1. Applied Economics for Business Management
Lecture #6

2. Lecture Outline Lecture outline Review Go over Homework Set #5
Continue production economic theory

3. Continue Factor – Product Case:
Output elasticity:
Output elasticity or the elasticity of production can be written as:

4. Example
This is a special type of production function called the Cobb-Douglas form.

5. Example
For this type of production function, the output elasticity is equal to the exponent.

6. Example

7. Stages of Production The production function can be divided into
three stages.
The definition of
each stage is
based solely on a
technological
basis without any
reference to
price (either
product price or
factor price).

8. Stages of Production
Boundary between Stages I & II:

9. Stages of Production
Boundary between Stages II & III:

10. Stages of Production
What’s happening in Stage I?

11. Stages of Production
What’s happening in Stage II?

12. Stages of Production
What’s happening in Stage III?

13. Stages of Production Which stage does the producer operate in?
Stages I and III are technically infeasible. For stage III,
of the input is negative. So an additional unit of input will decrease output.

14. Stages of Production The producer will not choose Stage I because
increases throughout Stage I.
Increasing implies that technical efficiency is increasing throughout Stage I. Consequently, it is best to produce beyond Stage I.
The rational or feasible stage of production is Stage II where diminishing returns occur
(as denoted by a positive but declining MPP).

15. Profit Maximization
First order condition(s):

16. Profit Maximization
What is the economic meaning of this first order condition?
First order condition(s):

17. Profit Maximization First order condition(s):
What is the economic meaning of this first order condition?
So the critical value for x1 occurs where the value of the marginal product for this input is equal to its price.

18. Profit Maximization 2nd order condition:
So profit maximization is occurring on the downward sloping portion of the MPP curve or the area of diminishing returns or diminishing MPP.
The negative second derivative verifies that the critical value is a rel max.

19. Example You are also given that
Find the profit maximizing level of input usage for x1.

20. Example  Use 2nd derivative to verify max:
 critical value represents a rel max

21. Example
What is the level of profit?

22. Example Why does this firm produce in the short run?
For this production period, the loss is less than fixed costs (recall FC = $100).
In the longer term, if the firm anticipates continued losses, it will decide to shut down.

23. Two variable factors and one product case:
In this case, we deal with 2 variable factors, a set of fixed factors, and a single output.
The production function looks like this:
or simply write the equation as

24. Isoquant
Technically, we are dealing with three dimensional space called a production surface. However, we can plot this surface in two dimensions by assuming one of the three variables is constant.
Most frequently held constant is output and the curve that is derived is called the isoquant.

25. Isoquant The isoquant is a curve which shows combinations of inputs,
which yield a specific and constant level of
output (y).
The isoquant is very similar to the indifference curve (from demand theory).

26. Isoquant
Like the indifference curve, the isoquant is downward sloping which illustrates the substitution of one input
while producing a specific
amount of output.
for the other

27. Isoquant
We can illustrate this substitution by examining the slope of the isoquant. Also, we can specifically measure the degree of substitution by calculating the slope at a specific point on the isoquant.

28. RTS
The rate of technical substitution (RTS) is a similar concept to the rate of commodity substitution (RCS) in demand theory.

29. RTS
or the slope of the production isoquant is equal to the negative ratio of marginal products.

30. RTS
or the slope of the production isoquant is equal to the negative ratio of marginal products.

31. Optimal combination of inputs in production:
There are two ways to illustrate the optimal combination of inputs in production for the factor-factor case:
(i) maximize output subject to a cost constraint and
(ii) minimize cost subject to an output constraint
How do we illustrate method (i)?

32. Optimal combination of inputs in production:
Assume a given production function:
Specify a given level of costs (a constraint):

33. Optimal combination of inputs in production:
Assume a given production function:
Specify a given level of costs (a constraint):
Objective function: maximize output subject to a cost constraint

34. Optimal combination of inputs in production:
First order conditions:
Solving for λ in the first two equations:

35. Isocost
┌slope of cost line
└slope of isoquant or
The cost line or budget line for production is called the isocost line.
The first order conditions state that the variable factors are combined in an optimal manner when the ratio of marginal products is equal to the ratio of factor prices. This optimal combination is called the least cost combination of inputs.

36. Second Order Condition
The second order condition is a bordered Hessian:
for maximum

37. Constrained Output Maximization
So for the case of constrained output maximization (where the constraint is costs), the optimal occurs where the ratio of marginal products is equal to the ratio of factor prices. This occurs where the isoquant is tangent to the isocost line.

38. Optimal Combination of Inputs in Production
The optimal combination of inputs can also be determined in the factor-factor case by constrained cost minimization.
For this case the objective function can be written as follows:

39. Optimal Combination of Inputs in Production
1st order conditions:
The least cost combination occurs where:
┌slope of isoquant
└slope of isocost

40. Optimal Combination of Inputs in Production
The least cost combination occurs where:
┌slope of isoquant
└slope of isocost
The least cost combination occurs at the tangency between isoquant and isocost.
This is the same conclusion as the case of constrained output maximization.

41. Optimal Combination of Inputs in Production
2nd order conditions:
for a minimum

42. Profit Maximization
Consistent with finding the optimal combination of inputs, is the question of determining the optimal level of input use.
To answer this question, we assume that the firm is a profit maximizer and has the following objective function:

43. Profit Maximization 1st order conditions:
The first order conditions state that for profit maximization inputs should be utilized such that the value of their marginal product is equal to the factor price.

44. Profit Maximization
2nd order conditions:

45. Example Assume perfect competition with output and input prices:
Also, assume also the firm wishes to spend $80 in production costs with fixed cost (FC) = $20.

46. Example 1st order conditions:
One way to find optimal combinations of inputs is constrained output maximization.
Set up the objective function as:
1st order conditions:

47. Example
From the 1st order conditions,
Substitute into 3rd equation:

48. Example
Substitute into 3rd equation:

49. Example 2nd order conditions:
 output is maximized subject to the given cost constraint when

50. Constrained Cost Minimization
λ can be interpreted as the change in output (y) given a $1 change in C (costs).
So λ can be interpreted as or the reciprocal of marginal cost.
The alternative formulation to solve for the optimal combination of inputs is constrained cost minimization.

51. Constrained Cost Minimization
For this method, the objective function is written as:
In this example, y0 is assumed to be 8.
1st order conditions:

52. Constrained Cost Minimization
First 2 derivatives
Solving these equations simultaneously yields
λ = 10 reflects the change in cost given a one unit increase in output (the constraint). So for this formulation, λ represents the MC.

53. Constrained Cost Minimization
Second order conditions:
 critical values will minimize costs subject to the output constraint