Scale and Substitution

Suppose that Josh wants to expand his business and mow more lawns. He could lease another lawnmower. This might enable him to work more steadily, because if one lawnmower runs out of gas or requires maintenance, he can use the other lawnmower. But leasing another lawnmower will not enable Josh to double the number of lawns that he mows.

The fact that doubling the number of lawnmowers will not double the number of lawns one person can mow is an illustration of the law of diminishing returns. When a production process requires many inputs (or "factors"), adding more of one input usually results in a less-than-proportionate increase in output.

Suppose that Josh keeps just one lawnmower, but he tries to double the number of lawns he mows by working longer. He will get tired, and he will find that working eight more hours does not enable him to mow eight more lawns in a day. That is another illustration of the law of diminishing returns.

Constant Returns to Scale

In theory, if you double all inputs in a production process, you should be able to double the output. That is called constant returns to scale.

In practice, businessmen tend to look at scale in terms of increases in some inputs but not in others. If they can double output without having to double all inputs, they say that there are economies of scale. For example, if Josh can double the number of lawns mowed by adding another worker and another lawnmower--but without having to add another pickup truck--then his business has economies of scale.

In general, suppose that you can produce x units of output with a given set of inputs. Now, suppose we double the level of inputs, and ask whether or not we get 2x units of output. We describe the returns to scale as follows:

OutputReturns to Scale
more than 2xeconomies of scale, or increasing returns
exactly 2xconstant returns to scale
less than 2xdiseconomies of scale, or diminishing returns

There is an argument to be made that any business ought to have constant returns to scale if you can identify all inputs and increase them proportionately. However, in practice, there are inputs, such as managerial supervision, that are nearly impossible to increase proportionately. For example, suppose that Josh's inputs consist of four workers (including himself), four lawnmowers, and one pickup truck. If he doubles all of those inputs, then the second pickup truck will have to go to work at a different neighborhood, and it will be more difficult for Josh to supervise the workers. Thus, Josh's business is likely to exhibit diminishing returns once he has to use more than one pickup truck.

Diminishing returns arise when an important factor or input is fixed, in that it cannot be increased along with other factors. Economists believe that just about any business is subject to diminishing returns at some point. However, many businesses have increasing returns at normal levels of output.

Substitution

Suppose that there are two types of lawnmowers--economy and deluxe. With the economy lawnmower, Josh can mow 800 lawns in a season. With the deluxe lawnmower, he can mow 840 lawns in a season. At \$25 a lawn, the deluxe lawnmower is worth \$1000 more for a season. If the economy lawnmower costs \$300 to lease and the deluxe lawnmower costs \$800 to lease, he should go for the deluxe lawnmower. However, if the deluxe lawnmower costs \$1500 to lease, Josh should stick with the economy lawnmower.

The decision of which lawnmower to use is an example of substitution. If the price is right, Josh will substitute the deluxe lawnmower for the economy lawnmower.

Another type of substitution involves capital and labor. Suppose that Josh runs his business with four workers (including himself) and four lawnmowers. What would make him use five workers and three lawnmowers, or vice-versa?

Suppose that a worker needs a lawnmower to be productive, and that lawnmowers sometimes break down. It might be worthwhile for Josh to have spare lawnmowers and fewer workers, so that workers never have to sit idle because of mechanical failure.

Alternatively, suppose that lawnmowers are very costly to lease, but that labor is inexpensive. Josh might want to have more workers than lawnmowers, so that when one worker is taking a break that worker's lawnmower is used by another worker. That way, lawnmowers never sit idle.

Josh's decisions about substitution between capital and labor will depend on two general factors.

1. Technology.

Different combinations of workers and lawnmower will produce different levels of output in terms of lawns mowed. The results of different combinations of labor and capital constitute the technology that is available to Josh.

2. Relative prices.

The wages of workers and the leasing cost of different lawnmowers help determine whether Josh wants to use more or less of either input. A higher hourly wage rate would lead Josh to use less labor and more capital. A higher leasing cost would lead him to use less capital and more labor. The ratio of the leasing cost to the wage rate is called a relative price.

Another important concept is called the elasticity of substitution. In economics, an elasticity is a measure of the amount by which quantities will adjust to a change in price. When the quantity adjustment is large, we say that the relationship is very elastic. When the quantity adjustment is small, we say that it is inelastic, which means not very elastic.

When the elasticity of substitution is high, it means that a small change in relative prices will cause a large change in the inputs used. For example, suppose that there are two brands of lawnmowers of similar quality that cost about the same. A small drop in the leasing cost of one brand would cause everyone with a lawnmowing business to switch to that brand. The elasticity of substitution between similar lawnmowers will be very high.

On the other hand, consider the elasticity of substitution between capital and labor. A small drop in the price of lawnmowers probably is not going to cause Josh to replace two workers with two lawnmowers. The elasticity of substitution between workers and lawnmowers within his business is likely to be low.

Suppose that the technology includes four types of inputs. There are experienced workers, who can mow lawns quickly, and there are inexperienced workers, who are less efficient. There are deluxe lawnmowers, which are fast, and there are regular lawnmowers, which are not so fast. You can use either type of worker with either type of lawnmower.

Experienced workers get paid w2, which is higher than the wage paid to inexperienced workers, w1. Deluxe lawnmowers cost r2 to lease, which is more than the cost of regular lawnmowers, which is r1.

1. What relative price would you use in computing the elasticity of substitution between inexperienced workers and experienced workers?

2. Do you think that the elasticity of substitution between inexperienced workers and experienced workers will be high or low? Explain, using the definition of elasticity.

3. What relative price would use in computing the elasticity of substitution between regular lawnmowers and experienced workers?

4. Do you think that the elasticity of substitution between regular lawnmowers and experienced workers will be high or low? Explain, using the definition of elasticity.

5. Suppose that a crew of 4 inexperienced workers using 4 regular lawnmowers can mow 30 lawns in a day. Suppose that 4 experienced workers using 4 deluxe lawnmowers can mow 35 lawns in a day. What can we say about the number lawns that can be mowed by a crew consisting of three experienced workers and one inexperienced worker, working with two deluxe lawnmowers and two regular lawnmowers? Can we say anything definite about what might be possible using five inexperienced workers and four deluxe lawnmowers?