Intermediate Microeconomics:
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