Returns to a Factor

As we learnt in the previous page, the production function can be written as:

$$Y=f\left(K,L,E,M\right)$$

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 paribus¹, 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.

¹The phrase “Ceteris Paribus” means “if nothing else changes,” or “holding everything else constant.”

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.

Table 3.1 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 $P\times Q$ . 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 $\left(P\times Q\right)$ , 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.

Table 3.2 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 why 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.

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 more than 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 a refinery. Let's say that the market for gasoline is growing, and you, as the owner of this refinery, would like to make more gasoline and make more money from selling it. So, what do you do? Well, if you want to make more gasoline, you will obviously need more crude oil. If you are currently operating your refinery 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 refinery process units longer, which means using more energy. It is relatively easy for you to do all of these things quite quickly: increasing the amount of crude 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 refinery. You are now running a 24 hour per day, seven day per week operation; your process units 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 gasoline? Well, you need to expand your refinery, increase the size (or number) of the process units, and build new buildings. This is something that takes a significantly longer period of time. You might be able to hire staff and buy crude easily, but building a new refinery unit 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 refinery, 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.