Topic 6: Labor Demand 2 part
This topic describes labor demand in the long run. The long run differs from the short-run in that all inputs are variable in the long run. (As noted earlier, the short run is defined as the period of time in which capital cannot be changed.)
Production function
For simplicity, we will assume that a firm produces output using two types of inputs: labor and capital. All of the results that we will derive generalize in a straightforward manner when there are many inputs. The analysis of the multi-factor case, however, requires mathematical tools beyond the scope of this course. For now, we can think of labor as representing all of the inputs that are variable in the short run and capital as representing all of the inputs that are fixed in the short-run. Under this assumption, we can define as firm's short-run production function as:
Q=f(L,K)
where: Q = quantity of output produced L = amount of labor input K = amount of capital input
The production function, f, is a mathematical function that provides the maximum quantity of output that can be produced for each possible combination of inputs used by the firm. A convenient way of representing this production function is through the use of a graph containing isoquant curves. An isoquant curve is a graph of all of the combinations of inputs that result in the production of a given level of output. (Note that the term "iso" means equal in Latin, thus the term "isoquant" literally means "equal quantity").
The diagram below contains a possible isoquant for a firm.
This isoquant suggests that the firm could produce 50 units of output per day using either 20 units of labor and 5 units of capital or 3 units of labor and 15 units of capital. In fact, any combination of labor and capital along this curve allows the firm to produce 50 units of output per day. Note that this curve is downward sloping because the firm can replace workers with machines or replace machines with workers and still manage to produce the same level of output.
In the diagram below, the firm can produce 50 units of output using any of the input combinations given by points: A, B, or C. What happens, though, at point D? Is the firm producing more or less output at point D than at point C. When students are first asked this question, a common response is: "You cannot answer this since point D involves using more capital but less labor than is used at point C." But, what happens if we compare points B and D. At point D, the firm is using more labor and more capital than it is using at point B. If it uses more of each input it can produce more total output (assuming productive efficiency). Thus, we know that this firm can produce more output at point D than at point B. Since the firm produces 50 units of output at points A, B, and C, the output level corresponding to point B is higher than at any of the points on the isoquant. More generally, we can state that any point that lies above and to the right of an isoquant curve corresponds to a higher level of output. Using similar logic, the level of output will be lower if the firm selects a combination of inputs that lies below and to the left of an isoquant (as at point E in this diagram).
An isoquant curve passes through each and every point in this diagram. Two additional isoquant curves have been added to form the diagram below.
The isoquant curves considered above tell us about the physical ways in which inputs can be combined to produce output. Notice that they do not tell us anything about the costs associated with alternative levels of input use. (This topic will be considered below.)
Marginal rate of technical substitution
The marginal rate of technical substitution of L for K (MRTSLK) is defined as the additional amount of capital needed to replace a unit of labor, holding output constant. Mathematically, the MRTSLK can also be expressed as:
Let's examine this expression carefully. The vertical bar at the end of this expression is a mathematical term that indicates that the expression to the left is evaluated given the condition stated at the bottom right of this line. This condition requires that Q = Q-bar (that's how this is pronounced). What this means, more intuitively, is that the expression to the left is evaluated only for points on a given isoquant (one corresponding to an output level of Q-bar). The term in parentheses is nothing more than the negative of the slope of a line connecting two points on the isoquant. Taking the limit of this expression as the change in L tends to zero, however, results in this being the negative of the slope of a tangent line to the isoquant.
The diagram below illustrates how the negative of the slope of the tangent line serves as a measure of the MRTS.
Law of diminishing MRTS
The law of diminishing MRTS states that the MRTS declines as the level of labor use rises along an isoquant. An equivalent way of stating this law is to state that isoquant curves are convex. Let's consider the intuition underlying this law.
This law suggests that it takes a large amount of capital to replace a unit of labor when capital use is high but little labor is used. As labor becomes more plentiful and capital becomes more scarce, however, less capital is required to replace an additional unit of labor. Roughly speaking, the law of diminishing MRTS indicates that it is relatively difficult to replace additional quantities of an input when the level of that input becomes relatively low. This seems to be characteristic of most production processes. Consider, for example, the situation on a farm. When a farm is highly mechanized and has only a small number of workers operating the farm equipment, a very large amount of capital would be required to replace a worker. If a firm, though, has many workers but few tools, the introduction of a small amount of capital (such as a tractor) can replace a relatively large number of workers.
An alternative derivation of the MRTS
Consider the following relationship:
This equation provides us with an approximate relationship between the change in each input and the change in output. An example might help to illustrates this. Suppose that the level of labor increases by 2 units when the marginal product of labor is 5. In this case, we'd expect to see output change by approximately 10 units. Similarly, if the MP of capital is 10, the addition of 3 extra units of output would cause output to increase by approximately 30 units. This relationship holds only approximately because changes in the level of labor or capital use result in changes in the MP of labor and capital (the law of diminishing returns is part of the explanation for this). The error will be small, though, when the changes in L and K are relatively small.
Suppose, we consider two points along an isoquant. Since output is constant (i.e., the change in Q is zero) along an isoquant, the relationship above suggests that:
Manipulating this expression a bit results in:
More precisely, since output was constrained to remain constant, this expression can be written as:
Taking the limit of this equation as the change in L becomes infinitesimally small, this becomes:
Notice that the approximate equality becomes an equality in the limit because the error in the approximation tends to zero as the changes in L and K become infinitesimally small.
A careful reader will note that the left-hand side of the equation above is equal to the definition of the MRTS. This tells us that the MRTS can also be expressed as the ratio of the MP of labor to the MP of capital. We'll make use of this result below.