Production Functions

In this lesson, we are going to talk about the supply curve and, if you will allow me to jump to the end for a second, the supply curve is based upon costs. So, if we want to try to figure out just what kind of prices producers charge, and how much (or how little) they can accept in exchange for their goods and stay in business, we need to think about what exactly creates the costs of doing business for a supplier (or seller or producer - remember, they are synonymous, the same thing.)

From here on, when talking about suppliers, I will use the term "firm," which is basically a form of shorthand for "an entity that makes things (or provides services) and sells them with the expectation of making a profit." This is what most firms are. As always, we can add a lot of fine print about exceptions and variations, but that does not help us with the illustration of basic concepts. It's better for all of us if we keep things simple. Of course, a firm could consist of an individual person, or an international mega-corporation with a workforce and budget that dwarfs several Eastern European countries, but they have the same basic goal: make things and sell them for enough money to be able to stay in business for another day, and pay the owners of the firm enough to compensate them for their investment.

A firm that does not turn a profit can only stay in business for as long as its owners can pump money into it, which is usually something like "not long." Firms exist to make a profit. More on just what a "profit" is later.

So, what do firms do to make a profit?

A firm takes things and converts them into products. Inputs are converted into outputs.

The inputs that go into making a product can form a list that can be almost endless. For example, if I want to do something as simple as sell orange juice, I need all of the following:

Oranges, water, machines to crush the oranges, electricity to run these machines, people to operate the machines, people to fix the machines, a building to house the machines, natural gas to heat the building in the winter, trucks to haul the orange juice to stores, gasoline to power those trucks, people to drive them, lawyers to make sure the company meets all regulations, paper to store records on, computers to process billing records, people to run those computers, furniture for those people to sit on, health care insurance for all those people, telephones to allow them to communicate, creative people to make advertisements for the orange juice, machines that make bottles out of plastic, plastic for the bottles, waste disposal systems for the factory and offices, cardboard for boxes for the plastic bottles of juice, glue to close those boxes, warehouses to store the boxes before they get loaded on trucks, and so on. This is not even considering the needs of the firms that are selling the juice to the end-user (that is, stores or restaurants), and it is not considering the needs of the firms that make the trucks or machinery, or provide the health care, or get the natural gas and crude oil out of the ground.

My point here is that analyzing the economics of running a business can be very complicated, and, as you may have noticed, economists like to simplify things. It makes the world an easier place to try to figure out, and lets us sleep at night.

So, to simplify things, we take the many and varied inputs into some production process, and divide them into a few simplified classes of inputs, for example:

• Labor (L): simply put, people.
• Materials (M): physical things that get used up to make products.
• Energy (E): electricity, petrol, and so on.
• Capital (K): machines and buildings (and may also include land). Does not get “used up” in production.

These four things are abbreviated L, M, E and K. The things produced, the outputs, are labeled Y. Some people use only K and L. Others use K, L and M. Since this course is part of a degree program focusing on energy use, it is important for us to view energy as a separate class of inputs, as we are often concerned about how we can maintain or maximize output while consuming less energy. It might be tempting to lump energy in with other "consumables" under the category of "materials" (M). However, in our orange juice example, we have physical materials like oranges and bottles and boxes, and we have "intangible" things we consume that are not an intrinsic, physical part of the final product. Mostly, this describes energy; we require energy to move machine parts or to add heat or to drive chemical reactions, but we do not sell the product based upon how much energy is in it. For this reason, it can be useful for us to break energy out into a separate class of input. This enables the entire field of "energy efficiency."

The relationship between the input and the outputs is called a “Production Function.”  Written in math terms, the production function is:

$$\mathrm{Y}=f(\mathrm{K},\mathrm{L},\mathrm{E},\mathrm{M})$$

The inputs K, L, E and M are called “factors of production.” The production function tells us the maximum amount of output (Y) that can be produced from a certain quantity of factors.

Returns to a Factor

The phrase “Ceteris Paribus” means “if nothing else changes,” or “holding everything else constant.”

The marginal product of a factor of production is written as delta Y/ delta F (where F is the factor in question). This tells us, ceteris paribus, how much the amount produced changes if we change the amount of one input. There is a general rule: if we hold everything else constant, but increase only factor i, then the increase in Y will get smaller with each additional unit of i employed. This is called the Law of Diminishing Returns. This is a bit like the idea of diminishing marginal utility.

For example, let us look at the following table, which lists output versus labor for a raspberry farm. The "marginal output" column refers to the extra amount of berries harvested by each additional worker, which can be written as delta Y/delta L, where Y = pounds of berries and L = number of workers.

Returns to a Factor

Number of workers	Total output, pounds of berries/day	Marginal output, pounds of berries/day
0	0	-
1	50	50
2	90	40
3	120	30
4	140	20
5	150	10

We can see that the amount of berries harvested increases with each extra worker, but the increase gets smaller and smaller for every extra worker.

Now, let’s expand the table to include prices for berries and for labor. Let’s say that a pound of berries can be sold in the farmer's market for $5, and a berry picker costs $60 per day to employ. Defining a couple of terms, revenue refers to the amount of money a firm brings in from selling things in the market. If the firm is selling goods that all have the same price, then revenue is simply the price for each good times the number of goods sold, or P x Q. Remember that "marginal" refers to "the extra amount from doing one more thing," so marginal revenue refers to the extra amount of revenue that is obtained from selling one more thing, or in this case, the amount of extra revenue that comes from adding an extra berry picker. The total cost of labor is how much the firms owner has to pay to the workers. If each worker makes the same amount, it is simply (P x Q), where P = wages = price of a worker, and Q = the quantity of workers employed. The marginal cost of labor is the additional cost of employing an additional worker.

Marginal Cost of Labor

# of Workers	Total output, pounds	Total revenue, (P x Q), dollars	Marginal revenue, dollars	Total labor cost, dollars	Marginal labor cost, dollars
0	0	0	-	-	-
1	50	250	250	60	60
2	90	450	200	120	60
3	120	600	150	180	60
4	140	700	100	240	60
5	150	750	50	300	60

So, we can compare the marginal revenue from labor to the marginal cost of labor. It pays to keep adding labor until we have 4 workers because the marginal cost of workers 1 through 4 is less than the marginal revenue that each extra worker generates. Each worker more than pays for himself. However, adding the 5^th worker does not make sense because he costs the farm more in wages ($60) than the farm can get from the fruits of his/her labor. (Sorry...)

So, if the farm has 4 workers and would like to expand production, it should probably spend its next dollar on something other than extra labor. It needs to invest in some other factor of production, such as capital, in the form of more land, or machines that can pick more berries, or give its workers a device that allows each one to pick more berries in a day.

This is the reason supply curves tend to slope upwards: the productivity from a factor of production decreases as its use increases. As I said before, this is the Law of Diminishing Returns. This happens for two reasons: the amount of output from each additional unit of a factor may get smaller, or the cost of an additional unit of input gets higher. The second effect happens when we have a situation of full-employment in a country: if everybody has a job, and a firm wants to add workers, then it has to lure workers away from other industries, and the only way that this can be done is to offer higher wages. So, returns decline because of lower productivity and higher costs.

Now it is your turn.

Assume the price of blueberries is 6 and the price of labor is 100. Fill in the table. Here, profits = total revenue minus total labor costs. What is the profit maximizing amount of labor to hire?

Output, revenue, and labor cost for a raspberry farm.

# of Workers	Total output, pounds	Total revenue (P x Q), dollars	Marginal revenue, dollars	Total labor costs, dollars	Marginal labor cost, dollars	Profits
0	0
1	70
2	130
3	180
4	220
5	250
6	260

Click here for answer

Output, revenue, and labor cost for a raspberry farm.

# of Workers	Total output, pounds	Total revenue (P x Q), dollars	Marginal revenue, dollars	Total labor costs, dollars	Marginal labor cost, dollars	Profits
0	0	0	-	-	-	0
1	70	420	420	100	100	320
2	130	780	360	200	100	580
3	180	1080	300	300	100	780
4	220	1320	240	400	100	920
5	250	1500	180	500	100	1000
6	260	1560	60	600	100	960

You can see that total profits at maximized at 1000, with 5 workers.

Equimarginal Principle

So, what is the perfect mix of capital, labor, materials and energy? Let us define something called the "marginal return to a factor," which is just a fancy way of saying "How much more money do we make from investing in another unit of some factor of production?" We want the marginal revenue (MR) from employing an extra unit of some factor to be greater than the marginal cost (MC), or as a ratio we want MR/MC to be larger than one. The higher that MR/MC is, the better.

Now, consider that there are four different factors of production: K, L, E, and M. Each one has a marginal return, or a ratio of MR/MC. If one of these factors has a higher MR/MC than any of the other factors, it makes sense to invest in that factor: you get more bang for your buck, as it were. However, given the idea of declining marginal returns, the more we spend on a factor, the lower the return, MR/MC. So in the above example, we have invested in labor until MR/MC got as close to 1 as possible. We want to invest, maybe, in machinery. But as you invest in more machinery, the MR/MC for machines gets closer to 1, which means then that investing in some other factor of production makes sense. Maybe you're starting to spot a pattern here: it makes sense for a firm to spend extra money on whatever factor has the best return, but eventually, all factors will have equal returns, because they will all basically be driven towards 1.

This is called the "equimarginal principle," which merely states that in an efficient firm with sufficient information, factors of production will be employed in some optimal mix such that the marginal return to each one is equal, and as close to 1 as possible. If one factor has a higher return, it makes sense to invest more in it, and if the marginal return is above 1, it makes sense to spend more, because each dollar spent returns more than one dollar in extra revenue.

Repeating myself a little bit, the general rule is that the marginal return to all factors will be the same. This is because if one factor gives you a better return, you will use more of it, and its return will drop to the same level as all other factors. This guides a business as to which factor they should employ more (or less) of, and tells them the ideal (most efficient mix) of factors.

Of course, this is a nice theoretical basis for running a business - just invest in more factors of production until they all have a marginal return equal to 1. If any of you have ever run a business, you know how simplistic this sounds. In real life, figuring out the returns to factors can be very difficult - assigning a benefit to every single thing in even a simple business, like our orange juice one, can be very difficult. How many lawyers do you need? How many trucks? How many drivers? How many shifts of workers? Should you generate your own electricity or buy it? Should you use local oranges or imported ones, and so on. I will repeat my often-mentioned caveats about models being (necessarily) simpler than real life.

Economies of Scale

Sometimes we want to make our businesses bigger. When you do this, you do not add just people, or just machinery, you add all factors. But how much should we add? The question is: Do we benefit from getting bigger?:

• If increasing inputs by x% increases output by  > x%, we have what are called “increasing returns to scale.”
• If increasing input factors by x% increases output by exactly x%, we have constant returns to scale.
• If increasing input factors by x% increases output by less than x%, we have decreasing returns to scale.

Firms generally try to be as big as possible without entering the phase of decreasing returns to scale

Long-term versus Short-term

Many people use the terms "short term" and "long term" quite loosely. In the context of economics, we have a more formal definition.

I will start by returning to our example of the orange juice factory. Let's say that the market for orange juice is growing, and you, as the owner of this factory, would like to make more juice and make more money from selling it. So, what do you do? Well, if you want to make more orange juice, you will obviously need more oranges. If you are currently operating your factory for a single 8-hour shift, Monday to Friday, you will need to add an extra shift, which means hiring more labor. You will need to run the machines longer, which means using more energy. It is relatively easy for you to do all of these things quite quickly: increasing the number of oranges you want to buy, and hiring some more workers, and using a bit more electricity are all things that you can accomplish in a few days.

However, let's imagine that you have expanded output as much as you possibly can from your juice factory. You are now running a 24 hour per day, seven day per week operation; your machines are running every hour at their fullest capacity, and you have all the workers you need to run those machines. What do you need to do if you want to sell more juice? Well, you need to expand your factory, buy more machines, and build new buildings. This is something that takes a significantly longer period of time. You might be able to hire staff and buy oranges easily, but building a factory and buying industrial equipment all have long lead times.

Think of the above two paragraphs in terms of factors of production. In the first case, where we were ramping up production in the existing factory, we were adding materials (M), labor (L) and energy (E). These could all be done easily and quickly. In the second case, we had to add buildings and machines, which fall into the category of capital (K). This takes a lot longer.

This is the crucial distinction between "short term"and "long term." In the short term, we can change three of our factors of production: materials, labor and energy, but we are stuck with the capital that we have. In the long term, we can build new factories. So, in the short term, L, E and M are "variable" factors, meaning that they can be changed, but K is fixed, or constant, meaning that it cannot be changed. In the long term, L, E, and M are, of course, variable, but so is K.

Summarizing:

• In the short term, capital is assumed to be constant, all other factors are variable.
• In the long term, all factors are assumed to be variable.
• In other words, the difference between “short-term” and “long-term” is the time required to change a firm’s capital.

The actual length of time that defines short-term versus long-term can be very different in different industries. If I am in the business of selling newspapers on street corners, my only capital is a rack to stand the newspapers on, and if business is very good and I need to expand to another corner, all I need to do is buy another newspaper rack, which can be done by going to a shelf shop. The transition between short- and long-term in this case is very short. At the opposite extreme, perhaps, is the nuclear power industry. If you want to build a new nuclear power plant, you are looking at a minimum time-frame of about 7-8 years from the beginning of planning to the start-up, in a best-case scenario. In other areas, we can look at the New York subway system, which is just now building the second Avenue line, which was initially proposed in the 1930s, although that might have something to do with the way governments operate... :-) More on that later in the course.

Try to think of the definition of long-term in a few different businesses. In a small accounting business the capital consists of computers. Adding more capital takes a couple of days, at most. If you want to sell pizza, it takes maybe 3 months to build or buy a building and install some ovens. In the auto industry, it might take two years to plan and build a new assembly plant, and it takes from 5 - 10 years for a large new power plant. The point being, short-term and long-term are not defined by some certain length of time, but are specified by how long it takes to add capital in the industry in question.

Take Aways

After working through the material on this page and reading the associated textbook content, you should be able to confidently:

1. explain what a production function is;
2. explain what a factor of production is;
3. define the usual factors of production employed in this course;
4. explain the meaning of the phrase “ceteris paribus”;
5. describe what the marginal product of a factor of production is;
6. describe the law of diminishing returns;
7. explain the concept of diminishing returns to a factor;
8. explain what is meant by “increasing returns to scale,” “decreasing returns to scale,” and “constant returns to scale”;
9. explain what companies should do if faced with increasing or decreasing returns to scale;
10. define the difference between long-term and short-term.