1. MICROECONOMICS Principles and Analysis Frank Cowell
The Firm: Basics
MICROECONOMICS
Principles and Analysis
Frank Cowell
October 2006

2. Overview... The environment for the basic model of the firm.
The Firm: Basics
The setting
The environment for the basic model of the firm.
Input require-ment sets
Isoquants
Returns to scale
Marginal products

3. The basics of production...
We set out some of the elements needed for an analysis of the firm.
Technical efficiency
Returns to scale
Convexity
Substitutability
Marginal products
This is in the context of a single-output firm...
...and assuming a competitive environment.
First we need the building blocks of a model...

4. Notation zi z = (z1, z2 , ..., zm ) q wi w = (w1, w2 , ..., wm ) p
Quantities zi
amount of input i
z = (z1, z2 , ..., zm )
input vector q
amount of output
For next presentation
Prices wi
price of input i
w = (w1, w2 , ..., wm )
Input-price vector p
price of output

5. Feasible production q £ f (z1, z2, ...., zm ) q £ f (z)
The basic relationship between output and inputs:
q £ f (z1, z2, ...., zm )
single-output, multiple-input production relation
The production function
This can be written more compactly as:
q £ f (z)
Note that we use “£” and not “=” in the relation. Why?
Consider the meaning of f
Vector of inputs
f gives the maximum amount of output that can be produced from a given list of inputs
distinguish two important cases...

6. Technical efficiency Case 1: q = f (z) Case 2: q < f (z)
The case where production is technically efficient
The case where production is (technically) inefficient
Case 2:
q < f (z)
Intuition: if the combination (z,q) is inefficient you can throw away some inputs and still produce the same output

7. The function  z1 q q >f (z) q <f (z) q =f (z) z2
The production function
Interior points are feasible but inefficient
Boundary points are feasible and efficient
q =f (z)
Infeasible points z2 z1
We need to examine its structure in detail.

8. Overview... The structure of the production function. The Firm: Basics
The setting
The structure of the production function.
Input require-ment sets
Isoquants
Returns to scale
Marginal products

9. The input requirement set
Pick a particular output level q
Find a feasible input vector z
remember, we must have q £ f (z)
Repeat to find all such vectors
Yields the input-requirement set
Z(q) := {z: f (z) ³ q}
The set of input vectors that meet the technical feasibility condition for output q...
The shape of Z depends on the assumptions made about production...
We will look at four cases.
First, the “standard” case.

10. The input requirement set
Feasible but inefficient z2
q = f (z)
Feasible and technically efficient
Infeasible points.
Z(q)
q < f (z)
q > f (z) z1

11. Case 1: Z smooth, strictly convex
Pick two boundary points
Draw the line between them
Intermediate points lie in the interior of Z.
Z(q) z¢
Note important role of convexity.
q< f (z)
q = f (z")
q = f (z')
A combination of two techniques may produce more output. z²
What if we changed some of the assumptions?

12. Case 2: Z Convex (but not strictly)
Pick two boundary points
Draw the line between them
Intermediate points lie in Z (perhaps on the boundary).
Z(q) z¢ z²
A combination of feasible techniques is also feasible

13. Case 3: Z smooth but not convex
Join two points across the “dent”
Take an intermediate point
Highlight zone where this can occur.
Z(q)
This point is infeasible
in this region there is an indivisibility

14. Case 4: Z convex but not smooth
q = f (z)
Slope of the boundary is undefined at this point.

15. Summary: 4 possibilities for Z
Standard case, but strong assumptions about divisibility and smoothness z1 z2
Almost conventional: mixtures may be just as good as single techniques
Only one efficient point and not smooth. But not perverse. z1 z2
Problems: the "dent" represents an indivisibility z1 z2

16. Overview... Contours of the production function. The Firm: Basics
The setting
Contours of the production function.
Input require-ment sets
Isoquants
Returns to scale
Marginal products

17. Isoquants { z : f (z) = q } Pick a particular output level q
Find the input requirement set Z(q)
The isoquant is the boundary of Z:
{ z : f (z) = q }
Think of the isoquant as an integral part of the set Z(q)...
If the function f is differentiable at z then the marginal rate of technical substitution is the slope at z:
Where appropriate, use subscript to denote partial derivatives. So
fj (z) ——
fi (z)
¶f(z)
fi(z) := ——
¶zi .
Gives the rate at which you can trade off one input against another along the isoquant, to maintain constant output q
Let’s look at its shape

18. Isoquant, input ratio, MRTSz1 z2
The set Z(q).
A contour of the function f.
An efficient point.
The input ratio
Marginal Rate of Technical Substitution
Increase the MRTS
z2 / z1= constant
MRTS21=f1(z)/f2(z)
The isoquant is the boundary of Z. z′
Input ratio describes one particular production technique. z° z2 z1
{z: f (z)=q}
Higher input ratio associated with higher MRTS..

19. Input ratio and MRTS
MRTS21 is the implicit “price” of input 1 in terms of input 2.
The higher is this “price”, the smaller is the relative usage of input 1.
Responsiveness of input ratio to the MRTS is a key property of f.
Given by the elasticity of substitution
∂log(z1/z2)
 
∂log(f1/f2)
Can think of it as measuring the isoquant’s “curvature” or “bendiness”

20. Elasticity of substitutionz2
A constant elasticity of substitution isoquant
Increase the elasticity of substitution...
structure of the contour map... z1

21. Homothetic contours z2 z1 O The isoquants
Draw any ray through the origin…
Get same MRTS as it cuts each isoquant. z1 O

22. Contours of a homogeneous function
The isoquants z2
Coordinates of input z°
Coordinates of “scaled up” input tz°
tz°
tz2
f (tz) = trf (z) z2 z°
trq q z1 O O z1
tz1

23. Overview... Changing all inputs together. The Firm: Basics The setting
Input require-ment sets
Isoquants
Returns to scale
Marginal products

24. Let's rebuild from the isoquants
The isoquants form a contour map.
If we looked at the “parent” diagram, what would we see?
Consider returns to scale of the production function.
Examine effect of varying all inputs together:
Focus on the expansion path.
q plotted against proportionate increases in z.
Take three standard cases:
Increasing Returns to Scale
Decreasing Returns to Scale
Constant Returns to Scale
Let's do this for 2 inputs, one output…

25. Case 1: IRTS z1 q z2 t>1 implies f(tz) > tf(z)
An increasing returns to scale function
Pick an arbitrary point on the surface
The expansion path… z2
t>1 implies
f(tz) > tf(z)
Double inputs and you more than double output z1

26. Case 2: DRTS z1 q z2 t>1 implies f(tz) < tf(z)
A decreasing returns to scale function
Pick an arbitrary point on the surface
The expansion path… z2
t>1 implies
f(tz) < tf(z)
Double inputs and output increases by less than double z1

27. Case 3: CRTS z1 q z2 f(tz) = tf(z)
A constant returns to scale function
Pick a point on the surface
The expansion path is a ray z2
f(tz) = tf(z)
Double inputs and output exactly doubles z1

28. Relationship to isoquants
Take any one of the three cases (here it is CRTS)
Take a horizontal “slice”
Project down to get the isoquant
Repeat to get isoquant map z2
The isoquant map is the projection of the set of feasible points z1

29. Overview... Changing one input at time. The Firm: Basics The setting
Input require-ment sets
Isoquants
Returns to scale
Marginal products

30. Marginal products ¶f(z) MPi = fi(z) = —— ¶zi .
Pick a technically efficient input vector
Remember, this means a z such that q= f(z)
Keep all but one input constant
Measure the marginal change in output w.r.t. this input
The marginal product
¶f(z)
MPi = fi(z) = ——
¶zi .

31. CRTS production function againq
Now take a vertical “slice”
The resulting path for z2 = constant z2
Let’s look at its shape z1

32. MP for the CRTS function
f1(z)
The feasible set q
Technically efficient points
f(z)
Slope of tangent is the marginal product of input 1
Increase z1…
A section of the production function
Input 1 is essential: If z1= 0 then q = 0
f1(z) falls with z1 (or stays constant) if f is concave z1

33. Relationship between q and z1
We’ve just taken the conventional case
But in general this curve depends on the shape of .
Some other possibilities for the relation between output and one input… z1 q z1 q

34. Key concepts Technical efficiency Returns to scale Convexity MRTS
Marginal product
Review
Review
Review
Review
Review

35. What next? Introduce the market Optimisation problem of the firm
Method of solution
Solution concepts.