BUS 525: Managerial Economics

Lecture 5

The Production Process and Costs

Activity Schedule:BUS525:2

  Originally scheduled classes Makeup classes (Announced by University) Makeup classes proposed (To be finalized in consultation with students)

Class Date Exams Cases

1 28 May

2 4 June

3 6 June

4 11 June Case 1

5 20 June Mid 1

6 23 July

7 25 July (Thu)

8 30 July   Case 2 

9 2 August (Fri)

10 13 August

11 16 August (Fri)

Mid 2

12 20 August Case 3

13 TBA Final  

Activity Schedule:BUS525:3

Originally scheduled classes Makeup class (Announced by University) Makeup classes proposed (To be finalized in consultation with students)

Class Date Exams Cases

1 29 May

2 5 June

3 12 June

4 19 June Case 1

5 26 June Mid 1

6 24 July

7 26 July (Fri)

8 31 July Case 2 

9 1 August (Thu)

10 14 August

11 16 August (Fri) Mid 2

12 21 August Case 3

13 TBA Final

OverviewI. Production 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

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

Table 5-1 The Production Function1-6

K L ΔL O MPL APLFixed Input

Variable Input Change in Labor

Output Marginal Product of Labor

Average Product of Labor

2 0 - 0 - -2 1 1 76 76 762 2 1 248 172 1242 3 1 492 244 1642 4 1 784 292 1962 5 1 1100 316 2202 6 1 1,416 316 2362 7 1 1,708 292 2442 8 1 1,952 244 2442 9 1 2,124 172 2362 10 1 2,200 76 2202 10 1 2,156 -44 196

Measures of Productivity: 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

Measures of Productivity: Average Product

• Average Product of Labor: A measure of the output produced per unit of input

• 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.

• 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 = 4, then the average product of capital is APK = [(16)0.5(4)0.5]/16 = 1/2.

Measures of Productivity: Marginal Product

• Marginal product : The change in total output attributable to the last unit of an input

• Marginal Product of Labor: MPL = Q/L– Measures the output produced by the last worker.– Slope of the short-run production function (with respect to

labor).• Marginal Product of Capital: MPK = Q/K

– 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).

Q

L

Q=F(K,L)

IncreasingMarginalReturns

DiminishingMarginalReturns

NegativeMarginalReturns

MP

AP

Increasing, Diminishing and Negative Marginal Returns

TP

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 .

The Profit Maximizing Usage of Inputs

∏ = R- C, Q = F (K,L), C = wL + rK∏ = P. F(K,L) - wL - rK

price its equalsproduct marginal of valueits wherepoint the toup used bemust input each i.e

wV andr V

wP. andr P.L

L)F(K, and K

L)F(K, Since

0wL

L)F(K,P.Lπ

0rK

L)F(K,P.Kπ

MPMPMPMP

MPMP

LK

LK

Lk

L P MPL VMPL WVariable Input Price of Output Marginal Product

of LaborP x MPl Unit Cost of Labor

0 $3 - - $4001 $3 76 $228 $4002 $3 172 516 $4003 $3 244 732 $4004 $3 292 876 $4005 $3 316 948 $4006 $3 316 948 $4007 $3 292 876 $4008 $3 244 732 $4009 $3 172 516 $400

10 $3 76 228 $40011 $3 -44 -132 $400

Table 5-2 The Value of Marginal Product of Labor

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.

Slope of Isoquant

L

K

MPMP

KL)/F(K,LL)/F(K,

dLdK or,

0] dQisoquant an along changenot dooutput Since[

0.L

L)F(K,.K

L)F(K, dQ

L)(K, F Qisoquants of Slope

dLdK

Marginal Rate of Technical Substitution (MRTS)

• The rate at which a producer can substitute between two inputs and maintain 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

K

LKL MP

MPMRTS

The Marginal Rate of Technical Substitution

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.

Q3Q2Q1

Increasing Output

L

K

Leontief Isoquants

• Capital and labor are perfect complements.

• Capital and labor are used in fixed-proportions.

• Q = min {bK, cL}• What is the MRTS?

Q3

Q2

Q1

K

Increasing Output

L

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

Q1

Q2

Q3

K

L

Increasing Output

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

LC1

L

KNew Isocost Line for a decrease in the wage (price of labor: w0 > w1).

C1/r

C1/wC0

C0/w

C0/r

C/w0 C/w1

C/r

New Isocost Line associated with higher costs (C0 < C1).

isocost of Slope

rw - Slope

)(r1 K

C rK wL

LrwC

Cost Minimization

Q

L

K

Point of Cost Minimization

Slope of Isocost =

Slope of Isoquant

A

C

Cost minimization

MRTSMPMP

KLK

L

KF(K,L)/LF(K,L)/

rw

by dividing

)(........F(K,L)QμH

)........(K

F(K,L)μrKH

)........(L

F(K,L)μwLH

][Q- F(K,L) rK wL H Lagrangian

21

3.......0

20

10

L)F(K, Q subject torK wLMinimizeL)F(K, Q rK, wL C

Cost Minimization

• Marginal product per dollar spent should be equal for all inputs:

• But, this is justrw

MPMP

rMP

wMP

K

LKL

rwMRTSKL

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.

Q0

0

A

L

K

C0/w1C0/w0 C1/w1L0 L1

K0

K1B

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

K L O FC VC TCFixed Input Variable Input Output Fixed Cost Variable Cost Total Cost

2 0 0 $2,000 $0 $20002 1 76 $2,000 400 $24002 2 248 $2,000 800 $28002 3 492 $2,000 1,200 $32002 4 784 $2,000 1,600 $36002 5 1100 $2,000 2,000 $40002 6 1,416 $2,000 2,400 $44002 7 1,708 $2,000 2,800 $48002 8 1,952 $2,000 3,200 $52002 9 2,124 $2,000 3,600 $56002 10 2,200 $2,000 4,000 $6000

Table 5.3 The Cost Function

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

0

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.

$

Q

FC

C(Q) = VC + FC

VC(Q)

Some Definitions

Average Total CostATC = AVC + AFCATC = C(Q)/Q

Average Variable CostAVC = VC(Q)/Q

Average Fixed CostAFC = FC/Q

Marginal CostMC = C/Q

$

Q

ATCAVC

AFC

MC

MR

Fixed Cost

$

Q

ATC

AVC

MC

ATC

AVC

Q0

AFC Fixed Cost

Q0(ATC-AVC)

= Q0 AFC

= Q0(FC/ Q0)

= FC

Variable Cost

$

Q

ATC

AVC

MC

AVCVariable Cost

Q0

Q0AVC

= Q0[VC(Q0)/ Q0]

= VC(Q0)

$

Q

ATC

AVC

MC

ATC

Total Cost

Q0

Q0ATC

= Q0[C(Q0)/ Q0]

= C(Q0)

Total Cost

O FC VC TC AFC AVC ATCOutput Fixed Cost Variable Cost Total Cost Average

Fixed CostAverage Variable

Cost

Average Total Cost

0 $2,000 $0 $2000 - -76 $2,000 400 $2400 $26.32 $5.26 $31.58

248 $2,000 800 $2800 8.06 3.23 11.29492 $2,000 1,200 $3200 4.07 2.44 6.5784 $2,000 1,600 $3600 2.55 2.04 4.59

1100 $2,000 2,000 $4000 1.82 1.82 3.641,416 $2,000 2,400 $4400 1.41 1.69 3.111,708 $2,000 2,800 $4800 1.17 1.64 2.811,952 $2,000 3,200 $5200 1.02 1.64 2.662,124 $2,000 3,600 $5600 0.94 1.69 2.642,200 $2,000 4,000 $6000 0.91 1.82 2.73

Table 5.4 Derivation of Average Costs

O ΔQ VC ΔVC TC ΔTC MCOutput Change in

OutputChange in

Variable CostTotal Cost Average

Fixed CostAverage Variable

Cost

Marginal Cost

0 - $0 - $2000 -76 76 400 400 $2400 400 $5.26

248 172 800 400 $2800 400 2.33492 244 1,200 400 $3200 400 1.64784 292 1,600 400 $3600 400 1.37

1100 316 2,000 400 $4000 400 1.271,416 316 2,400 400 $4400 400 1.271,708 292 2,800 400 $4800 400 2.811,952 244 3,200 400 $5200 400 2.662,124 172 3,600 400 $5600 400 2.642,200 76 4,000 400 $6000 400 2.73

Table 5.4 Derivation of Marginal Costs

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

Class Exercise– Total Cost: C(Q) = 10 + Q + Q2

– Find: – the variable cost function.– variable cost of producing 2 units.– fixed costs.– marginal cost function.– marginal cost of producing 2 units:

Long Run Cost: Economies of Scale

LRAC

$

Q

Economiesof Scale

Diseconomiesof Scale

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 multi-product cost function 2

22

12121, cQbQQaQfQQC

Economies of Scope

• C(Q1, 0) + C(0, Q2) > C(Q1, Q2).– 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.

Cost Complementarity

• The marginal cost of producing good 1 declines as more of good 2 is produced:

MC1Q1,Q2) /Q2 < 0.

• Example:– Bread and biscuits

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 C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2

f > aQ1Q2: Joint production is cheaper

Class Exercise IIC(Q1, Q2) = 100 – 1/2Q1Q2 + (Q1 )2 + (Q2 )2 • The firm wishes to produce 5 units of

good 1 and 4 units of good 2. • Find whether (a) cost complementarity exists?(b) the cost function demonstrateseconomies of scope?

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.