximum output that can be achieved with any given combination of inputs. An isoquant identifies all of the different combinations of inputs that can be used to produce one particular level of output.
What is a production function? How does a long-run production function differ from a short-run production function?
A production function represents how inputs are transformed into outputs by a firm. In particular, a production function describes the maximum output that a firm can produce for each specified combination of inputs. In the short run, one or more factors of production cannot be changed, so a short-run production function tells us the maximum output that can be produced with different amounts of the variable inputs, holding fixed inputs constant. In the long-run production function, all inputs are variable.
Explain why the marginal rate of technical substitution is likely to diminish as more and more labor is substituted for capital
As more and more labor is substituted for capital, it becomes increasingly difficult for labor to perform the jobs previously done by capital. Therefore, more units of labor will be required to replace each unit of capital, and the MRTS will diminish. For example, think of employing more and more farm labor while reducing the number of tractor hours used. At first you would stop using tractors for simpler tasks such as driving around the farm to examine and repair fences or to remove rocks and fallen tree limbs from fields. But eventually, as the number or labor hours increased and the number of tractor hours declined, you would have to plant and harvest your crops primarily by hand. This would take huge numbers of additional workers.
Isoquants can be convex, linear, or L-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 indicate that some units of one input can be substituted for a unit of the other input while maintaining output at the same level. In this case, the MRTS is diminishing as we move down along the isoquant. This tells us that it becomes more and more difficult to substitute one input for the other while keeping output unchanged. Linear 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 holding output constant. The inputs are perfect substitutes in this case. L-shaped isoquants imply that the inputs are perfect complements, and 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.
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 and returns to scale are completely different concepts, so it is quite possible to have both diminishing returns to, say, labor and constant returns to scale. Diminishing returns to a single factor occurs because all other inputs are fixed. Thus, as more and more of the variable factor is used, the additions to output eventually become smaller and smaller because there are no increases in the other factors. The concept of returns to scale, on the other hand, deals with the increase in output when all factors are increased by the same proportion. While each factor by itself exhibits diminishing returns, output may more than double, less than double, or exactly double when all the factors are doubled. The distinction again is that with returns to scale, all inputs are increased in the same proportion and no inputs are fixed. The production function in Exercise 10 is an example of a function with diminishing returns to each factor and constant returns to scale.
You are an employer seeking to fill a vacant position on an assembly line. Are you more concerned with the average product of labor or the marginal product of labor for the last person hired? If you observe that your average product is just beginning to decline, should you hire any more workers? What does this situation imply about the marginal product of your last worker hired?
In filling a vacant position, you should be concerned with the marginal product of the last worker hired, because the marginal product measures the effect on output, or total product, of hiring another worker. This in turn determines the additional revenue generated by hiring another worker, which should then be compared to the cost of hiring the additional worker. The point at which the average product begins to decline is the point where average product is equal to marginal product. As more workers are used beyond this point, both average product and marginal product decline. However, marginal product is still positive, so total product continues to increase. Thus, it may still be profitable to hire another worker.
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 one input can be reduced by 4 units as the other is increased by one unit, and output will remain the same.
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.
Many 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, because there are many ways to take advantage of greater specialization as the scale of operation increases. As the firm grows, the opportunities for specialization may diminish, and the firm operates at peak efficiency. If the firm wants to double its output, it must duplicate what it is already doing. So it must double all inputs in order to double its output, and thus there are constant returns to scale. 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.
Can an isoquant ever slope upward? Explain.
No. An upward sloping isoquant would mean that if you increased both inputs output would stay the same. This would occur only if one of the inputs reduced output; sort of like a bad in consumer theory. As a general rule, if the firm has more of all inputs it can produce more output
Why is the marginal product of labor likely to increase initially in the short run as more of the variable input is hired?
The marginal product of labor is likely to increase initially because when there are more workers, each is able to specialize in an aspect of the production process in which he or she is particularly skilled. For example, think of the typical fast food restaurant. If there is only one worker, he will need to prepare the burgers, fries, and sodas, as well as take the orders. Only so many customers can be served in an hour. With two or three workers, each is able to specialize, and the marginal product (number of customers served per hour) is likely to increase as we move from one to two to three workers. Eventually, there will be enough workers and there will be no more gains from specialization. At this point, the marginal product will begin to diminish.
Why does production eventually experience diminishing marginal returns to labor in the short run?
The marginal product of labor will eventually diminish because there will be at least one fixed factor of production, such as capital. As more and more labor is used along with a fixed amount of capital, there is less and less capital for each worker to use, and the productivity of additional workers necessarily declines. Think for example of an office where there are only three computers. As more and more employees try to share the computers, the marginal product of each additional employee will diminish.
Faced with constantly changing conditions, why would a firm ever keep any factors fixed? What criteria determine whether a factor is fixed or variable?
Whether a factor is fixed or variable depends on the time horizon under consideration: all factors are fixed in the very short run while all factors are variable in the long run. As stated in the text, "All fixed inputs in the short run represent outcomes of previous long-run decisions based on estimates of what a firm could profitably produce and sell." Some factors are fixed in the short run, whether the firm likes it or not, simply because it takes time to adjust the levels of those inputs. For example, a lease on a building may legally bind the firm, some employees may have contracts that must be upheld, or construction of a new facility may take a year or more. Recall that the short run is not defined as a specific number of months or years but as that period of time during which some inputs cannot be changed for reasons such as those given above.
Suppose that output q is a function of a single input, labor (L). Describe the returns to scale associated with each of the following production functions:
a. q = L/2. Let q′ be output when labor is doubled to 2L. Then q′ = (2L)/2 = L. Compare q′ to q by dividing q′ by q. This gives us q′/q = L/(L/2) = 2. Therefore when the amount of labor is doubled, output is also doubled. Hence there are constant returns to scale. b. q = L2 + L. Again, let q′ be output when labor is doubled. q′ = (2L)2 + 2L = 4L2 + 2L. Dividing by q yields q′/q = (4L2 + 2L)/(L2 + L) > 2. To see why this ratio is greater than two, note that it would be exactly two if q′ were to equal 2L2 + 2L, but q′ is larger than that, so the ratio is greater than two, indicating increasing returns to scale. c. q = log(L). In this case, q′ = log(2L) = log(2) + log(L), using the rules for logarithms. Then q′/q = [log(2) + log(L)]/log(L) = log(2)/log(L) + 1. This expression is greater than, equal to or less than 2 when L is less than, equal to or greater than 2. So this production function exhibits increasing returns to scale when L < 2, constant returns to scale when L = 2, and decreasing returns to scale when L > 2.
