1. 8. Production functions Econ 494 Spring 2013
For examples, use x1=hours of unskilled workers with shovels, x2=hours of skilled workers with machinery, y=ditches.

2. Agenda Production functions Properties of production functions
Concavity & convexity
Homogeneity
Also see:

3. Production functions
A primary activity of a firm is to convert inputs into outputs. (And then sell the output for a profit).
Example
Inputs ( 𝑥 𝑖 ): land, water, pesticides, machines, labor
Output ( 𝑦 𝑗 ): wheat, corn
For simplicity, inputs are often put into 2 categories:
Capital (𝐾): land, water, pesticides, machines
Labor (𝐿): labor

4. Production function
Economists represent the process of converting inputs to outputs with a production function of the form:
𝑦=𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛
𝑦 is the output
𝑥 𝑖 are inputs
𝑓() is the function that transforms inputs to outputs
𝑦 𝑤ℎ𝑒𝑎𝑡 =𝑓 𝑥 𝑙𝑎𝑛𝑑 , 𝑥 𝑤𝑎𝑡𝑒𝑟 , 𝑥 𝑝𝑒𝑠𝑡𝑖𝑐𝑖𝑑𝑒𝑠 , 𝑥 𝑚𝑎𝑐ℎ𝑖𝑛𝑒𝑟𝑦 , 𝑥 𝑙𝑎𝑏𝑜𝑟

5. Productive efficiency
There are many ways to produce 𝑦 𝑤ℎ𝑒𝑎𝑡 =100 bushels.
Little water, much water conservation technology
Much water, little water conservation
All possible combinations of inputs to outputs are represented by the production function
We assume that firms are efficient in production
The production function y= 𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 provides the maximum amount of output 𝑦 that can be produced for a given set of inputs

6. Properties of production functions
𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is defined only for 𝑥 𝑖 >0
𝑥 𝑖 are non-negative
Straightforward – negative inputs don’t make sense
𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is a single-valued function
𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is continuous and differentiable

7. 𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is a single-valued function
Remember, 𝑓(∙) gives us the maximum output for a given set of inputs.
A single-valued function means that for any given set of inputs, there can only be one level of output
Can’t have this.
If 𝑦=𝑓(𝑥𝑖) yields maximum 𝑦 for a given level 𝑥𝑖, then 𝑦 1 does not make sense.

8. 𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is continuous and differentiable
We rule out these possibilities:
We use the notation 𝑓(∙)∈ 𝐶 (𝑛) to denote that the function 𝑓 is 𝑛 times continuously differentiable over a given region.
We usually assume 𝑓(∙) is at least twice differentiable: 𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 ∈ 𝐶 (𝑛) .

9. Physical product relationships (2 inputs)
Total product
Average product
Marginal product

10. Total product Same as production function
To draw this in 2 dimensions, hold 𝑥 2 constant at 𝑥 (e.g. a fixed amount of land)
Suppose we vary 𝑥2.
How would we illustrate this?

11. Total product: Change 𝑥2

12. Marginal product
Marginal (physical) product of an input (e.g., 𝑥 1 ) is the additional output (𝑦) that can be produced by employing one more unit of that input (𝑥1) while holding all other inputs constant (𝑥2= 𝑥 2 0 ).
Marginal Product is slope of line tangent to Total Product.

13. Average product Average product AP maximized at 𝑥 1 1
Slope of a line from the origin to any point on TP.
AP maximized at 𝑥 1 1

14. Relationship between TP, AP & MP
Average product
Max at 𝑥 1 1
Marginal product
Max at inflection point 𝑥 1 0
MP equals AP at Max AP ( 𝑥 1 1 )
𝑀𝑃=0 at max TP ( 𝑥 1 2 )

15. Shape of production function What should a production fctn. look like?
Positive marginal productivity
Adding another unit of input will increase output
Don’t bother adding more inputs if it’s only going to decrease output.
“The more the merrier…”
Diminishing marginal productivity
Because other inputs are held constant, eventually the ability of an additional unit of input to generate additional output will begin to deteriorate.
“Too many cooks spoil the broth…”
We will be revisiting these conditions.
Now let’s look at the production fctn in 3-D.

16. The Production Surface
Horizontal “slices” are isoquants
Vertical “slices” are the 2-D TP graphs

17. Isoquants or Level Curves
An isoquant is a “horizontal slice” of the production function
Definition: All possible minimum input combinations that will produce a given level of output.
Consider the isoquants to be a view of the production surface from above.
When we hold x2 constant at x20, we get a 2-D total product curve.
When we hold y constant at y0, we get a 2-D isoquant.

18. Slope of isoquants: Rate of technical substitution (RTS)
Marginal rate of technical substitution (RTS or MRS or MRTS)
Shows the rate at which one input ( 𝑥 1 ) can be substituted for another input ( 𝑥 2 ), while holding output constant (𝑦= 𝑦 0 ) along an isoquant.
RTS is the slope of the isoquant evaluated at a particular point

19. Shape of isoquants: Derive RTS
The equation 𝑓 𝑥 1 , 𝑥 2 = 𝑦 0 represents one equation in two unknowns: 𝑥 1 and 𝑥 2 .
We can solve for one unknown in terms of the other: 𝑥 2 = 𝑥 2 ( 𝑥 1 ; 𝑦 0 )
For any 𝑥 1 , the function 𝑥 2 𝑥 1 gives us the value of 𝑥 2 such that the result will always be 𝑦 0 .
Substitute 𝑥 2 = 𝑥 2 𝑥 1 ; 𝑦 0 into 𝑓 𝑥 1 , 𝑥 2 = 𝑦 0 :
𝑓 𝑥 1 , 𝑥 2 𝑥 1 ; 𝑦 0 ≡ 𝑦 0

20. Shape of isoquants: Derive RTS
Slope of isoquant is:
𝑅𝑇𝑆= 𝑑 𝑥 2 𝑑 𝑥 1
From last slide:
𝑓 𝑥 1 , 𝑥 2 𝑥 1 ; 𝑦 0 ≡ 𝑦 0
Differentiate identity wrt 𝑥 1 :

21. Shape of isoquants: RTS – What does it mean?
RTS is the slope of an isoquant (evaluated at a particular point)
RTS represents the ability to substitute one input for another without changing output
Does RTS > 0 make sense?

22. Shape of isoquants: What if RTS > 0 ?
Use of 𝑥 1 beyond a (such as to b) would require more 𝑥 2 to produce 𝑦 0
Not rational
So…for a production function to make economic sense, RTS<0
Intuition: If both marginal products are positive, then more of both inputs (e.g. moving from a to b) must yield an increase in output, not same output.

23. Shape of isoquants: Diminishing RTS
Read Silb §3.5
Diminishing RTS:
Intuition: To get the same 𝑦 0 with less and less 𝑥 2 , you should have to use more and more 𝑥 1 .
Isoquants are convex
Slope from a  c gets less negative
If this were <0, then firms would only use one input. If the marginal benefits of x1 increased as x1 increased, why use x2?
We assert convexity because it’s the only assertion consistent with using multiple inputs.

24. Recall
Recall: A negative definite BH is a sufficient condition for quasi-concavity

25. Express 𝑑 2 𝑥 2 𝑑 𝑥 1 2 in terms of partial derivatives of 𝑓 𝑥 1 , 𝑥 2

27. Convexity of isoquants
Quasi-concavity of the production function and convex isoquants are consistent.
If the isoquants are convex (a sufficient condition), then the production function will be quasi-concave.
OR…
If we want convex isoquants, we need a quasi-concave production function.
Remember that a strictly concave production function (required for profit max problems) is also quasi-concave

28. Recap… Desirable properties for a production function:
Positive marginal product 𝑓 𝑖 >0
Diminishing marginal product 𝑓 𝑖𝑖 <0
Isoquants should have
Negative rate of technical substitution 𝑑 𝑥 2 𝑑 𝑥 1 <0
Diminishing RTS 𝑑 2 𝑥 2 𝑑 𝑥 >0
Diminishing marginal productivity is not even necessary for convex isoquants:
Convexity  how marginal evaluations change holding output constant.
Diminishing marginal product refers to changes in total output  changing levels of output.