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Managerial Economics & Business Strategy. Chapter 5 The Production Process and Costs. Production Analysis. Production Function Q = F(K,L)

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## Managerial Economics & Business Strategy

Chapter 5

The Production Process and Costs

### Production Analysis

• Production Function

• Q = F(K,L)

• The maximum amount of output that can be produced with K units of capital and L units of labor. The role of the manager is to make sure the firm operates on the production function

• Short-Run (SR) v. Long-Run (LR) Decisions

• In SR some variables fixed. The LR is a period long enough such that all inputs are variable

### Total Product

• Linear Production Function

• Perfect substitution among inputs (meaning, constant)

• Example: Q = F(K,L) = aK + bL

• Suppose: a = 10, b = 5, and K is fixed at 16 units in SR.

• Short run production function:

Q = 10(16)+5L= 160 + 5L

• Production when 100 units of labor are used?

Q = 160+500 = 660 units

### Total Product

• Cobb-Douglas Production Function

• Imperfect substitution (K and L are substitutable but not at a constant rate).

• Example: Q = F(K,L) = Ka Lb

• K is fixed at 16 units, let a = b = 0.5

• 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

### Total Product

• Leontief Production Function

• Substitution impossible. K and L are complements.

• Example: Q = F(K,L) = min(bK, cL)

• Let b=c=1. Let K=16 and L=100.

• Production when 100 units of labor are used?

16 units

### Marginal Product of Labor

• Measures the output produced by the last worker holding K constant.

• Discrete: MPL = DQ/DL (discrete)

• Continuous: Slope (derivative) of the production function (continuous)

• At times may see “increasing marginal returns” due to specialization, and “diminishing marginal returns.”

### Using Calculus to Find MPL

Linear Production Function

Cobb-Douglas Production Function

### Average Product of Labor

• APL = Q/L

• Measures the output of an “average” worker.

• Geometrically, APL is measured as the slope of a ray from the origin to a point on the TP curve.

Q

Slope=0, MP=0

Slope=.75, AP=.75

AP

L

MP

Stages of Production

Increasing

Marginal

Returns

Diminishing

Marginal

Returns

Negative

Marginal

Returns

Q=F(K,L)

### Productivity in SR

• Go to SR and LR production handout

### What’s more important? AP or MP

• Suppose in SR and K is fixed. How does a firm determine the optimal level of L to hire? How many workers are needed to maximize profits?

• If manager knows how much an additional worker contributes to revenues and how much an additional worker costs, then manager can decide if hiring the worker will increase the firm’s profits.

• Therefore, MP more important.

### Value Marginal Product of Labor

• Represent the additional revenue generated by employing an additional unit of L.

• VMPL = MR  MPL = dR/dQ  MPL.

• Note: In a perfectly competitive market, the price of the good is the marginal revenue.

• VMPL=P  MPL

• Similarly, for the value of the MPK, VMPK=P  MPK

### Find Profit-Maximizing Level of Labor

• Q=2(K)1/2(L)1/2.

• The company has already spent \$10,000 on 4 units of K, and due to current economic conditions, the company doesn’t have the flexibility to buy more equipment.

• W=\$100 per worker. P=\$200/unit.

• How much L? How much Q is produced? What are profits?

• L=16, Q=16, = 8400. Because \$8400 is less than FC of \$10,000, the firm is covering a portion of its FC (\$1600 of FC).

### Cost Analysis

Types of Costs

Fixed costs (FC)

Variable costs (VC)

Marginal costs (MC)

Total costs (TC)

Sunk costs

Need to understand costs to find profit-maximizing output

\$

C(Q) =VC+FC

VC(Q)

FC

Q

Total and Variable Costs (SR)

• C(Q): Minimum total cost of producing alternative levels of output:

• C(Q) =VC+FC

• VC(Q): Costs that vary with output

• FC: Costs that do not vary with output

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

Fixed and Sunk Costs

• You lease a building for \$5000 a month.

• This is a FC.

• After you pay, it is a sunk cost, unless some of it is refundable, then only the nonrefundableportion is sunk. Sunk costs are irrelevant to marginal decision-making that takes place during the month.

• If at the end of the month, the firm is losing money (negative economic profit), then paying the lease becomes a marginal decision, and shouldn’t renew

Fixed and Sunk Costs

• Problem from the book:

Fixed and Sunk Costs

• Assuming no re-sale value, how much of the liquor license is sunk?

• Should she accept the \$66,000 offer?

MC

\$

Q

More Cost Definitions (SR)

• 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 =

ATC

AVC

AFC

### Costs in SR

• Go to SR cost handout

MC

\$

ATC

AVC

Q

Fixed Cost

Q0(ATC-AVC)

= Q0 AFC

= Q0(FC/ Q0)

= FC

ATC

Fixed Cost

AFC

AVC

Q0

MC

\$

ATC

AVC

Q

Variable Cost

Q0AVC

= Q0[VC(Q0)/ Q0]

= VC(Q0)

AVC

Variable Cost

Q0

MC

\$

ATC

AVC

Q

Total Cost

Q0ATC

= Q0[C(Q0)/ Q0]

= C(Q0)

ATC

Total Cost

Q0

### Cubic Cost Function

• C(Q) = f + a Q + b Q2 + cQ3

• Marginal Cost?

• Calculus:

dC/dQ = a + 2bQ + 3cQ2

### 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

### Another Example

• C(Q) = 5000 + 20Q2 + 10Q

• Find AFC, AVC, ATC and MC when Q=10.

• C(10) = \$7100.

• AFC = \$500 per unit

• AVC=\$210 per unit

• ATC = \$710 per unit

• MC(Q) = 40Q + 10, thus MC(10)=\$410

### Inputs in the LR

• Goal: Minimize the costs of producing a given level of output. Finding the optimal level of K and L in LR.

• An isoquant shows 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.

• Go to SR and LR production handout

L

K

Increasing Output

Q2

Q3

Q1

L

### Linear Isoquants

• Capital and labor are perfect substitutes

• Constant slope (e.g., -3/4 always)

• Marginal Rate of Technical Substitution (MRTS): the rate of substitution between 2 inputs while maintaining the same level of output

• MRTS = MPL / MPK

• The negative of the slope of the isoquant.

Q3

K

Q2

Q1

Increasing Output

### Leontief Isoquants

• Capital and labor are perfect complements

• Capital and labor are used in fixed-proportions

### Cobb-Douglas Isoquants

• Inputs are not perfectly substitutable

• Diminishing MRTS

• Nonlinear, nonconstant slope

• Most production processes have isoquants of this shape

K

Q3

Increasing Output

Q2

Q1

L

### Isocost

• The combinations of inputs that yield the same total cost to the producer

• The above equation is just in y=mx + b form. The slope is the ratio of input prices (w/r). The ratio of input prices is the market rate of substitution.

### Isocost

K

• The combinations of inputs that cost the producer the same amount of money

• 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

C0

C1

L

K

New Isocost Line for a decrease in the wage (price of labor).

L

### Cost Minimization

• The additional output (i.e., marginal product) per dollar spent should be equal for all inputs:

• Expressed differently

### Cost Minimization

K

Slope of Isocost

=

Slope of Isoquant

That is,

MRTS = w/r

Point of Cost Minimization

Q=100

CLower

CHigher

L

### LR versus SR

K

C1 > C0, where C0 represents the costs of producing Q0 in LR.

C1

C0

C1: Cost of producing Q0 in SR

KLR

KSR*

Q0

L

LLR

LSR

### The Long-run and cost-minimization

• The important thing to remember is that the LR offers flexibility—the ability to substitute away from more expensive inputs to lower priced inputs (e.g., see the chapter Headline, “Winning the Battle versus Winning the War”).

### Cost-minimization problem

• A firm’s production process follows a Cobb-Douglas production process. At current production levels, an engineer has determined that the MPL is 60 units per hour and the MPK is 120 units per hour. The wage rate is \$8 per hour and the opportunity cost of capital is \$12 per hour. Is the firm producing its output at the lowest cost? If not, what changes should they make in the LR?

Relationship between LR & SR Costs

\$

ATC

ATC

ATC

LRAC

ATC

LRAC is “envelope” of SR ATC curves

Output (millions)

10

30

42

LR Costs and Economies of Scale

\$

LRAC

Economies

of Scale

Diseconomies

of Scale

Output

### Economies of Scale

• When AC decline as Q increases

• A reason to explain mergers

• Merger between HP and Compaq

### Merger or Selling off Subsidiary

• Merger makes sense if:

• Economies of scope exist (different product type)

• Economies of scale exist (same product type)

• Selling off an unprofitable subsidiary

• If economies of scope exist, then may not see a large reduction in total costs.

• Will see an increase in average costs.