One of the key behavioral relationships in macroeconomics is the production function, which specifies the relationship between the economy’s productive resources. The production function relates:
- The economy’s capital-labor ratio (how many machines, tools, and structures are available to the average worker), written K/L (K for capital and L for labor).
- The level of technology or efficiency of the labor force, written E.
- The level of real GDP per worker, written Y/L (Y for real GDP or total output and L for the number of workers in the economy).
An economist could write the production function in this general, abstract form:
This formula states that the level of output per worker is some function, F, of K/L and E; that is, the level of output per worker depends in a systematic and predictable way on the capital-labor ratio and the efficiency of labor in the current year. But this abstract form does not specify the particular form of this systematic and predictable relationship.
Alternatively, an economist could write the production function in a particular algebraic form, for example, the Cobb-Douglas form which is convenient to use:
This equation states: “Take the capital-labor ratio, raise it to the exponential power α, and multiply the result by the efficiency of labor E raised to the exponential power (1–α). The result is the level of output per worker that the economy can produce.” While this equation is more particular than the abstract form Y/L = F(K/L, E), it remains flexible: The parameter α could have any of a wide range of different values, and the efficiency of labor E could have any value. This Cobb-Douglas algebraic form still stands for a whole family of possible production functions. The values chosen for E and α will tell us exactly which production function is the real one and thus what the behavioral relationship is between the economy’s resources and its output. Once we know the values of E and α, we can calculate what the output per worker will be for every possible value of capital per worker.
Why do economists choose to write the production function in this particular algebraic form? Ease of use is the key reason. This form of the production function makes a lot of calculations much simpler and more straightforward than other forms.
That is worth explaining enough for us to digress down another level:
Tools: Working with Exponents: What is the point behind the use of the Cobb-Douglas production function, with its exponents?
Recall that exponents greater than 1 are a means of repeated multiplication. Thus 21 is 2 multiplied by itself once, that is, just 2; 22 is 2 times itself twice, that is, 2 × 2 = 4; 24 is 2 times itself four times, that is, 2 × 2 × 2 x 2 = 16.
Recall also that once one has decided that exponents are repeated multiplication, it is natural to decide that an exponent of 1/2 is a way of taking a square root. In fact, it becomes natural to take all fractions between 1 and zero as ways of taking various roots: 21/3 is the cube root of 2, 22/5 is the fifth root of two-squared, and so forth.
Whenever the Cobb-Douglas production function is used, the parameters that are the exponents will be between zero and 1.
In fact, in most applications of this production function the parameter α will be something like 1⁄2. Raising the capital-labor ratio to the α power is something like taking the square root of the capital-labor ratio. So the production function will state that output per worker is proportional to the square root of the capital-labor ratio. This function (a) is easy to calculate or look up for particular cases, (b) is one with which we have a lot of experience, (c) is an increasing function (so it fits the intuitive requirement that more capital is useful), and (d) is a function with diminishing returns—the higher the capital stock, the less valuable is the next investment in expanding the capital stock still further. Thus a lot of features that economists would like a sensible behavioral relationship between the capital-labor ratio and output per worker to have are already built into the Cobb-Douglas production function.
Moreover, there is an additional advantage to using the Cobb-Douglas production function. It makes calculating growth rates easy. Recall from Chapter 2 the rule of thumb for calculating the growth rate of a quantity raised to a power: The propor- tional change of a quantity raised to a power is equal to the proportional change in the quantity times the power to which it is raised.
Output per worker is proportional to the capital-output ratio raised to the power α. Thus if nothing else is changing and if we know the growth rate of the capital- labor ratio, we can immediately calculate the growth rate of output per worker. If α is 1⁄2 and if the capital-labor ratio is growing at 4 percent per year, then output per worker is growing at:
+ + + +
- Edit This File: http://www.typepad.com/site/blogs/6a00e551f08003883400e551f080068834/page/6a00e551f08003883401b8d296df0b970c/edit?saved_added=n
- This File: http://www.bradford-delong.com/a-behavioral-relationship-the-production-function.html
- How to Think Like an Economist—Lecture http://www.bradford-delong.com/how-to-think-like-an-economistlecture.html
- Optional Teaching Topic: How to Think Like an Economist... http://www.bradford-delong.com/how-to-think-like-an-economist.html
- Before Class Begins
- Economics Teaching Master