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Production. In this section we want to explore ideas about production of output from using inputs. We will do so in both a short run context and in a long run context. . Production function.

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### Production

In this section we want to explore ideas about production of output from using inputs. We will do so in both a short run context and in a long run context.

Production function

Here we will assume output is made with the inputs capital and labor. K = amount of capital used and L = amount of labor. The production function is written in general as Q = F(K, L),

where Q = output,and F and the parentheses are general symbols that mean output is a function of capital and labor.

The output, Q, from the production function is the maximum output that can be obtained form the inputs.

In production, we have said that firms have the ability to use both capital and labor.

When you consider the fact that capital is basically the production facility – the building, equipment, machines and the like – you can get the feeling that it is probably less easy to change the capital than it is to change the amount of labor used.

When you look at how long it takes to change the amount of capital in production, during that time when capital can not be changed in amount the time period of production is said to be the SHORT RUN. When all inputs can be changed we are in the LONG RUN.

Example

Say production of units of output follow the function

Q = 2KL. This means output is the multiplication of 2, the units of capital used, and the units of labor used. You can probably envision a table of numbers that puts units of labor across the top, units of capital down the side and inside the table is the output amount.

For example if we went down 1unit of capital and over to 2 units of labor and we would have output Q = 2(1)(2) = 4

Long run

Capital

On a curve we have different combinations of L and K that give the same amount of output. Curves farther out in the northeast direction have more output. Later we will say more about what the firm uses as a guide to choice of position in the graph. The position chosen will have implications for the amount of labor demanded.

Labor

Short Run

Capital

In the short run the firm would have a given amount of capital, say K* here. Production would occur along the dotted line.

K*

Labor

Short run/long run

The notion of a fixed or variable input is related to the time frame of production.

The short run is that period of time when at least one input is fixed in amount.

The long run is that period of time in which all inputs are variable.

As an example of this consider fast food in Wayne. About any store in town could remodel and increase floor space in about 3 months. So after 3 months we have the long run, all inputs can vary - even floor space. But less than three months is the short run because there is only so much floor space to use.

Say Q = 2KL. In short run say K=1.

Then if L = 1, Q = 2 and

if L = 2, Q = 4, and

if L = 3, Q = 6, and so on.

exampleCapital

K=1

Q = 6

Q = 4

Q = 2

Labor

1 2 3 and so on

Short run production function

Typically in the short run we use the graph here instead of the previous one. We put the variable input on the horizontal axis and the output amount on the horizontal axis. Implicitly we have the capital amount fixed at a level when we draw the short run production function.

Units of output Q

Labor amount

Short run production functionexample Q = 2KL

If K = 1 we have the production function Q = 2L. Some points would be if L = 1 Q = 2,

If L=2, Q=4 and so on. The graph is on the left here

Units of output Q

4

2

1 2

Labor amount

Short run Production example Q = sqrt(KL) when K = 4

General short run example

Here is a general short run production function. Notice 1) if labor input =0 output = 0, 2) initially output grows at an increasing rate when labor input rises, then 3) output grows at a decreasing rate (called diminishing returns), and 4) finally more labor may even make output start to decline.

Units of output Q

Labor amount

Malthus and diminishing returns

It has been suggested that enough production processes in the short run exhibit diminishing returns that we should take it seriously.

Malthus argued way back in late 1700’s that because of diminishing returns we would eventually starve to death. The fixed land (which places us in the short run) would eventually not be able to add food production at the rate at which the population increased.

In our model labor is the only variable input. In the real world there are many variable inputs. Technology has increased so much that Malthus has not been proven right, yet. Will he ever be proven right? Only time will tell (but I think NOT!)

General example continued

In the general example, the relationship between the labor used and the total product (TP), or output Q, is called the short run production function. Behind the scenes we assume there is a given amount of capital.

The marginal product of labor (MPL)is the additional output forthcoming from the additional unit of labor.

Note that as the units of labor increases the marginal product first increases, but then begins to diminish after more labor is employed.

example continued

The marginal product curve has the pattern it does because of the way the fixed input is used. Remember that the variable input is used in conjunction with only so much of the fixed input.

In the beginning, as more labor is added, specialization of labor can occur and increasing returns to labor can result, but eventually as more labor is added there will be less of the fixed input to work with and thus additions to output have to diminish.

The way output changes as the variable input is changed, with a given amount of a fixed input, is summarized with the phrase diminishing marginal product.

Example continued

The average product of labor (APL)is for each amount of labor the output produced divided by the labor amount.

The average product mimics, or follows, the marginal product. It is just a math thing.

Next let’s look at some graphs.

Definitions

APL = Q/L

MPL = change in Q / change in L

TP and MPL, APL

APL curve

Notes about MPL and APL

- Note
- When the MPL is above the APL the APL rises.
- When the MPL is below the APL the APL falls.
- The APL continues to rise while the MPL is falling only when the MPL is above the APL.

APL from the graph of TP

I have reproduced the general short run production function, also called the total product, TP, curve. I have also put out a “ray” line through the origin. Note at L1 we get Q1. The APL = Q/L so at the point shown APL1 = Q1/L1.

Also note as you go along the ray line from the origin to the TP the slope of the ray is Q1/L1.

Units of output Q

Q1

L1

Labor amount

So the slopeof the ray from the origin to the TP curve is the APL.

APL from the graph of TP

Note 1) as labor is first increased the ray lines are moving from right to left and since the slopes are getting bigger the APL is rising, 2) at L* the ray is tangent to the TP curve and we can see the APL is at a maximum, and 3) beyond Y* the ray lines have less steep slopes and thus APL is falling.

Units of output Q

Q1

L1

L*

Labor amount

MPL from the graph of TP

The MPL is the change in output divided by the change in labor. The MPL of an amount of labor is really the slope of the TP curve at the level of L used. A way to see the slope is to look at the slope of the tangent line at that point

Units of output Q

Q1

L1

Labor amount

MPL from the graph of TP

Crazy graph, I know. Before L* the tangent lines have slopes that get bigger. But when the curve switches from increasing at an increasing rate to increasing rate at a decreasing rate it is before L*. So MP reaches it peak and begins to diminish before AP has reached its peak. At L* MP = AP.

Units of output Q

Units of output Q

Q1

Q1

L1

L1

L*

Labor amount

Labor amount

Marginal analysis

It has been said economics is a science of marginal analysis. With this in mind, we see later MPL is the more interesting idea.

As an example of this, firms might ask, should another economist be hired?

By the way, when Q = 2KL and if K = 1, for example, we are in the short run. The APL = Q/L = 2L/L = 2, a constant, and MPL=2 as well.

You have to read pages 272-275 for a great example of why we focus more on margins than averages in economics.

Long Run - isoquants

In the long run all inputs can be varied. On the next slide you see what we call an isoquant. Along the curve the amount of output is the same, but we have different combinations of K and L.

Again say Q = 2KL. To get one isoquant you pick a level of output. Say we pick Q = 100. Then we have

100 = 2KL. Since capital is on the vertical axis we might re-express this function as

K = 100/2L = 50/L. If L = 1 K =50 to get Q = 100. If L = 2 K = 25 to get Q = 100. Isoquants have properties similar to indifference curves for consumers.

Marginal Rate of Technical Substitution - MRTS

Capital

Change in K

Change in L

On the next slide I will refer to a change with the use of a triangle.

slope =

Labor

The MRTS = absolute value of the slope.

MRTS

The slope of the curve at a point is

K/ L

Now, if the marginal product of an input is defined as the change in output divided by the change in the input, the slope can be manipulated to be:

K Q and since K = 1

L Q Q MPK

So the slope is MPL/MPK and is called the MRTS (in absolute value) and it is a measure of the rate at which inputs can be substituted and output remains the same.

A few slides back I showed an isoquant. I also put a tangent line at a point on the curve. The slope of a curved line at a point is really the slope of a tangent line at the point.

You will notice that as you move along the curved line from left to right that the slope of a tangent line gets smaller (in absolute value).

Isoquants for perfect substitutes in production will be straight, downward sloping from right to left, lines.

Isoquants for perfect complements are L shaped. The production process is often called a fixed proportion process. An example would be you need a computer and a computer operator in many cases.

LR – returns to scale tangent line at a point on the curve. The slope of a curved line at a point is really the slope of a tangent line at the point.

Remember in short run at least one input is fixed in amount. In LR all inputs can vary.

Returns to scale is idea that if all inputs are increased by a proportionate amount what happens to the amount of output. Since all inputs are changed it is a long run concept.

Example: We might be interested what would happen to output if all inputs were doubled.

It is not a done deal that if all inputs are increased in the same proportion that output will grow by that same proportion. But if it does the we say there are constant returns to scale.

LR – returns to scale tangent line at a point on the curve. The slope of a curved line at a point is really the slope of a tangent line at the point.

If inputs are all increased by a proportionate amount and the resulting output grows by more than this proportion, then increasing returns to scale exists.

If inputs are all increased by a proportionate amount and the resulting output grows by less than this proportion, then decreasing returns to scale exists.

A given production function may exhibit each of these returns to scale at different ranges of output, with increasing returns happening first, then constant and finally decreasing returns happening.

NOTE tangent line at a point on the curve. The slope of a curved line at a point is really the slope of a tangent line at the point.

Returns to scale is a long run concept when changing all inputs in the same proportion.

Diminishing returns is a short run concept when at least one input is fixed in amount and another input is changed in amount.

The two ideas are really not related in any general way.

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