Production and Costs

February 19, 2012

Production

A production function is the relationship by which inputs are

combined to produce output.

Production requires inputs, such as Labour and Machines. We

will denote labour with L and machines we will call capital and

give the letter K.

Mathematically we will say that the relationship between output

and inputs is:

Q = F (K, L)

Production: Short Run and Long Run

We can look at production in two time scales, short run and long

run.

Long run: shortest period of time required to alter the amounts

of every input.

Short run: period during which one or more inputs cannot be

varied.

An input whose quantity can be altered in the short run is called

a variable input.

One whose quantity cannot be alteredexcept perhaps at

prohibitive costwithin a given time period is called a ﬁxed input.

Production: Short Run

We will now look at a particular production function

Assume that Labour is a variable input

Capital is a ﬁxed input.

So: Q = F (K, L), where K is used to emphasize that capital is

ﬁxed in the SR

Picture might look something like this:

A Production Function

Q

60

Q=F(K,L)

40

10

0

4

8

12

L

Production

Properties of production function

A Production Function: Properties

Q

60

Q=F(K,L)

40

10

0

4

8

12

L

Production Properties

Properties of production function

Through the origin

Increasing inputs increases output

The production function exhibits diminishing returns

Initially adding inputs increases output at an increasing rate

Then adding further inputs increases output at a decreasing rate

Production: TP,AP,MP

Total Product= F (K, L)

Marginal Product is M P L =

increase in input

∂F (K,L)

,

∂L

The Average Product is AP L =

incremental output due to

F (K,L)

L

Production: TP,AP,MP

A Production Function: Total, Marginal and Average Product

Q

60

Q=F(K,L)

40

10

0

4

8

12

L

Production: Long Run

We will now look at long run production where both inputs are

variable

We will see that this looks very similar to consumer optimization

problems

Production: Long Run

The ﬁrst is to describe how diﬀerent input combinations generate

diﬀerent levels of output

In particular, our interest is going to be in understanding how we

can produce a given level of output with diﬀerent input

combinations

To be more concrete consider production function

Q(K, L) = 2KL

We can solve this equation out for a given level of output, say

Q = 16. Then

16 = 2KL ⇒ KL = 8 ⇒ K =

8

L

We can now say which input combinations produce Q = 16. For

example, K = 8, L = 1 or K = 4, L = 2

Production: Long Run

Graphically this is going to look a lot like consumer theory

We are going to draw production function in input space with K

on the vertical axis and L on the horizontal axis

We are then going to draw a curve connecting the diﬀerent input

combinations that generate a speciﬁc output level

Production: Isoquants

An Isoquant

8

4

Q=16

1

2

Production: Long Run

The slope of the isoquant tells us how we can substitute across

diﬀerent inputs to generate the same output

The slope is called the Marginal Rate of Technological

Substitution (MRTS) and is deﬁned as:

M RT S =

dK

dL

Production: Isoquants

An Isoquant: MRTS

ΔK

ΔL

Q=Q0

Production: MRTS

We can do this mathematically as with consumer theory. Let the

production function be F (K, L).

Let us vary all the inputs but keep the output level ﬁxed. Then:

dF (K, L) =

∂F (K, L)

∂F (K, L)

dK +

dL = 0

∂K

∂L

Then we can say that

∂F (K, L)/∂L

dK

=−

∂F (K, L)/∂K

dL

We know that

∂F (K, L)

= MPK

∂K

and equivalently for labour

∂F (K, L)

= MPL

∂L

Production: Returns to Scale

We are also interested in understanding how the size of an

operations aﬀects output.

In particular, we would like to know whether increasing the scale

of production is going to increase output in the same proportion

as the level of inputs.

A production function for which any given proportional change in

all inputs leads to a more than proportional