1. Managerial Economics & Business Strategy
Chapter 5
The Production Process and Costs

2. Overview I. Production Analysis II. Cost Analysis
Total Product, Marginal Product, Average Product.
Isoquants.
Isocosts.
Cost Minimization
II. Cost Analysis
Total Cost, Variable Cost, Fixed Costs.
Cubic Cost Function.
Cost Relations.
III. Multi-Product Cost Functions

3. Production Theory
Production theory serves as the basis for other business disciplines such as human resources, operations management, managerial accounting, and strategic management.
Successful businesses use an optimal amount and mix of production factors as factor and output markets change.

4. Production Analysis Technology – engineering know-how
Production Function
Q = F(K,L)
Q is quantity of output produced.
K is capital input.
L is labor input.
F is a functional form relating the inputs to output.
The maximum amount of output that can be produced with K units of capital and L units of labor.
Manager’s responsibility – use the production function efficiently – how much of each input to use,

5. SR and LR
Short-Run vs. Long-Run Decisions – in the SR some factors are fixed limiting input choice decisions.
The mix of variable factors can be changed in the SR.
In the LR is the production horizon in which all factors are variable.
In the simple two factor case if K is fixed in the SR then the only use decisions apply to L.
Q = f(L) = F(K*,L)

6. Production Function Algebraic Forms
Linear production function: inputs are perfect substitutes.
Where a and b are constants.
If capital is 5 times more productive than labor then the function becomes:
Q = 5K + L
If F(8,3) then 5(8) + 1(3), then Q = 43
Linear production functions do not obey the law of DMR

7. Linear Isoquants Capital and labor are perfect substitutes
Q = aK + bL
MRTSKL = b/a
Linear isoquants imply that inputs are substituted at a constant rate, independent of the input levels employed. Q3 Q2 Q1
Increasing Output L K

8. Production Function Algebraic Forms
Leontief production function: inputs are used in fixed proportions – a fixed proportion of K to L for output to increase.

9. Leontief Isoquants Capital and labor are perfect complements.
Capital and labor are used in fixed-proportions.
Q = min {bK, cL}
Since capital and labor are consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL). Q3 Q2 Q1 K
Increasing Output L

10. Production Function Algebraic Forms
Cobb-Douglas production function: inputs have a degree of substitutability.
Here the relationship between output and inputs is not linear.

11. Cobb-Douglas Isoquants
Inputs are not perfectly substitutable.
Diminishing marginal rate of technical substitution.
As less of one input is used in the production process, increasingly more of the other input must be employed to produce the same output level.
Q = KaLb
MRTSKL = MPL/MPK K Q3
Increasing Output Q2 Q1 L

12. Productivity Measures: Total Product
Total Product (TP): maximum output produced with given amounts of inputs.
Example: Cobb-Douglas Production Function:
Q = F(K,L) = K.5 L.5
K is fixed at 16 units.
Short run Cobb-Douglass production function:
Q = (16).5 L.5 = 4 L.5
Total Product when 100 units of labor are used?
Q = 4 (100).5 = 4(10) = 40 units

13. Productivity Measures: Average Product of an Input
Average Product of an Input: measure of output produced per unit of input.
Average Product of Labor: APL = Q/L.
Measures the output of an “average” worker.
Example: Q = F(K,L) = K.5 L.5
If the inputs are K = 16 and L = 16, then the average product of labor is
APL = [(16) 0.5(16)0.5]/16 = 1.

14. Productivity Measures: Average Product of an Input
Average Product of Capital: APK = Q/K.
Measures the output of an “average” unit of capital.
Example: Q = F(K,L) = K.5 L.5
If the inputs are K = 16 and L = 16, then the average product of capital is
APK = [(16)0.5(16)0.5]/16 = 1.

15. Productivity Measures: Marginal Product of an Input
Marginal Product of an Input: change in total output attributable to the last unit of an input.
Marginal Product of Labor: MPL = DQ/DL
Measures the output produced by the last worker.
Slope of the short-run production function (with respect to labor).

16. Productivity Measures: Marginal Product of an Input
Marginal Product of Capital: MPK = DQ/DK
Measures the output produced by the last unit of capital.
When capital is allowed to vary in the short run, MPK is the slope of the production function (with respect to capital).

17. Guiding the Production Process
Producing on the production function
Most difficult to achieve
Aligning incentives to induce maximum worker effort.
Employing the right level of inputs
When labor or capital vary in the short run, to maximize profit a manager will hire:
labor until the value of marginal product of labor equals the wage: VMPL = w, where VMPL = P x MPL.
capital until the value of marginal product of capital equals the rental rate: VMPK = r, where VMPK = P x MPK .

18. Marginal Product of an Input in a Cobb-Douglas Production Function
Q = F(K,L) = KaLb
then MPL = bKaLb-1
and MPK = aKa-1Lb
See page 167 for demonstration problem.

19. Increasing, Diminishing and Negative Marginal ReturnsQ L
Q=F(K,L) MP AP

20. Isoquants Assume management flexibility in the production process.
Illustrates the long-run combinations of inputs (K, L) that yield the producer the same level of output.
The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.

21. Cobb-Douglas Isoquants
Inputs are not perfectly substitutable.
Diminishing marginal rate of technical substitution.
As less of one input is used in the production process, increasingly more of the other input must be employed to produce the same output level.
Q = KaLb
MRTSKL = MPL/MPK K Q3
Increasing Output Q2 Q1 L

22. Marginal Rate of Technical Substitution (MRTS)
The rate at which two inputs are substituted while maintaining the same output level.
The manager must determine the efficient combination of resource use to max profits.

23. Isocost
The combinations of inputs that produce a given level of output at the same cost:
wL + rK = C
Rearranging,
K= (1/r)C - (w/r)L
For given input prices, isocosts farther from the origin are associated with higher costs.
Changes in input prices change the slope of the isocost line. K
New Isocost Line associated with higher costs (C0 < C1).
C1/r C0
C0/w
C0/r C1 L
C1/w K
New Isocost Line for a decrease in the wage (price of labor: w0 > w1).
C/r
C/w1 L
C/w0

24. Cost Minimization
Marginal product per dollar spent should be equal for all inputs (Equi-marginal rule):
But, this is just
So equilibrium for the producer is the point where the slope of the isoquant = slope of the isocost.

25. Equi-Marginal Rule Example
Suppose Ford uses 10 workers and 10 robots to produce one automobile. Suppose also the MPL = 10 cars/day and the MPK = 10 cars/day. If the wage = $40/hr and the cost of the robots is $100/hr is Ford using these two inputs in a cost-minimizing manner?
C = 10($40) + 10($100) = $1400
MPL/PL = MPK/PK
10/40 is not equal to 10/100
.25 > .10
Increase the number of workers to 12 so that the MPL decreases to 8 (ratio = .20), decrease the number of robots to 8 to increase the MPK to 20 (ratio = .20)
8/40 = 20/100
C = 12($40) + 8($100) = $1280

26. Cost Minimization K Q L Point of Cost Minimization Slope of Isocost =
Slope of Isoquant Q L

27. Optimal Input Substitution
A firm initially produces Q0 by employing the combination of inputs represented by point A at a cost of C0.
Suppose w0 falls to w1.
The isocost curve rotates counterclockwise; which represents the same cost level prior to the wage change.
To produce the same level of output, Q0, the firm will produce on a lower isocost line (C1) at a point B.
The slope of the new isocost line represents the lower wage relative to the rental rate of capital. K
C0/w1 A K0
C1/w1 B K1 L1 Q0 L0
C0/w0 L

28. Cost Analysis Types of Costs Short-Run Fixed costs (FC) Sunk costs
Short-run variable costs (VC)
Short-run total costs (TC)
Long-Run
All costs are variable
No fixed costs

29. Cost Function
The cost function provides the manager essential information needed to determine the profit maximizing level of output.
Summarizes the information about the production process.

30. Total and Variable Costs
C(Q): Minimum total cost of producing alternative levels of output:
C(Q) = VC(Q) + FC
VC(Q): Costs that vary with output.
FC: Costs that do not vary with output. $ Q
C(Q) = VC + FC
VC(Q) FC

31. Fixed and Sunk Costs
FC: Costs that do not change as output changes.
Sunk Cost: A cost that is forever lost after it has been paid.
The amount of fixed costs that cannot be recouped.
Decision makers should ignore sunk costs to maximize profit or minimize losses $
C(Q) = VC + FC
VC(Q) FC Q

32. Some Definitions $ Q Average Total Cost ATC = AVC + AFC ATC = C(Q)/Q
Average Variable Cost
AVC = VC(Q)/Q
Average Fixed Cost
AFC = FC/Q
Marginal Cost
MC = DC/DQ $ Q MC
ATC
AVC MR
AFC

33. Fixed Cost MC $ ATC AVC ATC Fixed Cost AFC AVC Q Q0 Q0(ATC-AVC)
= Q0 AFC
= Q0(FC/ Q0)
= FC MC $ Q
ATC
AVC
ATC
Fixed Cost
AFC
AVC Q0

34. Variable Cost MC $ ATC AVC AVC Variable Cost Q Q0 Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0) $ Q
ATC
AVC
AVC
Variable Cost
Minimum of AVC Q0

35. Total Cost MC $ ATC AVC ATC Total Cost Q Q0 Q0ATC = Q0[C(Q0)/ Q0]
Minimum of ATC Q0

36. Cubic Cost Function MC(Q) = a + 2bQ + 3cQ2 dC/dQ = a + 2bQ + 3cQ2
C(Q) = f + a Q + b Q2 + cQ3
Marginal Cost?
Memorize:
MC(Q) = a + 2bQ + 3cQ2
Calculus:
dC/dQ = a + 2bQ + 3cQ2

37. An Example Total Cost: C(Q) = 10 + Q + Q2 Variable cost function:
VC(Q) = Q + Q2
Variable cost of producing 2 units:
VC(2) = 2 + (2)2 = 6
Fixed costs:
FC = 10
Marginal cost function:
MC(Q) = 1 + 2Q
Marginal cost of producing 2 units:
MC(2) = 1 + 2(2) = 5

38. Another Example C(Q) = 45 + 6Q + 12Q2 Suppose Q = 20 MC = 6 + 24Q
AFC = 45/20 = 2.25
AVC = 6Q + 12Q2 /Q = Q = 246
ATC = Q + 12Q2/Q =
45/Q Q=

39. LRAC Function
The LRAC function is the envelope of all possible SRATC functions.
As output expands the firm must adjust its scale to be sure it is operating at least cost.

40. Long-Run Average Costs$
LRAC
Economies
of Scale
Diseconomies
of Scale Q Q*

41. Multi-Product Cost Function – Economies of Scope and Cost Complementarities
C(Q1, Q2): Cost of jointly producing two outputs.
General functional form:
MC1 = aQ2 + 2bQ1
MC2 = aQ1 + 2cQ2
If a > 0 then there are no cost complementarities in production.
If a < 0 then there are cost complementarities.

42. Economies of Scope C(Q1, 0) + C(0, Q2) > C(Q1, Q2). Example:
It is cheaper to produce the two outputs jointly instead of separately.
Example:
It is cheaper for Time-Warner to produce Internet connections and Instant Messaging services jointly than separately.
Dunkin Donuts

43. Cost Complementarity DMC1(Q1,Q2) /DQ2 < 0.
The marginal cost of producing good 1 declines as more of good two is produced:
DMC1(Q1,Q2) /DQ2 < 0.
Example:
Cow hides and steaks.

44. Quadratic Multi-Product Cost Function
C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
MC1(Q1, Q2) = aQ2 + 2Q1
MC2(Q1, Q2) = aQ1 + 2Q2
Cost complementarity: a < 0
Economies of scope: f > aQ1Q2
C(Q1 ,0) + C(0, Q2 ) = f + (Q1 )2 + f + (Q2)2
f > aQ1Q2: Joint production is cheaper

45. A Numerical Example: Yes, since a = -2 < 0
C(Q1, Q2) = Q1Q2 + (Q1 )2 + (Q2 )2
Cost Complementarity?
Yes, since a = -2 < 0
MC1(Q1, Q2) = -2Q2 + 2Q1
Economies of Scope?
Yes, since 90 > -2Q1Q2
Example pg 189.

46. Conclusion
To maximize profits (minimize costs) managers must use inputs such that the value of marginal product of each input reflects price the firm must pay to employ the input.
The optimal mix of inputs is achieved when the MRTSKL = (w/r).
Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters.