Let me preface this page by saying that I created my wiki in microsoft word. As I have attempted to make changes, the formatting on this page got really wacky. Nothing that I have done, even attempting to re-type it directly on this page, is working. I hope that it is not too difficult to read.

The Production Function
Production processes and costs are comprised of the production function and the cost function. The basic concept of the production function is the maximum amount of output from a known amount of inputs. The inputs are typically 2 of the factors of production, labor and capital. A mathematical equation has been developed that explains how the factors determine the production output. The basic equation is
Q =F(K,L).
This equation means that Q (output) is a function of K (capital) and L (labor). One of the most widely used production equations is the Cobb-Douglas Production Function. Its equation is as follows:
Y =F(K,L)A * K^a * L^b.
This function adds the effects of technology to the equation. Technology is represented by the “A” in the equation. If the sum of the exponents of “K” and “L” is 1, there are constant returns to scale. Additionally if the sum of the exponents is less than 1, there are diminishing returns to scale and if the sum is greater than 1 there are increasing returns to scale. Returns to scale refers to the effect on the output due to a proportionate change in the inputs (Bell). Increasing returns to scale mean that costs increase less in proportion to output and average costs decline as output increases. Decreasing returns to scale denote that costs increase more in proportion to output and average costs rise as output increases. Constant returns to scale indicate that costs increase/decrease in the same proportions to output.

A vital part of the production function conversation is the distinction between the short run and the long run. The short run is simply defined as a period of time in which there is no entry or exit and at least one of the inputs is fixed. The long run is the period of time in which the all of the inputs can be varied. For example, if a manager of a computer manufacturer is asked to increase production by 50%, there are several factors involved in that. In the short run the variable factors could be labor (working overtime) or purchasing additional raw materials. In the long run, new equipment might be required to be purchased and installed and workers trained in its use. This is something that most likely will take a long time and therefore is considered to be a fixed factor for the short run. In the long run, this equipment would be variable as would all of the factors. The fixed factors remain constant in the short run and can not be changed. Variable factors are able to be manipulated according to the output needs.

Another concept important to short run productivity is the law of diminishing marginal returns. This law states that while increasing one factor of production while holding the other factors fixed, eventually each additional input added will result in less output. For example, consider a factory that employs workers to produce a product. If additional workers are added, holding all other factors of production constant, at some point the added value of the additional worker will start to decrease. Why? There is only so much space in which the employees have to work and since no other factors are changing, e.g. adding more warehouse space, the employees are crowding each other and productivity decreases. Decreased productivity is counter productive to any firm. The goal is to maximize productivity in the most efficient way.

Isoquant curves are also a means to measure efficiency. An isoquant curve shows the most efficient way to combine labor and capital to get a specific output. The slope of the isoquant is equal to the marginal rate of substitution; this is the rate at which giving up labor is exchanged for one unit of capital holding output constant. Isoquant maps are graphs that show varying isoquants with different output levels. Isqoquant maps are often combined with isocost lines to show the big picture. Isocost lines show the combinations of capital and labor that have the same total cost. Below is a graphical representation of isoquants (red) and isocost (black) lines. Just as with indifference curves, isoquants do not intersect. Isoquants that are further to the right represent an increase in output. Isocost lines further to the right represent higher total cost.
isoquant-isocost.jpg

-graph taken from Prof. David Horlach, Middlebury College, Isoquant/Cost Analysis, 2001

The Cost Function
The cost function is the quantity of inputs multiplied by their prices. It can also be expressed as a simple linear equation,
C(x)= mx+b,
where "C" is the cost and "x" is the quantity of units. The cost function can be used to simplify the production process for the manager. It does this by quickly giving the manager necessary information to make decisions about output. There are several types of costs that make up the cost function. These costs are fixed, sunk, variable, marginal, and total. Fixed costs are costs that do not change as output changes. These costs are seen in the short run only. Sunk costs are costs that once paid, can not be recouped. An example of this would be costs associated with research and development. Total costs are the sum of all costs. If total costs are divided by the quantity produced, average cost is the resulting number. Average costs can also be calculated using variable and fixed costs. Variable costs are costs that can be changed as production necessitates.

Another key cost concept is marginal cost. Marginal cost is the cost of producing one more unit of output. An easy way to see the relationships of the costs is through graphic representation. Below is an example of the costs curves and how they relate to one another.
cost_curves.gif
-graph taken from Prof. Larry DeBrock, University of Illinois, Cost Lecture, 2006

The graph shows that the marginal cost curve intersects the average cost curve and the average variable cost curve at their minimum points. A common real world example to explain this has to do with grades. Say you get a "A" on your first test in a class. Your average grade is an “A”. Then your next grade is a “B”, your average will be pulled down. If you get a “C”, your average will then be pulled down more. Eventually your marginal grade will be equal to the average grade, at the lowest grade you have. Then if you start doing better, i.e. get “A”s on every test, your average will continue to increase. Marginal and average costs follow the same logic, thus the shape of the curves (Baye).

Economies of scale are also another important principle to note when discussing cost. Economies of scale refers to the concept that more output can be achieved a lower cost on a larger scale. Firms with large fixed costs (costs sustained even is production is zero e.g. land) of production will generally experience economies of scale. Capital intensive industries, such as automobiles or oil refineries are good examples. Once again a good way to visualize economies of scale is with a graph.

economies_of_scale.gif
-graph taken from tutor2u.net

The long run average cost curve has a U-shape. As output increases, costs fall, and a firm experiences increasing economies of scale. At the minimum point of the curve a firm will experience economies of scale and as costs rise in proportion to the increase in output there are diseconomies of scale.


Multiple Choice Questions
1. A mathematical statement of the way the inputs of the factors of production determine the production output is called:
a. demand curve
b. isoquant
c. production function
d. average revenue curve
The answer is C. By definition the production function is how much output is produced using the inputs of labor, capital, and often technology.
2. The short run is defined as 3-5 months.
a. true
b. false
The answer is B. There is no specific time frame that defines the short run. The short run is when at least one input is fixed and the rest variable.
3. Given the function F(8,5)= 8^.7 * 5^.3 Does this function exhibit returns to scale that are:
a. constant
b. increasing
c. decreasing
d. none of the above
The answer is A. A quick check of the exponents tells us that .7+.3=1, thus there are constant returns to scale.
4. The marginal rate of substitution is the:
a. the rate at which you can keep the same out put and vary inputs
b. 1/ slope of the isoquant
c. fixed costs/ units of output
d. none of the above
The answer is A. By definition the MRS is the rate at which giving up labor is exchanged for one unit of capital holding output constant.
5. In the equation, C(5)= 1.5(5)+ 500 what is the cost to produce one additional unit?
a. 1.5
b. 5
c. 500
d. 515
The answer is 1.5 because the slope (1.5) is equal to marginal cost, which is the cost to produce one more unit of output.

References
Baye, Michael R. Managerial Economics and Business Strategy. New York: McGraw-Hill Irwin, 2006.
Bell, Christopher R. **Economies of, versus Returns to, Scale: A Clarification**
The Journal of Economic Education Vol. 19, No. 4 (Autumn, 1988), pp. 331-335
Brue, Stanley L. **Retrospectives: The Law of Diminishing Returns** The Journal of Economic Perspectives Vol. 7, No. 3 (Summer, 1993), pp. 185-192
http://en.wikipedia.org/wiki/Production_function, retrieved September 10, 2007
http://economics.about.com/cs/economicsglossary/g/cobb_douglas.htm, "Definition of the Cobb-Douglas Production Function", retrieved September 10, 2007
http://www.economicswebinstitute.org/glossary/costs.htm, "Costs: A Key Concept in Economics", Valentino Piana, 2003, retrieved September 13, 2007
http://www.business.uiuc.edu/ldebrock/PTMBAe567/Lectures/Costs/costs.htm, Prof. Larry DeBrock, University of Illinois, Cost Lecture, 2006, retrieved September 13, 2007
http://s01.middlebury.edu/EC155A/Supplements/Lectures/isoquant-graphs.html#isoquant, Prof. David Horlach, Middlebury College, Isoquant/Cost Analysis, 2001, retrieved September 17, 2007
http://www.tutor2u.net/economics/revision-notes/a2-micro-economies-diseconomies-of-scale.html, retrieved September 17, 2007