1. BUS 525: Managerial Economics The Production Process and Costs
Lecture 5
The Production Process and Costs
Going behind the supply curve

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 Analysis Production Function
A function that defines the maximum amount output that can be produced with a given set of inputs
Q = F(K,L)
The maximum amount of output that can be produced with K units of capital and L units of labor.
Short-Run vs. Long-Run Decisions
Fixed vs. Variable Factors of Production
Managers are required to optimally choose the quantity and type of inputs to use in production process. E.g. employee mix, optimally substituting between labor and other inputs, as wages and other inputs prices change
Short run is defined as a time frame in which there are fixed factors of production
Long run is defined as the horizon over which the manager can adjust all factors of production
Fixed factors are the inputs the managers cannot adjust in the short run. Variable factors are the inputs a manager can adjust to alter production.
Short run Q = F(L) = F(K*, L), K is the fixed factor of production

4. 1-4
Refer to Table 5.1 Page 158
Up to 5th unit of labor MPL is increasing units
Remains constant with 6th unit
7-10th unit declines from units
From 11th unit TP falls and MPL becomes negative
Why?

5. Total Product
The maximum level of output that can be produced with a given amount of input
Cobb-Douglas Production Function
Example: Q = F(K,L) = K.5 L.5
K is fixed at 16 units.
Short run production function:
Q = (16).5 L.5 = 4 L.5
Production when 100 units of labor are used?
Q = 4 (100).5 = 4(10) = 40 units
Linear production function Q= aK+ bL, e.g Q = 4k + L
Leontief production function Q = min {bK, cL}, fixed proportion production functions, e.g. Word processing, Five typists Five computers: five papers; Five typists one computer: one paper

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

7. Marginal Productivity Measures
Marginal product : The 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).
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 long run, MPK is the slope of the production function (with respect to capital).
Marginal product : The change in total output attributable to the last unit of an input
Marginal product of a linear production function: Q= ak + bL
MPk = a; MPl = b
Marginal product of a Cobb-Douglas production function: Q= kaLb
MPk = aKa-1Lb ; MPl = bKaLb-1
MPL of the second unit of labor is 172 units

8. Class Exercise Given the Cobb-Douglas Production Function
Q = F(K,L) = K.5 L.5, K = 16, L=64
Find out the
(i) Average Product of Labor (APl),
(ii) Average Product of Capital (APk),
(iii) Marginal Product of Capital (MPl)
(iv) Marginal Product of Capital (MPk)

9. Increasing, Diminishing and Negative Marginal ReturnsQ
Q=F(K,L)
Increasing marginal returns to labor: Range of input usage over which marginal product increases.
Decreasing/diminishing marginal returns to labor: Range of input usage over which marginal product declines.
Negative marginal returns: Range of input usage over which marginal product is negative. AP L MP

10. Guiding the Production Process
Producing on the production function
Aligning incentives to induce maximum worker effort, tips, profit sharing.
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 .
Why tips? Profit sharing

11. The profit maximizing usage of inputs
∏ = R- C,
Q = F (K,L), C = wL + rK
∏ = P. F(K,L) - wL - rK

12. Refer to Table 5.2 Page 163
Up to 9th unit VMPL is greater than wage. Manager should hire upto ninth unit of labor
Why some firms have 5000 and others have 2 workers?
Notice that MPL is diminishing in the range

13. Isoquant
The 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.

14. Slope of Isoquant

15. Marginal Rate of Technical Substitution (MRTS)
The rate at which two inputs are substituted while maintaining the same output level.
The production function satisfies the law of diminishing marginal rate of technical substitution: As a producer uses less of an input, increasingly more of the other input must be employed to produce the same level of output
Add MRTS graph

16. 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
Inputs are perfectly substitutable for each other.

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

18. 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

19. 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

21. Cost Minimization K Q L Point of Cost Minimization Slope of Isocost =
Slope of Isoquant A C Q L

22. Cost minimization

23. Cost Minimization
Marginal product per dollar spent should be equal for all inputs:
But, this is just

24. Optimal Input SubstitutionK
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.
C0/w1 A K0
C1/w1 B K1
To minimize the cost of producing a given level of output, the firm should use less of an input and more of other inputs when that input price rises. L1 Q0 L0
C0/w0 L

25. Cost Analysis Types of Costs Fixed costs (FC) Variable costs (VC)
Total costs (TC)
Sunk costs
A cost that is forever lost after it has been incurred. Once paid they are irrelevant to decision making
Fixed costs: Costs that do not change with changes in output: include the cost of fixed inputs used in production
Variable costs: Costs that change with changes in output: include the cost of inputs that vary with output.
Example of sunk costs: Feasibility studies for projects

26. Refer to Table 5.3 Page 179

27. 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

28. Fixed and Sunk Costs FC: Costs that do not change as output changes.$
FC: Costs that do not change as output changes.
Sunk Cost: A cost that is forever lost after it has been paid.
C(Q) = VC + FC
VC(Q)
A decision maker/manager should ignore sunk costs to maximize profits or minimize costs
Buy a textbook for Tk. 500 that you do not need. The book seller does not allow refund of exchange
Someone wants to buy it for Tk. 200, what should you do? FC Q

29. Refer to Table 5.4 Page 181

30. 1-30
Refer to Table 5.5 Page 182

31. Some Definitions 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
Average fixed cost: Fixed costs divided by the number of units of output
Average variable cost: Variable costs divided by the number of units of output
Marginal costs: The cost of producing an additional unit of output.
Note that :
MC intersects ATC and the AVC at their lowest points. This implies that when MC is below an average cost curve AC is declining and When MC is above AC is rising
Grade in the course:
CGPA MG AGPA
B A G↑
B C G↓
Note further that
ATC and AVC curves get closer as output increases. Why? AFC declines as output increases.
AFC

32. 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
Relationship between productivity and cost curves: MC and AVC are mirror images of the MP and AP curves respectively. When MP is maximum MC is minimum when MP is diminishing MC goes up.
AVC Q0

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

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

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

36. 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

37. Economies of Scale $ LRAC Economies Diseconomies of Scale of Scale Q
Long-run costs: All cost are variable
Long-run average cost curve: A curve that defines the minimum average cost of producing alternative levels of output, allowing for optimal selection of both fixed and variable factors of production.
Economies of scale: Exists when long run average costs decline as output is increased.
Diseconomies of scale: Exists when long run average costs rise as output increased.
Constant returns to scale: Exists when long run average costs remain constant as output increased.
Economic versus accounting cost
Cost of production include not only the accounting costs but also the opportunities forgone by producing a given product.
Economies
of Scale
Diseconomies
of Scale Q

38. Multi-Product Cost Function
A function that defines the cost of producing given levels of two or more types of outputs assuming all inputs are used efficiently
C(Q1, Q2): Cost of jointly producing two outputs.
General multiproduct cost function

39. 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 Citycell to produce Internet connections and Instant Messaging services jointly than separately.

40. Cost Complementarity DMC1(Q1,Q2) /DQ2 < 0.
The marginal cost of producing good 1 declines as more of good 2 is produced:
DMC1(Q1,Q2) /DQ2 < 0.
Example:
Bread and biscuits

41. 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
When a is negative an increase in Q2 will lower MC1
C(Q1 ,0) + C(0, Q2 ) > C(Q1, Q2) = f + (Q1 )2 + f + (Q2)2 - f - aQ1Q2 - (Q1 )2 - (Q2 )2>0
f - aQ1Q2 >0

42. A Numerical Example: C(Q1, Q2) = 100 – 1/2Q1Q2 + (Q1 )2 + (Q2 )2
Cost Complementarity?
Yes, since a = -2 < 0
MC1(Q1, Q2) = -2Q2 + 2Q1
Economies of Scope?
Yes, since 90 > -2Q1Q2
The firm wishes to produce 5 units of good 1 and 4 units of good 2.
a=1/2<0, cost complementarities exist
f-aQ1Q2>0
100- 1/2x5x4 = 90 >0, Economies of scope exists
C(5,4) = = 131
C(5,0) = = 125
C(0,4) = = 116
Benefit of joint production = = $110

43. Conclusion
To maximize profits (minimize costs) managers must use inputs such that the value of marginal 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.

44. Mid1 Answers (a) p5; (b) p21; © p56; (d) p13; (e) p90; (f) p79.
(a)p46; (b) p53(price floor, surplus).
3. (a)p92; (b) ; © -0.91(inelastic demand); 0.156(substitute, inelastic); 0.32(normal good, inelastic);0.13(positively related, inelastic).
(a) Px = 115 – ¼ Qxd; (b) 12,800; © 16,200; (d) Consumer surplus decreases.
(a) p=300; (b) p=200; Q=1000; © p=225; Q=1250.
(a) No, demand is inelastic, revenue will fall; (b) Income must rise by 22.22%; © Sale of shoes will increase by 2.25%.